Université Paris VII – Denis Diderot Institut de Physique du Globe de

Transcription

Université Paris VII – Denis Diderot
Institut de Physique du Globe de Paris
Équipe de Sismologie – UMR CNRS 7154
Mémoire
présenté par
Yann CAPDEVILLE
pour obtenir
L’Habilitation à Diriger les Recherches
Contributions aux problèmes direct et inverse en sismologie
soutenu le 11/03/2011 devant le jury composé de Madame et Messieurs :
Marianne Greff (Présidente, IPGP)
Jean Virieux (Rappoteur, LGIT)
Yvon Maday (Rapporteur, Paris 6)
John Woodhouse (Examinateur, Oxford)
Claude Boutin (Examinteur, ENTPE)
Nikolai Shapiro (Examinateur, IPGP)
———————-
4 Place Jussieu, Tour 24, Case 89, 75252 Paris cedex 05 – Tél : 01.44.27.24.69 – Fax : 01.44.27.38.94
[email protected]
2
Table des matières
Résumé sur l’originalité des recherches
5
1
Exposé synthétique des recherches
7
1.1
Introduction au cadre de recherche . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.1.1
Type de données en imagerie sismique . . . . . . . . . . . . . . . . . . .
8
1.1.2
Le problème direct en sismologie . . . . . . . . . . . . . . . . . . . . .
9
1.1.3
Le problème inverse en sismologie . . . . . . . . . . . . . . . . . . . . . 12
1.2
Mon travail dans ce cadre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1
Contributions au problème direct . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2
Contributions au problème inverse . . . . . . . . . . . . . . . . . . . . . 22
1.2.3
Homogénéisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Références
38
2
39
3
Persectives de recherche à court et moyen termes
2.1
Perspectives à court terme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2
Perspectives à moyen ou long termes . . . . . . . . . . . . . . . . . . . . . . . . 41
CV, activité d’encadrement, liste des publications
43
3.1
CV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2
Activités d’encadrement, d’enseignement et administratifs . . . . . . . . . . . . 44
3.3
Collaborations et séjour à l’étranger . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4
Transfert de technologie, relations industrielles et valorisation . . . . . . . . . . 45
3.5
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3
4
TABLE DES MATIÈRES
3.6
3.5.1
Publications dans des revues de rang A . . . . . . . . . . . . . . . . . . 46
3.5.2
Brevet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.3
Communications orales et posters . . . . . . . . . . . . . . . . . . . . . 48
Recueil des articles et travaux significatifs . . . . . . . . . . . . . . . . . . . . . 50
3.6.1
Normal modes Born, 2005, GJI . . . . . . . . . . . . . . . . . . . . . . 51
3.6.2
SEM-Normal mode soupling, 2003, GJI . . . . . . . . . . . . . . . . . . 59
3.6.3
Stacked sources waveform tomography, 2005, GJI . . . . . . . . . . . . 93
3.6.4
Shallow layer correction, 2008, GJI . . . . . . . . . . . . . . . . . . . . 107
3.6.5
Non periodic homogenization, 1-D case, 2010, GJI . . . . . . . . . . . . 123
3.6.6
Non periodic homogenization, 2-D P-SV case, 2010, GJI . . . . . . . . . 137
TABLE DES MATIÈRES
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Résumé sur l’originalité des recherches
Le cadre de ma recherche est principalement centré sur la sismologie comme outil permettant
d’imager l’intérieur de la terre. Ce cadre m’a amené à contribuer aux études à la fois du problème direct et du problème inverse en sismologie. Pour ce qui est du problème direct, j’ai surtout
travaillé sur les méthodes de modes propres et sur les éléments spectraux. Mes principales contributions pour le problème direct ont été le développement d’un couplage entre les éléments spectraux et la méthode des modes propres pour la terre à l’échelle globale et le développement d’une
méthode basée sur les perturbations au premier ordre des modes propres proche des méthodes
d’adjoints utilisées pour le problème inverse. Ces travaux m’ont naturellement amené à m’intéresser aux méthodes de tomographie basées sur l’inversion des formes d’ondes complètes des
signaux sismiques. J’ai développé une méthode utilisant la méthode couplée éléments spectraux
- modes pour résoudre le problème direct et les perturbations de modes pour calculer les dérivées
partielles des formes d’ondes par rapport aux paramètres du modèle. L’originalité de l’approche
développée ici est liée à l’utilisation des formes d’ondes sommées pour toutes les sources plutôt
que les formes d’ondes individuelles comme cela est fait classiquement. L’intérêt de cette astuce
est de diminuer le coût de calcul de l’inversion de façon considérable. Finalement, les difficultés liées au maillage des modèles élastiques pour les éléments spectraux et celles liées à la
paramétrisation du problème inverse, rencontrées lors de ces travaux m’ont amené à m’intéresser
à l’homogénéisation de l’équation des ondes élastiques. Ma contribution dans ce domaine a été
le développement d’une méthode d’homogénéisation à deux échelles valable pour les milieux
non périodiques que sont les milieux élastiques géologiques. De mon point de vue, l’utilisation
de l’homogénéisation en sismologie est une avancée importante. Ces travaux permettent de faire
le lien entre le modèle tomographique (ne contenant que des grandes échelles) obtenu par une
inversion de formes d’ondes et le modèle élastique réel (contenant des petites échelles).
Mes perspectives de recherches sont nombreuses, mais la principale à moyen terme est de terminer les avancées sur l’homogénéisation en sismologie en développant un outil complet adapté
aux milieux géologiques 3-D et en l’appliquant à un certain nombre de problèmes en géophysique. Parmi ceux-ci, je vais m’intéresser à la génération anormale d’ondes S par les explosions, aux problèmes des hétérogénéités de petites échelles proches de la surface libre et à la
distinction de l’anisotropie induite de l’anisotropie intrinsèque. À long terme, mon objectif est
de contribuer au développement de l’imagerie sismique en reprenant mes travaux sur les sources
sommées en y incluant les résultats de l’homogénéisation.
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TABLE DES MATIÈRES
Chapitre 1
Exposé synthétique des recherches
1.1
Introduction au cadre de recherche
En sismologie, deux grands axes de recherche peuvent être dégagés : le premier concerne l’étude
des sources sismiques et l’autre l’étude des structures du sol. Pour le premier, on s’intéresse aux
signaux sismiques générés par un tremblement de terre afin de contraindre notre compréhension
de la rupture le long d’une faille sismique. Dans ce cas, on considère en général comme connu
le milieu géologique transportant les ondes sismiques pour ne s’intéresser qu’au processus qui
les a générées. Pour le deuxième, on s’intéresse à des signaux sismiques afin de contraindre le
milieu traversé par les ondes élastiques. Dans ce cas, on considère comme connu le processus qui
génère les ondes sismiques pour ne s’intéresser principalement qu’à la structure élastique ayant
servi de support aux ondes.
L’un ayant besoin de l’autre, ces deux axes de recherche ne sont bien sûr pas décorrélés. Ils ont
notamment en commun la résolution du problème direct (la propagation des ondes pour milieu
élastique et une source donnés) et les observations.
Dans ce cadre très général, mon travail s’est principalement axé sur la résolution du problème
direct et son application au problème de l’imagerie sismique par inversion de la forme d’onde.
Bien qu’ayant une signification plus spécifique dans le domaine de l’exploration pétrolière, j’emploierai le terme “imagerie sismique” pour désigner l’utilisation de données sismiques pour
retrouver des informations sur les propriétés du sous-sol par la résolution d’un problème inverse.
Dans cette introduction au contexte de mon travail de recherche, je vais d’abord brièvement parler des données utilisées dans ce domaine, puis des différentes méthodes utilisées pour résoudre
le problème direct et enfin du problème inverse. Cet ordre de présentation (d’abord les données,
puis le problème direct et enfin le problème inverse) peut sembler naturel dans le sens où les
observables conditionnent le problème direct qui lui même impose des contraintes sur la résolution du problème inverse. En pratique, l’interaction données-problème direct-problème inverse
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8
CHAPITRE 1. EXPOSÉ SYNTHÉTIQUE DES RECHERCHES
F IGURE 1.1 – Un exemple de donnée brute pour le séisme du 07/09/97,
Ms=6.8 au Venezuela. En bas, le grand
cercle entre la source et la station. À
droite, le signal enregistré sur la composante verticale de la station SSB.
est plus complexe que ce simple enchaînement. En effet, notre capacité à résoudre le problème
direct conditionne aussi le choix des données de même que notre capacité à résoudre le problème
inverse conditionne le problème direct et les données. Par exemple, le succès de l’utilisation des
temps d’arrivées des phases sismiques est, par bien des égards, dû à la rapidité et la simplicité
du problème direct et du problème inverse associés. Cette remarque étant faite, je m’en tiendrai
à une présentation simplifiée et rapide des données puis du problème direct et enfin du problème
inverse utilisé en sismologie pour les problèmes de tomographie et d’imagerie.
1.1.1
Type de données en imagerie sismique
La donnée de base en sismologie ou en exploration sismique est l’enregistrement des déplacements du sol (ou de leurs dérivées en temps) générés par un tremblement de terre, par une source
active (une explosion ou un tir de canon à air) ou par une source de bruit (un exemple de signal
brut enregistré par un sismomètre est donné figure 1.1). Ce type de données peut être enregistré
sur trois composantes ou seulement sur la composante verticale et dans des bandes de fréquences
variables selon le type de récepteurs. Cette donnée, la plus générale, est appelée la forme d’onde.
Elle contient l’information sur le sous-sol la plus riche disponible en sismologie.
Bien que ce soit de ce type de données que l’on pourrait extraire le plus d’informations sur le
milieu traversé par les ondes, la forme d’onde n’est jamais utilisée brute en pratique, du moins
jusqu’à présent. La principale raison de ce fait est que le problème inverse associé à la forme
d’onde brute est trop coûteux et trop complexe pour être envisagé pour l’instant (Ceci est néanmoins en train de changer pour la forme d’onde dans des bandes fréquences réduites, c’est à dire
pour des données filtrées). En général, les méthodes d’imageries sont basées sur des données
réduites pour lesquelles le problème direct et le problème inverse sont beaucoup plus simples
(du moins avec certaines hypothèses) que pour la forme d’onde brute. Ces données réduites sont
principalement constituées des temps d’arrivées des principales phases sismiques (ondes P, S, PP,
1.1. INTRODUCTION AU CADRE DE RECHERCHE
9
etc) pour les ondes de volumes, des vitesses de phases ou de groupe pour les ondes de surface
(ondes de Love, de Rayleigh, phases X) et des fréquences de modes propres de la Terre. Pour
l’exploration sismiques, ce ne sont pas forcement les temps d’arrivées qui sont utilisés directement mais plutôt la cohérence des signaux à différents offsets, mais l’idée est similaire que pour
les temps d’arrivées ou les vitesses de phases : c’est l’information de “phase” qui prime.
Enfin, un nouveau type de d’observables est apparu récemment avec les corrélations du bruit
sismique entre deux stations. Il a été montré que ces corrélations, dans le cas idéal, permettent de
retrouver les fonctions de green entre les deux stations (Shapiro & Campillo, 2004). Ces données
sont de plus utilisées pour faire de l’imagerie de la croûte, mais encore une fois seules les vitesses
de phases des ondes de surface identifiées sur les corrélations de bruit sont utilisées.
1.1.2
Le problème direct en sismologie
L’équation de base en sismologie est l’équation des ondes élastiques qui, si on néglige la gravité,
peut s’écrire comme,
ρ∂tt u − ∇ · σ = f ,
(1.1)
où u(x, t) est le vecteur déplacement du sol en tout point x et temps t, ∂tt la dérivée seconde en
temps, ∇ le vecteur gradient ( et ∇· la divergence) σ(x, t), le tenseur des contraintes et f (x, t)
la force externe résultant par exemple d’un tremblement de terre. Cette équation est associée à la
relation constitutive entre la contrainte et la déformation, qui dans le cas élastique peut s’écrire
σ = c : (u) ,
(1.2)
où (u) = 12 (∇u+ t ∇u) est le tenseur de déformation, t la transposition et c le tenseur élastique.
Pour compléter ces équations, on ajoute des conditions initiales sur le déplacement et la vitesse
ainsi que des conditions limites aux bords du domaine. Pour l’équation des ondes élastiques ou
acoustiques dans un cadre plus général (avec la gravité, l’atténuation intrinsèque, ...) on pourra
par exemple se référer à Dahlen & Tromp (1998).
Le système d’équation ci-dessus est un système d’équations aux dérivées partielles hyperboliques
qui ne présente pas de difficulté théorique particulière pour sa résolution numérique (sauf dans le
cas où les perturbations au potentiel de gravité ne peuvent être négligées en présence de liquide
comme c’est le cas dans la graine externe et pour les très longues périodes, voir par exemple
Dahlen & Tromp 1998 ou Chaljub & Valette 2004). Cependant, pour les applications pratiques
dans la terre, le nombre de longueurs d’onde à propager, le nombre de sources et récepteurs à
prendre en compte, la complexité des modèles géologiques (par modèle géologique j’entends ici
la description de (ρ, c) en fonction de la position) peuvent être tels que cette résolution se révèle
être extrêmement difficile, en particulier si on a besoin d’effectuer cette résolution rapidement
dans l’optique de résoudre le problème inverse. Il existe de nombreuses méthodes pour résoudre
ces équations et je ne vais brièvement parler que de certaines d’entre elles et surtout celles que
j’ai rencontrées, de plus ou moins près, dans mon travail.
10
– La méthode des rayons est sans doute un des outils les plus populaires en sismologie. Cet
outil est basée sur une approximation hautes fréquences de l’équation des ondes et permet de
modéliser très efficacement des données réduites comme les temps d’arrivées des premières
phases des ondes de volumes. Sa facilité d’interprétation, liés à sa similitude avec l’optique
géométrique, ainsi que sa grande efficacité expliquent cette popularité. Il existe de nombreuses
variantes et de très nombreuses applications de cette méthode. Une bonne introduction sur les
rais en sismologie est donnée par Slawinski (2009).
– La méthode des modes propres est aussi une méthode très populaire notamment en sismologie globale où les ondes de surface sont très utilisées. Elle consiste à calculer d’abord les
modes propres d’un modèle de terre en résolvant l’équation des ondes en fréquence sans second membre, puis à obtenir des sismogrammes synthétiques en les sommant. Contrairement
aux tracés de rayons qui sont basés sur une hypothèse hautes fréquences, les méthodes de
modes propres résolvent l’équation des ondes sous sa forme la plus complète (notamment en
incluant la perturbation au potentiel de gravité). Néanmoins, le calcul des modes propres ne
peut se faire efficacement que pour des milieux 1-D, c’est à dire pour des milieux stratifiés ou
à symétrie sphérique pour la terre globale. La méthode pour calculer les modes est en général
basée sur une décomposition spectrale horizontalement (des harmoniques sphériques pour la
terre globale), une résolution des équations sous forme forte verticalement et en fréquences.
Toute la difficulté consiste à trouver toutes les fréquences propres. Pour des détails sur ce type
de méthode, on pourra consulter par exemple Takeuchi & Saito (1972), Woodhouse (1988),
Dahlen & Tromp (1998) ou, pour une version simplifiée, ma thèse de doctorat (Capdeville,
2000) et les références qui s’y trouvent. Ces méthodes permettent de calculer très efficacement, pour un modèle de terre 1-D donné, les données réduites comme les courbes de vitesses
de phases et de groupes des ondes de surface.
– Les perturbations des modes propres en milieux faiblement hétérogènes. Comme il l’a été dit
ci-dessus, les calculs des modes propres ne sont efficaces que pour des milieux 1-D. Ceci est
largement suffisant si on ne s’intéresse qu’aux vitesses de phases et de groupe des ondes de
surface, et que l’on considère que la terre est suffisamment peu latéralement hétérogène pour
qu’un modèle moyen 1-D par couple source-récepteur constitue une bonne approximation.
Maintenant, si on s’intéresse à la forme d’ondes de certaines phases, que l’on considère que les
hétérogénéités latérales sont de faibles amplitudes, les méthodes de perturbations des modes
propres sont adaptées. Ces méthodes sont asymptotiques, elles sont d’autant plus précises que
l’amplitude des contrastes d’hétérogénéités est faible. Elles permettent de prendre en compte
la propagation des ondes dans un milieu 3-D à partir de la base des modes propres 1-D. Il existe
de nombreuses références sur ce sujet, par exemple Woodhouse & Dahlen (1978), Tanimoto
(1984), Romanowicz & Snieder (1988), Lognonné & Romanowicz (1990), Dahlen & Tromp
(1998), etc.
– les différences finies. Si on veut s’affranchir de la limitation de faible amplitude des hétérogénéités,
il n’existe pas beaucoup d’autre choix que d’utiliser des méthodes numériques pour résoudre
directement l’équation des ondes (ou l’équation de l’eikonal si on s’intéresse aux temps d’arrivées des ondes balistiques. Ce genre de méthode commence a être appliqué par l’industrie
pétrolière pour des migrations itératives). Une des méthodes les plus populaires et les plus
utilisées est sans doute la méthode des différences finies. Les différences finies, en temps ou
–
–
–
–
–
11
en fréquence, sont basées sur une discrétisation des équations du mouvement sous forme forte
(basée sur la version locale des équations de l’élasto–dynamique). Elles ont souvent été utilisées avec succès pour résoudre l’équation d’ondes à deux ou trois dimensions à l’échelle
régionale (voir par exemple Alterman & Karal 1968; Alterman et al. 1970; Boore 1972; Alford et al. 1974; Kelly et al. 1976; Virieux 1986 ou Moczo et al. (2007) et les nombreuses
références qui s’y trouvent. Certaines applications à deux dimensions (axisymétrique) pour
les ondes SH ont été un succès dans la Terre globale (e.g. Chaljub & Tarantola, 1997). Cette
méthode est très compétitive et très utilisée en géophysique académique et industrielle. Sa
principale limitation est sans doute la surface libre avec topographie qui est difficile à prendre
en compte avec précision avec cette méthode, comme c’est le cas pour la plupart des méthodes
résolvant l’équation des ondes sous forme forte.
les éléments frontières. Les éléments frontières (voir par exemple Bouchon & Sanchez-Sesma
2007) sont basés sur le théorème de représentation qui permet de ne discrétiser que les frontières d’éléments. Cette méthode est très efficace tant que les fonctions de Green nécessaires à
la méthode peuvent être calculées simplement.
les éléments finis Les éléments finis, quant à eux, sont basés sur une formulation faible (basée
sur la version intégrale des équations de l’élasto–dynamique) des équations, en temps ou en
fréquence, qui intègre naturellement les conditions aux limites telle que la surface libre et
donc parfaitement adaptés aux ondes de surface. Ces méthodes, avec des bases polynômiales
d’ordre faible, sont malheureusement peu précises, dispersives (Dupond, 1973; Backer, 1976;
Marfut, 1984) et finalement assez peu utilisées en géophysique.
les méthodes spectrales ou pseudo spectrales résolvent les équations des ondes sous forme
forte et sont basées sur un développement en Fourier ou en polynômes des solutions en espace (voir par exemple Gazdag (1981) ou Kosloff et al. (1990)). Ces méthode sont parmi les
plus efficaces pour des milieux élastiques relativement simples, mais souffrent de difficultés
en présence d’une surface libre avec topographie et des milieux complexes avec des discontinuités. Ces méthodes sont finalement assez peu utilisées en géophysique.
les éléments spectraux La méthode des éléments spectraux (Patera, 1984; Maday & Patera,
1989) est une méthode d’éléments finis qui utilise des bases polynômiales orthogonales d’ordre élevé, issues des méthodes pseudo-spectrales. Le choix de cette approximation spatiale
et d’une quadrature numérique adaptée conduit à des résultats remarquables de convergence
notamment dans l’approximation des systèmes elliptiques d’équations aux dérivées partielles
(Azaiez et al., 1994). Cette méthode, appliquée à l’équation de l’élasto–dynamique, donne
des résultats tout aussi remarquables dans le sens où toutes les ondes, depuis les ondes de
volume jusqu’aux ondes de surface, sont bien modélisées. De plus, pour des maillages hexaèdriques, cette méthode conduit par construction à une matrice de masse diagonale, ce qui
permet d’utiliser des schémas en temps explicite et d’une grande efficacité. Il existe de nombreuses références sur ce sujet (voir Priolo et al. 1994; Seriani & Priolo 1994; Faccioli et al.
1996; Komatitsch & Vilotte 1998; Komatitsch & Tromp 1999; Vai et al. 1999 ou Chaljub et al.
2007 et les références qui s’y trouvent).
Les méthodes de Galerkin discontinues ; Depuis peu, les méthodes de Galerkin discontinues
sont appliquée à l’équation des ondes en sismologie (voir, par exemples, Dumbser & Käser
(2006), Grote et al. (2006), Etienne et al. (2009)). Pour ce type de méthodes, à la différence des
12
méthode de type éléments finis comme les éléments spectraux, la continuité des solutions entre
les éléments n’est pas assurées par construction, mais par des échanges de “flux” (il existe
plusieurs choix de “flux” possibles). Ces méthodes ont l’avantage de pouvoir accommoder
naturellement les discontinuités du déplacement pour, par exemple, des interfaces solidesfluides ou de part et d’autre d’une faille en rupture. Cette méthode a aussi l’avantage d’être
bien adaptée aux maillages tétraédriques et de permettre les discontinuités d’ordre polynomial
d’un élément à l’autre.
Il existe de nombreuses variantes de toutes ces méthodes (en temps, en fréquence etc), mais, pour
l’instant, il n’existe pas de solution numérique universelle dominant toutes les autres et la plupart
sont couramment utilisées et ont leurs adeptes selon les applications.
1.1.3
Le problème inverse en sismologie
Comme il l’a été dit plus haut, un des objectifs de la sismologie est de retrouver des informations sur les propriétés structurales, de densité, élastiques, an-élastiques, etc, de la Terre. On
désigne cet objectif sous les noms de problème inverse, imagerie, tomographie et d’autres noms
encore selon les variantes des méthodes, selon les communautés scientifiques et les contextes
dans lesquels ces techniques sont utilisées. Comparé aux problèmes d’imagerie rencontrés dans
le domaine médical, le problème en sismologie présente bien des difficultés. En effet, un des
atouts de l’imagerie médicale est que les propriétés acoustiques du corps humain sont bien connues et varient très peu par rapport à celles de l’eau (voir table 1.1). Dans ce cas, en choisissant
une source d’ondes acoustiques pour laquelle la méthodes des rayons est valable, c’est à dire de
suffisamment hautes fréquences, le problème inverse pour repositionner les réflecteurs est trivial.
En effet, au vu des valeurs présentées table 1.1, considérer le corps humain comme homogène est
une bonne approximation. Dans ce cas, les rayons des ondes acoustiques sont de simples droites
entre les sources et les récepteurs et, connaissant la vitesse des ondes acoustiques, “l’inversion”
est une simple conversion du temps d’enregistrement en distance à la source. C’est le principe
de base de l’échographie médicale, technique qui permet d’imager l’intérieur du corps humain
sans avoir d’accès quantitatif aux propriétés élastiques de la zone imagée. Avec ces méthodes,
les réflecteurs (interface avec un contraste de vitesse) sont seulement positionnés spatialement.
Pour le cas de la terre, bien que l’idée soit la même que pour l’imagerie médicale, les choses sont
en pratique plus compliquées.
À très grandes échelles (plusieurs milliers de kilomètres), la principale difficulté vient des sources
sismiques et de la répartition des récepteurs. À ces échelles, seules les sources sismiques naturelles comme les tremblements de terre, et éventuellement quelques explosions nucléaires,
ont l’énergie nécessaire pour éclairer de telles distances. Par ailleurs, les récepteurs sismiques
ne sont placées que très irrégulièrement sur la Terre (très peu de stations au fond des mers et
sur terre, la plupart des stations sont placés dans l’hémisphère nord). En conséquence, même si
la terre à grande échelle est relativement bien connue et que tracer des rayons dans des modèles de référence (comme, par exemple, le modèle PREM, Dziewonski & Anderson 1981) est
relativement simple, les méthodes types réflections utilisées en échographie médicale sont dif-
materiaux
l’air
la graisse
les tissus
le cerveau
le sang
l’eau
13
vitesse (m/s)
330
1450
1540
1541
1570
1480
TABLE 1.1 – Vitesses des ondes acoustiques dans le corps humain en m/s
ficilement transposables à la terre à grande échelle. Les méthodes de tomographie (qui existent
aussi en imagerie médicale) basées sur un problème inverse plus sophistiqué qu’une simple conversion temps-distance sont alors utilisées. Cette technique n’est pas adaptée pour positionner
des réflecteurs, mais permet d’accéder à certaines propriétés élastiques de la terre. L’inversion se
fait le plus souvent avec les données réduites comme les temps d’arrivée des principales ondes
de volume balistiques et les vitesses de phase et de groupe, mais aussi avec la forme d’ondes
en association avec un problème direct approprié. De nombreux modèles de manteau ont été
ainsi obtenus avec les temps d’arrivées de certaines ondes balistiques en utilisant la méthode
des rais dans un modèle de référence à symétrie sphérique (il existe un nombre importants de
publications sur le sujet, dont, par exemple Romanowicz (2003), pour une revue sur les modèles tomographiques globaux). De nombreux modèles du manteau supérieur sont obtenus grâce
aux mesures des vitesses de phase des ondes de surface. Les formes d’ondes ont été utilisées
très tôt associées à un problème direct à une dimension (Woodhouse & Dziewonski, 1984), puis
associées à un problème direct 2-D (Li & Romanowicz, 1995). Enfin, les formes des ondes
commencent à être timidement utilisées à grande échelle associé à un problème direct 3-D (les
éléments spectraux) (Fichtner et al., 2009). Pour l’instant ces travaux utilisent des fonctions coûts
qui favorisent largement la phase du signal, et dans un certain sens, ne sont pas complètement
des inversions de forme d’onde au sens strict.
Pour les échelles locales (de quelques kilomètres à l’échelle de la croûte), d’un côté le problème
se rapproche des conditions de l’échographie médicale du point de vue des sources et des récepteurs, mais s’en éloigne du point de vue de la connaissance du modèle de référence. En effet,
à ces échelles, on utilise des sources actives (explosifs, canons à air, camions vibrateurs) contrôlées et un grand nombre de récepteurs bien répartis. Par contre les variations des propriétés
élastiques du sol à cette échelle sont importantes (voir table 1.2) et a priori mal connues. De ce
fait, la technique élémentaire consistant à simplement transformer l’axe des temps en distance
le long de rayons en ligne droite n’est pas aussi simple pour la terre que pour le corps humain.
Pourtant, dans le domaine de l’exploration sismique, il existe nombre de méthodes basées sur la
même idée : la migration. Il existe un très grand nombre de variantes de la méthode de migration
très utilisée en exploration pétrolière (par exemple, migration à zéro offset après “normal move
out” (Claerbout, 1985), la migration de Kirchoff (Schneider, 1978), migration par retournement
temporel (Baysal et al., 1983), etc) et cela depuis longtemps (voir, par exemple, Bednar (2005) et
14
matériau
eau
granite
grès
calcaire
argile
onde P (km/s) onde S (km/s)
1.5
5.6
2.9
1.4-4.3
0.7-2.8
5.9-6.1
2.8-3.0
1.0-2.5
0.4-1.0
TABLE 1.2 – Vitesses d’ondes P et S (en km/s) dans différents matériau de la croûte
.
les références qui s’y trouvent). Comme pour l’échographie, les techniques de migration permettent principalement de positionner des réflecteurs mais pas d’accéder aux paramètres élastiques
du sous-sol exploré. De plus, l’idée de la migration est d’utiliser la cohérence en temps des traces
entre différents offsets après la migration, et donc seule une information de temps d’arrivée du
signal sismique est utilisée.
Pour aller plus loin, il a été reconnu très tôt en exploration sismique que l’utilisation de la forme
d’onde complète associée avec un problème direct adapté à la géométrie et à des milieux fortement hétérogènes a un potentiel d’informations très important. Bien qu’une méthode de type
Monte Carlo pour l’inversion serait l’approche la plus complète pour résoudre ce type de problème, seule les inversions locales de type moindre carré (Tarantola & Valette, 1982) ont été
envisagées et testées sur des problèmes réalistes. Un pas significatif vers ce type d’imagerie a été
franchi lorsqu’il a été montré que le gradient de la fonction coût de problème inverse par rapport
aux paramètres du modèle peut être calculé de façon très efficace par la résolution d’un problème
direct et d’un problème adjoint (Lailly, 1983; Tarantola, 1984). Ce type de méthode peut être implementé en temps (voir par exemple Tarantola (1986), Mora (1987)) ou en fréquences (voir par
exemple Pratt & Worthington (1990)). Du aux problèmes de coût de calcul, de telles approches
et d’algorithme (problème des minimums locaux de la fonction coût de problème inverse), ce
type de méthodes n’a commencé à être envisageables pour des cas réels que récemment et des
résultats prometteurs sont montrés à 2-D et 3-D (voir Virieux & Operto (2009) pour une revue
de ce domaine).
D’une manière générale, il semble que l’objectif à long terme des communautés travaillant à
l’échelle globale et à l’échelle locale est bien d’aller vers l’inversion de la forme d’onde totale.
Cet axe de recherche a toujours été en avance dans le domaine de l’exploration pétrolière, mais
finalement, au moment où ces travaux commencent à aboutir, les communautés travaillant à
grande échelle, voire à l’échelle globale, se rapprochent de plus en plus des techniques utilisées
en exploration pétrolière.
1.2. MON TRAVAIL DANS CE CADRE
1.2
15
Mon travail dans ce cadre
Depuis le début de ma carrière en recherche, c’est à dire depuis mon stage de DEA, j’ai contribué
de manière plus ou moins significative à la plupart des étapes méthodologiques de la sismologie vue comme un outil pour imager la terre. J’ai divisé ces contributions en “Contributions au
problème direct” et “Contributions au problème inverse” bien que ces distinctions ne soient pas
tout à fait évidentes. Par exemple, la résolution du problème direct orientée vers la résolution du
problème adjoint pourrait se trouver dans la section “problème inverse”. Mon travail sur l’homogénéisation appliqué à l’équation des ondes élastiques ayant des applications aussi bien dans
la résolution du problème direct que du problème inverse, se trouve dans une section à part.
1.2.1
Contributions au problème direct
Les perturbations de modes propres au premier ordre : couplage de modes
Mon premier travail en sismologie a consisté à calculer l’effet d’un hypothétique point chaud
sur les ondes de surfaces longues périodes. Ce travail a été publié dans l’article Capdeville et al.
(2000). Les panaches mantelliques étant supposés constituer une hétérogénéité de relativement
faible contraste en termes de vitesses sismiques et de densité, une méthode de perturbation par
rapport à un modèle de référence sans le point chaud semble une bonne approche. Pour des
données longues périodes à l’échelle globale, pour un modèle de référence à symétrie sphérique
(1-D), la méthode des modes propres est très efficace. Si on réécrit les équations des ondes
présentées section 1.1.2 en fréquence comme :
L(ω)u = f
(1.3)
L(ω)u = −ω 2 ρu + Au
(1.4)
L(ωk )uK = 0 ,
(1.5)
avec
où, par exemple, Au = −∇ · σ dans le cas élastique sans gravité. La base de modes propres est
constituée de l’ensemble des solutions
où ωk est la fréquence propre associée aux modes uK . Ces solutions ne peuvent être calculées
de façon relativement simple uniquement pour les modèles de terre à symétrie sphérique, et
dans ce cas, les modes sont dégénérés. Ceci explique qu’il y ait plusieurs modes propres uK par
fréquences propres ωk , et c’est pour cette raison que l’on utilise deux indices K = (q, n, l, m)
et k = (q, n, l), où q est le type de solution (sphéroidale (Rayleigh) et toroidal (Love)), n est
l’ordre radial, l l’ordre angulaire et m l’ordre azimutal. Pour des modèles plus complexes brisant
cette symétrie, on ne cherche pas les modes de façon direct, mais comme une solution perturbée
d’un modèle de référence 1-D. Au premier ordre (approximation de Born au premier ordre), la
16
perturbation de déplacement δu due à la perturbation du modèle à symétrie sphérique δL s’écrit
comme :
Z
X
δu(rr , t) · v =
gkk0 (t)
RK (rr )LKK 0 (r)SK 0 dr ,
(1.6)
KK 0
V
où rr est la position du récepteur, v est un vecteur récepteur, RK (rr ) un terme récepteur associé
au mode K, SK 0 un terme source associé au mode K 0 , gkk0 un terme en temps qui n’implique que
les fréquences ωk et ωK 0 et dont les expressions peuvent être trouvées dans Capdeville (2005), et
le terme de couplage des modes K et K 0 ,
LKK 0 (r) = u?K (r) · δL(r)uK 0 (r) .
(1.7)
L’intégrale sur le volume de l’équation (1.6) peut être calculée grâce à une intégration numérique
ou par une méthode spectrale. Pour une hétérogénéité localisée comme un point chaud, il est
judicieux de passer par une intégration numérique car la méthode spectrale nécessiterait un degré d’expansion très élevé pour représenter correctement l’hétérogénéité. L’avantage de ce type
de formulation est que l’on peut examiner explicitement les couplages entre les différents harmoniques et les types de modes (Love et Rayleigh). Le désavantage est le coût calcul qui augmente de façon vertigineuse avec le nombre de mode (avec la fréquence à la puissance 4 par
point d’intégration). Pour ce qui est de l’effet de points chauds proprement dit, la conclusion de
ce travail est que cet effet est relativement faible et malgré la conclusion relativement optimiste
de l’article, il est clair qu’observer cet effet sur les données longues périodes est difficile.
Les perturbations de modes propres au premier ordre : problème adjoint
Après ce premier travail sur les modes, j’ai repris cette méthode 4 ans plus tard avec pour objectif de calculer les dérivées partielles des fonctions de Green par rapport au paramètre décrivant
le modèle de terre (dérivées de Fréchet) du problème inverse de forme d’onde avec les sources
sommées à l’échelle globale (voir section 1.2.2). En effet, pour un modèle de terre de départ à
symétrie sphérique l’approximation de Born au premier ordre est un calcul exact des dérivées de
Fréchet. Il est possible de reprendre le travail de la section précédente et de l’appliquer directement au problème du calcul des dérivées de Fréchet. Cependant, dans le cas de terre globale,
l’intégrale de l’équation (1.6) portant sur tout le volume de terre et plus seulement sur le volume
très réduit d’un panache mantellique, ce calcul devient vite extrêmement lourd. Pour réduire
ce coût de calcul, une idée assez simple est d’utiliser les résultats de Lailly (1983) et Tarantola (1984) montrant que le gradient de la fonction coût du problème inverse associé à la forme
d’onde peut être calculé avec deux problèmes directs seulement, un depuis la source et un adjoint
depuis les récepteurs et en convoluant ces deux champs de manière adéquate en tous points de la
terre. Ce résultat peut être facilement obtenu en partant des mêmes équations qui aboutissent au
couplage des modes équations (1.6) pour aboutir à :
Z Z +∞
δu(rr , t) · v = −
B(r, τ, rr ) · δL(r)F(r, τ − t)dτ dr ,
(1.8)
V
−∞
17
où le champ calculé depuis les récepteurs aux points d’intégrations est,
X
B(r, t, rr ) =
u?K (r)RK (rr )gka (t) ,
(1.9)
K
et celui depuis la source aux points d’intégrations est,
X
F(r, t) =
uK (r)SK gkb (t) ,
(1.10)
K
où gka et gkb sont des fonctions du temps, qui n’impliquent que la fréquence ωk , dont les expressions peuvent être trouvées dans Capdeville (2005). La perturbation de déplacement δv due à une
perturbation du modèle de terre donnant une perturbation δL de l’opérateur élastique se calcule
comme la convolution d’un champ direct F et d’un champ adjoint B, ces deux champs étant
calculés dans le modèle de départ à symétrie sphérique. Avec un tel algorithme, si fmax est la
2
à
fréquence maximum de la source, le coût calcul pour un point diffractant croit comme fmax
4
quand on couple les modes. Cet algorithme rend possible l’utilisation des
comparer avec fmax
modes propres pour calculer les dérivées partielles dans le modèle de départ à symétrie sphérique
pour l’inversion de forme d’onde. Ce travail est publié dans Capdeville (2005) où des exemples
de comparaisons dans le cas anisotrope avec les éléments spectraux sont donnés. Cet algorithme
peut aussi être utilisé comme problème direct au premier ordre. Les limites d’une telle utilisation
sont montrées dans Capdeville (2005).
Le couplage éléments spectraux avec les modes
Comme il peut l’être vu dans les exemples donnés dans Capdeville (2005), l’utilisation de la
méthode de perturbations au premier ordre pour la résolution du problème direct est très limitée.
En effet, les effets non linéaires (diffractions multiples) apparaissent rapidement sauf pour des
temps de propagation très courts ou des modèles 3D avec des amplitudes d’hétérogénéités très
faibles. À l’échelle globale, la croûte et la couche D” sont des zones suffisamment hétérogènes
pour rendre inutilisable les méthodes de perturbation, même à longue période, sauf pour les
modes de vibrations les plus graves. Pour la terre globale, une idée attrayante est d’utiliser une
méthode numérique directe non limitée sur le type d’hétérogénéité, mais numériquement lourde,
pour résoudre l’équation des ondes dans la croûte ou dans la couche D” et une méthode de
mode où la terre peut être considérée comme à symétrie sphérique. C’est sur cette idée de départ
qu’était basée la plus grosse partie de mon travail de thèse. Par chance, en 1997, au début de
ce travail, les éléments spectraux basés sur la quadrature de Gauss-Lebatto-Legendre venaient
d’être introduits en sismologie à l’IPGP (Komatitsch, 1997; Komatitsch & Vilotte, 1998) et Emmanuel Chaljub commençait sa thèse sur son application à l’échelle globale (Chaljub, 2000).
Les éléments spectraux étant la première méthode numérique directe capable de résoudre tous
les types d’ondes avec précision pour un modèle de terre complexe, le couplage éléments spectraux - modes s’est imposé naturellement. Les éléments spectraux sont basés sur la forme faible
de l’équation des ondes. Cette dernière peut être obtenue en multipliant l’équation (1.1) par une
18
fonction de déplacement admissible w, en intégrant sur le volume et en utilisant la formule de
Green pour le terme élastique. La forme faible des équations est alors, pour tout déplacement
admissible w,
(ρ∂tt u, w) + a(u, w) − hTΓ , wiΓ = (f , w) ,
(1.11)
où
(u, v) =
a(u, w) =
Z
Z
Ω+
et
u(x).v(x)dx,
(1.12)
[∇u : c : ∇w](x)dx
(1.13)
Z
(1.14)
Ω+
hTΓ , wiΓ =
(TΓ .w)(x)dx .
Γ
−TΓ est la traction appliqué par le domaine où les modes sont utilisés sur celui des éléments
spectraux. Les définitions de Γ, Ω+ etc sont données figure 1.2.
F IGURE 1.2 – Gauche : schéma en coupe de la terre avec la surface libre (∂Ω) de rayon rΩ ,
les zones solides (ΩS ) les zones fluides (ΩF ), les interfaces solides-solides (ΣF F ) et solidesfluides (ΣF S ). Droite : un exemple de décomposition de domaine où les éléments spectraux sont
utilisés dans le domaine Ω+ et les modes dans Ω− . Γ est l’interface de couplage. Il n’y a dans
cet exemple qu’une seule interface de couplage, mais il est possible d’en avoir plus notamment
pour les applications à la couche D” (voir figure 1.3).
Le couplage s’effectue par le terme de surface (1.14). La solution modale permet de calculer un
opérateur A tel que
TΓ = A(uΓ ) ,
(1.15)
où uΓ est le déplacement du côté des éléments spectraux sur l’interface de couplage. Cet opérateur est un opérateur Dirichlet to Neumann (DtN) car il transforme une condition un déplacement
en une condition limite en traction. Cet opérateur est global sur Γ. Une des principales difficultés
de ce couplage vient du fait que cet opérateur est calculé en fréquence alors que les éléments
spectraux sont ici utilisés en temps. Cette opérateur ne pouvant pas être numériquement calculé
sur une bande de fréquence infinie, une troncature en fréquence est introduite. Cette troncature
est équivalente à un filtre passe bas, et ce genre de filtre est acausal. L’opérateur causal A devient
19
donc numériquement acausal, ce qui devrait rendre le couplage impossible. La solution adoptée
ici est une régularisation avec un opérateur asymptotique de A vers les hautes fréquences qui
peut être passé en temps analytiquement. En pratique, le couplage donne de très bons résultats
comme le montrent les publications Capdeville et al. (2003) et Capdeville et al. (2003). Cette
méthode a été appliquée pour une étude des hétérogénéités dans la couche D” pouvant expliquer
les anomalies observées sur les ondes Sdiff (To et al. (2005), voir figure 1.3), pour des tests de
validations (Capdeville et al., 2002; Capdeville, 2005; Capdeville & Marigo, 2008; Romanowicz
et al., 2008; Panning et al., 2009) pour tester les modèles tomographiques (Qin et al., 2006; Qin
et al., 2008; Qin et al., 2009) et pour l’inversion de forme d’onde à l’échelle globale (Capdeville
et al., 2005).
F IGURE 1.3 – Schéma d’un couplage éléments spectraux - modes utilisé pour la couche D”. Les
éléments spectraux, utilisés pour la couche D”, sont pris en sandwich entre deux domaines de
solutions modales pour le manteau et la graine. Le maillage est représenté pour deux régions
sur 6 et les couleurs représentent les variations de vitesses d’ondes S du modèle SAW12d (Li &
Romanowicz, 1996)
Comparaison éléments spectraux - perturbations de modes
Une des difficultés importantes des travaux de développement de solutions numériques est liée
à leur validation. C’est un travail souvent difficile faute de solution de référence adéquate. Pour
20
la terre à l’échelle globale et dans des modèles à symétrie sphérique, la solution modale n’est
pas une solution analytique (elle est quasi analytique pour une boule homogène, voir Capdeville
(2000)), mais est d’une précision telle qu’on peut la considérer comme une solution de référence
vis à vis des éléments spectraux utilisés dans le même contexte. La solution modale a été utilisée
comme solution de référence pour plusieurs travaux dont Capdeville (2000), Capdeville et al.
(2003), Chaljub (2000), Chaljub et al. (2003), Chaljub & Valette (2004) ou Komatitsch & Tromp
(2002). Pour le cas latéralement hétérogène, les choses ne sont pas aussi simples. Si on se place
dans le cas de contrastes d’hétérogénéités latérales d’amplitudes tendant vers zéro, alors la méthode de perturbations de modes propres au premier ordre constitue, à quelques détails près, une
méthode exacte. Dans ce cas, les perturbations de modes propres sont une méthode de référence
pour les éléments spectraux et elle a été utilisée dans ce sens dans Capdeville (2005). Si les
contrastes d’hétérogénéités latérales deviennent non négligeables, c’est l’inverse, les éléments
spectraux, étant capables de capturer les effets de diffractions multiples, deviennent la solution
de référence pour tester les limites des perturbations de modes dans ce contexte. Cela été le cas
dans Capdeville et al. (2002) et Capdeville (2005).
Le renversement temporel en sismologie
J’ai participé aux travaux de thèse de Carène Larmat sur le retournement temporel en sismologie, notamment en écrivant un programme de retournement temporel, basé sur les modes propres,
pour des modèles de terre à symétrie sphérique. Ce programme est toujours utilisé et sert actuellement pour la thèse de Huong Phung sous la direction de Jean-Paul Montagner. En principe, le
retournement temporel est basé sur l’invariance de l’équation des ondes (1.1) avec le changement
t → −t. Si on considère un contour volume fictif V de contour ∂V , le théorème de représentation (Aki, 1981), écrit de manière simplifiée, permet de calculer le déplacement u en tout point
y de V comme :
Z
˙ (x)dx3
u(y) =
G(x, y)∗f
(1.16)
VZ
2
˙
+
G(x, y)∗T(u)(x)dx
(1.17)
∂V
Z
˙ : ∇G(x, y) · ndx2
+
u(x)∗c
(1.18)
∂V
Z
+
ρG(x, y) · u̇0 (x)dx3
(1.19)
ZV
+
ρĠ(x, y) · u0 (x)dx3
(1.20)
V
˙ l’opération de convolution
où G(x, y) est la fonction de Green d’un point x à un point y, ∗,
en temps et de produit scalaire, T(u) la traction due au champ u sur l’interface ∂V , n la normale sortante du contour ∂V , u̇0 et u0 les conditions initiales du champ de déplacement et de
vitesse. Le déplacement u(y) s’écrit comme une somme de contributions où (1.16) correspond
21
à la contribution des sources externes dans V , (1.17) aux contributions des tractions sur le contour ∂V , (1.18) aux contributions des déplacements sur ∂V , et (1.19) et (1.20) aux contributions
des conditions initiales dans V . Pour une expérience de retournement temporel, on considère un
problème direct pour une force ponctuelle dans V . On enregistre le déplacement u et la traction
T(u) sur le contour ∂V jusqu’à un temps tmax tel qu’il n’y ait plus d’énergie élastique dans V .
Si on inverse le temps à partir de tmax et qu’on injecte les enregistrements de déplacement et de
traction sur le contour ∂V , on peut montrer, en se basant sur le théorème de représentation, que
le champ d’ondes va se propager vers la source, puis re-focaliser avant de re-diverger (voir, par
exemple, Cupillard 2008) . C’est le retournement temporel tel que le pratique l’équipe de Mathias Fink (voir, par exemple, Fink 1992). Maintenant, si on applique le même principe à la terre,
il y a un problème. En effet, les récepteurs étant à la surface libre, le contour ∂V se confond avec
la surface libre. Dans ce cas, la traction enregistrée au bord T(u) et la traction de la fonction de
Green au bord c : ∇G(x, y) · n sont, par définition, nulles pour tous les temps et ne contribuent
pas au théorème de représentation. Par contre, si on ne considère pas l’atténuation, les conditions
finales ne seront jamais nulles dans V et ce sont elles qui permettent de faire du retournement
temporel dans ce cas. Par exemple, pour une simulation numérique, à un pas temps donné, il
suffit d’inverser le signe du champ de vitesse, puis de continuer la simulation pour observer une
re-focalisation à la source. Malheureusement, pour la terre, nous ne disposons pas de récepteur
dans le volume et le renversement temporel est impossible.
Pour autant, tout n’est pas perdu et on peut faire quelque chose proche du retournement temporel
dans la terre. En effet, en injectant le champ de déplacement enregistré au bord ∂V dans la ligne
(1.17) à la place de la traction, on observe une re-focalisation à la source. L’interprétation du
phénomène est la suivante : si on considère le problème de minimisation d’un système de forces
f (x, t) sur la surface libre tel que
C(f ) = (d − G ∗ f )2 ,
(1.21)
où d représente l’ensemble des données, ∗ une convolution à la fois spatiale et temporelle, G
l’ensemble des fonctions de Green depuis les points de la surface vers les récepteurs. L’objectif
est de trouver f tel que C(f ) soit minimum. Le calcul de l’inverse de G par rapport à l’opération
de convolution spatiale et temporelle n’étant pas trivial, on peut au moins calculer le gradient de
la fonction coût par rapport à f . Le gradient de C(f ) par rapport aux paramètres du problème
(c’est à dire la nappe de force f (r) où r appartient à la surface de la terre), est
∂C
|f =0 = 2GT ∗ d ,
∂f
(1.22)
où T est la transposition. G étant symétrique, c’est exactement ce qu’on a fait en injectant le
champ de déplacement enregistré au bord ∂V dans la ligne (1.17) à la place de la traction, on
minimise donc C(f ). Ceci permet d’expliquer la re-focalisation dans ce cas et également le fait
que le mécanisme au foyer peut être observé sur les images de re-focalisation. C’est l’équation
(1.22) qui est implémentée dans le programme basé sur la méthode des modes évoquée au début
de cette section.
22
1.2.2
Contributions au problème inverse
Après avoir travaillé sur le problème direct, notamment les éléments spectraux, il semble assez
naturel de s’intéresser au problème inverse. L’apport des éléments spectraux à la sismologie
étant une bonne modélisation de la forme d’onde complète et ayant pour ma part principalement
travaillé à l’échelle globale durant ma thèse et les premières années de mon post-doc à Berkeley,
le problème inverse de la forme d’onde à longue période et à l’échelle globale s’est imposé de
lui même.
Inversion de la forme d’onde complète : la méthode des sources sommées à l’échelle globale
Au moment où j’ai commencé ce travail (en 2002), l’utilisation des éléments spectraux était encore trop lourde numériquement pour envisager un problème inverse par couple source - station
comme cela est fait pour les temps d’arrivées. Pour réduire considérablement le coût de calcul d’un problème inverse de forme d’onde avec les éléments spectraux (ou une autre méthode
numérique) l’idée utilisée dans Capdeville (2005) est la suivante : les éléments spectraux peuvent modéliser la propagation des ondes de plusieurs sources en même temps sans augmenter le
coût de la simulation. Bien sûr, aux récepteurs, on enregistre la somme des sismogrammes de
chaque source prise séparément sans aucun espoir de pouvoir les séparer a posteriori. Si on ne
peut obtenir de cette manière les enregistrements aux récepteurs dus à chaque source individuelle, on peut par contre sommer les données réelles aux stations pour toutes les sources. Par ce
procédé, les données réelles sommées sont comparables à la simulation des sources sommées
et un problème inverse peut être construit sur cette observable secondaire. Les tests effectués
F IGURE 1.4 – Configuration sources - stations utilisée pour l’expérience d’inversion par la méthode des sources sommées avec des données réelles.
dans Capdeville (2005) montrent que cette idée peut donner de bons résultats et ceci est confirmé par les travaux utilisant la même idée mais à l’échelle de l’exploration sismique (Krebs
23
et al., 2009; Ben Hadj Ali et al., 2009a; Ben Hadj Ali et al., 2009b). Pour l’échelle globale, une
des grandes difficultés de la méthode est liée aux données manquantes. Ces dernières doivent
idéalement être remplacées par des données synthétiques calculées dans les modèles intermédiaires obtenus au cours du processus d’inversion itératif. Les données manquantes étant souvent
nombreuses, calculer ces synthétiques peut être un problème majeur. La solution adoptée par
Capdeville (2005) est de remplacer ces données manquantes par des synthétiques calculés au
premier ordre avec des perturbations de modes propres. Une inversion sur des données réelles
avec cette méthode a été tentée en 2003 en collaboration avec Y. Gung et B. Romanowicz. Les
résultats étant peu concluants, cette dernière n’a jamais été publiée, mais en voici une partie. La
F IGURE 1.5 – Statistique de la présence de données pour l’expérience. En ligne les événements,
en colonnes les stations. Un carré noir indique le présence de la composante Z de la donnée.
Il y a ici 84.6% de présence de données. Le graphique à gauche indique la présence ou non
d’évènements dans les 8 heures avant ou les 6 heures après l’évènement principal. La taille des
carrés correspond à la magnitude (le gros carré bleu correspond à une magnitude 4.5)
configuration sources - stations de l’expérience est montrée figure 1.4. Le point difficile est trouver un maximum de données pour cette configuration. Il apparaît que pour cette configuration,
24
Exemple de traces sommées pour la station CMB (Berkeley). La bande de période sélectionnée
est 160 s-300 s.
F IGURE 1.6 –
84.6% des données sont disponibles et la statistique de présence des données est montrée figure
1.5. Un exemple de donnée sommée est montré figure 1.6 pour la station CMB. Un résultat de
l’inversion après trois itérations est montré figure 1.7 et comparé avec le modèle tomographique
SAW24 (Mégnin & Romanowicz, 2000) tronqué au degré 8. Le modèle obtenu n’est pas complètement stupide, mais de nombreux problèmes sont visibles, notamment on peut voir la signature du maillage utilisé pour la paramétrisation du modèle (concentration des hétérogénéités
sur les coins du maillage). De mon point de vue, le problème principal de cette expérience est
que seules les hétérogénéités d’ondes S sont inversées et que, pour la forme d’onde, ce n’est plus
le paramètre dominant (les hétérogénéités anisotropes ont aussi un effet très important sur les
formes d’ondes). Ces effets constituent un bruit important pour ce problème inverse et polluent
les résultats. Pour un meilleur résultat, il faut donc, soit inverser un jeu de données massif (ce
qui est loin d’être le cas ici), soit inverser les paramètres manquants (comme l’anisotropie). C’est
25
ce deuxième chemin que je commençais à emprunter quand les problèmes de paramétrisations
verticales et les hétérogénéités dans la croûte m’ont amené à l’homogénéisation. J’espère revenir
à ce problème inverse prochainement.
F IGURE 1.7 – Comparaison de l’inversion de forme d’onde avec la méthode des sources sommées
(à gauche) avec le modèle tomographique SAW24 (Mégnin & Romanowicz, 2000) tronqué au
degré 8 (à droite).
Benchmark des méthodes tomographiques à l’échelle globale
La précision des éléments spectraux peut être mise à profit pour tester les méthodes tomographiques existantes et aussi tester les modèles tomographiques existants. C’est ce qui a été
fait durant la thèse de Yilong Qin et publié dans Qin et al. (2006), Qin et al. (2008) et Qin et al.
(2009).
Pour ce qui est de tester les méthodes tomographiques existantes, nous sommes partis du constat
qu’il existe actuellement de nombreuses méthodes tomographiques (une par groupe de recherche
dans ce domaine, ou presque). Ces méthodes ont produit de nombreux modèles, et ce qui est
26
remarquable est que ces modèles, même s’ils sont différents dans les détails, ont, pour la plupart,
leurs grandes lignes en commun (voir figure 1.8).
F IGURE 1.8 – Différents modèles tomographiques à l’échelle globale par différents groupes
(SAW24B16 : Berkeley ; S362D1 : Havard, SB4L18 : Scrippys ; S20RTS : Oxford ; Grand :
Haustin). Figure d’après B. Romanowicz.
Ces points communs entre les différents modèles existent-ils car ils correspondent à des structures réelles de la terre ou bien parce que ces méthodes ont toutes une base commune ? à moins
que ce ne soit un peu des deux. Dans le but de tenter de répondre à ces questions, dans le cadre du
projet européen SPICE, un test en aveugle a été développé. Il est constitué d’un jeu de données
synthétiques généré dans un modèle de terre réaliste et complexe grâce aux éléments spectraux.
Ces données devaient ensuite être utilisées par différents groupes (du projet SPICE et extérieur)
de façon aveugle (c.à.d sans rien connaître du contenu du modèle à retrouver) et ainsi déterminer les succès mais aussi les erreurs individuelles et communes de ces différentes techniques.
Le modèle de terre “réaliste” a été généré en collaboration avec V. Maupin (Université d’Oslo).
Cette base de données a eu un succès mitigé car seulement deux groupes ont testé leur méthode
sur ce modèle (un 3eme est en train de l’utiliser actuellement). Ces tests montrent d’abord une
certaine robustesse de ces méthodes mais permettent de quantifier plusieurs problèmes connus
de ces méthodes basés sur les ondes de surface, comme l’importance des corrections de croûte,
la perte de résolution en profondeur, l’étalement des hétérogénéités, la difficulté de retrouver la
vraie amplitude des hétérogénéités notamment pour l’anisotropie, etc. A posteriori, pour fournir
un meilleur test à la communauté sismologique, il aurait fallu fournir des bases de données pour
plusieurs modèles synthétiques (un facile, un plus complet et un difficile avec notamment des
hétérogénéités de petites échelles) et une bande de fréquence plus large (mais cela nécessiterait
beaucoup plus de ressources informatiques que ce que nous avons utilisé). Ces travaux sont publiés dans Qin et al. (2006) et Qin et al. (2008).
Un test proche du précédant consiste à tenter d’évaluer la qualité de certains modèles tomographiques avec les éléments spectraux. L’idée est cette fois de collecter un jeu de données réelles
de bonne qualité, de modéliser ces données dans différents modèles tomographiques grâce à SEM
et de comparer la qualité des prédictions des différents modèles. C’est cette idée qui a été utilisée
27
dans Qin et al. (2009). Si ce travail montre que tous les modèles testés améliorent la qualité de la
forme d’onde par rapport à un modèle de référence à symétrie sphérique, il apparaît aussi qu’il
très difficile de distinguer les différents modèles de cette manière. En effet, il apparaît d’abord
qu’aucun de ces modèles n’est adapté au calcul de la forme d’onde. Ceci est principalement dû
aux corrections de croûte qui sont apportées à ces modèles. En effet, la croûte est trop fine et trop
complexe pour être inversée avec ces méthodes et, pour éviter qu’elle ne pollue les modèles, son
effet est corrigé avec plus ou moins de succès grâce à un modèle de croûte a priori. Malheureusement, l’effet de cette croûte doit être pris en compte dans les simulations et il faut donc re- inclure
cet effet qui a été si difficile à enlever. Malheureusement, il est presque impossible d’inclure la
croûte dans les éléments spectraux de façon consistante avec la méthode de correction utilisée
dans les méthodes tomographiques (qui sont de plus différentes pour chaque modèle). De plus,
a posteriori, cette façon de tester les modèles n’est probablement pas très pertinente. En effet,
l’objectif de ces méthodes tomographiques n’est pas de bien prédire la forme d’onde mais de
retrouver l’information sur la terre. Si on recherche un modèle de terre exhaustif (c’est à dire
avec tout ce que l’on peut imaginer influencer la forme d’onde), un bon modèle de terre devrait
bien prédire les formes d’ondes. En pratique, ce n’est pas nécessairement le cas, par exemple, si
une méthode est uniquement faite pour retrouver les hétérogénéités d’onde P. Dans ce cas, même
si l’effet de l’onde P n’est pas dominant sur la forme d’onde pour les signaux sismiques pris
individuellement, on va sélectionner un grand nombre de données et les traiter de telle façon que
tous les autres effets (hétérogénéités d’ondes S, de densité etc) deviennent, par effet statistique,
le plus faible possible par rapport à celui des hétérogénéités de l’onde P. De cette façon on peut
retrouver les hétérogénéités de l’onde P sans pour autant expliquer beaucoup d’effets sur les données. De manière générale, le meilleur modèle d’un paramètre donné n’est pas forcement celui
qui explique le mieux les formes d’ondes, et, pour conclure par un exemple, il se peut qu’un très
bon modèle d’onde P explique moins bien les formes d’onde qu’un mauvais modèle d’onde S.
Utilisation de la forme d’onde des corrélations de bruit
Un domaine très à la mode ces dernières années concerne l’obtention de fonctions de Green
entre deux récepteurs en corrélant les données de bruit (sans séisme). Il a été en effet montré
que, si le bruit est idéal, ces corrélations du bruit sismique entre deux récepteurs permettent de
retrouver les fonctions de Green entre les deux stations sismiques utilisées (Lobkis & Weaver,
2001). Ce procédé est très séduisant car il permet d’obtenir des données sans séisme et il a déjà
été utilisé pour produire des images tomographiques à partir des vitesses de phases d’ondes de
surface (voir, par exemple, Shapiro et al. 2005). Pour l’inversion de forme d’onde, ces données
sont potentiellement très intéressantes pour la méthode des sources sommées car il n’existe aucune donnée manquante dans ce cas. L’objectif de ce travail était donc d’évaluer notre capacité à
modéliser ces “fonctions de Green” résultant des corrélations de bruit. Ce travail s’est fait dans
le cadre de la thèse de Paul Cupillard (soutenue en avril 2008). La principale difficulté de ce
travail vient du fait que le bruit réel est loin d’être idéal et que l’hypothèse d’une “vraie” fonction de Green n’est pas valable pour une propagation 3D. Par exemple, ces corrélations n’ont
pratiquement pas de modes harmoniques alors que les vraies fonctions de Green en ont. La pre-
28
mière partie de ce travail a été de montrer que l’on retrouve bien la forme d’onde dans un milieu
anélastique malgré les traitements non linéaires appliqués aux données (Cupillard & Capdeville,
2010). Du fait de la répartition du bruit réel, les corrélations de bruit ne donnent pas vraiment des
fonctions Green, mais plutôt des fonctions de green convoluées partialement et temporellement
par une source à déterminer. L’idée ici était de trouver un contour de stations autour de la station
source, de retourner temporellement le signal sur ce contour grâce au théorème de représentation
pour retrouver des conditions initiales les plus centrées possible sur la station source. Ces conditions initiales peuvent être considérées comme la source des corrélations centrée sur la station
en question. Il a été montré durant la thèse de Paul Cupillard que cette approche est possible
et les essais sur des données réelles ont été concluants. Cependant, le procédé est suffisamment
complexe pour que l’inversion des formes d’onde des corrélations par la méthode des sources
sommées ne soit pas envisagée pour l’instant.
1.2.3
Homogénéisation
Depuis 2006, mon activité principale de recherche est axée sur l’aspect multi-échelles de la
propagation d’ondes et de l’imagerie sismique. De 2006 à 2009, j’ai effectué ce travail dans le
cadre de l’ANR blanche MUSE que j’ai dirigé. Bien que je l’ai pour l’instant principalement
appliqué au problème direct, ce travail rentre aussi bien dans la catégorie du problème direct que
du problème inverse.
Pour le problème direct, l’idée est que d’importantes limitations et difficultés des méthodes numériques telles que les éléments spectraux sont liés aux échelles plus petites que la longueur
d’onde. Ces petites échelles imposent un maillage fin, souvent difficile à réaliser et impliquant un
coût de calcul très important. Une solution serait de remplacer les détails du milieu plus petits que
la longueur d’onde par un milieu équivalent effectif, mais il n’existait pas de solution pour trouver
ce milieu effectif jusqu’à présent. L’objectif de ce travail est de trouver le milieu effectif ainsi
que les équations effectives pour un modèle élastique et pour une bande de fréquences données.
Ces problèmes multi-échelles sont très étudiés et bien connus en mécanique des matériaux et
ont donné lieu à la théorie de l’homogénéisation des milieux périodiques. Nous appliquons et
étendons ces techniques au cas dynamique et non périodique. La principale difficulté apparaît
pour le cas non-périodique et pour les dimensions spatiales strictement supérieures à un.
Pour le problème inverse, les progrès dans le problème direct permettent maintenant d’aborder le
problème de l’inversion de la forme d’onde complète, mais, pour des raisons de coût, uniquement
dans une bande de fréquences limitée. Dans ces conditions, les modèles tomographiques obtenus
ne sont en aucun cas le “vrai” modèle, mais une version effective de ce modèle. Il est donc
primordial de comprendre la relation entre un vrai modèle et celui ’vue” par le champ d’ondes
dans cette bande de fréquences. L’homogénéisation est donc indispensable dans un but d’interprétation des modèles tomographiques. Pour un problème d’imagerie, nous pouvons utiliser
les résultats d’homogénéisations pour trouver une paramétrisation consistante avec la bande de
fréquences utilisée et ensuite pour trouver les modèles de terre a priori compatibles avec le mod-
29
èle tomographique obtenue par imagerie.
Principe
Le principe de base de l’homogénéisation s’établit pour les milieux périodiques (pour le cas dynamique, on peut se référer à Sanchez-Palencia (1980)). C’est dans ce cas que les résultats de
l’homogénéisation sont démontrés. Les cas qui nous intéressent sont bien sûr non périodiques
et l’extension de la théorie de l’homogénéisation du cas périodique au cas non périodique pour
l’équation des ondes élastique est ma principale contribution dans ce domaine. L’homogénéisation périodique repose sur l’introduction d’un petit paramètre
ε=
`
λmin
,
(1.23)
où ` est une grandeur caractéristique du milieu périodique et λmin la longueur d’onde minimum
du champ d’ondes dont l’existence est garantie si la source a une fréquence coin fmax (ou si les
données sont filtrées au delà de fmax ). On construit une série de modèles indexés par ε, ce qui
implique de renommer la masse volumique et le tenseur élastique ρε et cε . De même, la solution
que l’on cherche est aussi indexée sur ε et est nommée uε . On introduit une nouvelle variable
d’espace dite microscopique
x
(1.24)
y= .
ε
Bien que y et x ne soient pas indépendants au vue de la définition précédente, on les considère
comme tel dans la construction du problème homogénéisé. Le petit paramètre ε et la séparation
explicite des échelles va permettre la construction d’une série asymptotique de problèmes à deux
échelles indépendantes x et y qui va tendre vers un problème à une échelle quand ε tend vers
zéro. Pour cette construction, on transforme les gradients tel que
1
∇ → ∇x + ∇y ,
(1.25)
ε
où ∇x et ∇y sont les gradients par rapport à x et y respectivement, et on cherche une solution
comme
uε (x) = u0 (x, y) + εu1 (x, y) + ε2 u2 (x, y) + ...
(1.26)
où y est pris en x/ε. On construit une masse volumique et un tenseur élastique, dits de cellule,
qui ne dépendent pas de ε
ρ(y) = ρε (εx)
(1.27)
c(y) = cε (εx)
La construction (1.27) est triviale dans le cas périodique, mais c’est cette dernière qui constitue la
principale difficulté dans le cas non-périodique. En introduisant (1.25) et (1.26) dans l’équation
des ondes, on obtient une série d’équations que l’on doit résoudre une à une :
ρ∂tt ui − ∇x · σ i − ∇y · σ i+1 = f δi,0 ,
σ i = c : x ui + y ui+1 ,
(1.28)
(1.29)
30
où les σ i correspondent au même développement que (1.26) mais pour la contrainte, x et y aux
opérateurs déformations par rapport aux variables x et y respectivement. La résolution de ces
équations permet de montrer que,
– à l’ordre zéro, u0 ne dépend pas de la variable rapide y. C’est un résultat fondamental bien
connu en sismologie : le déplacement du sol ne dépend pas fortement de la structure locale
autour du récepteur (ce qui n’est pas le cas de la déformation ou la contrainte par exemple).
Ce déplacement à l’ordre 0 est solution d’une équation d’onde élastique classique mais pour
une masse volumique et un tenseur élastique effectifs. Le tenseur élastique effectif nécessite
le calcul d’un correcteur du premier ordre, solution d’une équation de type équilibre statique
(qui implique une résolution de type éléments finis) posée sur une cellule périodique que l’on
appelle problème de cellule.
– à l’ordre 1, les conditions limites doivent être changées.
– à l’ordre 2 et plus, l’équation effective à résoudre n’est plus une équation d’onde classique et
implique des dérivées spatiales d’ordres élevés.
Voilà pour le cas périodique. Pour le cas non périodique, toute la difficulté est de construire un
problème à deux échelles qui ait un sens. Pour cela, la première étape est d’introduire un filtre
passe bas F ε0 avec un nombre d’onde de coupure k0 (ε0 = (λmin k0 )−1 ) défini par l’utilisateur.
Ce filtre permet de séparer manuellement les nombres d’ondes macroscopiques (k < k0 ) des
nombres d’ondes microscopiques (k > k0 ). Ensuite, l’étape la plus difficile est la construction
des ρ et c de l’étape (1.27) mais dans le cas non périodique. Le cas périodique est idéal, car,
quelque soit la manipulation non linéaire appliquée à ρ et c, le résultat sera toujours périodique
et les deux échelles bien séparées. Pour le cas non périodique, c’est une autre paire de manches et
toute la difficulté est de bien construire les champs ρ et c tels que les échelles soient bien séparées.
Pour cela on distingue deux cas : le cas stratifié où on peut compter sur une solution analytique au
problème de cellule pour faire les bons choix (Capdeville & Marigo, 2007; Capdeville & Marigo,
2008; Capdeville et al., 2010a) et le cas général où cette solution analytique n’existe pas et ou
l’on doit s’en remettre à une construction implicite (Capdeville et al., 2010a; Guillot et al., 2010;
Capdeville et al., 2010b). Cette construction est donnée dans l’article Capdeville et al. (2010b)
en annexe.
L’homogénéisation des milieux stratifiés
Les milieux les plus simples pour la théorie de l’homogénéisation sont les milieux stratifiés, c’est
à dire les milieux pour les quels les variations rapides n’ont lieu que dans une seule direction.
C’est en effet le seul cas où il existe une solution analytique au problème de cellule. L’avantage
de cette solution analytique est qu’elle permet de connaître explicitement sur quelles quantités les
échelles doivent être séparées. Par exemple, dans le cas d’une barre de paramètre élastique E, la
solution analytique du problème de cellule permet de savoir que le paramètre élastique effectif E ∗
s’obtient comme la moyenne harmonique de E et non comme la moyenne arithmétique comme
on aurait pu le penser. Pour le cas non-périodique, si on utilise un filtre spatial passe bas F
pour séparer les échelles, il est intuitivement raisonnable d’utiliser E ∗ −1 = F(E −1 ) et c’est
31
effectivement ce qui est montré dans Capdeville et al. (2010a). Appliquée aux milieux stratifiés
dans Capdeville & Marigo (2007), cette théorie à l’ordre 0 permet de retrouver les résultats
de Backus (1962). Elles permet aussi de trouver les correcteurs des effets des structures fines
locales à la source et aux récepteurs. Pour ce qui est de la source, le correcteur d’ordre 0 donne
un résultat très proche de celui obtenu par Woodhouse (1981) pour une source placée de part et
d’autre d’une discontinuité élastique.
L’homogénéisation des milieux plus généraux
Notre principale avancée dans ce domaine ces deux dernières années a été l’extension de l’homogénéisation non périodique du cas 1D au cas 2D ou 3D (voir la figure 1.9). Cette extension
constituait la principale difficulté de ce travail, c’est à dire la construction implicite des champs ρ
et c mentionnés plus haut. Avec ce travail, nous sommes maintenant en mesure de trouver un milieu effectif pour n’importe quelles structures géologiques (voir (Capdeville et al 2010b, Guillot
et al 2010 et l’exemple donné figure 1.9). Le calcul de ces milieux effectifs dans le cas nonpériodique n’est pas simple et implique la résolution d’équations de type “équilibre statique”,
soit sur l’ensemble du domaine (de résoudre le problème de cellule sur une seule grande cellule), soit sur chaque sous-domaine d’un pavage du domaine (et de résoudre un grand nombre
de problèmes de cellules sur des petits domaines). Dans les deux cas, ces problèmes de cellules
impliquent un maillage tétrahédrique de la structure fine et un coût de calcul non négligeable.
Néanmoins ce calcul ne doit être fait qu’une seule fois pour un milieu donné. Ce travail a donné
lieu à trois publications (Capdeville et al 2010a, 2010b, Guillot et al 2010) et a impliqué le travail
d’un post-doc, Laurent Guillot, financé par l’ANR MUSE. Une demande de brevet a aussi été
déposée par le CNRS sur ce thème (Capdeville, 2009).
32
F IGURE 1.9 – a : Modèle élastique 2D Marmousi2. Les couleurs représentent les vitesses des
ondes P. Les traits gris les interfaces physiques.
b et c : Le modèle homogénéisé (valable pour une fréquence coin de la source jusqu’à 15Hz) est
un modèle complètement anisotrope (l’anisotropie totale est représentée en b) et sans interface
(lisse, comme on peut le voir pour les variations des ondes S en c), même si le modèle original
est isotrope et avec de multiples interfaces.
d : calculer une solution de référence dans le modèle original avec les éléments spectraux nécessite de générer un maillage complexe et coûteux (ce qui n’est possible que dans ce cas à 2D. Un
tel maillage n’est pas envisageable à 3D). Le calcul de référence de cet exemple a nécessité une
semaine de calcul sur 64 processeurs.
e : Le modèle homogénéisé étant lisse, le maillage est trivial. Le calcul de la solution homogénéisée n’a nécessité qu’une heure de calcul, toujours sur 64 processeurs (à comparer avec
les 7 jours de calculs nécessaires pour obtenir la solution de référence).
f : La solution de référence (noir) et la solution homogénéisée (pointillés rouges) pour un exemple de récepteur en profondeur sont en très bon accord. Pour comparaison, la solution calculée
dans un milieu lisse, obtenu en filtrant les vitesses des ondes S et P ainsi que la densité avec le
même niveau de détails que pour le milieu homogénéisé, est tracée en vert et n’est pas en accord
avec la solution de référence.
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Chapitre 2
Persectives de recherche à court et moyen
termes
2.1
Perspectives à court terme
Pour les 4 ou 5 prochaines années, mon objectif principal est de développer et de compléter
l’homogénéisation d’un milieu géologique dans toute sa complexité, en incluant les effets près
de la surface libre, les grandes variations de vitesses avec la profondeur etc, d’appliquer les outils
développés à certains problèmes d’intérêt ainsi que de fournir les programmes correspondants à
la communauté sismologique. Un projet de recherche dans ce sens a été déposé à l’ANR (ANR
mémé) et vient d’être accepté.
Dans le détail, l’objectif principal de ces prochaines années est d’abord de développer un outil
d’homogénéisation non-périodique pour les milieux 3-D. D’un point de vue théorique, peu de
difficultés sont attendues. En effet, la principale difficulté, c’est à dire passer du 1-D au 2-D, a été
levée et la solution adoptée à 2-D est aussi valable à 3-D. Du point de vue pratique, les choses
seront plus difficiles à 3-D qu’à 2-D dans la mesure où la résolution du problème de cellule
à 3-D peut être numériquement coûteux. La méthode numérique à développer pour obtenir un
outil d’homogénéisation 3-D optimisé nécessitera un travail important. Le second objectif est
de prendre en compte de façon consistante les petites hétérogénéités près de la surface libre.
Ce travail a déjà été fait pour le cas des milieux stratifiés (1-D) en se basant sur une méthode
de séries asymptotiques raccordées (Capdeville & Marigo 2008). Ce développement doit être
généralisé au cas 3-D et ce travail nécessite un effort théorique non négligeable. Un troisième
objectif est d’adapter la méthode à des milieux géologiques pour lesquels la longueur d’onde
dominante varie fortement avec la profondeur. Pour cela, le filtrage spatial utilisé doit pouvoir
varier avec la profondeur, de telle façon que le ratio de la longueur d’onde minimum du filtre
sur la longueur d’onde minimum du champ d’ondes soit le plus constant possible sur tout le
domaine. Une piste envisagée est l’utilisation d’une décomposition en ondelettes. Finalement,
39
40
CHAPITRE 2. PERSECTIVES DE RECHERCHE À COURT ET MOYEN TERMES
le 4ème objectif à court terme est de modifier le programme d’éléments spectraux de l’équipe
de sismologie pour l’adapter à l’homogénéisation (principalement au niveau des sources, des
récepteurs et de la surface libre). Une fois tous ces développement terminés, un effort sera fait
pour mettre ces programmes à la disposition de la communauté sismologique.
Pour ce qui est des applications, 3 projets sont envisagés.
Le premier concerne l’interaction d’une source avec la structure géologique locale. En effet, l’homogénéisation permet de calculer l’effet macroscopique d’une structure microscopique sur une
source sismique. Pour une explosion, ce correcteur permet de prendre en compte et d’expliquer
les conversions d’ondes P en S dans le champ proche de la source (voir un exemple figure 2.1).
Pour les explosions nucléaires, pour lesquelles une forte onde S peut être observée, cet aspect de
F IGURE 2.1 – Instantané de l’énergie du champ d’ondes généré par une explosion dans un milieux fortement hétérogène, notamment dans le champ proche de la source. Derrière l’onde P
balistique, une onde S apparaît clairement. Cette onde S est expliquée par une conversion d’onde
P en S dans le champ proche de la source.
l’homogénéisation est intéressant. Une collaboration avec les CEA va commencer sur ce sujet
et il est prévu de travailler sur la “Non-Proliferation Experiment” (NPE) de 1993. Au cours de
cette expérience, une explosion chimique de 1kilo tonne a été déclenchée dans le Nevada à une
profondeur comprise entre 300 et 400 m avec une bonne couverture instrumentale (les données
sont disponibles dans la base de données de l’IRIS). Pour cette expérience, il apparaît que le ratio
P/S est anormal, ce qui est encourageant pour notre objectif.
La seconde application envisagée concerne les hétérogénéités proches de la surface libre ou
d’interface solide-fluide dans le cadre de l’exploration pétrolière. En effet, ces hétérogénéités
ont une forte influence sur la qualité des campagnes d’exploration sismique principalement en
2.2. PERSPECTIVES À MOYEN OU LONG TERMES
41
raison de l’importante part d’énergie du champ d’ondes concentrée dans les ondes de surface.
L’homogénéisation, en permettant de trouver les paramètres élastiques effectifs des structures
complexes influençant les ondes de surface est une piste intéressante pour avancer sur ce problème. Un projet sur ce sujet avec Schlumberger va bientôt démarrer.
La troisième application concerne la distinction entre de l’anisotropie intrinsèque de l’induite
par effet de changement d’échelles. L’objectif de cette étude est d’évaluer si cette distinction
est possible et dans quel cas. Ce travail va commencer prochainement dans le cadre du projet
européen QUEST et en collaboration avec l’entreprise Microseismic.
A moyen terme, l’homogénéisation réserve sans doute beaucoup d’autres applications qui apparaîtront avec le temps. Par exemple, une application possible, discutée avec Romain Prioul
de Schlumberger, concerne le “upscaling” des mesures de puits. En effet, les vitesses des ondes
ainsi que l’orientation des fissures mesurées à petite échelle dans les puits doivent être mises à
l’échelle de la propagation des ondes sismiques (upscaling). Or, les techniques classiques pour
réaliser ce genre d’opération (par exemple, le filtrage de Backus) ne sont pas effectives du fait de
la nature 3-D du problème lié aux fissures. L’homogénéisation non périodique peut potentiellement résoudre ce problème.
2.2
Perspectives à moyen ou long termes
L’objectif à long terme de mon travail est toujours centré sur l’imagerie sismique. Depuis mon
travail sur l’homogénéisation, il me semble qu’une piste de recherche intéressante est de travailler sur une méthode d’imagerie en deux étapes. La première étape aurait pour objectif de
retrouver le modèle de terre “vu” par le champ d’ondes. Ce modèle tomographique est un modèle homogénéisé. Ce dernier est difficile à interpréter mais, dans le cas d’une couverture sourcestation idéale, il intègre toute l’information contenue dans les données de formes d’ondes. Du
point de vue d’une méthode d’imagerie, ce modèle n’est qu’une étape intermédiaire mais, en
pratique, il a un grand intérêt pour sa capacité de prédiction des données. En effet, pour un
source donnée et une bande de fréquences identique à celle utilisée pour l’inversion, les formes
d’ondes calculées dans ce modèle devraient être en accord excellent avec les données. Cet aspect est potentiellement intéressant pour des travaux sur la source sismique par exemple. Une
fois le modèle tomographique obtenu, la deuxième étape, c’est à dire le travail d’interprétation,
commence. Moyennant des informations a priori, il s’agit de trouver tous les modèles de terre
compatibles avec le ou les modèles tomographiques possibles précédemment obtenues. Une approche envisageable serait de tirer les modèles possibles par une méthode de Monte Carlo, de les
homogénéiser et de comparer le résultat avec celui de l’inversion. Les modèles à retenir seraient
dont le modèle homogénéisé serait proche du modèle tomographique obtenu à partir des formes
d’ondes. Un intérêt de cette approche en deux étapes est qu’aucune simulation de champ d’ondes
n’est nécessaire pour la deuxième étape.
Pour cet objectif à long terme, un bon cadre de travail me semble l’imagerie à l’échelle globale
42
CHAPITRE 2. PERSECTIVES DE RECHERCHE À COURT ET MOYEN TERMES
et à longue période. En effet, c’est un objectif relativement modeste dans le sens où les formes
d’ondes sont déjà relativement bien expliquées par les modèles de terre à symétrie sphérique,
et, qu’à longue période (par exemple 150 s et plus), la couverture des données est bonne et
stable. De plus, nos travaux (Qin et al., 2009) ont montré que, s’il existe un grand nombre de
modèles de terre globaux 3-D à longues périodes, il reste pas mal de chemin à parcourir pour
vraiment expliquer les formes d’ondes, et donc qu’il reste beaucoup d’informations à utiliser.
Le premier objectif est de trouver le modèle homogénéisé, probablement avec la méthode des
sources sommées avec une méthode de quasi-Newtown et un Hessian approximé calculé avec
des couplages de modes.
Chapitre 3
CV, activité d’encadrement, liste des
publications
3.1
CV
Identité
Yann Capdeville, 38 ans (date de naissance 01/06/1972).
Chargé de recherche 1ère classe au CNRS depuis octobre 2003.
Équipe de sismologie de l’Institut de Physique du Globe de Paris (UMR7154) jusqu’à fin août
2010.
Laboratoire de Planétologie et Géodynamique de Nantes (UMR6112) à partir de septembre 2010.
CV résumé
2003-2009
2000-2003
1996-2000
1996
1995
Chargé de Recherche au CNRS. Institut de Physique du Globe de Paris
Post-doctorat à l’Université de Berkeley, Californie, États Unis.
Thèse de Doctorat de l’Université de Paris 7.
Service militaire au CEA DAM.
DEA de géophysique interne de l’IPGP. Magistère de physique théorique d’Orsay.
Diplômes
Doctorat de 3ème cycle en Géophysique Interne, mention très honorable avec les félicitations du
jury. Université Paris 7. Septembre 2000.
Magistère de Physique fondamentale d’Orsay. Mention très bien.
Postes occupés
43
44
CHAPITRE 3. CV, ACTIVITÉ D’ENCADREMENT, LISTE DES PUBLICATIONS
Thèse 1996-2000
Département de sismologie de l’Institut de Physique du Globe de Paris. Direction Jean-Paul
Montagner et Jean-Pierre Vilotte.
Post-doctorat 2000-2003
Berkeley seismological Laboratory, Californie. Direction : Barbara Romanowicz. Boursier du
“Miller Institute”
CNRS 2003-2010
Chargé de recherche 1ère classe au CNRS depuis octobre 2003.
Équipe de sismologie de l’Institut de Physique du Globe de Paris (UMR7154)
3.2
Activités d’encadrement, d’enseignement et administratifs
– J’ai assuré de nombreux travaux dirigés durant ma thèse, au cours de laquelle j’ai été moniteur
durant deux années puis ATER la dernière année ;
– je contribue tous les ans à l’enseignement des stages de techniques analytiques donné dans le
cadre de la formation continue des enseignants du primaire et du secondaire ;
– J’ai encadré un petit nombre de stagiaires M2 (3 depuis le début de ma carrière).
– j’ai co-encadré deux thèses : Yinlong Quin (thèse soutenue en 2008), maintenant dans l’industrie chez Petroleum Geo-Services (PGS) et Paul Cupillard (thèse soutenue en 2008) actuellement en Post-doc à UC Berkeley ;
– J’ai encadré deux post-docs, un dans le cadre de l’ANR MUSE et un dans d’un contrat avec le
CEA ;
– J’ai coordonné l’ANR blanche MUSE (terminée fin 2009) durant laquelle a été développée
l’homogénéisation.
– Je vais coordonner l’ANR blanche mémé (Novembre 2010-Octobre 2014), qui regroupe les
projets à moyen terme autour de l’homogénéisation 3-D.
– j’ai dirigé ou co-dirigé l’organisation de 3 congrès scientifiques internationaux ;
– j’ai été membre du conseil d’administration de l’IPGP de 2005 à 2009.
– j’ai organisé les séminaires de l’équipe de sismologie de l’IPGP durant 7 ans.
– Je suis reviewer régulier des journaux Geophysical Journal International, Wave motion, et
parfois Physics of the Earth and Planetary Interiors.
3.3
Collaborations et séjour à l’étranger
– l’ANR blanche (MUSE, terminée en novembre 2009) que j’ai dirigée s’est déroulée dans le
cadre d’une collaboration entre l’IPGP, le laboratoire de Modélisation en Mécanique (LMM,
3.4. TRANSFERT DE TECHNOLOGIE, RELATIONS INDUSTRIELLES ET VALORISATION45
–
–
–
–
Paris 6), ainsi que EDF et Schlumberger. Ce projet est axé sur l’aspect multi-échelles de la
propagation d’ondes, du problème inverse en imagerie sismique et de la rupture.
Avec le CEA. Une collaboration avec le CEA est en cours. L’objet des travaux est le développement du couplage solide-fluide avec la méthode des éléments spectraux pour la propagation
des ondes T dans des modèles de terre avec océans 3D. Dans ce cadre, un post-doc hébergé par
l’IPGP, travaille sous la direction de Christian Mariotti (CEA) et moi-même. Un autre contrat
est en cours de négociation sur l’effet des hétérogénéités en champ proche d’une explosion.
Avec l’Université de Berkeley : un projet est terminé avec l’Université de Berkeley (lieu
de mon post-doctorat en 2000-2003). Ce projet concerne l’application de la méthode tomographique globale basée sur les éléments spectraux à l’échelle régionale. Ce projet a été financé
par le DOE (Département Of Energy américain). Du côté de Berkeley, ce projet implique Mark
Panning et Barbara Romanowicz. Du côté français, ce projet fait partie de la thèse de Paul Cupillard. J’ai passé l’été 2009 à Berkeley pour travailler sur le problème adjoint associé au
problème inverse en sismologie avec les éléments spectraux.
Dans la cadre du projet européen SPICE (2004-2007), un projet visant à tester certaines des
méthodes tomographiques est terminé. Le but est de développer un jeu de données généré dans
un modèle de terre réaliste et complexe grâce aux éléments spectraux. Ces données seront
ensuite utilisées par différents groupes (du projet SPICE et extérieur) de façon aveugle (c.à.d
sans rien connaître du contenu du modèle à retrouver). Mon travail avec l’étudiant Y. Qin et
avec V. Maupin (Université d’Oslo) était de construire le modèle de terre et de générer les
données.
Schlumberger. Une éventuelle collaboration entre Schlumberger Cambridge Research et le
CNRS est en cours de négociation sur l’utilisation des résultats de l’homogénéisation.
3.4
Transfert de technologie, relations industrielles et valorisation
Contrats de recherche :
– Je fait partie du training network européen QUEST (2010-2014). QUEST est une sorte de suite
de SPICE. C’est un projet important et qui implique dix institutions en Europe. Le thème de
ce training network est toujours axé sur la modélisation en sismologie.
– J’ai fait partie du training network européen SPICE (2004-2008). SPICE est un projet important financé à hauteur de 5 millions d’euros et qui implique quatorze institutions en Europe.
Le thème de ce training network était axé sur la modélisation en sismologie.
– J’ai coordonné un contrat ANR blanc (MUSE) financé à hauteur de 220 milles euros depuis
début 2007. Ce contrat implique le laboratoire de Modélisation en Mécanique (LMM), ainsi
que EDF et Schlumberger. Ce projet est axé sur l’aspect multi-échelles de la propagation
d’ondes du problème inverse et de la rupture.
– Je m’occupe d’un contrat de recherche entre le CEA et l’IGPG. Ce contrat court pour l’instant
jusqu’à fin 2010 (montant 61keen 2010). Une suite est en cours de négociation.
46
– Je vais coordonner un contrat ANR blanc (mémé) fin 2010 et pour 4 ans. Ce contrat implique
l’IPGP, l’école polytechnique, le CEA et Schlumberger.
Valorisation :
Une demande de brevet a été déposée par le CNRS sur la technologie liée à l’homogénéisation.
Des négociations entre le CNRS et Schlumberger sont en cours pour la cession d’une licence des
programmes de validations développés dans le cadre de l’ANR MUSE. L’idée est que Schlumberger puisse évaluer l’intérêt de l’homogénéisation non périodique pour des applications industrielles et d’éventuellement acquérir une licence du brevet si les tests sont concluants.
Par ailleurs, j’ai développé de nombreux programmes pour la sismologie (calculs de modes propres, sommation de modes propres, calculs des dérivées partielles de formes d’ondes, couplage
éléments spectraux modes, programme de retournement temporel, programme d’homogénéisation ...) qui sont actuellement utilisés un peu partout en sismologie.
3.5
3.5.1
Publications
Publications dans des revues de rang A
Une partie de mes publications se trouvent, sous forme pdf, sur la page :
http
://www.ipgp.fr/˜capdevil/biblio/biblio.html
Capdeville, Y. Guillot, L. and J. J. Marigo (2010) 2D nonperiodic homogenization to upscale
elastic media for P-SV waves. Geophys. J. Int., In press
Guillot, L., Capdeville, Y. and J. J. Marigo (2010) 2-D non periodic homogenization for the SH
wave equation Geophys. J. Int., Accepted
Cupillard, P. and Capdeville Y. (2010) On the amplitude of surface waves obtained by noise correlation and the capability to recover the attenuation : a numerical approach. Geophys. J. Int.,
doi : 10.1111/j.1365-246X.2010.04586.x
Capdeville, Y. Guillot, L. and J. J. Marigo (2010) 1-D non periodic homogenization for the 1wave
equation Geophys. J. Int., 181, 897-910
M. Panning, Y. Capdeville, B. Romanowicz (2009). Seismic waveform modeling in a 3D Earth
using the Born approximation : potential shortcomings and a remedy. Geophys. J. Int., Vol 177,
Issue 1, pp 161-178.
Y. Qin, Y. Capdeville, J.P. Montagner, L. Boschi , T. W. Becker (2009) Reliability of mantle tomography models assessed by spectral-element simulation Geophys. J. Int., Vol 177, pp 125-144.
B. Romanowicz, M. Panning, Y. Gung, Y. Capdeville (2008). On the computation of long period
seismograms in a 3-D earth using normal mode based approximations Geophys. J. Int., Vol 175,
Issue 2, pp 520–536
3.5. PUBLICATIONS
47
Y. Qin, Y. Capdeville, V. Maupin, J.P. Montagner, S. Lebedev, E. Beucler (2008) SPICE benchmark for global tomographic methods Geophys. J. Int., Vol 175, pp 598–616.
Capdeville, Y. and J. J. Marigo (2008) Shallow layer correction for Spectral Element like methods Geophys. J. Int., Vol 172, pp 1135–1150.
Capdeville, Y. and J. J. Marigo (2007) Second order homogenization of the elastic wave equation
for non-periodic layered media Geophys. J. Int., Vol. 170, pp 823–838, doi : 10.1111/j.1365246X.2007.03462.x
E. Chaljub, Dimitri Komatitsch, Yann Capdeville, Jean-Pierre Vilotte, Bernard Valette et Gaetano Festa (2007), in “Advances in Wave Propagation in Heterogeneous Media”, editor Ru-Shan
Wu and Valérie Maupin, " Advances in Geophysics" series, Elsevier, vol. 48, p 365-419.
Y. Qin, Y. Capdeville, V. Maupin and J.P. Montagner(2006). A SPICE blind test to benchmark
global tomographic methods Eos Trans. AGU, 87(46), 512.
Larmat, C., J.-P. Montagner, M. Fink, Y. Capdeville, A. Tourin, and E. Clévédé (2006), Timereversal imaging of seismic sources and application to the great Sumatra earthquake, Geophys.
Res. Lett., 33, L19312, doi :10.1029/2006GL026336.
Y. Capdeville (2005) An efficient Born normal mode method to compute sensitivity kernels and
synthetic seismograms in the Earth. Geophys. J. Int. Volume 163, Issue 2, pp 639-646.
Y. Capdeville, Y. Gung and B. Romanowicz (2005) Towards Global Earth Tomography using the
Spectral Element Method : a technique based on source stacking. Geophys. J. Int., Vol 162, Issue
2, pp 541-554.
A. To, B. Romanowicz, Y. Capdeville and N. Takeuchi (2005). 3D effects of sharp boundaries at
the borders of the African and Pacific Superplumes : Observation and modeling ? Earth Planet.
Sci. Lett., Vol 233, Issues 1-2, pp 137-153
Y. Capdeville, B. Romanowicz and A. To (2003). Coupling Spectral Elements and Modes in a
spherical earth : an extension to the “sandwich” case. Geosphys. J. Int. Vol 154, pp 44-57
Capdeville Y., E. Chaljub, J.P. Vilotte and J.P. Montagner (2003). Coupling the Spectral Element
Method with a modal solution for Elastic Wave Propagation in Global Earth Models. Geophys.
J. Int., Vol 152, pp 34–67
Chaljub, E. Capdeville, Y. and Vilotte, J.P. (2003) Solving elastodynamics in a fluid–solid heterogeneous sphere : a parallel spectral element approximation on non–conforming grids. J. Comp
Phy. Vol. 187,2 pp. 457-491
Capdeville Y., C. Larmat, J.P. Vilotte and J.P. Montagner (2002) Numerical simulation of the scattering induced by a localized plume-like anomaly using a coupled spectral element and modal
solution method. Geoph. Res. Lett. VOL. 29, No. 9, 10.1029/2001GL013747
Capdeville Y., E. Stutzmann and J. P. Montagner (2000). Effect of a plume on long period surface
waves computed with normal modes coupling. Phys. Earth Planet. Inter., Vol 119 pp. 57–74.
Montagner, J. P., E. Stutzmann and Y. Capdeville (1995). Hotspot detection from seismological
48
data. In D. Anderson, S. Hart and A. Hofmann (Eds.), Plume 2, Terra Nostra, Vol 3, pp. 103-106.
3.5.2
Brevet
Capdeville (2009) “Procédé de détermination d’un modèle élastique effectif.” Brevet FR 09
57637.
3.5.3
Communications orales et posters
Y. Capdeville, L. Guillot and J. J. Marigo (2009). 2D/3D Elastic model up-scaling for the wave
equation based on non-periodic homogenization. Annual meeting of the Society of Exploration
Geophysics (SEG), Houston.
Y. Capdeville, L. Guillot and J. J. Marigo (2008). High order non-periodic homogenization for
wave propagation in complex 1D and 2D media. AGU (American Geophysical Union) Fall meeting, San Francisco.
Y. Capdeville, L. Guillot and J. J. Marigo (2008). High order nonperiodic homogenization for
seismic wave propagation in layered media and for the 2D SH case. 8th. World Congress on
Computational Mechanics (WCCM8)5th European Congress on Computational Methods in Applied Sciences and Engineeering (ECCOMAS 2008) June 30 July 5, 2008, Venice, Italy
Y. Capdeville (2008). Toward full waveform tomography of the global Earth using the Spectral
Element method : a technique based on source stacking. ORFEUS meeting, June 2008. (Invited
presentation).
Y. Capdeville, L. Guillot and J.J. Marrigo (2007). Multi-scale issues and two scale homogenization solutions for the direct and inverse problems in seismology for layered earth model and
beyond. AGU Fall Meeting, San Francisco. (oral)
L Guillot, Y Capdeville, JJ. Marigo (2007). Homogenization of the SH-Wave Equation in 2D
Heterogeneous Media AGU Fall Meeting, San Francisco. ( poster)
Y. Capdeville, L. Guillot and J.J. Marrigo (2007). Multi-scale issues and two scale homogenization solutions for the direct and inverse problem in seismology. SPICE-CIG join workshop,
Jackson NH, USA. (Invited presentation).
Y. Capdeville, L. Guillot and J.J. Marrigo (2007). Second Order Homogenization and Shallow
Layer Correction for Seismic Wave Propagation in Layered Media and Its Application to the
Spectral Element Method. ICTCA Meeting, Crete, Greece (oral)
Y. Capdeville, J.P. Montagner (2005). Normal modes practicals. SPICE workshop, Smolenice,
Slovak Republic. (invited presentation)
3.5. PUBLICATIONS
49
Y. Capdeville (2005). A tomographic method for the global Earth based of the Spectral Element
Method. Séminaire invité ETH, Zurich.
Y. Capdeville (2005). Vers une méthode tomographique pour la Terre à globale basée sur les
éléments spectraux. Séminaire invité LGIT, Grenoble.
A. To, B. Romanowicz, Y. Capdeville and N. Takeuchi (2004). Evidence for a Sharp Lateral
Boundary at the Southern Border of the Pacific Superplume. AGU Fall Meeting, San Francisco.
( poster)
Y. Capdeville, Y. Gung and B. Romanowicz (2004) Tomography of the global Earth based on the
Spectral Element Method AGU Fall Meeting, San Francisco. ( poster)
Capdeville, Y. ; Romanowicz, B. ; Gung, Y. (2003) Global seismic waveform tomography based
on the spectral element method. EGS General Assembly, Nice 2003 ( oral)
Capdeville, Y. ; Gung, Y ; Romanowicz, B. (2002) The Coupled Spectral Element/Normal Mode
Method : Application to the Testing of Several Approximations Based on Normal Mode Theory
for the Computation of Seismograms in a Realistic 3D Earth. AGU Fall Meeting, San Francisco.
( oral)
Capdeville, Y. ; Romanowicz, B. ; To, A. (2002) A sandwich of spectral elements between modal
solutions for seismic forward modeling in realistic D" layer models. EGS General Assembly,
Nice 2002 ( oral)
Capdeville, Y. ; Romanowicz, B. (2001) A coupled method of spectral elements and modal solutions for seismic forward modeling in realistic D" layer models. AGU Fall Meeting, San Francisco. ( poster)
To, A. ; Romanowicz, B. ; Capdeville, Y. (2001) Towards Forward Modeling of 3D Heterogeneity at the Base of the Mantle. AGU Fall Meeting, San Francisco. ( poster)
Vilotte, J. ; Chaljub, E. ; Capdeville, Y. ; Ampuero, J. (2001) Recent Developments in Computational Seismology Using the Spectral Element Method AGU Fall Meeting, San Francisco. ( oral)
Capdeville Y. ; Chaljub E. ; J.P. Vilotte and J.P. Montagner (2000). Wave Propagation in Global
Earth Models Using the Coupled Method of Spectral Elements and Modal Solution : Two Examples of Typical Applications AGU Fall Meeting, San Francisco. ( oral)
Capdeville Y. ; Chaljub E. ; J.P. Vilotte and J.P. Montagner (2000). Coupling Spectral Elements
and Modal Solution : a New Efficient Tool for Numerical Wave Propagation in Laterally Heterogeneous Earth Models. AGU Fall Meeting, San Francisco. ( poster)
Capdeville Y. ; Chaljub E. ; J.P. Vilotte and J.P. Montagner (1999). Numerical Simulation of 3D
Wave Propagation in Laterally Heterogeneous Synthetic and Tomographic Earth Models with a
Mixed Method of Modal Solution and Spectral Elements AGU spring Meeting, Washington DC.
( oral)
Capdeville Y. ; Chaljub E. ; J.P. Vilotte and J.P. Montagner (1999). 3D Wave Propagation in a
50
Spherical Earth Model Mixed Spectral Element and Modal Summation Method AGU Fall Meeting, San Francisco. ( poster)
Chaljub E. ; J.P. Vilotte and Y. Capdeville (1998). 3D Wave Propagation in Spherical Earth Model
using the Spectral Element Method. AGU Fall Meeting, San Francisco. ( poster)
Capdeville Y. ; E. Stutzmann and J. P. Montagner (1997). Effect of a mantle plume on surface
waves and normal modes. In IASPEI 1997, Thessaloniki, Greece, Abstracts, pp 245. ( poster)
3.6
Recueil des articles et travaux significatifs
Geophys. J. Int. (2005) 163, 639–646
doi: 10.1111/j.1365-246X.2005.02765.x
An efficient Born normal mode method to compute sensitivity
kernels and synthetic seismograms in the Earth
Y. Capdeville
Departement de sismologie, Institut de Physique du Globe de Paris, Paris, France. E-mail: [email protected]
SUMMARY
We present an alternative to the classical mode coupling method scheme often used in global
seismology to compute synthetic seismograms in laterally heterogeneous earth model and
Frechet derivatives for tomographic inverse problem with the normal modes first-order Born
approximation. We start from the first-order Born solution in the frequency domain and we
use a numerical scheme for the volume integration, which means that we have to compute the
effect of a finite number of scattering points and sum them with the appropriate integration
weight. For each scattering point, ‘source to scattering point’ and ‘scattering point to receivers’
expressions are separated before applying a Fourier transform to return to the time domain.
Doing so, the perturbed displacement is obtained, for each scattering point, as the convolution
of a forward wavefield from the source to the scattering point with a backward wavefield from
the scattering integration point to the receiver. For one scattering point and for a given number
of time steps, the numerical cost of such a scheme grows as (number of receivers + the number
of sources) × (corner frequency)2 to be compared to (number of receivers × the number of
sources) × (corner frequency)4 when the classical normal mode coupling algorithm is used.
Another interesting point is, when used for Frechet kernel, the computing cost is (almost)
independent of the number of parameters used for the inversion. This algorithm is similar to
the one obtained when solving the adjoint problem. Validation tests with respect to the spectral
element method solution both in the Frechet derivative case and as a synthetic seismogram
tool shows a good agreement. In the latter case, we show that non-linearity can be significant
even at long periods and when using existing smooth global tomographic models.
Key words: Fréchet derivatives, Global seismology, normal modes.
1 I N T RO D U C T I O N
The normal mode solution of the wave equation is well known and
widely used in global seismology when dealing with spherically
symmetric models in the 20 s and longer period range. When dealing with laterally heterogeneous models, a standard method is the
first-order perturbation of the normal mode basis obtained in a spherically symmetric reference earth model. This approximation is only
valid for weak enough lateral heterogeneities and for short enough
time-series. This ‘weak enough’ condition, which is closely related
to the ‘short enough’ time condition, is not clearly defined and should
be checked in each particular situation. Most of the long wavelength
global tomographic models are based on methods derived from the
Born solution in the normal modes framework. The computations
involve the coupling of the modes obtained in the spherically symmetric reference model due to the 3-D structure. Depending on the
roughness of the 3-D model and of the corner frequency of the
source, the number of modes that need to be coupled can be very
large and, therefore, computationally intensive. In result,the complete Born solution is rarely used and less computationally intensive
C
2005 The Author
C 2005 RAS
Journal compilation 51
methods based on approximations to the Born approximation have
been developed for practical uses. Among incremental approximations to the Born approximation, we can quote the very popular ‘path
average’ approximation (Woodhouse & Dziewonski 1984) in which
perturbations due to the average spherically symmetric model on
the source–receiver path is taken into account. This approximation
provides only 1-D sensitivity kernels, but is very efficient. A better
approximation is NACT (Li & Tanimoto 1993; Li & Romanowicz
1995), an asymptotic method, in which is included the cross coupling between different dispersion branches and provides 2-D sensitivity kernels. These different approximations are very interesting
because their efficiency allows one to compute the large number
of synthetic seismograms required for tomography in a acceptable
CPU time. It is nevertheless interesting to go beyond these approximations and use the full Born approximation to include effects such
as focusing or component conversions due to lateral heterogeneities.
In this paper we present an efficient normal mode Born technique
in which the full mode coupling is present but not explicitly. In this
approach, lateral heterogeneities (of a model for synthetic seismograms, or of a spatial parameter for Frechet kernels) are discretized
639
GJI Seismology
Accepted 2005 July 28. Received 2005 July 21; in original form 2005 January 14
640
Y. Capdeville
The first-order perturbation displacement wavefield expression in
the frequency domain is (Woodhouse 1983; Tanimoto 1984),
R K L K K (r)SK dr,
2
δu(rr , ω) · v = −
(6)
iω ωk − ω2 ωk2 − ω2
K K V
as a finite number of scattering points over an integration mesh. The
perturbation (or partial derivative) of displacement due to a lateral
heterogeneity is computed with the convolution of a direct wavefield from the source to a scattering point with a backward wavefield
for the receiver to the scattering point. This scheme is significantly
more efficient than the explicit mode coupling solution and does not
require to specify an angular coupling range l as it is usually the
case. This solution is equivalent to the one obtained when solving
the adjoint problem (e.g. Lailly 1983; Tarantola 1984, 1988; Mora
1987; Geller & Hara 1993; Pratt et al. 1998; Tromp et al. 2005) and
similar to the one used by Tanimoto (1990) for surface waves on a
membrane.
We will first introduce the classical mode coupling approach and
then focus on this alternative approach. We will then present practical and numerical considerations before presenting some validation
tests with respect to the spectral element Method (SEM) solution.
where R K = u K (rr ) · v is the receiver term for component v, SK =
(u K , f) the source term and
L K K (r) = u∗K (r) · δL(r)u K (r) .
We will first explain how, starting from eq. (6), δu(r r, t) is usually
computed in seismology, and then we will propose an alternative.
2.1 A classical way
Performing an inverse Laplace–Fourier transform of eq. (6), we
obtain
gkk (t)
R K L K K (r)SK dr ,
(8)
δu(rr , t) · v =
2 T H E O RY
Let us first introduce some notations that will be used in that paper,
such as the dot product:
u·v=
u i vi ,
(1)
K K
gkk (t) =
where ∗ denotes the complex conjugate and V the earth volume.
The displacement u in the Earth has to be solution of the following
wave equation
(3)
where f is the external source term which represent the earthquake
and L is the anelasto-dynamic operator. In the frequency domain,
Lu = −ω2 ρu + Au,
(4)
where A is the anelatic operator and ρ is the density. In the following,
we will assume that the external source f is a point source in space
located at r s and a step function in time.
We consider L = L0 + δL, where L0 is the anelasto-dynamic
operator in a reference earth model and δL a perturbation with
respect to L0 . For the sake of simplicity, only perturbation of the
anelastic tensor will be considered and, therefore, δL = δA.The
density perturbation can nevertheless be included with no extra difficulties and practical expressions including density effect are given
in Appendix A. Because the normal mode solution is well known
in spherically symmetric models, the reference model is often (if
not always) chosen spherically symmetric for the normal mode perturbation scheme. The displacement wavefield is also written under
the form u = u 0 + δu, where L0 u0 = f and, to the first order, it can
be easily shown that
L0 δu = −δLu0 .
ωk 2 [1 − cos(ωk t)][1 − cos(ωk t)]
H (t) .
ωk2 ωk 2 (ωk2 − ωk 2 )
(9)
When k = k in the previous equation, a first-order Taylor expansion
has to be performed on g kk , which results in a term proportional
to time t (secular term). To compute the volume integral in eq.
(8), two approaches have been considered. In the first one, the lateral heterogeneities are expanded over the spherical harmonics basis (e.g. Woodhouse & Dahlen 1978; Woodhouse & Girnius 1982;
Tanimoto 1986), which allows the horizontal part of the volume
integral to be computed analytically using Wigner 3-j symbols
(Edmonds 1960), or asymptotically, using the stationary phase approximation (Romanowicz 1987). In the second approach, (e.g.
Snieder 1986; Snieder & Romanowicz 1988; Li & Tanimoto 1993;
Capdeville et al. 2000) the volume integral of eq. (8) is performed
numerically and it can be seen as a sum over a finite number of
scattering points. RK , SK and L KK are usually expanded over the
generalized spherical harmonics basis (Phinney & Burridge 1973),
and using the generalized spherical harmonics summation theorem,
the sum over m can be suppressed to reduce the amount of computing. In practice, synthetic seismograms are only computed down to
a given corner frequency, which makes it possible to define a maximum angular degree l max up to which the sum over k and k must be
computed. Knowing that the number of eigenfrequencies below the
corner frequency of the source is proportional to l 2max , for a single
integration point, the number of floating point operation grows as
l 4max as the corner frequency increases if all the modes are coupled. In
practice, not all mode couplings are considered, which reduces the
number of computations but also reduces the accuracy. These computations have to be performed for each pair of source and receiver,
therefore the numerical cost for a single scattering point increase as
Ns × Nr × l 4max , where Nr and Ns are the number of receivers and
sources, respectively.
(2)
V
Lu = f,
V
with
i=1,3
and the inner product
u∗ (r) · v(r) dr ,
(u, v) =
(7)
(5)
We name u K and ω k the set of eigenfunctions and eigenfrequencies,
solutions of L0 u = 0 with the index k = (q, n, l) and K = (k,
m), where q can take two values, one for spheroidal modes and
one for toroidal modes, n is the radial order, l the angular order
and m the azimuthal order. Because of the spherical symmetry of
the reference model, the eignefrequencies ω k do not depend on the
azimuthal order.
2.2 An alternative to the classical way
In the rest of this article, we will use a numerical integration scheme
to compute the volume integral of eq. (6). Using the definition of
52
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2005 The Author, GJI, 163, 639–646
C 2005 RAS
Journal compilation Normal modes kernel
L KK , eq. (6) can be rewritten as
δu(rr , ω) · v
∗
u K (r)SK
u K (r)R K
2
· δL(r)
dr, (10)
2
=−
2
ωk − ω2
V
K
K iω ωk − ω
and in the time domain
B(r, τ ) · δL(r)F(r, τ − t) dτ dr ,
δu(rr , t) · v = −
(11)
V
where
B(r, t) =
u∗K (r)R K
sin(ωk t)
H (t) ,
ωk
(12)
u K (r)SK
1 − cos(ωk t)
H (t) ,
ωk2
(13)
K
and
F(r, t) =
K
Now, either because h k (ω) is multiplied by a source term that has a
corner frequency or either because numerical computation will stop
at the Nyquist frequency, h k (ω) is always multiplied by a function
f (ω) with a given corner frequency ω c . Even if it is often done in
practice, we do not have
+∞
sin ωk t
h k (ω) f (ω)e−iωt dω = f (ωk )
H (t) .
(18)
ωk
−∞
Indeed, eq. (18) is true only if f (ω)e−iωt can be extended to an
analytic complex function on the entire complex plane, which is
not the case for most of the usual bounded filters with a frequency
cut-off. In general, we have
+∞
(1)
(2)
h k (ω) f (ω)e−iωt dω = f h k (t) + f h k (t) ,
(19)
−∞
where H is the Heaviside function. We therefore have, for each
integration point, a convolution of a forward wavefield (F) from the
source to the scattering point and a backward wavefield (B) from
the receiver to the scattering point. For each scattering point of the
integration mesh, the B and F fields can be computed independently
for each source and receiver. Therefore, for a single scattering point
and for a constant number of time steps, the number of computations
increases as (Ns + Nr ) × l 2max , which make this process numerically
more interesting than eq. (8). A practical expression of eq. (11) is
derived in appendix A.
The same result can be obtained directly starting form eq. (5). We
name G(r; r s, t) the Green’s function solution of
L0 u(r, t) = δ(rs − r)δ(t)I ,
(14)
where I is the identity tensor. Noting that eq. (5) is the same as eq.
(14) with a different source term, we obtain:
δu(rr , t) = −
G(rr ; r, τ ) · δLu0 (r, t − τ ) dτ dr ,
(15)
V
which is equivalent to eq. (11). This result is also equivalent to
the one obtained when solving the adjoint problem (Lailly 1983;
Tarantola 1984, 1988; Mora 1987; Pratt et al. 1998; Tromp et al.
2005).
3 N U M E R I C A L C O N S I D E R AT I O N S
with
(1)
f h k (t)
sin ωk t
H (t) ,
ωk
(20)
(2)
f h k (t)
=
+∞
−∞
1
2ωk2
f (ω) − f (−ωk ) −iωt
f (ω) − f (ωk )
+
e
dω .
ωk − ω
ωk + ω
(21)
The distribution of eigenfrequencies (an example is given Fig. 1)
allows, for a given corner frequency, to determine a maximum angular degree l max and for each l a maximum radial order n max (l)
(1)
after which f h k terms are always equal to zero. This is not the case
(2)
for terms f h k , even if it decays rapidly for (l, n) outside of [0,
l max ] × [0, n max (l)]. A practical solution is to take an [0, l] × [0,
n] window bigger than [0, l max ] × [0, n max (l)] with a taper weighting of terms outside of [0, l max ] × [0, n max (l)]. This is nevertheless
an approximation that introduce errors that depend on the source,
heterogeneities and receiver geometrical configurations.
(2)
In classical approaches f h k can be often neglected because it has
an effect only near t = 0. In our case, when the scattering point is
close to the source or to the receiver, the influence of the second
term becomes very important around t = 0 for one of the two fields
F or B. Because of the convolution, this error is spread to the whole
signal as it can be seen on the example given in Fig. 2. In a realistic
application there are always lateral heterogeneities around the source
(2)
and the receiver. Neglecting f h k using the approach presented in
0.08
0.06
Angular frequency (Hz)
Sums over k of eqs (12) and (13) cannot be computed numerically
without truncations. Indeed, index k is a summary of indexes q, n and
l. q can only take two values but both the radial order n and angular l
order lie in [0, +∞[. In practical cases, the source term has a corner
frequency ω c or the signal will be filtered with a filter with a corner
frequency ω c . In that case, time expressions eqs (12) and (13) have
to be rewritten to allow sum truncations with an acceptable accuracy.
Here we will present in detail the case of the time expression in eq.
(12), but a similar operation can be performed for eq. (13) with little
k t)
differences. The sin(ω
H (t) in eq. (12) has been obtained with the
ωk
Fourier transformation of
1
1
1
1
h k (ω) = 2
+
.
(16)
=
ωk + ω
ωk − ω2
2ωk2 ωk − ω
C 2005 The Author, GJI, 163, 639–646
C 2005 RAS
Journal compilation = f (ωk )
and
3.1 Sum truncations
Using the Cauchy theorem, it can easily shown that
+∞
sin ω K t
h k (ω)e−iωt dω =
H (t) .
ωk
−∞
641
0.04
0.02
0
0
100
50
Angular order
(17)
53
Figure 1. Some Rayleigh eigenfrequencies in PREM (Dziewonski & Anderson 1981).
642
Y. Capdeville
0
2000
-400
4000
-200
0
time (s)
8000
6000
200
used to solve the wave equation at the global scale with the SEM
(Chaljub 2000; Chaljub et al. 2003). An example of such a mesh
is given in Fig. 4. Many other choices for this numerical integration could have been made, but the availability of this integration
scheme and also the good accuracy with no over sampling at the
poles justify this choice. The sphere is first divided into Ne nonoverlapping deformed cubic elements. The ‘cubic sphere’ proposed
by Sadourny (1972) and further extended by Ronchi et al. (1996)
allows such a meshing of a spherical surface by decomposing it into
six regions of identical shape, which can be mapped onto a cube
face. To obtain the meshing of a spherical shell, spherical surfaces
are connected radially. In each element, the numerical integration is
performed using the Gauss–Lobatto–Legendre (GLL) quadrature in
each Cartesian direction. In each of these Cartesian direction, the
polynomial basis of degree N is built using the Lagrange polynomial
associated with GLL points. If we name
d(r) = B(r, τ ) · δL(r)F(r, τ − t) dτ ,
(22)
400
600
Figure 2. Forward term F (13) computed for a Dirac source time function
for a scattering point at less than a degree from the source. The filter f is
cosine filter with a taper from 200 s to 80 s. In dotted line is plotted the
result obtained when only f h 1k of eq. (19) is used and in plain line is plotted
the result obtained when both terms of eq. (19) are used. The lower plot is
a zoom around the origin time of the upper plot. One can see that including
2
f h k terms indeed affects only the early time on the trace, but the effect is
significant around t = 0. Taking into account this effect is important to obtain
a good result when convolution eq. (11).
then numerical approximation of eq. (11) is
δu(rr , t) · v = − d(r) dr
V
−
0.1
0
-0.1
-0.2
3000
4000
5000
time (s)
6000
7000
8000
(23)
where r = Fe (ξ) with Fe the transformation function from the reference cube to the deformed cube number e, J e the Jacobian of the
transformation Fe and ρ iN the 1-D integration weights associated
with GLL point number i among N + 1. For more details about
the integration scheme we refer to Komatitsch & Vilotte (1998)
and Chaljub et al. (2003). Our experience with spectral elements
and with the method presented here shows that a good accuracy is
achieved with two minimum wavelength per element of degree 8.
This sampling may need to be increased in some cases like a sharp
variation of the elastic property within an element. Discontinuities
of elastic properties or eigenfunction derivatives are accurately integrated only if they match element interfaces.
In this paper, the integration is performed as explained above by
computing d(r) for every scattering point of the integration mesh
but heavy computational optimization can be performed here. First,
not all scattering points contribute significantly to δu. Indeed, the
sensitivity of a wave for a given source and receiver configuration
is primarily focused on the Fresnel zone and this can be used to reduce computations. Second, as proposed by Zhao & Chevrot (2004),
a 2-D mesh of 1-D Green’s functions can be computed and stored
and finally used to compute the contribution of all the required integration points using interpolations. Indeed, each component of
eqs (12) and (13) or (A11) and (A12) only depends on the source
depth and an epicentral distance (between the source and the integration point or between the integration point and the receiver).
Therefore, all the terms can be pre-computed on a 2-D mesh with a
good spacial sampling rate and stored in a database. Some elements
of the database will be later loaded to compute eq. (11) for a particular source–receiver configuration.This solution requires some
heavy bookkeeping, but this is probably necessary to be effective
when dealing with higher frequencies and body waves to compute
a large number of kernels.
0.2
2000
d(ri jk )|J e (ξi jk )ρiN ρ Nj ρkN ,
e=1 i, j,k
0.3
-0.3
1000
Ne 9000
Figure 3. Differential traces (difference between traces computed in the 3D model model and traces computed in the 1-D reference model) computed
with (solid line) and without (dotted line) the terms f h 2k and b h 2k in the
3-D tomographic SAW24B16 model for an epicentral distance of 70 degree
along the equator. The source has 150 s corner frequency, a depth of 19 km
(2)
and an origin time at 1000 s. The effect of neglecting f h k can be important
especially for the R1 scattered train (around 3000 s)
this paper can be an important issue, knowing that the scattered
wavefield is especially sensitive to structure below the source and
(2)
below the receiver. An example of the effect of neglecting f h k in
the 3-D tomographic model SAW24B16 (Mégnin & Romanowicz
2000) is given Fig. 3.
3.2 Integration mesh
3.3 Frechet derivatives
The volume integral of eq. (11) needs to be computed numerically.
In order to do so, we use here the same integration scheme as the one
So far we have discussed solutions for a single model perturbation δL, which is typically the case when computing synthetic
54
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2005 The Author, GJI, 163, 639–646
C 2005 RAS
643
as the solutions used to solve the wave equation in both cases are
completely different, there is a good chance that both solutions are
correct if the two solutions match. We first performed a validation
test in the Frechet kernel case. We define a single and simple spatial
parameter p: a vertical cylinder of 10◦ of diameter from the surface
to the 660 km transition zone. The amplitude on the elastic parameter contrast in the cylinder has a Gaussian shape. To compute the full
waveform partial derivative with respect to this cylindrical spatial
parameter p with SEM, we use a simple finite difference formula:
∂g(0)
g(δp) − g(0)
,
∂p
δp
(26)
where g(δp) represent the waveform at a given seismic station computed with SEM in the earth model PREM plus δ p. In practice we
perform two SEM runs, one in the reference model (PREM) and
one in the 3-D model built with PREM plus a 3-D structure of shape
p and a small amplitude, and then take the difference. In that test,
anelasticity is turned off. Fig. 5 presents a comparison of SEM and
normal modes results for two elastic parameters, δ0000 and the
real part of δ++00 (see eq. A2). The agreement between the two
methods is very good.
The second test is performed in the tomographic model
SAW24B16 for the source–receiver locations presented in Fig. 6.
SAW24B16 is a Vs model and, for this test Vp and ρ perturbations
Figure 4. Typical integration mesh used for the volume integration of
eq. (11). Two regions of the cubic sphere have been removed for this figure
for representation purpose.
1
2
SEM
normal modes
0.5
seismograms. When dealing with Frechet derivatives, the solution δu has to be computed for each spatial parameter p of the
parametrization used in the inversion process. Each parameter p can
be either global as a spherical harmonic or local as a spherical spline.
For the sake of simplicity, let us say that we are working with a single elastic parameter (e.g. Vs ) and let δL1 be the unit perturbation
operator for which this single elastic parameter is 1 everywhere.
We name p(r) the spacial variation of the elastic parameter corresponding to p, then the perturbation of the elastic operator for the
parameter p is δL p (r) = p(r)δL1 . We, therefore, have
Ne ∂u(rr , t)
p(ri jk )d1 (ri jk )|J e (ξi jk )ρiN ρ Nj ρkN ,
·v=−
∂p
e=1 i, j,k
where
d1 (r) =
(24)
1
0
0
−0.5
−1
−1
0
5000
time (s)
10000
−2
0
5000
time (s)
10000
Figure 5. Examples of partial derivatives waveforms (vertical component)
with respect to the cylindrical structure p computed with SEM (solid line)
and the normal modes Born algorithm presented in this paper (dotted line),
for elastic parameter δ0000 (left plot) and the real part of δ++00 (right
plot). The source corner frequency is 150 s and at 19 km depth, the receiver
is at an epicentral distances of 90◦ and the cylinder p is on the great circle
path and at the same distance from the source and the receiver.
B(r, τ ) · δL1 (r)F(r, τ − t) dτ .
(25)
In practice, computing d1 (ri jk ) represents most of the CPU time
required for the whole computation, and, as they are common to all
the parameter p, they need to be computed only once. This makes
the computation of all the Frechet derivatives independent of their
number. Of course, this technique can be used to compute different
type of kernels (time arrivals, phase velocity, etc.).
4 VA L I D AT I O N S
In order to test and validate this work, we will compare results obtained with this Born solution and the one obtained by the SEM.
More specifically, we will use the coupled SEM–normal mode
method (Chaljub et al. 2003; Capdeville et al. 2003): the SEM will be
used in the mantle and the normal mode solution in the core, which
will remain spherically symmetric. Both of these methods contain
approximations and, therefore, none of them can be considered as
an absolute reference with respect to the other one. Nevertheless,
C 2005 The Author, GJI, 163, 639–646
C 2005 RAS
Figure 6. Source station path used for the test performed in SAW24B16
tomographic model. The source is at 19 km depth. The background map
represents SAW24B16 velocity perturbations at 200 km depth.
644
Y. Capdeville
long time-series (e.g. R2) are used. Of course, this is not a concern
when building partial derivatives.
0.06
SEM
normal modes
0.03
5 D I S C U S S I O N A N D C O N C LU S I O N S
We have presented a solution to compute synthetic seismograms in
3-D global models earth and Frechet partial derivatives for global
tomography based on the Born approximation in the normal mode
framework. The main interest of this solution is to reduce the number of computations with respect to classical approaches by avoiding
to explicitly have to compute the mode coupling due to the presence of lateral heterogeneities. Indeed, the numerical cost of the
scheme proposed here grows as (number of receivers + the number
of sources) × (corner frequency)2 to be compared to (number of
receivers × the number of sources) × (corner frequency)4 when the
classical normal mode coupling algorithm is used. This approach,
which involves the convolution of direct wavefield form the source
to a scattering point with a backward wavefield from the receiver to
the scattering point, is very similar to adjoint problem approaches
used to compute the gradient in seismic inverse problem (Lailly
1983; Tarantola 1984, 1988; Mora 1987; Geller & Hara 1993; Pratt
et al. 1998; Tromp et al. 2005). The CPU time required for this
approach as presented here can be significantly improved by storing
2-D propagation functions and then use interpolations to compute
the direct and backward wavefield at the different scattering points
of the 3-D integration mesh as proposed by Zhao & Chevrot (2004).
This improvement can be very useful especially if the scheme is
used to compute time arrival kernels for body waves, which require
high frequencies and therefore a large number of modes.
Our comparison with the SEM have shown a good agreement for
the Frechet derivatives case. When used to compute synthetic seismograms, differences can be significant, even in a long period and in
smooth 3-D models. These differences are due to non-linear effects
and have, therefore, less to do with the scheme used than with the
Born approximation. It should be noted anyway that there is still one
drawback to this technique in that case. Indeed, classical mode coupling calculations allow to correct eigenfrequencies as explained by
Woodhouse (1983), which allow to perform some approximate nonlinear correction. This cannot be done with the method presented
here. Finally, one should always worry about non-linear effect especially when working on R2 or long epicentral distance R1 phase
velocities and amplitudes, even at long periods.
Applications of this work should be found for the computation
of different type of Frechet kernels from time arrival kernels to full
waveform kernels for global earth tomography.
0
−0.03
−0.06
2000
4000
6000
time (s)
8000
10000
Figure 7. Differential waveforms computed by SEM (solid line) and by the
normal modes perturbation technique presented in this paper (dotted line).
The source–receiver configuration is given Fig. 6. The model is SAW24B16
with an amplitude of velocity contrast divided by 10 to avoid most of the
non-linear effects. The corner period of the source is here 100 s.
1
PREM
SAW24, with SEM
SAW24 with Born
R1
R2
0
−1
2000
4500
1
7000
0.7
R2
R1
0.2
0
−0.3
−1
3500
4000
4500
time (s)
−0.8
5000 6500
7000
7500
8000
time (s)
Figure 8. Top: Vertical component synthetic waveforms computed in
PREM (solid line), SAW24B16 with SEM (dotted line) and in SAW24B16
within the Born approximation (dashed line) for the same source–receiver
configuration as in Fig. 7. The two lower plots are zooms on R1 and R2
phases of the same seismograms. Non-linear effects are significant on R2.
The differences between SEM and normal modes computation observed in
Fig. 7 are small compared to the non-linear differences observed here.
AC K N OW L E D G M E N T S
I thank Paco Shánchez–Sesma for a fruitful discussion which indirectly initiated this work. I also thank Barbara Romanowicz for
many interesting discussions about this topic. Finally I thank two
anonymous reviewers for helping to improve the manuscript. Computations were performed with IDRIS, CINES and IPGP IBM computer facilities.
have been linearly linked to Vs . In order to avoid most of the nonlinear effects that will be present in the SEM simulation we first
perform a run in a SAW24B16 model with an amplitude of velocity
contrasts divided by 10. The result is presented in Fig. 7. The agreement is good, but not perfect for the amplitudes. This difference can
be accounted for by the difference in the attenuation scheme. When
we use SAW24B16 with its real amplitude, non-linear effects become important as can seen in Fig. 8, especially for the R2 Rayleigh
phase, even for these long period (100 s and above). The conclusion of this last test is that, when using Born to compute synthetic
seismograms, non-linear effects can be important and it should be
checked if this approximation is valid, even at long periods when
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APPENDIX A: PRACTICAL EXPRESSIONS
Considering only a perturbation of the anelastic tensor δc and using an integration by part, eq. (10) can be rewritten
∗
K (r)SK
(r)R K
K2
: δc(rd ) :
drd ,
2
δu(rr , ω) · v =
2
ωk − ω2
V
K
K iω ωk − ω
(A1)
where K the deformation tensor corresponding to the mode u K . In the following, generalized spherical harmonics expansion will be used to
simplify expressions. To do so, it is useful to use the contravariant canonical component of δc
δαβγ η =
Cαi
Cβ j Cγ k Cηl δci jkl ,
(A2)
i jkl
as well as for the deformation tensor associated to mode u K
αβ
K (r) =
Cαi
Cβ j K ,i j (r),
(A3)
ij
where C is

0
 1
√
(Ciα ) = 
 2
√1
2
1
0
0
−1
√
2
0
−i
√
2


.

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57
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Y. Capdeville
αβ
Generalized spherical harmonics expansion of R K , S K and K are well known:
Rkα (rr )Ylαm (θr , φr ) ,
R K (rr ) =
(A5)
α=−1,1
SK (rs ) =
Skα (rs )Ylαm (θs , φs ) ,
(A6)
α=−2,2
αβ
αβ
α+β m
K (rd ) = E k (rd )Yl
(θd , φd ),
(A7)
Y lαm
R αk
αβ
S αk
is the generalized spherical harmonics, the expression of
and
can be found in Woodhouse & Girnius (1982) and where E k
where
can be easily evaluated and can be found, for example, in Tanimoto (1986). We define
(rd )R K
K2
,
(A8)
b E(rd , ω) =
ωk − ω 2
K
which in the time domain and with canonical component gives
η αβ sin(ωk t)
α+β m
αβ
Rk E k (r )Ylηm (θr , φr )Yl
(θ, φ)
H (t).
b E (rd , t) =
ωk
Kη
(A9)
At this point, the spherical harmonics summation theorem is useful to suppress the sum over m. We have (Li & Tanimoto 1993):
m=l
YlN m (θs , φs )YlN m (θr , φr ) = ei N γsr PlN N (cos(βsr ))ei N ξsr ,
(A10)
m=−l
where P lNN are the generalized Legendre functions. If the index s is related to the ‘source’ location and r to the ‘receiver’ one, then the angle
−ξ sr is the backazimuth at the receiver, π − γ sr the azimuth at the source and β sr is the angular epicentral distance. Using this theorem,
eq. (A9) becomes
η αβ sin(ωk t)
η α+β
αβ
Rk E k (rd ) Pl
(cos(βr d )) ei(α+β)γdr +iηξdr
H (t).
(A11)
b E (rd , t) =
ωk
kη
Similarly, we define
η αβ
1 − cos(ωk t)
α+β η
αβ
Sk E k (rd ) Pl
(cos(βsd )) ei(α+β)ξsd +iηγdr
H (t).
f E (rd , t) =
ωk2
kη
Finally, knowing that i Ciα
Ciβ = δαβ and i Ciα Ciβ = eαβ with


0
0 −1


1 0 
(eαβ ) = 
0
.
−1 0 0
Eq. (A1) gives
δu(rr , ω) · v =
V
bE
αβ
(A12)
(A13)
(rd , τ )δαβγ η (rd ) f E ικ (rd , t − τ )eγ ι eηκ dτ drd .
(A14)
αβγ ηικ
For a density perturbation, using the following generalized spherical harmonic expansion for the eigendisplacement,
UαK (rd ) = Ukα (rd )Ylα m (θd , φd ) ,
and defining
bU
α
(rd , t) =
(A15)
Rkη Ukα (rd ) Plη α (cos(βr d )) eiαγdr +iηξdr sin(ωk t)ωk H (t) ,
(A16)
kη
fU
α
(rd , t) =
Skη Ukα (rd ) Plα η (cos(βsd )) eiαξsd +iηγdr
kη
the displacement perturbation is
δu(rr , ω) · v = − δρ(rd )
V
bU
α
1 − cos(ωk t)
H (t) ,
ωk2
(A17)
(rd , τ ) f U α (rd , t − τ ) dτ drd .
(A18)
α
58
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2005 The Author, GJI, 163, 639–646
C 2005 RAS
Journal compilation Geophys. J. Int. (2003) 152, 34–67
Coupling the spectral element method with a modal solution
for elastic wave propagation in global earth models
Y. Capdeville,1,∗ E. Chaljub,1,2, † J. P. Vilotte1,2 and J. P. Montagner1
1 Département
2 Département
de Sismologie, Institut de Physique du Globe de Paris, 4 Place Jussieu, 75252-Paris cedex05, France
de Modélisation Physique et Numérique, Institut de Physique du Globe de Paris
Accepted 2002 July 2. Received 2002 April 11; in original form 2001 June 5
SUMMARY
We present a new method for wave propagation in global earth models based upon the coupling
between the spectral element method and a modal solution method. The Earth is decomposed
into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only
spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous
shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust.
In the outer shell, the solution is sought in terms of the spectral element method, which stem
from a high order variational formulation in space and a second-order explicit scheme in
time. In the inner sphere, the solution is sought in terms of a modal solution in frequency
after expansion on the spherical harmonics basis. The spectral element method combines the
geometrical flexibility of finite element methods with the exponential convergence rate of
spectral methods. It avoids the pole problems and allows for local mesh refinement, using a
non-conforming discretization, for the resolution of sharp variations and topography along
interfaces. The modal solution allows for an accurate isotropic representation in the inner
sphere. The coupling is introduced within the spectral element method via a Dirichlet-toNeumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized
spherical harmonics. The inverse transform in space and time requires special attention and an
asymptotic regularization. The coupled method allows a significant speed-up in the simulation
of the wave propagation in earth models. For spherically symmetric earth model, the method is
shown to have the accuracy of spectral transform methods and allow the resolution of wavefield
propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis.
Key words: body waves, global wave propagation, modal solution, spectral elements, surface
waves, synthetic seismograms.
1 INTRODUCTION
The availability of continuously increasing large data sets of high quality broad-band digital data from global instrument deployments and
global data centres, like IRIS or GEOSCOPE, have been critical in fostering global seismology. Improvement of seismic earth models must
incorporate new numerical methods that allow the computation of detailed waveforms and provide insights into the physics of the wave
propagation in models with a broad range of heterogeneity scales. This new challenge takes place at the same time as the advent of parallel
computers. Progress in computational seismology is now bringing direct numerical simulation of wave propagation back into the heart of
global seismology.
In the past decades much effort has been expanded to improve images of the large-scale variations of velocity heterogeneities in the
Earth’s mantle with the aim of inferring some information on the thermal and chemical heterogeneities and to relate them to dynamic processes.
Arrival times (Vasco & Johnson 1998; van der Hilst et al. 1997, 1998; Grand et al. 1997), differential traveltimes of surface phases and body
waves (Su et al. 1994; Liu & Dziewonski 1998; Masters et al. 1996), normal mode coefficients (Resovsky & Ritzwoller 1999), and waveform
analysis (Li & Romanowicz 1996) have been used in global tomography providing increasing evidence that the Earth’s mantle is laterally
∗ Now
† Now
34
at: Berkeley Seismological Laboratory, UC Berkeley, CA, USA
at: Department of Geosciences, Princeton University NJ, USA
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2003 RAS
35
heterogeneous in a broad range of scales, with coherent large-scale patterns. Tomographic models, derived from global or regional studies,
agree reasonably well, at least between 100 and 600 km, on the long wavelength structure, i.e. the first 12 degrees in spherical harmonics
with a spatial resolution of the order of 1500 km at the surface (Dziewonski 1995; Ritzwoller & Lavely 1995). Regional tomographic models
tend, however, to predict somewhat higher power spectral density at shorter wavelengths than can be resolved in global studies (Chevrot et al.
1998). An increasing number of indirect studies suggest the importance of small-scale heterogeneities both in the upper (Aki 1981; Kennett
1997; Kennett & Bowman 1990) and lowermost part of the mantle (Lay 1995; Ding & Helmberger 1997; Lay et al. 1997, 1998; Pulliam &
Sen 1999; Kendall & Silver 1996; Matzel et al. 1996, Vinnik et al.1996).
The impact of the theoretical assumptions used in the methods for computing synthetic seismograms has not yet been fully assessed
but recent studies by Clévédé et al. (2000) and Igel et al. (2000) have pointed out the importance of the theoretical approximations in
the tomography models. Direct numerical simulation (DNS) methods are now required to: understand the physics of wave propagation in
heterogeneous earth models; assess the impact of the theoretical approximations used in practice to compute the synthetic seismograms; assess
the actual resolution of the tomographic models and of the seismic records with respect to small-scale heterogeneities. However, in contrast
to the rapid progress of computational seismology in exploration geophysics, DNS methods are less advanced in global seismology in part
because of the large scale of the system and of the pole problems associated with the spherical coordinates system.
The principal methods used today in global seismology for the computation of synthetic seismograms are based upon spectral transform
methods (STM) (Gelfand et al. 1963; Newman & Penrose 1966; Takeuchi & Saito 1972; Phinney & Burridge 1973; Wu & Yang 1977), which
expand the solution, and the physical variables, into vector spherical harmonics defined on a spherical reference configuration. These methods
are exceptionally accurate and stable for smooth functions with a uniform resolution on the sphere and provide an elegant solution to the
so-called pole problem.
Among these methods, normal mode summation methods have long been used in global seismology (Dahlen & Tromp 1998; Lognonné
& Clévédé 2000). The normal modes of a reference spherical configuration, and the corresponding eigenfrequencies, are computed as
intermediate functions for the computation of the seismograms at the surface. Asphericity and lateral heterogeneity distributions are then
considered as perturbations. Perturbation theories (Dahlen 1969; Woodhouse 1980; Valette 1987; 1989; Lognonné 1991; Dahlen & Tromp
1998; Lognonné & Clévédé 2000) rely on two approximations: small lateral variations of amplitude and small lateral gradients. The most
sophisticated perturbative normal mode methods (Lognonné & Romanowicz 1990; Lognonné 1991; Lognonné & Clévédé 1997, 2000)
requires the coupling of modes along the same dispersion branch while the multiple scattering is resolved up to the accuracy limit of the actual
order of the perturbative expansion. Even for only a second order expansion, these methods face serious computational limits when dealing
with dominant periods of less than 50 s. For practical analysis, high frequency (asymptotic) approximations (Romanowicz & Roult 1986;
Romanowicz 1987; Li & Tanimoto 1993) are combined with a Born approximation (Snieder & Romanowicz 1988; Tromp & Dahlen 1992;
Pollitz 1998; Friederich 1998) in the computation of synthetic seismograms. High frequency approximations assume that the wavelengths
of the heterogeneities remain large compared to the dominant propagating wavelength while first order Born approximations assume that
heterogeneities can be considered as secondary sources. Despite continuous theoretical improvements and the many successes of refined
asymptotic theories, especially for surface waves, these methods face serious theoretical problems when finite frequency effects on the whole
waveform are important. Higher order asymptotic approximations become rapidly computationally expensive and can not deal with localized
zones of high velocity contrasts.
Other spectral transform methods directly solve the elastodynamics equation in the frequency domain, bypassing the intermediate step of
calculating the normal modes. For spherical earth models, the method developed by Friederich & Dalkolmo (1995) integrates the strong form
of the elastodynamics equation expanded spatially on the vector spherical harmonics basis. The synthetic seismograms are then obtained after
summation over the spherical harmonics and a numerical inverse Fourier transform. Another interesting method based on a weak variational
formulation of the elastodynamics equation in the frequency domain was proposed by Hara et al. (1993) and Geller & Ohminato (1994).
This method makes use of a weighted residual formalism (Finlayson 1972; Zienkiewicz & Morgan 1983), in which boundary and continuity
conditions becomes natural boundary conditions with the help of appropriate surface integrals. Using a Galerkin approximation, the trial and
the weight functions are chosen to be the tensor product of linear splines in the radial direction and generalized spherical harmonics basis
functions. Cummins et al. (1994a,b) and Geller & Ohminato (1994) used the direct solution method (DSM) to compute synthetic seismograms
for laterally homogeneous media in spherical coordinates. In this case, there is no coupling between the toroidal and the spheroidal modes.
Due to the degeneracy of the spherically symmetric case, the algebraic system can be decomposed into smaller separate banded subsystems
for spheroidal and toroidal displacements that are solved for each angular and azimuthal order. For laterally heterogeneous models, all angular
and azimuthal orders become coupled and the computational requirement increases drastically. This implies some approximations such as a
truncation of the coupling between angular orders, the restriction to axisymmetric lateral variations (zonal) (Cummins et al. 1997), or the use
of Born approximations (Takeuchi et al. 2000).
All these methods, based on global spectral transforms, usually solve the elastodynamics problem in frequency and do not provide direct
information on the wave propagation. Moreover, spectral transform methods involve an expansion of the variables into a non local spherical
harmonics basis as well as the computation of the spatial derivatives in this basis. The major drawbacks are therefore: the need for high
order spherical expansions when the lateral variations become rough and spatially localized; the difficulty to incorporate mesh refinement;
the inherent computational complexity and the communication overheads. Despite recent improvements (Kostelec et al. 2000), the transform
requires O(N 3 ) operations, where N is the number of points in latitude. On distributed memory architectures, the global structure of the
communication between processors involved in this transform imposes an overhead which becomes critical as N is increased.
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2003 RAS, GJI, 152, 34–67
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Y. Capdeville et al.
A limited number of studies have been recently conducted toward direct numerical simulation of seismic wave propagation in spherical
earth models. They are based upon finite differences or pseudo-spectral methods. Finite differences (FD) have been widely used in exploration
seismology. They however suffer from grid dispersion near strong gradients and require a large number of grid points to achieve the expected
accuracy (Kelly et al. 1976). Balancing the trade-off between numerical dispersion and the computational cost turns out to be quite difficult
(Virieux 1986; Dablain 1986; Bayliss et al. 1986; Levander 1988) and critical in global seismology (Alterman et al. 1970). Higher order
methods, like the pseudo-spectral methods, enable to achieve the expected accuracy using fewer grid points. Maximum efficiency is obtained
when using fast Fourier transforms but for more general boundary conditions the set of truncated Fourier series are often replaced by a set
of algebraic polynomials (Kosloff et al. 1990; Fornberg 1996), Chebychev or Legendre, with the drawback of introducing non uniform grid
points. This limitation can lead to severe stability problems that can be partly overcome with the help of generalized coordinates (Fornberg
1988; Kosloff & Tal-Ezer 1993). Finite differences and pseudo-spectral methods are based on a strong formulation of the elastodynamics
initial boundary value problem and face severe limitations for an accurate approximation of free-surface and continuity conditions which
is crucial for long term simulation of surface and interface waves. Finite differences and pseudo-spectral methods have been used in global
seismology under rather restricted approximations: axisymmetric P and SV (Furumura et al. 1998, 1999) or SH (Igel & Weber 1996; Chaljub
& Tarantola 1997) wave propagation avoiding the inherent pole problems associated with spherical coordinates. The major problem with
these grid-based methods is that, unlike global spectral transform methods, they fail to provide an isotropic representation of scalar functions
on a sphere. Latitude-longitude grid points become clustered near the poles leading to severe CFL conditions. This problem can be partly
circumvented when using latitude–longitude pseudo-spectral solvers (Merilees 1973; Fornberg 1995) in conjunction with suitable spectral
filters (Jakob-Chien & Alpert 1997; Yarvin & Rokhlin 1998) or when resorting to a multidomain approach (Sadourny 1972; Heikes & Randall
1995; Ranić et al. 1996). However in the latter case, one is faced with the non trivial problem of how to glue together the different coordinate
systems. This implies overlapping regions between the mesh subdomains which put severe stability conditions. These techniques have not yet
received much attention in the seismological community.
Another direction, that was recently explored (Chaljub 2000; Chaljub et al. in preparation), is to resort to the spectral element method
(Patera 1984; Maday & Patera 1989; Bernardi & Maday 1992). This method which stems from a variational formulation of the elastodynamics
equations (Priolo et al. 1994; Faccioli et al. 1997; Komatitsch & Vilotte 1998), combines the geometrical flexibility of conventional finite
element methods with the exponential convergence rate associated with spectral techniques. It provides an optimal dispersion error and an
accurate representation of surface and interface waves. The first problem in applying the spectral element method in global seismology is the
paving of a 3-sphere using hexahedra, a natural element for efficient sum-factorization techniques. This was solved (Chaljub 2000; Chaljub
et al. 2001) by first the tilling of embedded spherical interfaces in six quadrangular regions using the central projection (Sadourny 1972; Ronchi
et al. 1996) of a cube onto the circumscribed spherical interface, then by inscribing a cube at the centre of the sphere with a smooth transition
between the six faces of the inner cube and the innermost spherical interface (Chaljub 2000). Such a discretization is shown to provide an
almost uniform tilling of the spherical interfaces and to avoid the pole problems and the singularity at the centre of the sphere. Moreover,
all the transformations can be defined analytically, and perturbed to map aspherical geometries or interface topographies. In contrast with
finite differences, the spectral element method provides a natural setting for handling the connection between the six regions produced by
the central projection. Spectral element can then be formulated in conjunction with a hybrid variational formulation (Chaljub 2000; Chaljub
et al. in preparation) allowing: an efficient coupling between solid and fluid domains at the core mantle boundary; a non conforming domain
decomposition and mesh refinements with the help of the mortar method (Bernardi et al. 1994; Ben Belgacem & Maday 1994; Ben Belgacem
1999). Such mesh refinement flexibility is crucial for an accurate representation of localized heterogeneities within a global model while
reducing the computational cost. Coupled with an explicit second order predictor-corrector scheme in time, the method can be very efficiently
parallelized on distributed memory architectures as shown in Chaljub et al. (in preparation). Even though this method should become one
of the leading method for direct numerical simulation in global seismology, it still requires front end computational resources for realistic
problems, typically 128 processors with more than 90 GB distributed memory for a dominant period of less than 30 s, which still preclude its
use in a day to day basis.
It is worth noting that no one scheme can be expected to be optimal for the entire range of applications we might wish to consider in the
context of global seismology. Within the actual limitation of the computer resources, beside few front end infrastructures, it is of importance
to explore the possibility of combining the respective advantages of the spectral transform and spectral element methods for exploring some
realistic 3-D problems. We present here a new method that couple spectral element and normal mode summation methods. Such a coupled
method actually allows the computation of synthetic seismograms and wavefield propagation in laterally heterogeneous earth models down
to periods of 30 s on a medium size parallel architecture, typically a cluster of 32 processors and 32 GB distributed memory. The underlying
idea is that, with regard to the actual resolution of the broad-band records, it is quite reasonable to focus on the effect of lateral heterogeneities
confined in an outer shell while the corresponding inner sphere is still approximated as spherically symmetric. Depending on the problem in
hand, the outer heterogeneous shell can be mapped as the whole mantle, down to the CMB, or restricted to some portion of the upper-mantle
or to the crust. In this approach, a modal solution is sought for the inner sphere while the high-order spectral element solution is retained for
the heterogeneous outer shell. The coupling between the spectral element method, formulated in space and time, and the modal summation
method, formulated in frequency and wavenumber, requires some original solution. Within the spectral element method, the a coupling is
introduced via a dynamic interface operator, a Dirichlet-to-Neumann (DtN) operator, which can be explicitly constructed in frequency and
generalized spherical harmonics. The inverse transform in space and time requires however an optimal asymptotic regularization. Such a
coupling allows a significant speed-up in the simulation of the wavefield propagation and the computation of synthetic seismograms in laterally
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37
heterogeneous earth models. It is of interest to note here that in the coupled method the outer shell can be aspherical providing that the coupling
interface remains a spherical interface. The interface can then be chosen to be an artificial coupling interface below the CMB. This still reduces
significantly the computational requirements since the stringent CFL constrains come from the core discretization in the full spectral element
method.
In this paper, we focus on the construction of the DtN operator which is the crucial part of the method and applications are restricted to
spherical earth models for which we do have solutions to compare with. Applications to laterally 3-D earth models are part of two companion
papers (Capdeville et al. 2002, in preparation). The paper is organized as follows. We first introduce the coupling method in a simple 1-D
example where the explicit calculation of the DtN operator is provided with several numerical examples to assess the accuracy of the method.
We then explicit the construction of the DtN operator in the 3-D case in conjunction with the spectral element method developed in Chaljub
(2000) and Chaljub et al. (in preparation). Several numerical tests are performed in order to illustrate the accuracy of the method and the
efficiency of the asymptotic regularization of the DtN operator. Then the method is used to compute the synthetic seismograms in the case of
the PREM model. In conclusion, we outline some ongoing extensions especially for the study of the CMB zone.
2 PROBLEM STATEMENT AND EQUATION OF MOTION
We consider a spherical earth domain Ω, with the boundary ∂Ω and internal radius rΩ , see Fig. l. The domain is decomposed in solid and
fluid parts, Ω S and Ω F respectively, where the fluid part typically includes the outer core. We assume here for sake of simplicity that each
of these subdomains is spherical with no interface between two fluid subdomains. The solid–solid interfaces are denoted SS , the solid-fluid
interfaces SF and the set of all the interfaces .
For a spherical non rotating earth model, the linearized momentum equation, around an initial hydrostatic state of equilibrium, is given
by:
∂ 2 u(r, t)
− Hu(r, t) = f(r, t),
∂t 2
where H is the elasto-gravity operator which, when neglecting the perturbation of the earth gravity potential, is given by:
ρ(r)
Hu(r, t) = ∇ · [τ (u) − u · ∇τ 0 ] − ∇ · (ρu)g.
(1)
(2)
Here ρ is the initial density, g is the unperturbed gravity acceleration, f is a generalized source term and τ the initial reference hydrostatic
stress. The perturbed displacement u and the associated stress perturbation τ have to satisfy the following continuity relations:
0
[|u(r, t)|] = 0 ∀r ∈ SS ,
[|u(r, t) · n(r)|] = 0
(3)
∀r ∈ S F ,
(4)
[|τ (u)(r, t) · n(r)|] = 0 ∀r ∈ ,
(5)
with the initial conditions
∂u
u(r, 0) = 0;
(r, 0) = 0 ∀r ∈ Ω,
∂t
and the following free surface condition
(6)
τ (r, t) · n(r) = 0
(7)
∀r ∈ ∂Ω,
where n (r) denotes the unit outward normal.
Figure 1. The earth model Ω, with boundary ∂Ω and internal radius rΩ . The solid and fluid parts are Ω S and Ω F respectively. SS and SF denote respectively
the set of all the solid–solid and solid-fluid interfaces. The spectral element domain is Ω+ while the domain of modal summation is Ω− with Γ denoting the
artificial interface between Ω+ and Ω− .
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In the solid parts Ω S , the elasto-gravity operator can be rewritten as:
Hu(r, t) = ∇ · τ (u) − ∇ · (ρu)g + ∇(ρu · g),
(8)
where for a linearly elastic medium, the Lagrangian stress perturbation is simply defined as:
τi j (u)(r, t) = di jkl (r)u k,l (r, t),
(9)
with u k,l = ∂u k /∂ x l and d ijkl is the elastic tensor with all the major and minor symmetries.
In the following, the lateral heterogeneities of interest are restricted within an outer spherical shell Ω+ and only radial heterogeneities
are retained within the inner sphere Ω− . The interface between these two domains will be denoted Γ. Even though such a partitioning
might appear quite artificial, it has some practical interest. With regard to the actual resolution of the observations, it is quite reasonable,
when studying the mantle lateral heterogeneities, to assume only radial heterogeneities within the earth core. The artificial boundary Γ can
therefore be chosen at the core-mantle boundary or below the CMB. In the latter case, the earth model can be aspherical as far as the coupling
boundary remains spherical. When focussing to the strong lateral heterogeneities within the crust, or the upper lithosphere, the interface Γ
can be set somewhere within the mantle. The aim is to take advantage of the high order variational formulation, in space and time, only
within the domain Ω+ , where strong lateral heterogeneities are assumed to be important depending on the problem in hand, while taking
advantage of the modal summation method in Ω− . This leads to a significant speed up in the computation of seismograms while retaining
information on the complicated wave propagation within the laterally heterogeneous parts. The coupling between the variational formulation
of the elastodynamic problem, formulated in space and time, and the modal summation method, formulated in frequency and wavenumber,
requires special attention. We first consider a simple 1-D scalar example to introduce the methodology and then extend it to more realistic
3-D problems.
3 A 1-D EXAMPLE
We first consider, Fig. 2, a simple 1-D domain Ω of length L. The domain is decomposed into two subdomains Ω− = [0, L 0 ] and Ω+ = [L 0 , L].
The interface between the two domains Γ = {L} is oriented by the unit outward normal nΓ , directed from the domain Ω− toward Ω+ . The
interface Γ may be related toa physical discontinuity or not. The elastic medium is characterized by the density ρ(x), the elastic parameter λ(x)
and the sound speed c(x) = λ(x)/ρ(x). The source condition is taken as a collocated point force, at x = xs ∈ Ω+ , with f = Fδ(x −xs )g(t −t0 )
where δ denotes the Dirac distribution, g a time wavelet centred at t 0 and F the amplitude of the force.
The problem involves only a scalar displacement field in the x-direction, u (x, t) = u(x, t)ex and the only non zero stress component
τ (x, t) = τ xx (x, t) = λ(x)∂u(x, t)/∂ x. We introduce the restrictions u + and u − of the displacement u to Ω+ and Ω− respectively as
u + (x, t) = u(x, t), ∀x ∈ Ω+ and u − (x, t) = u(x, t), ∀x ∈ Ω− .
The scalar wave equation ∀ x ∈ is:
∂
∂u(x, t)
∂ 2 u(x, t)
−
λ(x)
= f (x, t),
(10)
ρ(x)
∂t 2
∂x
∂x
with the initial conditions:
∂u
(x, 0) = 0,
u(x, 0) = 0 and
∂t
and at both ends, a free-boundary condition is assumed:
∂u(x, t) = 0,
T (0, t) = − λ(x)
∂x (11)
(12)
x=0
−1
1
Figure 2. The 1-D domain Ω: the modal solution method is used in Ω− and the spectral element method in Ω+ . The discretization of the domain Ω+ in six
regions is also represented. The dotted lines denote the Fe=4 transformation of the reference segment into Ω4 . The Gauss–Lobatto–Legendre (GLL) integration
points are shown with the black squares in the reference element = [−1, 1] for a polynomial degree N = 8.
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2003 RAS, GJI, 152, 34–67
T (L , t) = λ(x)
∂u(x, t) = 0,
∂ x x=L
39
(13)
where T (x, t) = τ · n is the traction defined on the interface at position x with the unit outward normal vector n. Across the solid–solid
interface Γ, the solution has to satisfy the continuity conditions of both displacement and traction:
−
[|u(L 0 , t)|] = u +
Γ (t) − u Γ (t) = 0,
(14)
[|T (L 0 , t)|] = TΓ+ (t) − TΓ− (t) = 0,
(15)
where
u±
Γ (t) =
and TΓ± (, t) =
lim u(L 0 ± , t)
→0,>0
lim T (L 0 ± , t),
→0,>0
In order to illustrate the coupling between the spectral element and the modal summation methods in this simple 1-D case, the problem
will be solved using a spectral element approximation within the Ω+ and a modal summation method within Ω− . The coupling between the
two solutions requires an explicit matching of the traction and displacement at the interface Γ.
3.1 Variational formulation
The spectral element approximation of the scalar wave equation in the subdomain Ω+ , is first outlined. The solution u + , i.e. the restriction of
the solution u to the domain Ω+ , is searched in the space H 1 (Ω+ ) of the square-integrable functions with square-integrable generalized first
derivative. Introducing a time interval, I = [0, T ], the solution belongs to the space Ct of the kinematically admissible displacements:
Ct = {u(x, t) ∈ H 1 (Ω+ ) : Ω+ × I → IR}.
(16)
+
The variational formulation of the problem (10)–(11) reads: find u ∈ Ct , such that ∀t ∈ I and ∀w ∈ Ct
∂ 2u+
w, ρ 2 + a(w, u + ) = (w, f ) + w, T + (u + )Γ ,
∂t
(w, ρu + )|t=0 = 0,
w, ρ∂t u + |t=0 = 0,
(17)
(18)
(19)
where (· , ·) is the classical L inner product with
(w, u) =
w · u d x.
2
(20)
Ω+
The bilinear form a(· , ·) defines the symmetric elastic strain-energy inner product :
∂w ∂w
d x,
τ (w) : ∇w d x =
λ
a(w, w) =
∂x ∂x
Ω+
Ω+
and the bilinear form ·, ·Γ is defined as:
w, T + (u + )Γ = w · T + (u + ) dx = (w · T + (u + ))| L 0 .
(21)
(22)
Γ
The last term involves the traction along the interface between the two subdomains Ω+ and Ω− . Taking into account the continuity
conditions (14), and assuming that we are able to compute the solution that characterize the response of the domain Ω− to a prescribed
+
−
displacement u −
Γ = u Γ on the interface Γ, the interface operator A will relate the traction T on Γ to that imposed displacement:
A : TΓ− (t) = A(u −
Γ )(t),
(23)
+
where A is defined in terms of a convolution. With the continuity conditions (14), the interface operator relates directly the traction T to the
displacement u + on that side of the interface as a result of the dynamic behaviour of the domain Ω− :
(24)
A : TΓ+ (t) = A u +
Γ (t).
The interface operator A is a Dirichlet to Neumann (DtN) operator which, for a given displacement on the interface (Dirichlet condition),
provides the associated traction (Neumann condition) on the interface. Such an operator was first introduced for absorbing boundary conditions
(e.g. Givoli & Keller 1990; Grote & Keller 1995). For the domain Ω+ , the DtN operator A, defined on Γ, acts as a dynamic boundary condition
in traction.
3.2 Spatial discretization
In order to solve numerically the above problem, the physical domain Ω+ is discretized, Fig. 2, into n e non-overlapping elements Ωe , e =
e
1, . . . , n e , such that Ω = ∪ne=1
Ωe . For each segment Ωe , an invertible local geometrical transformation Fe is defined which map the reference
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40
segment = [− 1, 1] into Ωe , i.e. x(ξ ) = Fe (ξ ), where ξ is a local reference coordinate system defined on . In the 1-D case, Fe is trivial
to define, as well as its inverse.
In the following, the upper-script + , for the restriction of u to the domain Ω+ , is dropped unless some possible confusion, and u he denotes
the restriction of u to the element Ωe .
Associated with the spatial discretization of Ω+ , a piecewise-polynomial approximation of the kinematic admissible displacements is
introduced:
(25)
C Nh = u h ∈ Ct : u h ∈ H 1 (Ω+ ) and u eh ◦ Fe ∈ IP N () ,
where IPN () is the set of the polynomials defined on and of degree less than or equal to N, and ◦ is the functional composition symbol.
The discrete solution u h is therefore a continuous function such that its restriction to each element Ωe can be represented in terms of local
polynomials after transformation to the reference element . The spectral element approximation can be characterized by the total number
of elements n e and the polynomial degree N on the reference element . The discrete variational problem then reads: find u h ∈ C Nh , such that
∀t ∈ I and ∀w h ∈ C Nh :
∂ 2uh
h
(26)
w , ρ 2 + a(w h , u h ) = (w h , f ) + w h , A(u h )Γ ,
∂t
(w h , ρu h )|t=0 = 0,
(27)
w h , ρ∂t u h t=0 = 0.
(28)
The discretization is completed by the definition of a discrete inner product: a Gauss–Lobatto–Legendre (GLL) quadrature allows the
computation of each element integral after transformation into the reference element . This quadrature defines N + 1 integration points,
{ξ iN , i = 0, N }, as the zeros of the polynomial (1 − ξ 2 )P N where PN is the Legendre polynomial of degree N defined on . The polynomial
basis on can now be defined by the Lagrange polynomials {h Ni , i = 1, N }, associated with the GLL points, such that h Ni (ξ j ) = δ ij , where δ
is the Kronecker symbol. The value of the displacement at any arbitrary point ∈ Ωe is obtained through a Lagrange interpolation,
u eh (x) =
i=N
h iN (ξ )u ie ,
(29)
i=0
where x = Fe (ξ ). The discrete displacement u h is defined uniquely by its values u ie at the collocation points Fe (ξi ) where ξ i are the GLL
integration points on .
Expanding w h as in (29), the discrete variational problem (26) is shown to be equivalent to the set of algebraic equations ∀t ∈ I:
M
∂ 2U
= Fext − Fint (U ) + FDtN (U ),
∂t 2
(30)
where U is the displacement vector of all the displacements associated to the total number of integration points, e.g. defined as the set of all
the integration points defined at the element level. The mass matrix M, the internal force vector Fint (U ), the source term Fext results from the
assembly of the element contributions Me , Fint,e and Fext,e according to the connectivity of each element with
Miej =
e=n
e
N
e=1 l=0
Fiint,e =
e=n
e
N
h iN (ξl )ρ(ξl )h Nj (ξl )∂ξ Fe (ξl )ωl ,
λ(ξl )
e=1 l=0
Fiext,e =
e=n
e
N
e=1 l=0
∂h nj
∂ξ
(ξl )
N ∂h N
i
i=0
∂ξ
(31)
(ξl )u ie
−1
∂ξ Fe (ξl )
ωl ,
(32)
h iN (ξl ) f e (ξl )∂ξ Fe (ξl )ωl ,
(33)
where ωl are the integration weights associated with the GLL quadrature. It is worth to note here that by construction the element mass
matrix, and therefore the global mass matrix, is diagonal leading to fully explicit schemes in time and great computational efficiency. The
same procedure can be apply to the DtN interface contribution which lead to a simple collocated expression in this simple 1-D example.
The algebraic ordinary differential system (30) is solved in time using a classical Newmark scheme (Hughes 1987) with an explicit
predictor-corrector scheme. Such a time stepping is conditionally stable and requires that
t ≤ tC = C
dx
,
α
(34)
where dx is the size of the smallest grid cell, α the wave velocity and C the Courant number. The average size of a spectral element for a fixed
size scales as n −1
e while the minimum grid spacing between two Gauss–Lobatto–Legendre points within an element occurs at the edge and
−2
scales as N −2 . Therefore the critical time step tC ∝ O(n −1
) and a compromise has to be found between the degree of the polynomial
e N
approximation and the number of elements. For the 1-D example, an empirical value of C = 0.84 has been found to provide stable results.
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3.3 Construction of the DtN operator in the frequency domain
The construction of the DtN operator requires the solution of the scalar wave equation in Ω− assuming a Dirichlet boundary condition on the
+
interface Γ, i.e. u −
Γ (t) = u Γ (t) . This problem can be solved in the frequency domain, i.e.
∂ û − (x, ω)
∂
λ(x)
= 0,
(35)
ω2 ρ(x)û − (x, ω) +
∂x
∂x
with the following boundary conditions:
∂ û −
(x, ω) = 0 for x = 0,
λ(x)
∂x
(36)
+
(ω)
=
û
(ω)
for
x
=
L
,
û −
0
Γ
Γ
and where û is defined as:
+∞
û(x, ω) =
u(x, t)e−iωt dt,
(37)
−∞
For a given frequency, the problem (35) has two solutions with only one satisfying the free boundary condition at x = 0. We denotes this
solution U (x, ω) and the corresponding displacement and traction on the coupling interface, Γ, D(ω) where T (ω) defined as:
∂U (ω).
(38)
D(ω) = U |Γ (ω) and T (ω) = λ
∂ x Γ
Solutions of the problem in Ω− are therefore of the form û(x, ω) = a(ω)U (x, ω), where a(ω) is an excitation coefficient that may be determined
from the Dirichlet boundary condition, corresponding to the continuity condition,
û +
Γ (ω)
, ∀ω ∈ d = {ωn , n ∈ IZ},
D(ω)
where d is the set of the eigenfrequencies ωn for which D(ωn ) = 0. The actual traction on the coupling interface is given by:
T (ω) +
T̂ + (ω) = T̂ − (ω) = a(ω)T (ω) =
û (ω).
D(ω) Γ
a(ω) =
By definition, the DtN operator Â in the frequency domain is:
T (ω)
Â(ω) =
, ∀ω ∈ d .
D(ω)
Transforming back the traction in the time domain, using the causality, leads to:
+∞
t
TΓ+ (t) =
A(t − τ )u +
A(t − τ )u +
Γ (τ )dτ =
Γ (τ ) dτ,
−∞
(39)
(40)
(41)
0
3.4 Computing the operator DtN in the time domain
The inverse transform of the DtN operator Â is not straightforward due to the singularities at all the frequencies in d . The discrete spectrum,
associated with these singular contributions, have first to be isolated from the continuous spectrum. Noting An the Cauchy residual of Â at
the eigenfrequencies ωn , and using the continuity of T
dD −1
.
(42)
An = lim (ω − ωn )Â(ω) = T (ωn )
ω→ωn
dω ωn
The numerical computation of the derivative can be done using finite difference formula or more accurately using the kinetic energy as detailed
in the Appendix A:
dD 2ωn I1 (ωn )
,
(43)
=−
dω ωn
T (ωn )
where
I1 (ω) =
L0
ρ(x)U 2 (x, ω) dx,
(44)
0
and therefore to
T 2 (ωn )
An = −
.
2ωn I1 (ωn )
In practice, this requires an accurate computation of the kinetic energy I 1 .
The DtN operator Â can be written as
An
An
Â(ω) =
+
Â(ω)
−
.
n ω−ω
n ω−ω
n
n
a
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b
66
(45)
(46)
42
The inverse Fourier transform can be computed using Cauchy’s theorem for the singular contributions (46.a) and a regular inverse numerical
Fourier transform for the continuous contribution (46.b).
∞
An
1
iωn t
A
ie
H
(t)
+
(47)
Â(ω)
−
eiωt dω .
A(t) =
n
n ω−ω
n
2π −∞
n a
b
where H (t) is the Heaviside function. It is worth noting here that both contributions are in fact non causal and that the causality is only
retrieved by summing up the two contributions.
3.5 Time regularization of the DtN operator
Due to the spectral-element time-discretization, the allowed maximum frequency is restricted to the Nyquist frequency ω N . The corner
ˆ
frequency ωc (ωc < ω N ) of the source–time function, defines implicitly a truncated DtN operator, Âc (ω) = Â ◦ (ω),
where
1 for |ω| ≤ ωc
ˆ
.
(48)
(ω)
=
0 for |ω| > ωc
Since (t) is not a causal filter, Âc is non causal and the inverse transform of the traction becomes:
∞
)(t)
=
Ac (t − τ )u +
TΓ+ (t) = (Ac ∗ u +
Γ
Γ (τ )dτ,
(49)
0
but u +
Γ (t) comes from the spectral element solution and is known only up to the current time step. The DtN operator has therefore to be
regularized
Âr (ω) = Â(ω) − Ĉ(ω),
(50)
where Ĉ is an asymptotic approximation of Â. At high frequencies, Âr (ω) 0 and Âr ◦ ˆ Âr and the causality problem is avoided, and
the traction can be computed as
+
TΓ+ (t) = Ar ∗ u +
(51)
Γ (t) + C ∗ u Γ (t),
a
b
where the convolution term (51a) is now well defined and the last term can be computed analytically.
Following Barry et al. (1988), Ĉ can be constructed from the asymptotic solution of (35). We consider
−
u k (x)(iω)−k ,
û (x, ω) = e−iω(x)
(52)
k≥0
and for the stress:
τˆ− (x, ω) = e−iω(x)
τk (x)(iω)−k+1 .
(53)
k≥0
Assuming the normalization û +
Γ (ω) = 1, we have
1 pour k = 0
and Γ = 2nπ, n ∈ IZ.
u k |Γ =
0 pour k = 0
(54)
The second-order hyperbolic eq. (35), is equivalent to a first-order hyperbolic system of equations in displacement and stress. Noting
û − (x, ω)
,
(55)
y(x, ω) =
τˆ− (x, ω)
the first-order system becomes:
∂y
= S(x, ω)y,
∂x
with
0
λ(x)−1
.
S(x, ω) =
−ρ(x)ω2 0
(56)
(57)
The solution y can be developed as
y(x, ω) = e−iω(x)
yk (x, ω)(iω)−k ,
(58)
k≥0
with
yk (x, ω) =
u k (x)
iωτk (x)
.
(59)
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In order to solve (56) as a perturbation problem, S is now developed in powers of (iω)−1 ,
Sk (x, ω)(iω)−k .
S(x, ω) =
43
(60)
k≥−1
In the 1-D case, only S−1 is different from zero,
0
(iωλ(x))−1
S−1 (x, ω) =
.
iωρ(x) 0
(61)
Combining (55)–(56), (58) and (60), we get
(S−1 + I)y0 = 0,
k = −1
k≥0
(S−1 + I)yk+1 = yk −
(62)
j=k
Sk− j y j ,
(63)
j=0
where I is the identity matrix and “ ” denotes the x derivative. Eq. (62) has a non-trivial solution if and only if |S−1 + I| = 0 which means
√
2 − ρλ = 0. The outgoing wave solution is = − ρ/λ and from (62), we get
τ0 (x) = ραu 0 (x).
(64)
where α is the sound wave velocity. This condition is similar to the para-axial condition of order 0 of (Clayton & Engquist 1977). After some
manipulations, eq. (63), for k = 0, gives
α
(65)
τ1 (x) = −(ρα) u 0 (x) + ραu 1 (x),
2
and, with c1 = −(ρ α) α, we obtain for k = 1,
α α
(ρα)
c1 − c1
u 0 (x) − (ρα) u 1 (x) + ραu 2 (x).
(66)
τ2 (x) =
2
ρα
2
Setting x = L 0 in (64)–(66) and with the help of (54), we get a fourth order approximation of Ĉ:
Ĉ(ω) =
k=2
ck × (iω)−k+1 ,
(67)
k=0
with
c0 = ρα,
c1 = −
αc0
,
2
c2 =
α
2
c1 c0
− c1 ,
2c0
where ρ, α and their derivatives are computed at x = L −
0.
3.6 Validation tests of the coupled method
To validate the method, we performed several tests in two heterogeneous examples. In all cases, we compare the displacements, at three
positions, obtained when using a full spectral element method and the coupled method. The difference between the two solutions, amplified
by 100, is also shown.
In the first test, Fig. 3, the heterogeneity in Ω− is smooth and can be represented using a fourth-order polynomial. All the terms in the
expansion C are different from zero as shown in Fig. 4. The first term of the expansion ∝(iω), is similar to a time derivation. When subtracted
from Â, the regularized operator Âr is closed to a filtered Dirac shape. The second term of the expansion ∝(iω)0 , is similar to a Dirac term.
After substraction, the regularized Âr get a filtered Heaviside shape. The third term of the expansion, ∝(iω)−1 , is similar to an integration.
When subtracted, all the non causal signal is now removed. The agreement between the coupled method and the spectral element solution,
Fig. 5, is very good even at the coupling interface. The error is also shown as a function of the order of the expansion used to compute C in
Fig. 6. For this smooth example, the third order term does not really improve the solution when compared to the second order approximation
since the error is already of the order of the accuracy of the spectral element time stepping.
In a second test, we consider a rougher heterogeneity distribution, Fig. 7. In this case, the third order approximation is clearly more
accurate even if the improvement is not that impressive due to the simplicity of the example.
It is worth mentioning here, that even for these simple examples the coupled method leads to an interesting computational speed-up.
Indeed, in the last example, Fig. 7, the wave speed in Ω− leads to a severe CFL condition for the full spectral element method. In contrast,
the coupled method is only controlled by the far less stringent CFL condition in Ω+ . Such a computational speed-up will become clear when
solving the elastodynamics equations in spherical earth models.
4 3-D CASE
We now consider the case of a 3-D spherical earth domain Ω with the surface boundary Ω. The lateral heterogeneities are restricted to be
within the outer solid shell Ω+ , which may be composed itself of several subdomains, while the inner spherical domain Ω− is assumed to be
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68
44
13
11
9
rho (kg m−1)
v (m s−1)
7
5
0
1000
2000 3000
x (m)
4000
5000
Figure 3. 1-D smooth heterogeneous model with L 0 = 3000 m.
3
0.3
(a)
(b)
0
0.0
−3
1000
1500
0.01
−0.3
2000 1000
(c)
2000
1500
2000
(d)
0.00
−0.01
1000
1500
0.01
0.00
1500
−0.01
2000 1000
time (s)
Figure 4. (a): A(t) ∗ g(t) computed for the 1-D example of Fig. 3. The source-time wavelet, g, is a Gaussian of width 30s, centred in t = 1500 s. (b):
(A(t) − C(t)) ∗ g(t) after applying the first order regularization term. The Gaussian derivative shape has disappeared, and only a Gaussian shape remains. (c):
(A(t) − C(t)) ∗ g(t) after applying the second order regularization term. Now only a filtered Heaviside shape remains. (d): (A(t) − C(t)) ∗ g(t) after applying
the third order regularization term. A causal signal is now retrieved.
radially heterogeneous with solid and liquid subdomains. The coupling interface Γ between the outer shell and the inner spherical domain
may be a physical interface, like the CMB, or an artificial interface somewhere within the mantle or the outer core.
The original elastogravity problem, within the outer shell Ω+ , is solved, in space and time, using the variational spectral element
approximation. The explicit construction the DtN operator, on the coupling interface Γ, involves a modal solution within the inner spherical
domain Ω− . We focus on the actual coupling between the spectral element and the modal solution methods, and only sketch out the spectral
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1
45
(a)
0
−1
1
(b)
0
−1
1
(c)
0
−1
0
1000
2000
3000
4000
time (s)
Figure 5. Displacement computed for the smooth heterogeneous example of Fig. 3 and recorded at x = L 0 (a), x = 3960 m (b) and x = L (c). The time-source
function is a Ricker (second derivative of a Gaussian) of 30 s width and centred at t 0 = 400 s. The modal solution for the whole domain is shown in solid
line. The difference with the coupled solution, amplified by a factor 100, is shown in dotted line. The amplitude is normalized with reference to the maximum
displacement recorded at x = L.
element variational formulation in Ω+ within the context of a conforming approximation, i.e. point-wise continuity conditions on the solid–
solid SS and the solid-fluid SF interfaces associated with a conforming spatial discretization of Ω+ . In practice however, a more sophisticated
non conforming spectral element approximation is used allowing mesh refinements both in the radial and lateral directions in order to resolve
accurately the velocity and the geometry structures of realistic earth models. For a more thorough presentation of the non conforming spectral
element approximation, in the context of global earth models, the readers are referred to Chaljub (2000) and Chaljub et al. (in preparation),
where an efficient parallel implementation is also discussed in some details.
4.1 Variational approximation
We look now for a solution in the space of the kinematically admissible displacements,
C t = {u(r, t) ∈ H 1 (Ω+ )3 : Ω+ × I → IR3 },
(68)
The problem to be solved is: find u+ ∈ C t , such that ∀t ∈ I = [0, T ] and ∀w ∈ C t
∂u
ρ 2 , w + a(u+ , w) − T+
Γ , w Γ = (f, w),
∂t
(w, ρu+ )|t=0 = 0,
w, ρ
(70)
∂u+ = 0,
∂t t=0
(71)
where (· , ·) is the classical L 2 inner product. The symmetric bilinear form a(· , ·) is now given by
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2003 RAS, GJI, 152, 34–67
(69)
70
46
1
SEM
error 1
100
error 2
100
error 3
100
0
−1
500
1000
1500
2000
time (s)
Figure 6. Displacement recorded at x = 3960 m for the heterogeneous example of Fig. 3 with the same source–time function than Fig. 5. The solution obtained
when using only the spectral element method is shown in solid line. The Dotted lines represent the difference, amplified by a factor 100, with the solution
obtained using the coupled method with a first order regularization (error 1), a second order regularization (error 2), and a third order regularization (error 3).
1
(a)
SEM
error 1
error 2
error 3
15
(b)
100
100
100
13
0
11
rho (kg m−1)
v (m s−1)
9
−1
500
1000
1500
2000
0
1000
2000 3000
x (m)
4000
5000
time (s)
Figure 7. (a) the same experiment as in Fig. 6 but for a rougher heterogeneity distribution shown on the left. The displacement is recorded at x = 3960 m.
a(u+ , w) =
Ω+
[τ (u+ ):∇(w)] dx +
Ω+
ρ sym{(w · g)∇ · u+ − u+ · ∇(w · g)} dx,
(72)
and
+ + TΓ · w dx,
TΓ , w Γ =
(73)
Γ
where sym denotes the symmetric part and T+
Γ = τ · n|r =rΓ+ is the traction on the spherical coupling interface Γ.
The continuity conditions across the coupling interface Γ, depend whether Γ is a solid–solid or solid–liquid interface:
−
T+
Γ (r, t) = TΓ (r, t),
−
u+
Γ (r, t) = uΓ (r, t), if Γ is a solid–solid interface,
u+
Γ (r, t).nΓ (r)
=
u−
Γ (r, t)
(74)
· nΓ (r), if Γ is a solid–fluid interface,
and
are the restrictions of T and u to Γ in Ω± . Assuming a solution of the elastogravity problem within the domain Ω− , for
where
−
a prescribed Dirichlet boundary condition along the interface Γ, the DtN operator A, that relates the displacement u−
Γ to the traction TΓ ,
can be constructed. Taking into account the continuity equations, the DtN operator, for solid–solid coupling interface, is therefore:
T±
Γ
u±
Γ
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47
+
A : T+
Γ (r, t) = A uΓ (r, t) ,
(75)
and for a solid–fluid interface,
+
A : T+
Γ (r, t) = A uΓ (r, t) · n(r) .
(76)
4.1.1 Spatial discretization
The first step is to define, in a Cartesian reference coordinate system, a spatial discretization of the outer shell Ω+ into n e non overlapping
e
elements Ωe such that Ω = ∪ne=1
Ωe . The spectral element method puts the additional constraint that the spherical grid must be based on
hexahedra. Furthermore, the spherical mapping should be as regular as possible and such that the elements of the mesh are not too distorted
and the sampling of the grid-points as uniform as possible.
To satisfy the first constraint, we must be able to tile a 2-sphere with quadrangles for each spherical interface of Ω+ . This is achieved
using the central projection of a cube onto its circumscribed sphere. Such a cubic-gnomic projection has been introduced by Sadourny (1972)
and further extended by Ronchi et al. (1996) as the ‘cubed sphere’. With the help of the central projection, the 2-sphere is decomposed into
six regions isomorphic to the six faces of a cube. By choosing the coordinate lines within each region to be arcs of the great circles (see Fig. 8)
six coordinate systems, with identical metrics, are obtained free of singularity.
With a constant angular distance between each great circle, a regular meshing of the six regions is defined in terms of deformed squares
with uniform edge width. The surface mesh is quite uniform with a maximum distortion not exceeding 30 per cent Chaljub (2000). The
construction of a 3-D mesh inside the spherical shell is then straightforward by simply radially connecting the quadrangles of two concentric
discretized spherical interfaces. These interfaces can be mapped onto physical interfaces associated with radial discontinuities within the
mantle. As a result, the spherical shell is discretized into regular hexahedra (see Fig. 9) and the associated geometrical transformation is
known analytically (Chaljub et al. in preparation). It can be easily extended to take into account an interface topography. We refer to Chaljub
et al. (in preparation) for more details and for the extension to the discretization of the whole sphere.
For each element Ωe , an invertible geometrical transformation Fe can be defined allowing one to map the reference cube × × into Ωe , with = [−1, 1], such that x(ξ) = Fe (ξ) where ξ = (ξ1 , ξ2 , ξ3 ) defines a local coordinate system associated with the reference unit
cube.
Following the same steps as in the last section, we introduce a piecewise-polynomial approximation of the kinematic admissible displacements C t :
(77)
C Nh = uh ∈ C t : uh ∈ H 1 (Ω+ )3 and ueh ◦ Fe ∈ [IP N ()]3 ,
In the reference cube, [IP N ()]3 is taken as the space generated by the tensor product of the polynomials of degree ≤N in each of the
three Cartesian directions. The discrete inner products involved in the variational formulation are constructed as the tensorial product of
the 1-D Gauss–Lobatto–Legendre formulae in each of the local Cartesian directions ξ 1 , ξ 2 , ξ 3 . This defines a grid of (N + 1)3 quadrature
points. The piecewise polynomial approximation ueh of u is defined using the Lagrange interpolation associated with the grid composed of
the Gauss–Lobatto–Legendre integration points. The Lagrange interpolants are therefore the tensor product of the 1-D Lagrange interpolants.
North
West
East
South
Figure 8. The Gnomic projection: great circles used to mesh one region of the ‘cubic sphere’.
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48
Figure 9. (a) a split view of the six regions produced by the gnomic projection which map the six faces of a cube inscribed inside the sphere to the surface of
the sphere. (b) a gathered view of the six regions of the sphere. Each region has its own coordinates system and the coordinate transformations can be calculated
analytically.
At the end of the discretization, the problem is reduced to an system of ordinary differential eq. (30) in time which is solved by a classical
Newmark algorithm.
4.2 DtN operator construction in the spectral and frequency domain
The construction of the DtN operator requires the solution of the elasto-gravity problem in the inner sphere Ω− , assuming a Dirichlet boundary
+
−
+
condition on the interface Γ: u−
Γ = uΓ for a solid–solid coupling or uΓ · nΓ = uΓ · nΓ for a solid–liquid coupling. For a spherically symmetric
inner domain, the classical modal solution is computationally attractive. In the following, we outline the modal solution in the case of a
solid–solid coupling. A similar solution can be found for the solid–liquid coupling case and only the explicit expression of the DtN operator
in this case is given.
In the following, the domain Ω− is assumed to have a spherical symmetry. The problem can be formulated in the generalized spherical
harmonics base Phinney & Burridge (1973), eα , α ∈ {−, 0, +}, defined as
1
e0 = er , e± = √ (∓eθ − ieφ )
2
where er , eθ , eφ are unit vectors in the r , θ and φ directions. The fully normalized generalized spherical harmonics are denoted Y α,m (θ, φ)
where is the angular order, m the azimuthal order and α the component in the eα basis. The forward Legendre transformation of a given
vector field u ( r , t) is defined by
2π π
u(r, θ, φ, t) · Y ,m (θ, φ) sin θ dφ dθ.
(78)
u,m (r, t) =
0
0
α
where Y ,m is a tensor with components [Y ,m ]αα = Y,m
δαα in the eα basis and Y ,m its conjugate. The inverse Legendre transform is
therefore
u(r, t) =
u,m (r, t) · Y ,m (θ, φ),
(79)
,m
where the sum over m extends from − to .
The momentum equation in the frequency domain is,
−ω2 ρ(r)u− (r, ω) − H(r)u− (r, ω) = 0.
(80)
The solutions have to be regular at the origin r = 0, and must satisfy the Dirichlet condition at the coupling interface Γ:
+
u−
Γ (r, ω) = uΓ (r, ω).
(81)
−
Taking into account the spherical symmetry of Ω , we seek for a solution of the form
u− (r, ω) = d,m (r, ω) · Y ,m (θ, φ).
(82)
and the corresponding stress vector, defined on concentric spherical surfaces as T(r, ω) = τ (r, ω) · er , is written as
T− (r, ω) = T,m (r, ω) · Y ,m (θ, φ).
(83)
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It is worth mentioning here that there is a one to one relationship between this parametrization and the more classical one used for normal
modes (e.g. Gilbert 1971), where the solution is expressed as
0
u− (r, ω) = [U,m (r, ω)er + V,m (r, ω)∇1 − W,m (r, ω)(er × ∇1 )]Y,m
(θ, φ),
(84)
0
= ζ U,m ,
d,m
(85)
where ∇1 is the gradient operator on the unit sphere. In the eα -basis, we do have,
ζ γ
−
d,m
= √ (V,m − i W,m ),
2
ζ γ
+
= √ (V,m + i W,m ),
d,m
2
where γ = ( + 1) and ζ = (2 + 1)/4π.
Inserting (82) into (80) leads to two independent systems of second order differential equations in r degenerated for the azimuthal order
m. The first system, spheroidal, is of order two, while the second, toroidal, is of order one (Takeuchi & Saito 1972; Saito 1988; Woodhouse
1988). For a given and a fixed frequency ω, they are six independent solutions, four spheroidal ones and two toroidal ones, but only three
of them satisfy the regularity condition at the origin : two spheroidal ones and one toroidal. These solutions are denoted q d (r , ω) with q =
{1, 2, 3}.
The set of solutions {q d (r , ω), q = 1, 2, 3} defines a complete basis for the displacement in Ω− . The solution of the elastogravity
eq. (80) can therefore be written as:
u−
(86)
q a,m (ω)q d (r, ω).
,m (r, ω) =
q
where {q a ,m (ω)}, with (, m) ∈ IN × [−, ], are the excitation coefficients which have to be determined according to the Dirichlet boundary
condition imposed on the coupling interface Γ.
In the following, we note D and T the tensors defined in the eα -basis by,
[D ]q,α (ω) = q dα (rΓ , ω),
[T ]q,α (ω) = q tα (rΓ , ω),
(87)
and ã,m the vector of components [ã,m ]q = q a,m . Taking into account the Dirichlet condition associated to the solid–solid coupling, we get
−1
ã,m (ω) = u+
,m (r Γ , ω) · D (ω)
∀(, m, ω), ω ∈ d ,
(88)
d
is the set of all eigenfrequencies for which D is singular. It is worth noting here that, in contrast to the earth free-oscillation
where
problem, d is defined as the set of all the eigenfrequencies corresponding to an homogeneous Dirichlet boundary condition, see
Appendix B.
When ã,m (ω) is known, the traction on the surface Γ, on the Ω+ ’s side, can be easily found using the continuity conditions:
−
T+
,m (r Γ , ω) = T,m (r Γ , ω) = ã,m (ω) · T (ω),
(89)
and therefore
+
−1
· T (ω).
T+
,m (r Γ , ω) = u,m (r Γ , ω) · D (ω)
(90)
This explicitly defines the DtN operator:
+
T+
,m (r Γ , ω) = A (ω) · u,m (r Γ , ω)
∀ω ∈ d ,
(91)
where in the frequency–spectral domain
t
A (ω) = D −1
(ω) · T (ω)
∀ω ∈ d ,
(92)
t
and denotes the transposition. The DtN operator can be shown to be symmetric (see Appendix B) and therefore, the transposition in expression
(92) can be dropped.
In the case of a solid—fluid coupling interface Γ, there is only one solution for (80) which satisfy the regularity condition at the origin.
The tensors D and T reduce now to scalars D and T . The expression of the DtN operator in the eα -basis is given by:
[A (ω)]αα = D−1 (ω)T (ω)δ0,α δ0,α ∀ω ∈ d .
(93)
4.3 Time and space regularization of the DtN operator
The spectral element approximation requires the interface traction T+
Γ to be known in space and time,
t
T+
A (τ ) · u+
,m (r Γ , t − τ )dτ · Y ,m (θ, φ).
Γ (r, t) =
,m
(94)
0
The sum over and m is the inverse Legendre Transform and the time integral involves a convolution. Numerically, the DtN operator requires
a truncation both in the spherical harmonics expansion and in frequency. In order for the truncated problem to be well posed, it has to be
regularized both in time and space. Such a regularization follows the same steps than in the 1-D example.
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50
The singularity contributions have first to be isolated as in (46). The Fourier transform can then be computed using Cauchy’s theorem
for the singular contributions and a regular Fourier transform for the continuous part. However, due to the truncation in frequency, A (t) has
to be regularized in order to circumvent the same causality problem that was encountered in the 1-D example. This is done with the help of
an asymptotic approximation C (ω), for the high frequencies, of the DtN operator A:
Ar (ω) = A (ω) − C (ω).
(95)
where C (ω) is given by:
C (ω) =
jmax
c, j × (iω)− j+1 ,
(96)
j=0
and the coefficients c, j can be explicitly computed, see Appendix C for details. In practice, a third order approximation, jmax = 2, is required.
The regularized DtN operator Ar (ω) is now causal and its inverse transform in space and time presents no difficulty. The traction on the
coupling interface can be computed as:
τ
t r
+
+
+
+
T+
(r,
t)
=
A
(τ
)
·
u
(r
,
t
−
τ
)dτ
+c
·
v
(r
,
t)
+
c
·
u
(r
,
t)
+c
·
u
(r
,
τ
)dτ
· Y ,m (θ, φ),
(97)
Γ
,0
Γ
,1
Γ
,2
Γ
,m
,m
,m
,m
Γ
,m
0
0
where v+ is the first time derivative of u+ . As shown in Appendix C, c,0 corresponds to the Sommerfeld operator and does not depend on ,
i.e. c,0 is local in space, leading to:
τ
t r
+
+
+
+
(r,
t)
=
c
·
v
(r
,
t)
+
A
(τ
)
·u
(r
,
t
−
τ
)dτ
+
c
·
u
(r
,
t)
+c
·
u
(r
,
τ
)dτ
· Y ,m (θ, φ).
(98)
T+
0,0
Γ
Γ
,1
Γ
,2
Γ
,m
,m
,m
Γ
Γ
,m
0
0
The inverse Legendre transform in eqs (94) or (98) requires to truncate the infinite sum over and m. The choice of the truncation max is
based directly on the dispersion curves Fig. 10 which provides an upper angular order. Knowing the corner frequency of the spectral element
approximation, a critical c can be determined such that there is no eigenfrequency smaller than the corner frequency. Far enough from the
source, the wavefield has a maximum angular degree of c and therefore max = c provides an accurate approximation for the coupling. Such
an estimation is only valid when the source is not too close to the coupling interface. When this is not the case, the choice max = 2c has
been found empirically to provide an accurate approximation.
Even though there is no causality problem induced by the space truncation, such a truncation is only well-posed for the lower-order
spherical harmonics. The problem may be circumvented, as shown by Grote & Keller (1995), with the help of a spatially regularized operator
As = A − S :
T+
Γ (r, ω) =
max +
=0 m−
+
As (ω) · u+
,m (r Γ , ω) · Y ,m (θ, φ) + S(ω) · u (rΓ , ω),
(99)
where S (ω) may be any computationally efficient approximation of the DtN operator with the property: Su, uΓ < 0, where denote the
imaginary part. In practice for S (ω), one can take the Sommerfeld operator:


ρβ
0
0


S (ω) = iω 
(100)
ρα
0
0
.
0
0
ρβ
(a)
0.010
Pulsation (rad s−1)
Pulsation (rad s−1)
0.010
0.008
0.006
0.004
0.002
0.000
Toroı̈dal eigenfrequencies
(b)
Sphéroı̈dal eigenfrequencies
0.008
0.006
0.004
0.002
0
10
20
30
40
50
Angular order
60
0.000
70
0
10
20
30
40
50
60
70
Angular order
Figure 10. The sphéroidal (a) and the toroidal (b) eigenfrequencies of the DtN operator computed for a homogeneous sphere.
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Figure 11. The spectral element mesh used for the homogeneous sphere model. The regions 1 and 4 have been removed for visibility.
1
0.5
Amplitude
0.8
0
0.6
0.4
−0.5
0.2
−1
0
400
time (s)
800
0
0
0.005
0.01
Frequency (Hz)
0.015
Figure 12. The source–time function used for the homogeneous sphere model (a) and the corresponding source spectrum content (b). The central frequency
is 3.2 × 10−3 Hz (321 s) and the corner frequency is 8 × 10−3 Hz (125 s).
where α and β are the P- and S-wave velocities, respectively . The Sommerfeld operator is an absorbing boundary operator corresponding to
the first order para-axial approximation Clayton & Engquist (1977). Therefore, all spurious waves, with a spatial spectrum of angular order
greater than max , will not be transmitted by the regularized operator but actually absorbed, an interesting and stabilizing property. It is worth
noting here that S (ω) = iωc,0 . Therefore the space regularization is already taken into account by the time regularization and (98) can be
used directly to compute T+
Γ (r, t) truncating the sum over to the appropriate max .
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t=1264s
t=414s
P
Source
P
t=1551s
t=600s
X
P
P
R
t=1035s
P
t=1864s
P
Figure 13. The wavefield propagation in a cross-section of the homogeneous sphere model for an explosive source. The snapshots represent the x-component
of the displacement field inside Ω+ , the spherical shell domain of the spectral element method, at different time steps of the propagation. The displacements in
Ω− , the inner sphere of the modal summation method, are not shown here. The accuracy of the coupling method can be assessed here looking at the transmitted
wavefield.
5 VA L I D A T I O N S T E S T S O F T H E 3 - D C O U P L E D M E T H O D
In this section, synthetic seismograms for a simple homogeneous sphere and the spherical earth model PREM Dziewonski & Anderson (1981)
are computed and compared with the solution obtained by the summation of the free oscillations in order to test the accuracy of the method.
In a homogeneous sphere, normal modes radial eigenfunctions are known analytically and eigenfrequencies are known up to the computer
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accuracy. In a spherically symmetric earth model, normal modes are known with a very good precision. Therefore, in both cases, the normal
mode solution is a very accurate reference solution which allows for the validation of the coupled method.
In the homogeneous sphere test, only a solid–solid Γ interface is included. For the more realistic PREM test, the coupling interface Γ is
set to be the CMB and includes a solid–liquid coupling.
5.1 Wave propagation in a homogeneous sphere
We consider a homogeneous solid sphere with a radius rΩ = 6371 km, a density ρ = 3000 kg m−3 , a P-wave speed α = 8 km s−1 and an
S-wave speed β = 6 km s−1 . The position of the solid–solid coupling interface Γ is set to rΓ = 3471 km. The overall thickness of the outer
spherical shell Ω+ is 2500 km. Each of the 6 regions of Ω+ are discretized with 8 × 8 elements horizontally and 2 elements vertically (see
Fig. 11), with a total number of 768 elements.
In each element, the polynomial approximation is of degree N = 8, which set a total number of 559 872 grid points in Ω+ . The shortest
distance between two grid points is 38 km, which, for a Courant number of 0.4, leads to a maximal time step of 1.9 s. In the numerical
experiment, the actual time step is 0.5 s which is quite conservative.
For the source, the lowest period of the wavelet time function has been set to 125 s which, for a wave speed of 5 km s−1 (the approximate
Rayleigh wave group velocity in this model), insures less than two wavelengths per element. The wavelet time function, Fig. 12, is a Ricker
with a central frequency 1/312 Hz and a corner frequency of 1/125 Hz. The source is an explosion of amplitude 1020 kg m s−2 and three
different source depths are used:
(i) source A: at 398 km depth. The source is very closed compared to the wavelength of the free surface and provides a good test for the
spectral-element accuracy in the case of strong surface waves.
(ii) source B: at 1048 km depth.
(iii) source C: at 2298 km depth. The source is now very closed to, compared to the wavelength, the coupling interface Γ and provides a
good test for the accuracy of the coupling when Γ is in the near field range.
6000
4000
R
2000
X
time (s)
8000
Vertical component
P
-135
-90
-45
0
45
90
135
Epicentral distance (degrees)
Figure 14. The vertical component of seismograms, recorded at the surface, as a function of the epicentral distance from the source B. Different phases are
pointed out here: Rayleigh (R), X phase (X) and P wave (P).
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In all these numerical experiments, max = 37, but for sake of security, we choose max = 47 for source A and B and max = 119 for source C,
in order to capture with the near field. The explosion has been chosen for its symmetric radiation pattern, but a test with non diagonal moment
tensors is also presented to check the transverse component and a more realistic source.
Snapshots of the wave propagation in Ω+ are shown in a cross-section, Fig. 13, for the source B. At t = 660 s, the central time of the
source–time function is 400 s, the P-wave start to be absorbed by the DtN operator, without any visible spurious reflection. At 1035 s, the pP
wave start now to be absorbed by the DtN operator. At 1551 s the P-wave emerges from Ω− into Ω+ . The X phase, a superposition of higher
spheroidal modes, is strongly excited by the source B. The Rayleigh wave, which is weakly excited due to the depth of the source, is however
visible on the snapshots.
Traces of the vertical component of the displacement, are also recorded on the surface as a function of the epicentral distance, Fig. 14.
Despite the depth of the source, the Rayleigh wave is clearly visible. The Rayleigh wave is almost non-dispersive for this frequency source
range: only a small dispersion is observed, and accurately modelled, for the lowest frequencies as expected theoretically. The X-phase, which
corresponds in terms of body waves to a superposition of P-wave subsurface reflections, is strongly excited. This phase is quite dispersive as
expected theoretically and the dispersion is accurately modelled. Finally the P-wave is clearly observed as well as some several multiples PP,
PPP, . . . It is worth noting that these multiples tend slowly towards the X phase.
No spurious phase can be observed on Figs 13 and 14. However, in order to assess the accuracy of the simulation, a direct comparison
between the results obtained with the coupled method, for the three source positions, and the reference solution obtained by the normal modes
summation is shown in Fig. 15. In each case the residual, i.e. the difference between the two solutions, has been multiplied by a factor ten.
The agreement between the two solutions is indeed very good, with less than 1 per cent of error. Surface waves, which are strongly excited in
the case of the source A, are very accurately modelled by the spectral element method. The body waves (P- and S-waves) recorded directly on
the coupling interface Γ are also well modelled,an especially difficult test for this method and clearly shows that the DtN operator accurately
Receiver on surface
(a)
(b)
source A (398 km depth)
source A (398 km depth)
12
2
0
0
−12
Receiver on
−2
source B (1048 km depth)
4
0
0
source B (1048 km depth)
Amplitude (
m)
1.5
−1.5
−4
source C (2298 km depth)
2
3
0
0
−2
0
5000
time (s)
−3
10000 0
normal modes
coupled method
residual * 10
source C (2298 km depth)
5000
10000
time (s)
Figure 15. The vertical component of the displacement, at an epicentral distance of 90◦ , recorded on the surface (a) and on the coupling interface Γ (b). The
reference normal mode solution is drawn with the solid line, the coupled method solution is displayed with the dotted line and the residual between the two
methods, amplified by a factor 10, is displayed with the bold dotted line. The method is very accurate and the maximum relative error is less than a few percents.
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Amplitude (
m)
(a)
(b)
Receiver on surface
Receiver on
50
coupled method
normal modes
residual * 10
80
55
25
0
−20
−25
−120
0
5000
−50
10000 0
Time (s)
5000
10000
Figure 16. The transverse component of the displacement, at an epicentral distance of 49◦ , recorded on the surface (a) and on the coupling interface Γ (b).
The source is at 1048 km depth, for a moment tensor with Mrr = Mθ θ = Mφφ = Mr θ = 0 and Mr φ = Mθ φ = 1. The reference normal mode solution is
drawn with the solid line, the coupled method solution is displayed with the dotted line and the residual between the two methods, amplified by a factor 10, is
displayed with the bold dotted line.
captures the response of the inner sphere for all the phases. The same accuracy is achieved on the transverse component with a moment tensor
(Mrr = Mθ θ = Mφφ = Mr θ = 0 and Mr φ = Mθ φ = 1) using the source B location (Fig. 16).
5.2 Wave propagation in PREM
We consider the spherically symmetric reference earth model PREM. The coupling interface Γ is set at the CMB. The spectral element outer
shell Ω+ includes the whole mantle and the crust. This is, in fact, quite a challenging problem for the spectral element method due to the crustal
Mortar
Figure 17. The non conforming mesh used for the PREM example: below the 660 km interface, the mesh has been derefined by a factor two in the horizontal
direction. The colour indicates the density structure of the model. The regions 1 and 4 have been removed here for visibility.
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56
(b)
(a)
1
1
0.8
Amplitude
0.5
0.6
0
0.4
−0.5
−1
0.2
0
500
time (s)
1000
0
0
0.005
0.01 0.015 0.02 0.025
Frequency (Hz)
Figure 18. The source–time function used for the PREM example (a) and the corresponding spectrum content (b). The corner frequency is 1/48 Hz.
structure inherent to the PREM model. The crust is characterized by slow wave velocities and the presence of very thin piecewise homogeneous
layers with sharp elastic property contrasts: 15 and 10 km for the two uppermost layers. In the vertical direction, each of these concentric
layers has to match with an element boundary in order to correctly approximate those contrasts. This drastically reduces the minimum size of
the elements, in the vertical direction, and therefore put stringent constraints on the time step which must fulfill the CFL condition. Since each
of these layers is piecewise homogeneous, the problem can be partly improved when using a very low vertical polynomial degree, a degree 2,
within these crustal layers while retaining a degree 8 in the lateral directions. The time step imposed by the vertical structure of the crust in
PREM is of the order of 0.29 s. The slow wave velocities in the crust imply laterally a good spatial resolution in order to accurately represent
the surface waves.
The mesh used in this example was built to match the nine surfaces of discontinuity in the mantle that are included in PREM. The mesh is
non-conforming (Chaljub et al. in preparation) in the vertical direction. In the lateral direction, 32 elements are used in the upper-mantle and
16 elements in the crust. The non conforming interface is set at the 660 km transition zone. Such a mesh allow for a uniform CFL condition
all through the mantle. This mesh could theoretically support a corner frequency of 1/45 Hz, but the actual corner frequency used in this
example is 1/48 Hz, Fig. 18. The source is still explosive, for sake of simplicity, and its location is at 169 Km.
The vertical component of the displacement recorded at the surface, for PREM, is shown on Fig. 19 as a function of time and of the
epicentral distance. In order to identify some of the body-wave phases, time arrivals of various phases have been computed by an asymptotic
ray tracing, as shown on Fig. 20 and superimposed on the synthetic seismograms of Fig. 19. The comparison between the time arrivals, derived
from a high frequency approximation and a full waveform modelling, derived from a direct numerical simulation, is not that obvious due to
finite frequency effects on the whole waveform, especially for body waves. Actually the notion of the ‘arrival time’ only makes sense within
the high frequency approximation. As a matter of fact, as for the P-wave, the ray tracing arrival time points sometimes to the maximum of
the phase amplitude, in which case the agreement appears quite good, while sometimes it points to the minimum of the phase amplitude, or
to a change of sign, mixing of different phases. Obviously full waveforms obtained by direct numerical simulation do contain much more
information than traditional arrival time analysis based on asymptotic theories. However for the subject of this paper, such a comparison does
not allow to assess the accuracy of the coupled method. A more thorough discussion of these results in the case of PREM and other laterally
heterogeneous earth models will part of a forthcoming paper.
The accuracy can be assessed by a direct comparison between the synthetic seismograms obtained by the proposed direct numerical
simulation and the classical normal modes summation method. Such a comparison is shown on Figs 21 and 22 where displacements are
recorded at the free surface and on the coupling interface, actually the CMB, for three different epicentral distances. The residual between the
two solutions, amplified by a factor 10, does show a very good agreement, even when the displacement is recorded on the coupling interface.
This clearly show that the DtN operator, corresponding in this case to a solid-fluid interface, accurately model the response of the earth core,
for all the wave phases.
Interestingly enough, one should note here that if the PREM example was solved by a full spectral element approximation (Chaljub et al.
in preparation), the geometry and the velocity structures of the inner core would put stringent constraint on the time step. This is completely
removed here in the coupled method since the core domain is now solved by a modal summation technique. Typically for large problems, such
as the PREM example, the construction of the DtN operator and the coupling represents less than 13 per cent of the whole CPU time. With the
mesh used in this example and a corner frequency of 1/45 Hz, a simulation over 3050 s, e.g. 12 200 time steps, requires 30 hr on a cluster of
32 processors Pentium IV 900 MHz with 16 GB distributed memory and further minor optimizations should allow to reduce by a factor 1/3
the requested memory.
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1000
time (s )
2000
3000
Radial component
57
Figure 19. The vertical component of the seismograms computed in PREM and recorded at the surface, as a function of the epicentral distance. Time arrivals
of some body waves computed by a ray tracing has been superposed and can be identified using Fig. 20.
4200
PcS(ScS)3
3200
PcS(ScS)2
Time (s)
P5
P4
(PKIKP)2
PPS
(PcP)4
PKKS
PKKP
2200
PcSScS
P3
(PcP)3
PS
PKS
PP
PcPPcS
PKP
(PcP)2
PKIKP
PKiKP
PcS
1200
PcP
P
200
150
100
50
Epicentral distance (degrees)
Figure 20. Time arrivals of some body waves in PREM computed with a ray tracing.
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82
0
58
(a)
(b)
20 degrees, vertical component
8
5
0
0
−8
−5
3
0
0
90 degrees, longitudinal component
Amplitude (
m)
2
−2
−3
2
4
0
0
−2
0
1000
2000
Time (s)
−4
3000 0
1000
2000
Time (s)
3000
Figure 21. Displacement recorded at the free surface in the PREM example. The vertical (a) and longitudinal (b) displacements are plotted for three epicentral
distances. The modal solution (reference) is plotted with a solid line while the coupled method solution is displayed with a dotted line (superposed to the
previous one). In a bold dotted line, the residual, multiplied by 10, between the two solutions is also shown on the same figures.
Fully 3-D validation tests are very difficult to set up because a reference solution is missing in that case. Nevertheless some comparisons
with method based upon approximation can be performed in the condition of validity of the approximation. Such a comparison with first
order normal mode perturbation (Born approximation) has been attempted in Capdeville et al. (2002) and shows very good agreement when
the heterogeneity velocity contrasts is weak.
6 CONCLUSIONS
A new method which couple a spectral element with a modal summation method for simulating 3-D seismic wave propagation in non rotating
elastic earth models has been presented. The strategy is to decompose the earth model into a 3-D heterogeneous outer shell and an inner
sphere that is restricted to being spherically symmetric. The two domains are therefore connected through a spherical coupling interface that
may be either a physical interface or an artificial one. Depending on the problem, the outer shell can be mapped either as the whole mantle or
restricted to some portion of the upper-mantle or the crust.
In the outer heterogeneous shell, the solution is sought in terms of the spectral element method based on a high order variational
formulation in space and a fully second-order explicit time discretization. Within the inner sphere, the solution is sought in terms of a modal
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Figure 22. Displacement recorded at the CMB which here is the coupling interface Γ for the PREM example. The vertical (right) and longitudinal (left)
displacements are plotted for three epicentral distances. The modal solution (reference) is plotted with a solid line while the coupled method solution is
displayed with a dotted line (superposed to the previous one). In a bold dotted line, the residual, multiplied by 10, between the two solutions is also shown on
the same figures.
summation in the frequency domain after expansion of the space variable into the generalized spherical harmonics basis. The spectral element
method combines the geometrical flexibility of classical finite element method with the exponential convergence rate associated with spectral
techniques. Classical pole problems are avoided by tilling the spherical interfaces with six rectangular regions that can be easily mapped to
rectangles. This is accomplished using the central or gnomic projection. The six regions can be further divided into quasi-uniform rectangular
elements. The surface discretizations are then connected radially to build the 3-D mesh of the outer shell. An essential feature of the spectral
element method is that it can resolve sharp localized variations, as well as topographical features along the interfaces. High resolution can
be obtained using mesh refinements with a non conforming discretization. The coupling with the modal summation method is achieved
in the spectral element method by introducing a dynamic coupling operator, the DtN operator, which can be explicitly constructed in the
frequency/wavenumber domain. This allows to significantly speed up the computation. The key point here is the inverse transformation in the
space-time domain of the coupling operator. This requires special attention and a suitable asymptotic regularization. This is fully detailed for
both a simple 1-D example and 3-D earth models.
The effectiveness of the method is then demonstrated for both the simple 1-D example and 3-D cases: a homogeneous elastic sphere and
the PREM model. For spherically symmetric earth model, the method is shown to have most of the accuracy of spectral transform methods.
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2003 RAS, GJI, 152, 34–67
84
60
However, the method has the advantage of allowing the resolution of wavefield propagation in a 3-D laterally heterogeneous model for the
same computational cost and without any perturbation hypothesis. Applications to 3-D laterally heterogeneous earth models has already been
performed and will be detailed in a forthcoming paper. The method has been highly parallelized on distributed memory architecture, both
for the spectral element and the modal summation part, as detailed in Capdeville (2000), Chaljub (2000) and Chaljub et al. (2001). The
coupled method allows to speed up the computation and makes the simulation of seismic wavefield in earth models down to 30 s possible
today on a medium size parallel architecture such as a cluster of 32 processors with 32 GB of memory. This is in contrast with the more
demanding full spectral element approximation, as detailed in Chaljub et al. (2001), even though the last approach is more general. The gain
in memory and time computing obtain by using the coupled method compared to full spectral elements is strongly dependent on the earth
model, location on the DtN and frequency range. In cases presented in this paper, the gain obtain by the coupled method is the not sufficient
to justify the coupled method (less than a factor of two on both memory and time computing). But there are cases, like the simulation in
3-D D models at period under 10 s (Capdeville et al. in preparation), where the coupled method is the only option with a gain of more than
10 in memory (the machines necessary to do the same thing with only spectral elements are not available), which fully justified the coupled
method.
It is worth noting that no one scheme can be expected to be optimal for the entire range of applications we might wish to consider in the
context of seismology. The method presented here is therefore a first step toward efficient direct numerical simulation methods for wavefield
propagation and synthetic seismograms for some applications in global seismology. Extensions are already actually under study. Among them,
it is worth mentioning the incorporation of realistic surface topographies and the implementation of a spectral element strip between two
domains in which the solution is sought via a modal summation technique. The latter will allow the study of highly localized heterogeneities
in the vicinity of some interfaces like the core–mantle boundary at relatively high frequencies (Capdeville et al. in preparation).
ACKNOWLEDGMENTS
The authors wish to acknowledge enlightening discussions with B. Valette, F.J. Sánchez–Sesma, C. Bernardi and Y. Maday. This work has
been partially funded by the IT programme of INSU (CNRS). The computations were made using the computational resources of the DMPN
(IPG Paris), with the help of P. Stoclet and G. Moguilny, and of the CINES in Montpellier, South of France.
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APPENDIX A: COMPUTATION OF THE CAUCHY RESIDUALS
The analytical computation of the Cauchy residuals An in eq. (42) requires the evaluation of the derivative of D at ωn . Using the continuity
of T at the frequency ωn ,
dD −1
.
lim (ω − ωn )Â(ω) = T (ωn )
ω→ωn
dω ωn
(A1)
An expression of this derivative can be directly obtained from the weak form of the eq. (35). The solution U (x, ω) of (35) satisfies ∀w ∈ C
L0
L0
∂U ∂w
∂U
dx.
(A2)
= −ω2
ρ(x)U (x)w(x) dx +
λ(x)
λ(x)w(x)
∂ x L0
∂x ∂x
0
0
With w = U and noting I 1 the kinetic energy and I 2 the elastic strain energy,
I1 (ω) =
L0
ρU 2 (x, ω) dx,
0
I2 (ω) =
0
L0
λ(x)
2
∂U
(x, ω)
∂x
dx,
becomes
∂U
= −ω2 I1 (ω) + I2 (ω).
(x, ω)
λ(x)U (x, ω)
∂x
L0
(A3)
For ω = ωn , U (x, ωn ) = 0 and therefore
−ωn2 I1 (ωn ) + I2 (ωn ) = 0.
(A4)
Furthermore, (A3) follows the Rayleigh’s stationary principle, e.g. Takeuchi & Saito (1972) or Dahlen & Tromp (1998), and for all perturbations
∂U (x, ωn ) of U (x, ωn ):
−ωn2 ∂ I1 (ωn ) + ∂ I2 (ωn ) = 0.
(A5)
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Computing the derivative of (A3) along ω, we obtain
∂ I1
∂ 2U
∂ I2
∂U ∂U
= −2ωI1 − ω2
+U
+
.
λ(x)
∂ω ∂ x
∂ x∂ω L 0
∂ω
∂ω
63
(A6)
Setting ω = ωn in the last equation, we finally obtain for x = L 0 ,
2ωn I1 (ωn )
dD ,
=−
dω ωn
T (ωn )
(A7)
and therefore
An = −
T 2 (ωn )
.
2ωn I1 (ωn )
(A8)
APPENDIX B: DTN OPERATOR AND THE RADIAL FUNCTIONS
For practical reasons, it is of interest to express the operator DtN in terms of the radial functions (U , V , W ), as defined in the eq. (84), with
the corresponding stress vectors (T U , T V , T W ), as classically used in surface wave seismology and normal modes theory (e.g. Gilbert 1971).
(1)
(1)
(2)
(2)
The three independent solutions of (82) in Ω− , regular in r = 0, are denoted according to (84) as (U , V , 0), (U , V , 0) and (0, 0, W ).
Using (85) we obtain

(1)
(1)
(1) 
U
κ V
κ V


(2)
(2)
(2)
(B1)
D = ζ 
U
κ V 
 κ V
,
−iκ W
0
iκ W
and for the stress vectors

(1)
(1)
(1) 
TU,
κ TV,
κ TV,


(2)
(2)
(2)
T = ζ 
TU,
κ TV, 
 κ TV,
,
−iκ TW,
0
iκ TW,
√
˜ is straightforward to compute:
Where κ = γ / 2 = (( + 1)/2). The inverse of D̄

(2)
(1)
i 
−U
U
W


−1
1
˜ =
−2V (2) κ 2V (1) κ
D̄
0 


2 κ ζ
(2)
(1)
−i
U
−U
W
(B2)
(B3)
where = V U − V U . This last equation is undefined for ω ∈ d , the set of all the eigenfrequencies of Ω− corresponding to
a rigid displacement condition: (rΓ , ω) = 0 or W (rΓ , ω) = 0. As in the normal mode case, there are two types of eigenfrequencies: the
spheroidal ones corresponding to the zeros of (rΓ , ω); the toroidal ones corresponding to the zeros of W (rΓ , ω). Performing the product
of (B3) with (B2), the following expression for A is obtained:
#
$


(1) (2)
(2) (1)
(1) (2)
(1)
(1) (2)
(2) (1)


TU U − Tu2 U /κ
TV U − TV U
TV U − TV U
1
0 −1


#
$
#
$
#
$



1 
2κ T (2) V (1) − T (1) V (2) 2 T (2) V (1) − T (1) V (2) 2κ T (2) V (1) − T (1) V (2)  + TW  0
A =
(B4)
0
0
V V U U V V  2W 
.
2 


#
$
−1
0
1
(1) (2)
(2) (1)
(1) (2)
(2) (1)
(1) (2)
(2) (1)
TV U − TV U
TV U − TV U
TU U − TU U /κ
(1)
(2)
(2)
(1)
where the radius r is set to rΓ .
In the case of a spherical symmetry, the radial eigenfunctions and the eigenfrequencies can be computed with the help of the minors
corresponding to the solutions basis (Takeuchi & Saito 1972; Saito 1988; Woodhouse 1988). In terms of these minors, the DtN operator has
the following expression:


√


m 3 (rΓ , ω)
m 1 (rΓ , ω)/ 2
m 3 (rΓ , ω)
1
0 −1

√
√ 
1
(r
,
ω)
y
6 Γ


−m 6 (rΓ , ω)/ 2
(B5)
A(ω) =
0
0 .
2m 4 (rΓ , ω)
−m 6 (rΓ , ω)/ 2
 + 2y5 (rΓ , ω)  0
2m 2 (rΓ , ω) 
√
−1
0
1
m 1 (rΓ , ω)/ 2
m 3 (rΓ , ω)
m 3 (rΓ , ω)
where m 1 , . . . , m 6 denotes the minors corresponding to the spheroidal solutions and y 5 , y 6 the radial eigenfunctions and the stress vector
corresponding to the toroidal solution, as defined by Saito (1988). The dependence have been omitted for sake of simplicity. The symmetry
of A results directly from the fact that m 1 = −m 6 .
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64
APPENDIX C: ASYMPTOTIC EXPANSION OF THE DTN OPERATOR IN THE 3-D
CASE
Following Takeuchi & Saito (1972), we seek a solution of (80) in the form of (82) for the displacement and of (83) for the stress vector. This
leads to solve: for the solid–solid case, two 1-D coupled first-order differential equations in r for the spheroidal solutions and one for the
toroidal solution; for the solid–fluid case, a 1-D coupled first-order differential equation in r. In both cases, the procedure follows the same
steps than in the 1-D example of Section 3.5.
For sake of simplicity, only a linearly isotropic medium will be considered here. Extension to transversely isotropic medium
is quite
straightforward and does not lead to any specific problem. The P-wave and S-wave velocities are denoted here respectively α = (λ + 2µ)/ρ
√
and β = µ/ρ, where λ and µ are the elastic Lame parameters.
C1 Solid–fluid coupling
This involves only a scalar differential equation and therefore this case is very close to the 1-D example. The only difference is that the
expansion in (iω)−k of S(x, ω) matrix has now more terms. Following Saito (1988),
(+1)
1
− r1
2 − ω2 ρr 2
ρα
S(r, ω) =
.
(C1)
1
−ω2 ρ
r
S(r, ω) is expanded in (iω)−1 as
1
0
− r1
2
iωρα
S(r, ω) =
iω +
0
iρω
0
0
1
r
a
iω
0
+
0
0
1
.
iω
(C2)
where a = ( + 1)/ρ r 2 and S−1 , S0 and S1 are here equal to zero.
Eq. (62) gives
2 −
1
= 0.
α2
(C3)
This last equation has two solutions. Keeping only the outgoing waves, we obtain = −1/α. Furthermore, (62) gives
T0 (r ) = c0 u 0 (r ),
(C4)
with c0 = ρα.
After some algebra, (63) leads for k = 1
c0
c0
T1 (r ) = α − +
u 0 (r ) + c0 u 1 (r ),
2
r
c
(C5)
c
and u 0 (r ) = − 2c00 u 0 (r ). We note c1 (r ) = α(− 20 +
T2 (r ) = c2, (r )u 0 (r ) +
c0
).
r
and for k = 2,
ρα
− c0 u 1 (r ) + c0 u 2 (r ),
r
(C6)
where
c2, (r ) =
α
2
−c02 a − c1 −
c1 c0
c1
+
2 c0
r
.
(C7)
Setting r = rΓ , we obtain the first three terms of the asymptotic expansion of C00 ,
C00 = c0 (rΓ )iω + c1 (rΓ ) + c2, (rΓ )
1
+ O((iω)−2 ).
iω
(C8)
The remaining components Cαα for α = 0 or α = 0 are equal to zero. The first two terms of C00 do not depend upon which means that
these operators are local in space in contrast to the third term c2, .
C2 Solid–solid coupling
This case involves two uncoupled systems of equations, one for the SH waves (toroidal) and one for the P − SV waves (spheroidal).
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65
C2.1 Toroidal case
Once again, the equation to solve is a scalar one and the procedure is the same as in the previous case.
1
2
S(r, ω) =
µ
r
−ρω2 +
µ
r
,
− r2
(C9)
where = ( − 1)( + 2) and S can be expanded as
1
2
0
0
0
0 1
iωµ
r
S(r, ω) =
.
+ µ
iω +
0 − r2
0 iω
iωρ
0
r
(C10)
For k = 0, one gets
= −
1
β
and T0 (r ) = t0 u 0 (r )
(C11)
with t 0 = ρ β.
For k = 1,
T1 (r ) = t1 u 0 (r ) + t0 u 0 (1)
with
β
t1 = −
2
t1
t0
+4
r
(C12)
For k = 2, one finally gets
t0
u 1 (r ) + t0 u 2 (r ),
T2 (r ) = t2 u 0 (r ) − β t0 + 2
r
(C13)
with
t2 =
β
2
µ t1 t0
t1
+
− t1 − 2
r
2 t0
r
.
(C14)
C2.2 Spheroidal case
In this case, the dimension of the solution space is now 2. Let us first introduce here the following parameters:
4
λ2
−µ ,
a = 2 λ + 2µ −
γ = ( + 1),
r
λ + 2µ
γ
2µ
λ2
2µ
− 2 , d =1−
,
b = 2 λ + 2µ −
r
λ + 2µ
r
λ + 2µ
µ
.
e = γ
λ + 2µ
We have

d
r



−ρω2 + a
S(r, ω) = 

 − γ

r
1
λ + 2µ
d
−
r
0
aγ
2
− er
which can be expanded as

0
(iωρα 2 )−1
iωρ
0

S(r, ω) = 
0
0

0
0
e
r
aγ
2
2
r
−ρω2 + b
0
0
0
iωρ

0

γ 


r 
1 

µ
2
−r
(C15)


 
d
0
0 er 0
0
r

 0 −d
 aiω
γ 
0
0


r
r +
 iω +  γ
 
(iωρβ 2 )−1 
− r 0 r2 0   0
aγ
e
0 − r 0 − r2
0
iω
2
Looking for a solution of the form
yk (x, ω)(iω)−k ,
y(r, ω) = e−iω(x)
0
0
0
0
0
aγ
iω
2
0
b iω

0

0 1

0 iω
0
(C16)
(C17)
k≥0
with
yk (x, ω) = t (u k (r ), iωTu,k (r ), vk (r ), iωTv,k (r )),
(C18)
where t denotes the transpose operator.
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2003 RAS, GJI, 152, 34–67
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66
Eq. (62) leads to
1
1
− 2 = 0.
− 2
α
β
This last equation has four solutions
1
= ± ,
α
1
= ± .
β
(C19)
Keeping only the outgoing waves, we get two solutions. The first solution is = −α −1 for r = rΓ and
u (1) = 1
Tu(1) = a10 iω + a11 + a12 (iω)−1
v (1) = c10 + c11 (iω)−1 + c12 (iω)−2
Tv(1) = b10 iω + b11 + b12 (iω)−1
(C20)
The second solution is = −β −1 for r = rΓ and
u (2) = c20 + c21 (iω)−1 + c22 (iω)−2
v (2) = 1
Tu(2) = b20 iω + b21 + b22 (iω)−1
Tv(2) = a20 iω + a21 + a22 (iω)−1
(C21)
For the first solution, = −α −1 , eq. (62) gives
(1)
(1)
Tu,0 (r ) = a10 (r )u 0 (r ),
(1)
with a 10 = ρα, and also v (1) (r ) = T v,0 (r ). This implies that c10 = b10 = 0.
Now eq. (63), for k = 1, leads to
(1)
(1)
(1)
Tu,1 (r ) = a11 (r )u 0 (r ) + a10 u 1 (r ),
(1)
(1)
(1)
(1)
v1 = c11 (r )u 0 (r ),
Tv,1 (r ) = b11 (r )u 0 (r ),
ρα 2 β 2
(e + γ ),
r (α 2 − β 2 )
c11 =
with
α
a10 d
a10
,
+2
2
r
and for k = 2, eq. (63) gives
a11 = −
(1)
b11 =
(1)
Tu,2 (r ) = a12 (r )u 0 (r ) + · · · ,
(1)
b11
αe
.
+
a10
r
(1)
v2 = c12 (r )u 0 (r ) + · · · ,
(1)
(1)
Tv,2 (r ) = b12 (r )u 0 (r ) + · · · ,
where ‘. . . ’ stands for the remaining terms which take a zero value for r = rΓ , and are therefore only relevant for the third order approximation
of the solution of (63). The coefficients a 12 , b12 and c12 can be explicitly computed:
γ b11
a10 c11 e
α
a10
a11 d
a11 −
−
+a+
,
a11 +
a12 = −
2
2a10
r
r
r
ραβ 2
a10
2c11
a10
a11 e
2b11
aγ
b12 =
−
c
−
−
b
+
c
a
+
b
+
−
,
10
11
11
11
11
r (α 2 − β 2 )
2a10
r
2a10
r
r
2
2b11
aγ
b12
1
a
a11 e
b11
+
−
,
+
− 10 b11 +
c12 =
a10
ρ
2a10
r
r
2
Setting r = rΓ provides the asymptotic expansion of the first solution on Γ.
For the second solution, following the same steps it can be shown that
a20 = ρβ,
b20 = 0,
c20 = 0,
and
β
a20 d
ρα 2 β 2
b21
βγ
a20 + 4
, b21 =
(e + γ ), c21 =
.
−
a21 = −
2
r
r (α 2 − β 2 )
a20
r
with finally
e b21
β
a
2a21
a20 c21 γ
a21
+
− b −
,
− 20 a21 +
a22 = −
2
2a20
r
r
r
db21
aγ
ρα 2 β
a20
dc21
a20
a21 γ
b22 =
a
+
b
+
−
,
−
c
+
−
b
−
c
20
21
21
21
21
r (β 2 − α 2 )
2a20
r
2a20
r
r
2
b12
1
a
a21 γ
db21
aγ
+
− 20 b21 −
b21
+
−
c22 =
a20
ρ
2a20
r
r
2
Having characterized the asymptotic expansion of the two spheroidal solutions, they still have to be combined in order to obtain the
asymptotic expansion of C. First, the asymptotic expansion of can be readily computed as
= V (1) U (2) − V (2) U (1) = −1 − c11 c22 (iω)−2 + O((iω)−3 ).
(C22)
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67
and
Tu(2) V (1) − Tu(1) V (2)
(C23)
, = a10 iω + a11 + (a12 + c11 c21 a10 − c11 b21 ) (iω)−1 + O((iω)−2 ),
T (2) V (1) − Tv(1) V (2)
1 (C24)
C 01 = v
, = √ b11 + (b12 − c11 a21 − c12 a20 ) (iω)−1 + O((iω)−2 ),
√
2
2
T (1) U (2) − Tv(2) U (1)
1 (C25)
C 10 = v
, = √ b21 + (b22 − c21 a11 + c22 a10 ) (iω)−1 + O((iω)−2 ),
√
2
2
1
T (1) U (2) − Tv(2) V (1)
,=
a20 iω + a21 + (a22 + c11 c21 a20 − c21 b11 ) (iω)−1 + O((iω)−2 ).
(C26)
C 11 = v
2
2
The full asymptotic expansion of C is obtained by adding the toroidal solution. One can check that C 01 = C 10 , and that the first two terms of
C 00 and C 11 are local in space (independent of ).
C 00 =
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2003 RAS, GJI, 152, 34–67
92
Geophys. J. Int. (2005) 162, 541–554
doi: 10.1111/j.1365-246X.2005.02689.x
Towards global earth tomography using the spectral element method:
a technique based on source stacking
Y. Capdeville,∗ Y. Gung and B. Romanowicz
Department of Earth and Planetary Science and Seismological Laboratory, University of California at Berkeley, CA, USA
SUMMARY
We present a new tomographic method based on the non-linear least-squares inversion of
seismograms using the spectral elements method (SEM). The SEM is used for the forward
modelling and to compute partial derivatives of seismograms with respect to the model parameters. The main idea of the method is to use a special data reduction scheme to overcome the
prohibitive numerical cost of such an inversion. The SEM allows us to trigger several sources
at the same time within one simulation with no incremental numerical cost. Doing so, the
resulting synthetic seismograms are the sum of seismograms due to each individual source for
a common receiver and a common origin time, with no possibility to separate them afterward.
These summed synthetics are not directly comparable to data, but using the linearity of the
problem with respect to the seismic sources, we can sum all data for a common station and a
common zero time, and we perform the same operation on synthetics. Using this data reduction
scheme, we can then model the whole data set using a single SEM run, rather than a number
of runs equal to the number of events considered, allowing this type of inversion to be feasible
on a reasonable size computer.
In this paper we present tests that show the feasibility of the method. It appears that this
approach can work owing to the combination of two factors: the off-path sensitivity of the
long-period waveforms and the presence of multiple-scattering, which compensate for the loss
of information in the summation process. We discuss the advantages and drawbacks of such a
scheme.
Key words: global seismology, inverse problem, spectral element method, tomography.
Global seismic tomography is one of the most powerful tools to study
the earth’s interior structure (for a recent review, see Romanowicz
2003). Its basic principle involves two steps. The first step is a forward modelling step, in which a starting 1-D or 3-D model is chosen,
and a seismic wave propagation theory is used to compute a synthetic data set to be compared with real observations. Depending on
the approach, the observed data set can consist of time domain waveforms, that is, entire seismograms or portions thereof, or extracted
‘observables’ such as body wave traveltimes or surface wave phase
velocities, collected from earthquake records at seismic stations distributed around the globe. In the second step, an inverse problem
is solved, in which perturbations to the starting model are sought
in order to explain differences between the synthetic and observed
data sets. The procedure can be iterated until convergence.
∗ Now at: Département de sismologie de l’Institut de Physique du Globe de
Paris, Paris, France. E-mail: [email protected]
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2005 RAS
93
Practically all forward modelling approaches used so far to account for 3-D structure at the global scale are based on first order
perturbation theory (Born approximation), which is limited in its
domain of validity to weak lateral seismic velocity contrasts. Even
within this theory, further approximations are considered, such as,
in the context of normal mode perturbation theory, the Path Average approximation (PAVA, Woodhouse & Dziewonski 1984), which
limits the sensitivity to the average 1-D structure beneath the great
circle path containing the source and the receiver, and is therefore
strictly only a good approximation for the analysis of phase velocities of fundamental mode surface waves. In the context of the
analysis of body wave traveltimes, the approximation often used is
ray theory, which is an infinite frequency approximation and distributes the sensitivity to structure uniformly along the infinitesimal
ray path. These standard approximations carry with them another
drawback, which is that only a fraction of the information contained
in whole seismograms can be utilized, well-separated body wave
phases in the case of ray theory (except diffracted waves), fundamental mode surface wave phase velocities and mode frequency
shifts for modes that are well separated in the frequency domain,
in the case of normal mode perturbation theory. The advantage of
541
GJI Seismology
Accepted 2005 May 16. Received 2005 March 2; in original form 2004 July 16
542
Y. Capdeville, Y. Gung and B. Romanowicz
Capdeville et al. 2003a; Chaljub et al. 2003; Capdeville et al. 2003b).
This method has the advantage of being able to model with accuracy the entire seismogram at any location with respect to the source,
without any a priori assumptions on the velocity contrasts within
the Earth. The availability of this new tool allows us to address the
issue of full waveform tomography. Obviously, the main difficulty
is the computing power required, which may be so large that the
inverse problem would not be solved in practice with this tool for
many years to come. We will investigate the inverse problem in the
framework of classical non-linear least squares formalism (Tarantola & Valette 1982). This choice, in contrast to full space search
approaches, already limits the type of models that can be obtained,
but is arguably a reasonable approach at the global scale. We first
show that, even with a least squares inversion technique, the complete inverse problem is numerically too expensive to be solved
with presently accessible computers. Then, we present an approach,
based on a specific data reduction scheme, which makes this problem more tractable. We illustrate the feasibility and potential of this
approach through several synthetic tests.
these approaches is that they are very fast computationally. Recently, the introduction of more sophisticated, higher order approximations to Born seismograms has allowed to perform tomographic
inversions of complete long-period seismograms, containing waveforms of mixed body wave phases, diffracted waves, fundamental
and higher mode surface waves, with more accurate forward modelling step and structure sensitivity kernels. For example, the Nonlinear Asymptotic Coupling Theory (NACT, Li & Tanimoto 1993;
Li & Romanowicz 1995) considers across- branch mode coupling to
zeroth order asymptotically, resulting in 2-D broadband sensitivity
kernels appropriate for body waves and diffracted waves, leading
to the first global 3-D models based entirely on whole seismogram,
long-period waveforms (Li & Romanowicz 1996; Mégnin & Romanowicz 2000; Gung & Romanowicz 2004). Still, sensitivity is
limited to the 2-D vertical plane containing the source and the receiver. An extension to a higher order asymptotic approximation
(Romanowicz 1987), which allows the introduction of sensitivity to
the third dimension (focusing/defocusing), is still computationally
effective, but restricted in validity to relatively smooth heterogeneity
and observations away from the source and its antipode (Capdeville
et al. 2002) or fundamental mode surface waves (Zhou et al. 2004).
On the other hand, broadband kernels have recently been introduced
for the computation of long-period body wave traveltimes. Most recently, with the advent of more powerful computers, full Born computations have started to be put in practice (e.g. Zhao et al. 2000),
however, at the global scale, the corresponding broadband kernels
have only been used for tomography based on traveltimes (Montelli
et al. 2004), again, limiting the type of information utilized in seismic records, and therefore the sampling of the earth’s interior that
can be obtained.
The main advantage of the approximations currently used in tomography is that their computational speed is fast enough to allow
inversion within a reasonable time frame (several days to several
weeks). There are also serious drawbacks, especially as we seek to
constrain increasingly finer details in the models. Because source
station distribution is limited around the globe, restricting the data
set to a few well-isolated body wave phases in the seismogram limits the sampling within the earth, leaving large gaps in areas inaccessible in other ways than through the illumination by multiply
reflected/converted phases (in the vertical direction), or scattered
waves interacting with lateral heterogeneity (in the horizontal direction). Also, the large data processing effort involved in measuring
traveltimes or phase/group velocities is somewhat disproportionate
with the limited sampling obtained—which is not likely to improve
significantly unless major data collection efforts such as USArray
of Earthscope are systematically extended to the whole globe, including the ocean floor.
Therefore, we look to future progress in mantle tomography
through the combined use of full time domain seismograms and
an accurate wave propagation theory in a 3-D earth, with fewer limits of validity. Until recently, the latter has not been available. In
recent years, progress has been made in two directions: the development of higher order perturbation theory in the context of normal mode theory (e.g. Lognonné 1989; Lognonné. & Romanowicz
1990), as well as numerical approaches (e.g. Cummins et al. 1997).
While higher order perturbation theory is currently being explored
as a possible tool for mantle tomography (e.g. Millot-Langet et al.
2003), we will here consider a numerical approach, the Spectral Element Method (SEM), recently introduced in seismology in Cartesian Geometry (Komatitsch & Vilotte 1998; Komatitsch & Tromp
1999), and in spherical geometry for global earth scale applications (Chaljub 2000; Capdeville 2000; Komatitsch & Tromp 2002;
2 N U M E R I C A L C O S T O F S O LV I N G
T H E I N V E R S E P RO B L E M W I T H S E M
Our aim is to find an Earth model with the minimum number of
parameters that can explain our seismic data set, as well as data not
used in the inversion but obtained under similar conditions. By Earth
model, we mean the 3-D variations of elastic parameters, anelasticity
and density. Let us assume that we wish to solve the inverse problem
using a classical least square inversion (Tarantola & Valette 1982)
and with a complete modelling theory (i.e. the SEM applied to the
wave equation). Let p be the set of parameters which describe our
model. The data set d is comprised of seismic time traces of Ns
events recorded by Nr three component seismometers yielding 3
× Ns × Nr time series. We call g the forward modelling function
that allows us to model the data for a given set of model parameters:
d = g(p). In our case g represents the SEM, which is able to compute
a precise set of synthetics in any given model. The inverse problem
has to minimize the classical cost function ,
t
−1
(p) = t [g(p) − d]C−1
d [g(p) − d] + (p − p0 )C p (p − p0 ),
(1)
where p 0 is the a priori value of the model parameters, C d and C p
are the covariance matrices of data and model parameters respectively. If g is a nonlinear function, the minimum, or the closest local
minimum to the starting model, of , can be found by the Gauss–
Newton method iterative process (Tarantola & Valette 1982). Given
the model at iteration i, we can obtain model at the iteration i + 1:
−1 −1
pi+1 = pi + t Gi C−1
d Gi + C p
t
−1
(2)
Gi C−1
d (d − g(pi )) − C p (pi − p0 ) ,
where G i is partial derivative matrix
∂g(p)
.
Gi =
∂p p=pi
(3)
Usually, the forward problem is solved using first-order approximations such as, for example, the Born approximation within the
normal mode framework (e.g. Woodhouse & Dziewonski 1984) or
arrival time Frechet kernels (Dahlen et al. 2000). This leads to a
linear relation matrix (G 0 ) between the set of parameters and the
synthetic data. In that case, only one iteration of (2) is required
and the partial derivative matrix is built in the forward theory and
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2005 RAS, GJI, 162, 541–554
Spectral element tomography
computed at a relatively low numerical cost. Some tomographic approaches are slightly non-linear (e.g Li & Romanowicz 1996) but
are still based on the Born approximation. They also have the advantage of providing naturally the partial derivative matrix with no extra
numerical cost, at least at the a priori model stage (i = 0). When
the SEM is used, the partial derivative matrix or kernels cannot be
computed naturally. We must use a ‘brute force’ finite difference
formulation.
At iteration i of the inversion scheme (3), the line l of the partial
derivative matrix G i is given by
∂g(p)
g(pi + δpl ) − g(pi )
,
(4)
∂pl p=pi
δpl
where p l is the parameter vector set in which only the component
l is nonzero. Therefore, in order to compute the partial derivative
matrix, one needs to compute synthetic data for each seismogram,
which involves Ns runs, and for each parameter of the model. This
will obviously be the most expensive part of the inversion.
Let us estimate the computing time of such a tomography. Here
we assume that only G 0 is needed and that it can be used throughout
the iteration of (2). This assumption is probably valid if the starting
model is not too far from the solution and if the problem is only
weakly non-linear (Note that if the problem is strongly non-linear,
the least squares inversion would probably not converge toward the
right solution anyway). Obviously, the determination of G 0 will
dominate the computation time. Let us assume that we wish to build
a model with a lateral resolution equivalent to a degree 12 spherical harmonic expansion, which is indeed a modest objective given
that current global tomographic models consider expansions up to
or beyond degrees 20–24. We consider 10 vertical parameters and
invert only for one elastic parameter (S velocity), which means that
our model is roughly described by 3000 parameters. We also assume
that very long-period waveforms (160 s and above) that will be used
here are sensitive up to degree 24 in the sense of a spherical harmonics expansion. This is true in theory, but in practice it is not obvious
that the effect of the highest degree is large enough to overcome
the background noise. One may have to use higher frequencies to
obtain a good result which means our estimation will be optimistic,
since numerical cost increase as a power four with the frequency
(for a given number of parameters and a given trace length). Let
us assume that our data set is comprised of seismograms for 100
events recorded at a large number of receivers (the exact number is
not relevant to the numerical cost). Finally, let us assume that we
can simulate a 3-hr waveform at a frequency cut-off of 1/160 Hz for
a single event in 1 hr, which is roughly what we can do now using
a state-of-the-art 16 processors PC cluster. With such a hardware,
computing the partial derivative matrix would take 100 × 3000 ×
1 hr 34 yr. Of course, using a 100 or 1000 times faster computer would reduce the computing time to several weeks, but even if
such a computer exists nowadays (i.e. the Earth Simulator, Japan),
it would require to use 100 per cent of the machine’s capacity for
weeks. Therefore, such an approach is not realistic for the moment.
The seismic exploration community has faced such a problem for
waveform tomography, and the Gauss—Newton method to solve the
inverse problem is not used in practice because it requires to compute
the partial derivative matrix explicitly. Instead, the gradient method
is used, which does not require to compute the partial derivative
matrix. Indeed, it has been shown that the gradient of the cost function (t G i C−1
d (d − g(p i ))) in eq. (2), can be computed with only
two forward modelling computations per source using the adjoint
problem (Lailly 1983; Tarantola 1984; Tarantola 1988, or more recently, Pratt et al. 1998; Tromp et al. 2005), which considerably
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reduces the number of forward modelling runs despite the fact that
the gradient method requires a larger number of iterations than the
Gauss—Newton method to converge. The solution can perfectly be
applied to our problem but we would not do so in this article. We will
investigate another solution that can be cumulated with the adjoint
problem to compute gradients. So when reading the next sections,
keep in mind that the numerical cost can be even more reduced than
what we describe here.
3 D AT A R E D U C T I O N
We propose in this paper to use two SEM properties to reduce the
numerical cost of a non-linear least square inversion with this tool.
First, because the SEM computes the wave field at any location of
the Earth, the numerical cost of an inversion is independent of the
number of receivers. Second, it is possible in SEM (and in most other
direct solution methods) to input several sources in the scheme, to
trigger them simultaneously, without increasing the numerical cost.
Of course, the resulting traces on the receivers side will be the sum
of the traces due to each individual source and there is no possibility
to separate them once the computation is done. If we cannot recover
the individual synthetic seismograms after the computation, we can
perform an equivalent stack of data for common seismometers assuming a common origin time for all the events, and use that reduced
data set instead of traces of individual events, as the stacked data
are directly comparable to the stacked synthetic seismograms. This
operation is possible thanks to the linearity of the wave equation
with respect to seismic sources which means that computing traces
for one seismometer for each source separately and then summing
them is equivalent to computing one trace of all the sources triggered
simultaneously. This data reduction scheme allows us to model the
whole data set with one SEM simulation with respect to Ns when
traces for a common station are not stacked. Finally, note that summing the traces with a common zero time is not necessary (i.e. the
sources can be staggered in time), but it is used here to provide a
simpler explanation.
If we apply this to our example of Section 2, the 34 yr of computation reduce to 4 months. Of course, this data reduction is not without
drawbacks and some information that is contained in independent
seismograms will be lost in the summation process. However, we
hope that this loss of information will be compensated by the fact
that we are able to use all the information present in a long time
series for each trace.
4 VA L I D AT I O N T E S T S
In this section, we present several numerical experiments to assess
the robustness of the inversion when the stack data reduction is
applied. These tests are circular tests in the sense that the ‘data’
to be inverted are generated with the same forward theory as the
one used to invert. We name the model used to generate the data
to be inverted the input model or the target model. These tests only
provide information on the ability of the process to converge toward
the solution under some circumstances (e.g. amplitude of velocity
contrast, data coverage, presence of noise etc.) They do not provide
any information on the behaviour of the inversion in the case of
an incomplete theory, like, for example, how an isotropic inversion
would map an anisotropic medium or how high degree horizontal
spherical harmonics components (or equivalent) would leak or alias
in a low degree inversion. Nevertheless, these tests provide valuable
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Figure 1. Two examples of meshes of the sphere used to parametrize the velocity model: one with 274 free parameters (left) and one with 2610 free parameters
(right).
corresponds to a spherical harmonic degree 16 horizontally in the
upper mantle and a spherical harmonic degree 8 horizontally in the
lower mantle. In practice, this parameterization may not be a good
choice, because parameters at the corner of elements have a different spatial spectral content than parameters at the centre of an
element. However, for the tests presented here, as the input model
is represented on the same mesh as the inversion mesh, this choice
does not affect the results.
information on the feasibility of the process. At least if the process
failed in these tests, there is little chance that it will ever succeed.
In the following tests, no damping is applied (C d = I and
C−1
p = 0) so that the least squares inversion process is simply a
Gauss—Newton method to invert g:
−1 t
pi+1 = pi + t Gi Gi
Gi (d − g(pi )) .
(5)
In order to limit the numerical cost of these experiments, only G 0
will be computed and will be used instead of G i at iteration i. We
will see that this approximation does not hinder the convergence,
at least for these tests. Note that if the starting model is spherically
symmetric, normal mode perturbation theory would provide an exact
solution for G 0 (Woodhouse 1983) and will be computationally
more efficient. In practice we already use that possibility, but this
work will be presented in a later publication. Of course, this normal
mode perturbation approach is only an option when the starting
model is spherically symmetric, which may not be desirable with
the present level of sophistication in tomography. Nevertheless, an
interesting possibility for 3-D starting models may be to combine
the adjoint problem solution mentioned in Section 2 to compute an
accurate gradient of the cost function and normal mode perturbation
theory to compute the approximate Hessian (t G i G i ).
4.2 Experiments setup and input models
For computation cost reasons, the following experiments have been
carried out with the small mesh (274 free parameters) and only one
elastic parameter has been inverted (S velocity). We choose a realistic source-receiver configuration of 84 well-distributed events
recorded at 174 three-component stations of the IRIS and GEOSCOPE networks (Fig. 2). The corner frequency used here is
1/160 Hz and each trace has a duration of 12 000 s. For each test, the
starting model is the spherically symmetric PREM (Dziewonski &
Anderson 1981). The partial derivatives matrix G 0 is therefore the
same for all tests and requires 275 SEM runs to be built, which is
reasonable in terms of numerical cost (11 days using our hardware
example, Section 1).
Two input models will be used. For both of them, the reference
background model is PREM to which a 3-D Vs velocity contrast
field is added. This 3-D Vs velocity contrast field is generated on the
same mesh as the one that will be used for the inversion (Fig. 1 left).
The first model is named BIDON (Fig. 3) and is a very simple model:
all the parameters are set to zero except one in the upper mantle and
one in the lower mantle. The amplitude of the velocity fluctuations
is large (9 per cent) compared to what we expect for the Earth for
such a long spatial wavelength. On Fig. 3 (left) we plot a depth cross
section of the model and on Fig. 3 (right) we plot the Vs velocity as
function of the parameter number of the mesh (from 1 to 274). This
1-D representation does not provide a precise idea of what a map of
the model would actually look like, but it gives accurate information
on the precision of the inversion, which a geographical map does
not. The parameter indexes are sorted such that the lower mantle
is predominantly on the left side of the plot and the upper mantle
predominantly on the right side of the plot to give some information
4.1 Parametrization
Instead of spherical harmonics or block parametrization, we use
a piecewise-polynomial approximation description based on our
spectral element discretization (Sadourny 1972; Ronchi et al. 1996;
Chaljub et al. 2003). The sphere is discretized in non-overlapping
elements and each of these elements can be mapped on a reference
cube. On the reference cube, a polynomial basis is generated by the
tensor product of a 1-D polynomial basis of degree ≤N in each
direction. The continuity of the parametrization between elements
is assured. More details on this discretization mesh can be found in
Chaljub et al. (2003). Fig. 1 presents two examples of meshes on the
sphere used for this parameterization with a polynomial degree over
elements N = 2. The first mesh (left) has 274 free parameters and
roughly corresponds to a spherical harmonic degree 8 horizontally
in the upper mantle and a degree 4 horizontally in the lower mantle. The second mesh (right) has 2610 free parameters and roughly
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Events (84), Receivers (174)
Figure 2. Sources (stars) and receivers (diamonds) configuration used to test the inversion process in this article. A total of 84 earthquakes recorded over
174 three components stations are used.
335km depth
0.1
0.08
0.0000
0.0899
dVs/Vs
0.06
0.04
1780km depth
0.02
0
-0.02
0.00
0
100
parameter number
200
0.08
dVs/Vs
Figure 3. Earth model BIDON. Only two parameters have a velocity contrast with respect to the spherically symmetric reference model (PREM). Left panel
shows maps at two different depths and on the right is shown a 1-D representation on the model where the Vs velocity contrast is plotted as a function of the
parameter number (from 1 to 274).
about the location of potential errors when looking at these plots.
The second model is named SAW6 (Fig. 4) and is more realistic
than BIDON. This model is derived from the tomographic model
SAW24B16 (Mégnin & Romanowicz 2000), truncated at degree 6
and mapped on the 274 parameter mesh (Fig. 1 left). The maximum
amplitude velocity contrast is much lower (about 3 per cent) than in
BIDON which is typical of long-wavelength mantle heterogeneity. In
this case all the parameters have non-zero values has it can be seen
on the right plot of Fig. 4.
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4.3 Test in BIDON model
Stacked data are generated with SEM in the model BIDON and are
inverted following the inversion scheme presented in this paper.
The results of the first three iterations of inversion are shown in
Fig. 5. The first iteration already gives a velocity contrast very close
to the correct value for the two parameters with non-zero velocity
contrast, but for the other ones the result is very noisy. The second
iteration gives a much better result and the third one has converged
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75km depth
-0.014
0.000
dVs/Vs
0.02
0.014
0
2800km depth
-0.02
0
100
parameter number
ii
-0.021
0.000
200
0.021
dVs/Vs
Figure 4. Earth model SAW6. This model is derived from the tomographic model SAW26B16 (Mégnin & Romanowicz 2000). Maps at two different depths
(left) and a velocity contrast of each parameters as a function of the parameter number (right) are represented.
toward the correct result. This first experiment is satisfactory and
shows that the process can work, at least in simple models. The
fact that the first iteration is relatively far from the correct model
is interesting because it means that a method based on the first
order Born approximation would give a very poor result in that
case. The non-linearity is here strong enough to justify a non-linear
scheme, but it is weak enough to allow the convergence toward
the right solution and not toward a wrong local minimum model,
and this without updating the partial derivative matrix G i at each
iteration.
iteration 1
0.1
0.05
0
dVs/Vs
-0.05
0.1
iteration 2
0.05
0
-0.05
0.1
4.4 Test in SAW6 model
We now perform the same test but with data generated in the more
realistic model SAW6. Results of the first two iterations of the inversion are shown Fig. 6 and already present a good convergence
toward the input model for the second iteration. This faster convergence compared to the first test can be explained by the lower
velocity contrast of the input model, which implies smaller nonlinear effects. All model parameters, from the lower mantle to the
surface, are well retrieved.
iteration 3
0.05
0
-0.05
0
50
100
150
parameter number
200
250
Figure 5. Inversion results for the three first iterations for data generated
in the model BIDON (Fig. 3). The velocity contrast with respect to PREM is
plotted as a function of the parameter number. The input model is accurately
retrieved after three iterations.
4.5 Test in SAW6 model with noisy data
slope from −175dB and −165dB in the 100 to 300-s period range)
and for each event-station pair, stack them and then add them to
the synthetic data. The result of the inversion is shown Fig. 7. The
noise affects the results of the inversion, but the scheme is still
able to retrieve the target model correctly. The deepest parameters
of the model are the most affected by noisy data, which is not a
surprise knowing the poor sensitivity to deep layers of long-period
data. Fig. 8 shows that, despite the noise, the inversion is able to
retrieve a model that explains the data far beyond the noise level.
The fact that we are able to fit the data so well, even though the
The purpose of this experiment is to assess the noise sensitivity
of the inversion scheme. This kind of test reflects how stable the
inversion is, and in this experiment, we are not in a favourable case.
Indeed, data with periods 160 s and above have a very poor depth
resolution, and to obtain very good results with such an experiment,
one should use higher frequency data or decrease the number of
vertical parameters. We nevertheless perform the test with SAW6
input model again, but this time synthetic noise is added to the data.
To do so, we generate a random noise corresponding to a realistic
background noise in this frequency band (the noise spectrum is a
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iteration 1
dVs/Vs
0.02
0
-0.02
0
100
200
iteration 2
SAW6 (target model)
inversion result
residual
dVs/Vs
0.02
0.01
0
-0.01
-0.02
-0.03
0
100
parameter number
200
Figure 6. Inversion results for the two first iterations for data generated
in the model SAW6 (Fig. 4). After two iterations, the inversion result match
very well the input model for all model parameters.
model is not perfect, is also due to the lack of depth sensitivity of
long wavelength data.
4.6 Test in SAW6 model with only one station
We perform an extreme test to assess the ability of the process to
recover information in the case of very poor data coverage. To do so,
we perform a test with only one receiver, the GEOSCOPE station KIP
in Hawaı (Fig. 9). This time, no noise is added to traces. The input
model is SAW6 (Fig. 4) and the results of the inversion are shown
in Fig. 10, in the 1-D representation for iterations 1, 5 and 10. The
output model for the first iteration is very far from the input model
and at this point it seems that the inversion scheme has no chance to
recover it. However after a large number of iterations, the process
finally converges toward the input model. It is impressive that the
process is still able to converge without updating the partial derivative matrix G 0 at any iteration. The conclusion of this experiment is
that, what allows us to retrieve the input model is not only the wide
off path sensitivity of the theory, but also the non-linearity or, in
other words, the multiple scattering. Indeed, a Born theory with no
geometrical approximation has the same wide off path sensitivity as
a direct solution method like SEM, but would give a wrong model
(similar to the one which is obtained at iteration one). Clearly, the
inversion is in that case highly unstable and a very high data precision is required to allow the inversion to converge toward the right
model. An application to real data would be a disaster due to the
presence of noise or, equivalently, of physical processes not included
in the theory (anisotropy, attenuation, effect of atmospheric pressure
etc.).
4.7 Test in SAW6 model with moment tensor errors
So far, a perfect knowledge of the source location, origin time and
moment tensor have been assumed. When applying the method to
real data, this will not be the case and significant errors on the source
parameters can be expected. In order to partly address this issue, we
perform a test where the a priori moment tensors are not well known
but we keep the locations and origin times perfectly known. This
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reflects the fact that, at least at very long period, the location and
origin time errors are small compared to the wavelength. In this
test we generate data in the SAW6 model with each component of
each a priori moment tensor perturbed by a random coefficient lying between −30 per cent and +30 per cent. The moment tensors
used to generate the partial derivatives and to compute the forward
modelling part of the inversion are, therefore, not the ones that have
been used to generate the synthetic data to be inverted. The result of
the inversion after three iterations is shown Fig. 11. The scheme can
clearly not retrieve the input model. The unknown moment tensors
create a large noise that can not be overcome with a reduced number
of data. A solution to this problem can be to increase the number
of data, and therefore, because the number of stations can not be
significantly increased, to use multiple stacked data sets. Another
solution is to invert also for the moment tensor at the same time as
Vs velocity. In this case, a difficulty due to the stacked data set, is
that, for sources close to each other, only the sum of these moment
tensors can be retreived, but not individual moment tensors. If the
primary goal of the inversion id to retrieve Vs field, an accurate sum
of the moment tensors of sources very close to each other is enough.
Indeed, an accurate sum of the moment tensors of sources very close
to each will give a correct prediction of the stacked displacement
at stations, which is all what we need for a Vs tomography with
stacked data. Now, if we are also interested in individual moment
tensors, a solution can be to separate sources in the time domain by
introducing time delays between close sources. Doing so, different
sources located at the same place will have a different effect on
stacked data. In this example, we will only focus on retrieving Vs
field. In the case of sources very close to each other, we therefore
wish to invert only for the sum of the moment tensors. In order to
to so, we generate partial derivatives of individual components of
moment tensors. The Hessian matrix (t G i G i ) for moment tensors
only is then build, an eigenvalue analysis of this matrix is performed
and only the 75 per cent larger eigenvalues are keept. This is equivalent to a damping that removes the instabilities, but it only affects
the moment tensor inversion part. Of course the choice of 75 per
cent is not precise and will prevent us from explaining the signal
perfectly. Therefore, we expect a small error due to this choice to
spread into the Vs inverted field. We finally invert for the Vs field at
the same time as the moment tensors cleaned from its 25 per cent
lower eigenvalues. Fig. 12 shows the result of the inversion. Thanks
to the inversion for the moment tensors, we are able to retrieve very
well the input model. The remaining errors are due to the 75 per cent
choice in the number of kept eigenvalues for the source inversion.
Indeed, this choice is not optimum and some signal that should be
explained by the moment tensors is not and slightly degrades the Vs
inversion.
5 T O WA R D R E A L C A S E S : D E A L I N G
W I T H M I S S I N G D AT A
Thanks to the success of all preliminary tests of this inversion
scheme, we have already started to work on real data to get a preliminary long-period model. The results of this work will be presented
in a later publication. When working with real data, a problem with
this kind of approach immediately appears: the missing data. Indeed
when trying to gather data for a reasonable number of events (let’s
say around 50) recorded at a large number of stations (around 80),
there are always between 10 and 20 per cent of missing data whatever the configuration is. The reason why the data are sometimes
not available at a given station varies from case to case, but there are
very few stations that have 100 per cent availability for 50 events.
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Input model (saw6)
Output model (iteration 3)
75km depth
75km depth
-0.014
0.000
0.014
-0.014587
2800km depth
-0.021
0.000
0.000000
0.014587
2800km depth
0.021
-0.017905
dVs/Vs
0.000000
0.017905
dVs/Vs
dVs/Vs
0.02
0.01
0
-0.01
iteration 3
target model (saw6)
-0.02
-0.03
0
50
100
150
parameter number
200
250
Figure 7. Inversion results for third iteration for data generated in the model SAW6 (Fig. 4) but with noise added to synthetic data. The two left maps show
the input model (SAW6) at two different depths and the two right maps show the result of the inversion (output model) at the same depths after three iterations
when synthetic noise is added to the synthetic data. The lower plot shows the Vs velocity contrast on the input and output model as a function of the parameter
number. Note that the deep parameters are more affected by the noise than the one close to the surface. The model is nevertheless correctly retrieved by the
inversion.
dt = da + dm .
When combining the 80 stations, even for large events (magnitude
from 6.5 to 7) we end up with about 10–20 per cent of missing
data. For our inversion scheme, missing data is a problem as it is
easy to generate the sum of all the data in one run but impossible
to remove some of them without computing each missing source
individually. Since almost all sources are missing at least at one
station, removing missing data would require to perform a simulation for each source and it seems we are back at our starting
point.
However, the sum of all the data d t can be separated into the sum
of missing data d m and the sum of available data d a :
(6)
The total direct problem can also be separated into missing and
available synthetic parts:
gt (p) = ga (p) + gm (p).
(7)
The main difficulty is that there is no way to compute efficiently the
partial derivatives matrices of g a and g m , therefore trying to solve
100
ga (p) = da
(8)
or
gt (p) − gm (p) = da
(9)
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KIP, vertical displacement
interation 1
0.04
0.1
noise
starting residual (without the noise)
residual after 2 iterations
0.02
0
0.05
-0.02
-0.04
iteration 5
0.04
0
dVs/Vs
normelized amplitude (with 3D trace)
549
0.02
0
-0.02
-0.05
-0.04
iteration 10
0.04
saw6 (target)
0.02
-0.1
0
2000
4000
time (s)
6000
8000
-0.02
Figure 8. Here is plotted, at KIP station: the noise (dotted line) added to
the synthetic vertical component data before inversion; the starting residual
signal without the noise (solid line) which is difference between the synthetic
data when noise is not added yet and the synthetic seismogram in the reference model (PREM); finally the residual (solid bold) after two iterations that
is the difference between the synthetic in the obtained model after inversion
and the starting model. We see the scheme is able to go beyond the noise
level (the amplitude of the last residual is smaller than the noise level).
is not an option. On the other hand, solving
gt (p) = da + gm (p)
(10)
is possible because the partial derivatives matrix of this last problem
only depends on the sum of all the data, the missing and available
ones. Nevertheless, since the right hand side of the last equation
depends on p, it requires an iterative scheme. This scheme is only
interesting if we do not update the partial derivative matrix at each
iteration of the iterative scheme, but since we can expect only a small
number of missing data, this should not be a problem. The available
solutions for g m (p) are:
(1) 0,
(2) Synthetics in the spherically symmetric model (g m independent of p) with normal mode summation,
Events (50),
inverted model
residual
0
-0.04
0
100
Parameter number
200
Figure 10. Inversion results with only one station (KIP, see data coverage
Fig. 9) for data generated in the model SAW6 (Fig. 4) for iterations 1, 5 and
10. The convergence is slow, but after 10 iterations, the output model match
the input model.
(3) Synthetics in p with normal mode summation first order perturbation and
(4) Synthetics in p with the spectral element method.
The first two solutions are numerically inexpensive but probably not
very good, depending on the amount of missing data. The third one is
probably a good compromise between numerical cost and precision
and the last one is perfect but expensive. An equally good solution
for a finalized model may be to use normal mode perturbation theory
during the iterative process of the inversion and spectral elements
synthetics at the last iteration.
In order to test this solution, we generate a data set in the model
SAW6 (Fig. 4) with missing data. To do so, we first compute synthetic
seimograms from each source individually with 84 runs. We then
select randomly the missing data among each source receiver pair.
The selected data are not used in the construction of the stacked
Receiver (KIP)
Figure 9. Data coverage used in the single station test. The same number of events (84, plotted as diamonds) as in the other tests is used, but only one station
(KIP) is used.
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approach is that it allows us to investigate full waveform tomography
now, without waiting to have the computing power to use a classical
approach. There is also a data selection and processing advantage.
Phase identification, time picking, phase velocity measurements are
not required for waveform inversions which saves a lot of time and
also minimizes human error.
Clearly, there are also drawbacks to this approach. One of them
is that some information is lost in the data reduction. However all
tomographic methods use some data reduction scheme. For example, travel time tomography uses only a limited number of arrival
times per trace (often only one or two). Here, we use traces of
12 000-s duration, which represents a lot of information, even with
a 160 s corner period. We, therefore, hope that these long traces can
overcome the loss of information due to the stacking. Nevertheless
it is important to keep in mind that information is lost when that
data is stacked. If only a single stacked data set is used, there is a
limit on number of sources, after which adding new sources does
not bring anything new. This limit is not obvious to address and
depends on the corner frequency and the length of the signal that is
used. However when this limit is reached, the only way to get more
information about the model (e.g. to improve the resolution) will be
to use multiple stacked source data sets. Note that the process has
the advantage to allow to move gradually toward the classical case
(no data stacking) by splitting the data set into two or more data
subsets as a function of the computing power available. Through
this process, we will end up eventually with the classical case where
all the sources are considered individually.
Another drawback is that the process does not allow us to select
some time windows on traces to enhance some part of the signal with
respect to others, such as separating body wave packates from each
other and from surface waves (e.g. Li & Romanowicz 1996). The
body waves have small amplitude but contain information about the
lower mantle whereas surface waves have a large amplitude but do
not contain information about the lower mantle. As surface waves
will dominate the stacked signal, there is little chance to recover the
lower mantle before the upper mantle is very well explained. We
have seen that it is not a problem in our tests, but this is because
we exactly know what to invert in order to explain exactly the upper
mantle and therefore once the upper mantle is explained, the lower
mantle is easy to retrieve. In a real case, it may be much more
difficult, since we do not know for sure what elastic parameters
are required to explain surface waves well enough, and therefore
to be able to access body waves and information about the lower
mantle.
This last point leads to another difficulty that we will face in
future work. What physical parameters (elastic, anelastic, density,
etc.) do we need to invert for and at what resolution to explain our
data set correctly? The resolution issue is not obvious: a too low
resolution for a given data frequency content will lead to aliasing
and a too high resolution may lead to an unstable inversion scheme,
as our data set may not have the information to resolve all the parameters, but also to a prohibitive extra numerical cost. The number
of physical parameters is also a difficult question. Are Vs fluctuations enough to explain our data set? Probably not. Do we need
Vs, Vp, density, anisotropy, 3-D anelastisity, perturbations in source
parameters? What is the relative sensitivity of our data set to those
parameters? All these questions will need to be addressed in future
work.
Finally, it is well known that the choice of the type of least squares
inversion can strongly constrain the possible model. This is not really
a choice as we cannot afford the numerical cost of a more general
inversion scheme based on random exploration of our parameter
0.1
dVs/Vs
0.05
0
-0.05
-0.1
iteration 3
target model (saw6)
0
50
100
150
parameter number
200
250
Figure 11. Inversion results for third iteration for data generated in the
model SAW6 (Fig. 4, bold line) with error on a priori moment tensors. To
generate the data, a random coefficient lying between −30 per cent and
+30 per cent has been applyed to each component of each moment tensor. The result of the inversion (without inverting for sources) after three
iterations (thin line) doesn’t match the input model.
data set. We perform here a rather extreme case with 35 per cent of
missing data (our experience with real cases have shown that it is
possible to gather data set with 15–20 per cent of missing data when
working with 50 sources and 90 vertical component receivers). We
then perform three inversions. For the first one, the missing data
are replaced with synthetic seimograms computed in the starting
spherically symmetric model. This solution is numerically interesting because the scheme is still explicit: the data set completed
with the synthetics of missing data does not depend on the obtained
model. The drawback is that the final model cannot be accurate as
the signature of the starting model will always be present and can
not be corrected. The result of such an inversion is given Fig. 13.
As expected, the result is noisy, but the main features of the input
model are still retrieved. In the second inversion, the missing data are
replaced by synthetic seismograms computed in the current model
(x i ) with the Born approximation in the normal modes framework
(Capdeville et al. 2000; Capdeville 2005). This time the scheme
becomes implicit as the data set completed with the synthetics of
missing data depends on the current model. The result of such an
inversion is given Fig. 14. The result is much better than for the
previous test but still not perfect. This is expected as the first order
Born approximation is not very accurate especially when time series
are long and non-linear effect cannot be neglected anymore. Finally
we perform a last test in which the missing data are replaced by synthetics computed in the current model with SEM. This solution is
CPU time consuming as it requires to compute each source individually. To perform such a test and to lower the numerical cost we start
from the model obtained at the last iteration of the previous test and
we perform only one extra iteration. The result is given Fig. 15 and
shows a very good result. Some more iterations would be required
to obtain the same precision as the one we get when no data are
missing, but this result is already accurate enough with respect to
noise coming from other effects (noise, error on moment tensors).
6 D I S C U S S I O N A N D C O N C LU S I O N S
In this paper, we have presented a way to perform non-linear full
waveform inversion at the global scale, using the spectral element
method as a forward modelling tool. This method is based on a noncoherent trace stacking at a common receiver for a common source
origin time. The data reduction allows us to simulate the whole data
set in a single Spectral Elements run and, therefore, to reduce the
number of computations by a factor equal to the number of sources
with respect to a classical approach. We have presented preliminary
tests which show very promising results. The main advantage of the
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2005 RAS, GJI, 162, 541–554
Input model (saw6)
75km depth
75km depth
-0.014
0.000
0.014
-0.014
2800km depth
-0.021
0.000
0.000
551
0.014
2800km depth
0.021
-0.021
dVs/Vs
0.000
0.021
dVs/Vs
dVs/Vs
0.02
0.01
0
-0.01
iteration 3
target model (saw6)
-0.02
-0.03
0
50
100
150
parameter number
200
250
Figure 12. Same test as the one presented Fig. 11 with error on a priori moment tensors, but this time a moment tensor inversion joint with the Vs inversion
is performed. The two left maps show the model that has been used to generate that data (SAW6, input model) at two different depths and the two right maps
show the result of the inversion (output model) at the same depth after three iterations. The lower plot shows the Vs velocity contrast on the input and output
model as a function of the parameter number. When moment tensors are inverted as the same time as Vs , the inversion is able to retrieve correctly the input
model despite large errors on a priori moment tensors.
space. Nevertheless the question of what we may be missing due to
this least square inversion scheme, that is, what are the error bars on
the obtained model values, will also need to be addressed in future
work.
BSL contribution #05–10. Computations have been performed on
‘seaborg’, the supercomputer of the NERSC (California), ‘zahir’ the
supercomputer of the IDRIS (France), ‘regatta’ the supercomputer
of the CINES (France) and at the DMPN (IPGP, France) computing
facility.
The authors would like to thank to Jeroen Ritsema for helping with
the manuscript. Many thanks to V. Maupin, P. Sanchez–Sesma, E.
Beucler and many others for helpful discussions. We also thank T.
Tanimoto and a anonymous reviewer for very useful comments. It is
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103
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552
Input model (saw6)
75km depth
75km depth
-0.014
0.000
0.014
-0.0106
2800km depth
-0.021
0.000
0.0000
0.0106
2800km depth
0.021
-0.0227
dVs/Vs
0.0000
0.0227
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-0.02
-0.03
0
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100
150
parameter number
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250
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Input model (saw6)
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0.000
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-0.0136
2800km depth
-0.021
0.000
0.0000
553
0.0136
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-0.0228
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target model (saw6)
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-0.03
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250
Figure 14. Same as Fig. 13 with 35 per cent of missing data, but solution adopted to deal with the missing data in this test is to replace the missing data by
synthetics computed in the current model (3-D) with the born approximation. They are updated during the iterative inversion. The result is good but not perfect,
which is expected with this solution for missing data.
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SAW6
iteration 4
residual
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0.03
0
50
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Figure 15. Same as Fig. 13 with 35 per cent of missing data. This time the missing data are replaced by synthetics computed in the current model with SEM
at the iteration 4 starting from the iteration 3 of the precedent test (Fig. 14). The result (dotted line) is in a good agreement with the target model.
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Geophys. J. Int. (2008) 172, 1135–1150
doi: 10.1111/j.1365-246X.2007.03703.x
Shallow layer correction for Spectral Element like methods
Y. Capdeville1 and J.-J. Marigo2
1 Équipe
de sismologie, Institut de Physique du Globe de Paris, CNRS, France. E-mail: [email protected]
de Modélisation en Mécanique (UMR 7607), Université Paris VI, France
2 Laboratoire
SUMMARY
Today’s numerical methods like the Spectral Element Method (SEM) allow accurate simulation
of the whole seismic field in complex 3-D geological media. However, the accuracy of such a
method requires physical discontinuities to be matched by mesh interfaces. In many realistic
earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical
cost. In this paper, we address a limited aspect of this problem: an earth model with a thin
shallow layer below the free surface in which the elastic and density properties are different
from the rest of the medium and in which rapid vertical variations are allowed. We only consider
here smooth lateral variations of the thickness and elastic properties of the shallow layer. In
the limit of a shallow layer thickness very small compared to the smallest wavelength of the
wavefield, by resorting to a second order matching asymptotic approximation, the thin layer
can be replaced by a vertically smooth effective medium without discontinuities together with
a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows
to accurately take into account complex thin shallow structures within the SEM without the
classical mesh design and time step constraints. Corrections at receivers and source—when
the source is located within the thin shallow layer—have been also derived. Accuracy and
efficiency of this formulation are assessed on academic tests. The stability and limitations of
this formulation are also discussed.
Key words: Numerical approximations and analysis; Surface waves and free oscillations;
Computational seismology; Wave propagation.
1 I N T R O D U C T I O N A N D M O T I VAT I O N S
Over the last 10 yr, computational seismology has made important
progress allowing accurate computation of the whole wavefield in
complex 3-D models. In particular, the Spectral Element Method
(SEM) (e.g. Priolo et al. 1994; Faccioli et al. 1996; Komatitsch
& Vilotte 1998) has been shown to be very effective for large scale
seismology (e.g. Komatitsch & Tromp 2002; Capdeville et al. 2003;
Chaljub et al. 2003). The SEM is a high-order variational method
based on hexahedra-type of mesh which, like the finite element
method, can accurately take into account discontinuities of the elastic and density properties when discontinuities are matched by a
mesh interface. Such a constraint leads to a lack of geometrical flexibility. Designing a cubic element mesh discretization of an earth
model that honours discontinuities can be a difficult process that
leads to small size elements. When using an explicit time marching
scheme, such small elements impose a small time step in order to
respect the stability condition, increasing dramatically the simulation cost. This is typically the case when using crustal models built
on several thin layers located just below the free surface. In that
case, the mesh discretization can hardly be done. So far, only crude
approximations to this problem have been used. A first one is just to
remove the crustal model where the mesh is too difficult to design
and to replace it by a simpler model for which the mesh design is
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easier. A second solution is to design a mesh that does not honour the crust model discontinuity interfaces, especially the bottom
Moho discontinuity interface where it gets too thin. In that case, the
discontinuity interfaces do not match with a mesh interface and can
be located within an element. In both solutions the accuracy cannot be warranted, and in the latter case it becomes mesh dependent
and difficult to predict. To illustrate this problem, let us consider a
spherically symmetric sphere, of the same radius (6371 km) as the
earth, and two models.
(i) A fully homogeneous, model 1, with V s = 6 km s−1 , V p =
8 km s−1 and ρ = 3000 kg m−3 .
(ii) A homogeneous model with a 20 km thin slow layer below
the surface, model 2, with V s = 3 km s−1 , V p = 8 km s−1 and ρ =
3000 kg m−3 , and the same properties as model 1 below 6351 km.
This simplistic example is quite representative of some of the difficulties associated with more realistic earth models like when using
PReliminary Earth Model (PREM, Dziewonski & Anderson 1981),
for the mantle, together with 3-D crustal models, like CRUST2.0
(Bassin et al. 2000) or 3SMAC (Nataf & Ricard 1996).
Simulations of a 100 s corner period wavefield are done using the
coupled mode-SEM (Capdeville et al. 2003). The domain for the
normal mode solution is defined as the domain from the centre up to
4371 km while the SEM solution domain is defined as the domain
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Accepted 2007 November 30. Received 2007 October 4; in original form 2007 July 20
1136
Y. Capdeville and J.-J. Marigo
Figure 1. 2-D vertical cross-section of one region of meshes 1 (top) and 2 (bottom) used to propagate waves in, respectively, model 1 and 2 (see text). The
element edges are in solid line and the integration point mesh is plotted with dashed lines. mesh 1 is made of two layers of elements and mesh 2 is made of
three. The last element layer of mesh 2 is vertically so thin (20 km) compared to the 990 km edge size of the other two element layers that a zoom is needed to
see it. The polynomial degree is 8 in each direction.
2b for the discretization of model 2. In that case, the discontinuity interface of model 2 is roughly approximated within the upper element
layer. This allows to relax the time step restriction associated with
mesh 2 but fail to approximate accurately the discontinuity interface.
Actually, this last approximation is commonly used in practice when
considering crustal models in earth global models, (e.g. Komatitsch
& Tromp 2002). To assess the implications of these two choices in
terms of accuracy, synthetic seismograms have been computed for
two source locations, a deep one (source 1, 160 km deep) and a
shallow one (source 2, 10 km deep). The source–receiver epicentral
distance is 90◦ and the source is an arbitrary double couple moment
tensor.
In Fig. 2, the synthetic seismograms corresponding to the reference solution, that is, computed for model 2 using mesh 2, and to
the first crude approximation, that is, using model 1 with mesh 1,
are compared. Synthetics are computed for the deep source location.
For both vertical and transverse components, this type of approximation (ignoring the shallow layer) is clearly not accurate, especially
for surface waves which exhibit a strong phase shift. It is quite remarkable that such a small layer can have such a large effect on
seismograms. In Fig. 3, same comparison is performed but now for
the second type of approximation, that is, using model 2 with mesh
2b . Even though the phase shift of the surface waves is now slightly
reduced, it is still quite large. Finally, in Figs 4 and 5, the same comparisons are performed but now in the case of the shallow source
location. Similar conclusions can be drawn, but these time differences in amplitude can also be clearly noted and not only for surface
waves.
First this relatively extreme example shows how large the effect
of a shallow structure can be, especially on surface waves phase
velocity, even if the shallow structure is much thinner than the
from 4371 km up to the surface. Three different SEM meshes have
been considered.
(i) Mesh 1 (Fig. 1, top), corresponds to the discretization of model
1 with two elements of 1000 km thickness and a polynomial approximation of degree eight in each direction. For the minimum propagating wavelength considered here (roughly 600 km outside of the
shallow layer), this corresponds to a two wavelength sampling for
each element. Mesh 1 is slightly oversampling the wavefield.
(ii) Mesh 2 (Fig. 1, bottom), corresponds to the discretization of
model 2. The 20 km thin layer below the surface is actually matched
by an element layer. Vertically, mesh 2 has now three elements vertically, two of 990 km and one of 20 km size.
(iii) Mesh 2b is a relaxed version of mesh 2, where the upper elements have now a vertical thickness of 250 km instead of
20 km. For this mesh, the upper thin layer below the surface is now
contained in the upper layer of elements and the discontinuity is not
honoured by a mesh interface.
Stability condition for the second-order Newmark explicit time discretization leads to a time step of about 1.5 s for mesh 1 and 0.05 s
for mesh 2. It is worth to note here the drastic impact of the mesh 2
discretization: computing time for mesh 2 is expected to be 30 times
larger than for mesh 2. Even though this situation may be slightly
improved when using a different degree for the vertical polynomial approximation in the upper layer, mesh 2 simulations will still
be much more expensive than mesh 1 simulations. Computing time
limitations lead in practice to fall back on simplistic approximations
among which we present two. In the first one, the upper thin layer
is simply ignored and model 2 is approximated using model 1 with
mesh 1. A second approximation is to resort to a mesh that does not
match the discontinuity interface, this is the case when using mesh
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Journal compilation Shallow layer correction for SE like methods
R1
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R2
Z component
T component
G1
0
1000
2000
G2
3000
4000
5000
6000
7000
8000
Time (s)
Figure 2. Comparison between reference seismograms computed in model 2 (solid line) using mesh 2 and seismograms computed in model 1 using mesh 1
(dashed line). The source is a moment tensor at 161 km depth and the epicentral distance is 92 degree. R1 and R2 phases are the minor and major arc Rayleigh
surface wave trains. G1 and G2 phases are the minor and major arc Love surface wave trains. The source origin time is 500 s (this is the case for all seismograms
presented in this paper).
R1
R2
Z component
T component
G1
0
1000
2000
G2
3000
4000
5000
6000
7000
8000
Time (s)
Figure 3. Same configuration and reference solution as for Fig. 2. The second solution (dashed line) is this time computed in model 2 but using mesh 2b. mesh
2b is very close to mesh 2 but with a thickness of the last spherical layer of elements of 250 km instead of 20 km for mesh 2 such that the physical discontinuity
of model 2 is not matched by an element boundary.
wavelength. This fact has been known for long by seismologists
working on large scale seismic imaging for whom shallow structure corrections are a constant issue (e.g. Montagner & Jobert 1988;
Marone & Romanowicz 2007). Second, the results of this simple
example clearly illustrate that the crude approximations classically
used for simulation of complete wavefield propagation in earth models incorporating crustal models are not accurate. This problem is
very close to the two scale homogenization problem used to compute effective media and effective wave equations, (see, for example
Leonach & Grover 2000; Capdeville & Marigo 2007, for an application to the wave equation in layered media), but here small scales
remain localized close to the free surface only. Such a boundary layer
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type problem has been widely study in composite material mechanics (Dumontet 1986; Abdelmoula & Marigo 2000), but very little
in the wave propagation context (e.g. Boutin & Roussillon 2006).
In the static context, the goal of the works simply consists in adding
a first order boundary layer corrector to the leading term. To derive
those correctors, a widely used technique is the matched asymptotic expansions. This formal but constructive technique is well
known in the framework of fluid mechanics (to study the boundary layers of the flow near an obstacle), but was introduced more
recently in solid mechanics (Nguetseng & Sanchez-Palencia 1985;
Sanchez-Palencia 1987). The mathematical justification of this approach and convergence results have been obtained for some model
1138
R1
R2
Z component
T component
G1
0
1000
2000
G2
3000
4000
5000
6000
7000
8000
Time (s)
Figure 4. Same as Fig. 2 but for a 10 km depth source. The source is in the shallow layer.
R1
R2
Z component
T component
G1
0
1000
2000
G2
3000
4000
5000
6000
7000
8000
Time (s)
Figure 5. Same as Fig. 3 but for a 10 km depth source. The source is in the shallow layer.
tests, the range of applications is not limited to this scale and can
be applied, for example, at the scale of a basin with a thin sediment
layer on the top.
problems based on Laplace-type equation with rapidly oscillating
bulk coefficients or rapidly oscillating boundary conditions (Allaire
& Amar 1999; Amar 2000). In this paper, we address the problem of
the effective behaviour of a thin shallow layer on the wave equation.
To keep the equations simple, we shall consider here the case of
an infinite half-space rather than a spherically symmetric medium.
However, the results obtained here can be directly extended to spherical geometry with only slight or even no modifications. We first recall the classical procedure used to solve the wave equation in such
a medium using the spectral expansion in the horizontal directions.
We then solve the problem by removing the shallow layer and finding the effective boundary conditions using a matching asymptotic
approach. The solutions in the spectral-frequency domain are then
transformed back to the space-and-time domain. The SEM implementation is provided and finally accuracy tests at the global scale
are shown together with a discussion of stability and validity limitations. Even if the global scale is chosen here for the validation
2 THEORETICAL DEVELOPMENT
2.1 Preliminary: solving the wave equation in 1-D media
We choose here to work with an infinite half-space with material
properties only varying with the vertical axis (layered medium).
Working with a spherically symmetric medium would provide very
similar and often exactly the same results but with some unnecessary
heavier hand calculus. For the flat geometry, it is convenient to use a
cylindrical coordinate system, where any position x can be written
x = (z, r, φ) = r cos(φ)x̂ + r sin(φ)ŷ + z ẑ where x̂, ŷ and ẑ are the
Cartesian unit vectors and z the vertical axis. We also introduce the
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1139
following notation, for any vector v, its horizontal part is noted with
an index 1: v1 = v − vz ẑ with vz = v · ẑ the vertical part.
If gravity and anelasticity are not taken into account, the wave
equation can be written
with a 0 (z) = F(z)/C(z), a 1 (z) = A(z) − F 2 (z)/C(z) and
1
0
L
.
t Sk (z, ω) =
k 2 N − ω2 ρ
0
ρ ü − ∇ · σ = f,
(1)
σ = c : (u),
(2)
Note that these two problems do not depend on m, which explains
the fact that U k , V k . . . have no m subscript. The free boundary
conditions are [q Y(z, ω)] j∈Bq = 0 where [Y] j is the component j
of the Y vector, B s = {2, 4} and B t = {2}. As z → −∞, q Y k
must vanish.
To obtain synthetic seismograms, the classical procedure is to find
a complete set of normal modes solving (9). For a given solution type
q and wavenumber k, only a discrete number of frequencies ω qkn
are solutions. Once the eigenfunction basis is known, a synthetic
seismogram is found by expanding the solution for a given source
on this normal mode set:
1
u(x, ω) =
uqkmn (x)
uqkmn .f dΩ dk, (12)
2
2
Ω
k qnm ω − ωqkn
where ρ is the density, u the displacement field, ü the acceleration
field, σ the stress tensor, f the source force, c the fourth order elastic
tensor, : the double indices contraction and (u) = 12 (∇u+T ∇u) the
strain tensor withT the transpose operator. A free surface boundary
condition is imposed at the surface (t = σ · ẑ = 0, where ẑ is the
vertical axis unit vector) and the solution must vanish as z goes
to minus infinity. We assume that f both depends upon time and
space.
In layered transverse isotropic media, the 21 independent coefficients of the elastic tensor c(z) reduce to five, for example, the
classical A(z), C(z), F(z), L(z) and N(z) elastic parameters. To take
advantage of the layered model assumption, we use the classical
spectral expansion of the displacement and traction on a horizontal
plane in the horizontal direction (e.g. Takeuchi & Saito 1972):
u(x, ω)
[Uk (z, ω)Pkm + Vk (z, ω)Bkm + Wk (z, ω)Ckm ] dk,
=
k
(3)
t(x, ω)
= σ(x, ω) · ẑ,
=
k
(5)
with
Pkm = Ykm (r, φ)ẑ,
we have
.f dΩ = f (ω)M : qkmn
(xe ),
uqkmn
Ω
= f (ω)M : ∇uqkmn
(xe ),
(4)
[TU k (z, ω)Pkm + TV k (z, ω)Bkm + TW k (z, ω)Ckm ] dk,
m
where u qknm is a mode, [e.g. u snkm = U k (ω kn ) P km + V k (ω kn ) B km for
a spheroidal mode], Ω dΩ the volume integration,∗ is the complex
conjugate. Moreover, if f is a double couple point source located in
x e with a time history f (t),
f(x, ω) = − f (ω)M · ∇δ(x − xe )
m
(6)
Bkm
(7)
uS H =
1
(ẑ × ∇1 )Ykm (r, φ),
k
(8)
where k is the horizontal wavenumber, m the azimuthal wavenumber, ∇ 1 is the surface gradient vector, Y km (r , φ) = J m (kr )eimφ ,
where J m is the order m Bessel function of the first kind. Introducing (3) and (5) in the wave eqs (1) and (2) without the source
term f, we obtain two independent systems of equations of the
form:
∂z q Y k (z, ω) = q Sk (z, ω)q Y k (z, ω).
(9)
The index q can take two values, q = s for the spheroidal (or PSV ) problem and q = t for the toroidal (or SH) problem. We have
T
s Y k = (Uk , TU k , Vk , TV k ) for the spheroidal problem and t Y k =
T
(Wk , TW k ) for the toroidal problem,
⎛
⎞
1
ka0
0
0
C
⎜ −ω2 ρ
0
0
k⎟
⎜
⎟
(10)
s Sk (z, ω) = ⎜
1 ⎟
⎝ −k
⎠
0
0
L
0
−ka0 k 2 a1 − ω2 ρ
0
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2008 The Authors, GJI, 172, 1135–1150
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(14)
(15)
m
k
Ckm =
(13)
where M is the (symmetric) earthquake moment tensor and qkmn =
(u qkmn ).
For the next two sections, the wavefield will often be decomposed
into two parts, one spheroidal (P-SV ) and one toroidal (SH), u =
uPSV +uSH where
(Uk Pkm + Vk Bkm ) dk,
(16)
u P SV =
k
1
= ∇1 Ykm (r, φ),
k
(11)
Wk Ckm dk.
(17)
m
Similarly, we can write t = tPSV + tSH . The following properties can
be easily demonstrated and will be useful to compute the solutions
back in the space domain from the spectral domain:
∇1 · Ckm = 0,
(18)
∇1 × Bkm = 0,
(19)
∇12 · Pkm = −k 2 Pkm ,
(20)
Using (18) and (19) we have
∇1 · u S H = 0,
(21)
∇1 × u1P SV = 0.
(22)
The following classical identity will also be useful: for any regular
enough vector v(x) we have
∇12 v1 = ∇1 (∇1 · v1 ) − ∇1 × (∇1 × v1 ).
(23)
1140
Figure 6. Sketch showing variation with z of one of the original medium property. The medium property shown here can stand for the density or elastic wave
velocities.
2.2 Matching asymptotic expansions in the shallow layer
the free surface (see Fig. 6) as well as the S (y) operator:
Because the following development is the same for each frequency,
for each wavenumber and for the spheroidal or toroidal cases, the
dependency of vectors and operators with respect to q, k and ω is
dropped in this section. Because S operators fully define the elastic
(As , C s , F s , Ls , N s ) and density (ρ s ) parameters of the model, we
will often refer to S as an ‘earth model’.
We now assume the elastic properties and density are constant or
smoothly varying with depth everywhere in the model, except in a
thin layer just below the surface where they are varying rapidly with
z as described in Fig. 6. The top of the model is in z = a, the bottom
of the shallow layer at z = b 0 and, therefore, the layer thickness is
H 0 = a − b 0 . We assume the layer to be thin with respect to the
0
minimum wavelength of the wavefield: ε0 = λHmin
1. This defines
the ‘real model’ Sε0 .
In the following, we assume to be given a smooth model Ss (z)
defined on [0, a] and a shallow layer model SH (z) defined on [b, a]
of thickness H = a − b. Ss is ‘smooth’ in the sense that it doesn’t
contain any small scales and SH can vary rapidly in [b, a]. We build
H
Sε , with ε = λmin
, such that
S(y) = Sε (εy + b).
ε
S (z) =
Ss (z) if z < b,
S H (z) if z ≥ b.
The main idea of this procedure is to solve the wave equation with
a heavy numerical method in the smooth model Ss which is a much
simpler task than in the original model and to use an asymptotic
matching condition to compute the missing upper boundary condition. To find this missing upper boundary condition, we introduce
two vectors,
(i) Y ε solution of
∂z Y ε (z) = Ss (z)Y ε (z).
(27)
ε
(ii) D solution of
∂ y D ε (y) = εS(y)D ε (y).
(28)
The boundary conditions for (27) and (28) are a free surface boundary condition for D ε at y = ya , Y ε must vanish as z → −∞ and
the two solutions must match in an overlapping area which will be
specified later. Y ε has a ε dependency through this last boundary
condition even if its wave equation (27) doesn’t depend on ε. In
(28), the ε appears because of the transformation ∂z → 1ε ∂ y in the
original wave equation (9).
We use two asymptotic expansions for Y ε and D ε :
(24)
This defines a series of models Sε as ε (or b) varies. What we call
the ‘real model’ is just a particular case of this series obtained for
ε = ε 0 (or b = b 0 ).
In practice only the real model Sε0 is known. Ss is a priori not
given and has to be built depending on the properties of Sε0 below
b 0 . For example, if the model below the depth b 0 is constant as a
function of z, a valid (i.e. that doesn’t depend on ε) construction is:
Sε0 (z) if z < b0 ,
s
(25)
S (z) =
Sε0 (b0− ) if z ≥ b0 ,
ε0
(26)
Y ε (z) =
D ε (y) =
∞
i=−1
εi Y i (z),
(29)
εi Di (y).
(30)
∞
i=−1
The sums start at i = −1 because we will see later that the first
terms non-trivialy equal to zero starts at i = −1. Introducing these
expansions into (27) and (28), for each i, we obtain
(b0− ) is the value of Sε0
where S
just below b 0 . For less simple cases,
the Ss operator corresponds to a simpler model than the original
model that can be any smooth prolongation of the model below the
thin layer as long as it does not depend on ε. SH is easily built by
scaling S H0 (z) = Sε0 (z) for z in [b 0 , a].
We introduce a new variable y = z−b
, a zoom on the shallow
ε
layer, such that y = 0 at the bottom of the layer and y = ya = Hε at
∂z Y i = Ss Y i
(31)
∂ y Di+1 = SDi .
(32)
The boundary conditions are
112
[Di ] j∈B = 0 for y = ya ,
(33)
Y i → 0 as z → −∞.
(34)
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2008 The Authors, GJI, 172, 1135–1150
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Journal compilation Di and Y i must match somewhere.
(35)
For the third boundary condition, we assume there exists a large
negative y where the solution D ε (y) is still valid as well as Y ε (εy+b)
such D ε (y) = Y ε (εy + b). In the limit when ε tends towards zero,
even if − y is large, ε y + b is close to a and, therefore, a Taylor
expansion for Y i around a can be used:
∞
(y − ya ) j ∂ j Y i
Y I [a + ε(y − ya )] =
εj
(a).
j!
∂z j
j=0
(36)
ε
Introducing (36) in (29) and imposing that the two solutions D and
Y ε match for very large − y, we find the matching conditions:
i
(y − ya )i− j ∂ i− j Y j
(a) = 0.
lim D (y) −
y→−∞
(i − j)! ∂z i− j
j=0
i
(37)
(38)
Knowing that D is constant in y and D = Y (a), the last equation can easily be integrated:
y
D 1 (y) = D 1 (ya ) +
S(y ) dy Y 0 (a).
(39)
0
0
0
ya
The matching condition for i = 1 gives
lim D 1 (y) − (y − ya )∂z Y 0 (a) − Y 1 (a) = 0.
y→−∞
Knowing that S (y) =S (εy + b) for y < 0, using (31) for i = 0 and
(39), the matching condition (40) gives
with
ya
X1 (y) = −
S(y ) − Ss (a) dy .
(41)
(42)
For i = 1, (32) gives
(44)
Integrating the last equation, we find
y
S(y )[X1 (y ) − X1 (0)] dy Y 0 (a)
D 2 (y) = D 2 (ya ) +
+
+
ya
y
ya
y
ya
C
(y − ya )S(y ) dy Ss (a)Y 0 (a)
S(y ) dy Y 1 (a).
2008 The Authors, GJI, 172, 1135–1150
C 2008 RAS
Journal compilation Knowing that S(y) = Ss (εy + b) for y < 0, using (31) for i = 0 and
(45), the matching condition (46) gives
D 2 (ya ) + X2 (0)Y 0 (a) + X1 (0)Y 1 (a) − Y 2 (a) = 0,
(47)
with
X2 (y) =
ya
−
{S(y )[X1 (y ) − X1 (0)] − [X1 (y ) − X1 (0)]Ss (a)} dy .
(48)
To obtain the last equation, we have assumed that ∂ z S (a) Ss (a)
which implies we have been able to build a smooth model that does
not vary much in the thin shallow layer with respect to its absolute
value. This assumption is easy to meet for the earth. Nevertheless, when this assumption is not valid, one can with some algebra
but no real difficulty complete the last expression. Finally, using
[D 2 (ya )]i∈B = 0, (47) gives the boundary conditions for Y 2 in
z = a:
2 Y (a) i∈B = X2 (0)Y 0 (a) + X1 (0)Y 1 (a) i∈B .
(49)
s
We are now able to solve eqs (31) and (32) for order 0, 1 and
2. Nevertheless, it is not very convenient for methods like SEM to
solve different orders one after another. Instead of doing so, it is
more interesting to somehow solve for the sum of all orders. To do
2
2
so, we solve for Ŷ and D̂ , solutions of (27) and (28), respectively,
with the free boundary condition at the surface and regularity of the
solution as z → −∞ but with the following matching condition:
2
a−b
2
D̂
(50)
= I − εX̂2 (0) Ŷ (a),
ε
X̂2 (y) = X1 (y) + εX2 (y).
(45)
113
(51)
One can check that
2
Ŷ = Y 0 + εY 1 + ε 2 Y 2 + O(ε 3 ),
2
Eq. (41) makes the link between the two solutions D 1 and Y 1 . Using
[D 1 (ya )]i∈B = 0, it gives the boundary conditions for Y 1 in z = a:
1 (43)
Y (a) i∈B = X1 (0)Y 0 (a) i∈B .
(46)
D̂ = D 0 + εD 1 + ε 2 D 2 + O(ε 3 ).
y
∂ y D 2 = SD 1 .
The matching condition for i = 2 is
1
lim D 2 (y) − (y − ya )2 ∂z22 Y 0 (a) − (y − ya )∂z Y 1 (a)
y→−∞
2
−Y 2 (a) = 0.
with
(40)
s
D 1 (ya ) − Y 1 (a) + X1 (0)Y 0 (a) = 0,
1141
y
Eqs (31) and (32) have now to be solved for each i with the
boundary conditions (33) and (34) and the matching conditions (37).
For i < −2, we obtain Di+1 = Y i+1 = 0.
For i = −2, (32) gives ∂ y D −1 = 0. With the boundary conditions
and matching condition, the only solution to the previous equation is
D −1 = 0, which also impose Y −1 = 0.
For i = −1, (32) gives ∂ y D 0 = 0. D 0 is, therefore, constant in the
shallow layer. The matching condition gives D 0 = Y 0 (a). These
expected results show that the shallow layer has no effect on the
solution Y at order 0. Neglecting the shallow layer by just ignoring
it and replacing it by the smooth model is, therefore, an order zero
approximation.
For i = 0, (32) gives
∂ y D 1 = SD 0 .
Shallow layer correction for SE like methods
(52)
(53)
The source and the receiver are often located in the shallow layer
2
2
and it is useful to know D̂ [(z − b)/ε] as a function of Ŷ (z). It can
be shown that, for z > b,
z−b
z−b
2
2
D̂
= I − ε X̂2 (0) − X̂2
Ŷ (z),
(54)
ε
ε
For double-couple sources, it is useful to know the first derivative
of the solution with respect to z in the shallow layer:
z−b
∂z D̂ 2
ε
z−b
2
= Sε (z) I − ε X̂2 (0) − X̂2
Ŷ (z) + O(ε 3 ). (55)
ε
The expressions obtained here are very similar to the ones obtained by Capdeville & Marigo (2007) which indicates that the
transition from this shallow layer correction to the more general
homogenization case should be straightforward.
1142
2.3.2 Receiver correction
2.3 From the horizontal spectral domain
to the space domain
Most of the time, the receivers are located on the free surface and,
therefore, a shallow layer correction must be applied. This time, the
solution needed is D̂(ya ) knowing Ŷ(a) given by (50). Up to the
order 1, using (50) for the first component in the toroidal case, and
the first and third component in the spheroidal case, we find, for a
receiver located in x r ,
In order to use the results obtained in the previous section with numerical methods based on a space formulation, one needs to convert
them from the horizontal spectral domain to the space domain. This
has to be done for the boundary condition in any case and for the
source and the receiver when they are located in the shallow layer.
In the following, we name u and t the displacement and traction
corresponding to the Ŷ solution and uc and tc the displacement
and traction corresponding to the D̂ solution corrected for the shallow layer effect. More generally, all quantities (vector components,
strain tensor . . .) corresponding to the D̂ solution are noted with a
‘c ’ upper script.
1
uc (xr , t) = u(xr , t) + ε X a0
(0) (∇1 · u1 ) (xr , t) ẑ,
(64)
where X 1a0 is given in Appendix A. It can be seen that, to order 1 in
ε, only the vertical component of the spheroidal case has a non-zero
correction. The order 2 correction for the receiver has not been used
in this paper.
2.3.1 Boundary conditions
2.3.3 Sources located in the shallow layer
We explicit here the spectral to space domain conversion only for
the first order of the toroidal case. The complete case is given in
Appendix A. The matching condition (50) gives
1
(I − ε t X̂1k )t Ŷ k (a) = 0,
(56)
If the source is located in the shallow layer, a correction also needs
to be applied. As shown in eq. (15), the moment tensors is applied
to the strain tensor. Using (54) and (55) in y = (z e − b)/ε, where
z e is the vertical location of the source, we can find Mc : ∇uc (x e )
as a function of u. Mc is the ‘real’ moment tensor used to compute
the solution in the original model underlying Sε . As for the receiver
correction, we limit the asymptotic expansion to the order 1. Only
the toroidal case is developed here, the spheroidal case is given in
Appendix B. We have, for any symmetric M:
2
which, using (11) and (42) can be written
TW k (a, ω) = ε k 2 X 1N (0) − ω2 X ρ1 (0) Wk (a, ω),
where
X 1N (y) = −
ya
M : ∇u = M11 : 11 + M1z · (∇1 uz + ∂z u1 ) + Mzz uz ,
(58)
y
X ρ1 (y)
[N (y ) − N s (a)] dy (57)
ya
=−
[ρ(y ) − ρ (a)] dy
s
where 11 = 12 (∇1 u1 + T ∇1 u1 ),
(59)
M1z = T (Mr z , Mφz , 0),
y
⎛
with N (y) = N ε (εy + b), ρ(y) = ρ ε (εy + b). Using (3), (5), (20)
and returning to the time domain, for all x belonging to the free
surface, to the first order we have:
t S H (x, t) = −ε X 1N (0) ∇12 u S H (x, t) − X ρ1 (0) ü S H (x, t) .
(60)
M11
and, therefore, for all x belonging to the free surface, we have
t S H (x, t) = ε X 1N (0) ∇1 × ∇1 × u1 (x, t) + X ρ1 (0) ü S H (x, t) .
t = A (u, ü).
(67)
uc = Wkc Ckm ,
(68)
we have
Mc : ∇uc = Mc11 : c11 + Mc1z · ∂z uc1 ,
= Wkc Mc11 : ∇Ckm +
(61)
1 c
T Mc · Ckm ,
L εe W k 1z
(69)
(70)
where L εe = L ε (z e ) and the relation ∂z Wkc = L1εe TWc k has been used.
Using (54) in the toroidal case, we find, to the first order
(62)
Note that the problem may seem to be not completely cleared because of the last term of the last equation still depends on uSH .
Nevertheless, it is shown in Appendix A that the exact same term
appears in the spheroidal part of the solutions and, therefore, it is
safe to replace uSH by u here.
As mentioned earlier, the complete expression of the traction on
the free boundary is given in Appendix A, and has the form, for all
x belonging to the free surface and all times t,
ε
⎞
0
⎟
0⎠.
0
Mr φ
Mφφ
0
Mrr
⎜
= ⎝ Mφr
0
(66)
Considering the following toroidal displacement for any k and m,
In practice, the toroidal part and spheroidal part of a wavefield cannot be accessed and only the total wavefield u is known. If u is
used instead of uSH in (60), because ∇ 21 u = ∇ 21 uSH + ∇ 21 uPSV and
∇ 21 uPSV is in general non-zero, there will be an unwanted contribution of the spheroidal part of the displacement on the toroidal
traction. To avoid this problem, the identity (23) and the property
(21) can be used to show that
∇12 u S H = −∇1 × ∇1 × u1
(65)
1
Wkc = Wk − εL e X eL
∂ z Wk ,
(71)
1
1
TWc k = L e ∂z Wk − ε k 2 X eN
Wk ,
− ω2 X eρ
(72)
where
X 1L (y) = −
ya
y
1
1 (y ) − s (a) dy ,
L
L
1
X eL
= X 1L (0) − X 1L
(63)
Because the last equation is a boundary condition in traction that
depends upon the displacement, it is called a Dirichlet to Neumann
condition (DtN).
ze − b
,
ε
1
= X 1N (0) − X 1N
X eN
114
ze − b
,
ε
C
(73)
(74)
(75)
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1
X eρ
= X ρ1 (0) − X ρ1
ze − b
,
ε
(76)
with L(y) = L ε (εy + b). Using (3), (20), (23), the property (21)
and the symmetry of the moment tensor, we can show that
SH
Mc : c S H = M01z · 1z
−ε
+εω2
1
X eρ
Ls
1
X eN
M0 · (∇1 × ∇1 × u1 )
L s 1z
1
SH
M01z · u S H − ε X eL
L s Mc11 : ∂z 11
,
(77)
Ls
where M01z = L εee Mc1z and L se = L s (z e ). It appears clearly here that,
for a moment tensor source, there is an order 0 effect of the shallow
layer through the Lse /Lεe ratio. The M0 moment tensor (see Appendix B for complete expression) is the order 0 apparent moment
tensor. If located outside of the shallow layer, we have M0 = Mc
but not if source is located in the shallow layer. Including the time
dependence f (t) of the source (77) can be rewritten as a force vector,
f(x, t) = F [ f (t)δ(x − xe )],
(78)
where for any scalar function g(x, t)
F (g) = M01z · ∇1 g ẑ − ε
−ε
1
X eρ
L se
1
X eN
∇1 × ∇1 × M01z g
L se
1
M01z g̈ − ε X eL
L se Mc11 ∇1 g.
(79)
To be complete, the spheroidal part of the solution needs to be
included in the source correction. This is done in Appendix B.
2.4 Spectral element implementation
We now consider a bounded domain Ω that can be a part of the
infinite half-space or the whole Earth (in the spherical geometry
case) and its boundary ∂Ω. In the infinite half-space, we consider
that ∂Ω is limited to the free surface and that a solution is found for
the other faces of the domain (like absorbing boundaries). The SEM
solves the wave equation under the weak form, that is, the solution
u, for all admissible displacements w and all times t must satisfy
(ρ ü, w) + a(u, w) − t, w∂ Ω = (f, w)
(80)
with (w, ρu)| t=0 = 0 and (w, ρ u̇)|t=0 = 0, where (·, ·) is the classical
L2 inner product, the symmetric elastic bi-linear form
a(u, w) =
∇u : c : ∇w dx
(81)
Ω
and
t · w dx.
(82)
t, w∂ Ω =
∂Ω
Usually the term t, w ∂
vanishes because of the free boundary
condition. In our case, we have
(83)
t, w∂ Ω =
Aε (u, ü) · w dx,
∂Ω
where Aε is defined in eq. (A16).
If the source happens to be located in the shallow layer, the source
term (f, w) can be computed from eq. (B11).
The classical spectral element approximation can then be applied
without any major difficulty, at least for the first order. The boundary
condition term requires to compute terms with gradient and curl on
a surface, like for example, for the toroidal contribution:
∇1 × ∇1 × u1 , w∂ Ω =
(84)
(∇1 × w1 ) · (∇1 × u1 ) dx,
∂Ω
C
2008 The Authors, GJI, 172, 1135–1150
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1143
where the Green’s theorem has been used. In flat geometry, computing terms like (84) is straightforward, but in spherical geometry this
is less simple. Indeed, because of the pole singularities, the spherical coordinates cannot be used. Instead, the local coordinates of
the ‘cubed sphere’ are used (Sadourny 1972; Ronchi et al. 1996).
These local coordinates system are non-orthogonal curvilinear systems leading to less simple expressions than in the flat case. Nevertheless, with some patience, these expressions can be retrieved
and computed without any numerical difficulty. The second order
terms are potentially problematic. Indeed, as presented in Appendix A, the DtN operator Aε leads to terms in (81) which involve third
order spatial derivatives, which is not compatible with C 0 spectral
elements. Even if most of the terms of t, w ∂
can be symmetrize
(see Appendix A) and the problem of the third order spatial derivatives can be worked out by introducing auxiliary variables (Givoli
2004; Givoli et al. 2006) or by using a spatial filtering to remove
unphysical small spatial frequencies of u before using it in Aε , none
of these solution is used here. Instead, we commit a variational crime
(Strang & Fix 1973) by implementing the second order DtN term
as it is given in Appendix A, which appears to give an accurate solution, at least for cases shown in this paper. Nevertheless, proper
solutions to this problem will be explored in future works.
For the time evolution, a classical explicit Newmark time marching scheme is used. With the introduction of the DtN boundary
condition (83), because of its dependency on acceleration, this time
marching scheme becomes implicit. This problem is similar to the
one obtained when coupling a normal solution with the SEM through
a DtN operator (see Capdeville et al. 2003). The order 1 acceleration
terms can be included in the mass matrix, but the order 2 terms have
to be treated implicitly. Nevertheless, this modification is minor and
localized on the surface of the mesh. From the stability point of
view, the DtN boundary condition can be a problem depending on
the elastic, the density contrast and on the thickness of the shallow
layer. For example, the order 1 acceleration terms, once included
in the mass matrix, can lead to a locally negative mass matrix for
mesh points localized on the surface. Obviously, such a case cannot
be stable. Depending on the mesh design, on the thickness, on the
elastic and on the density properties of the shallow layer, all terms
of Aε potentially lead to instabilities when ε gets large enough. For
order 2 terms, this stability issue is probably linked to the variational
crime mention earlier. Nevertheless, this stability problem has not
been an issue for any of the cases presented in this paper.
3 VA L I D AT I O N T E S T S
All the validation tests presented in this paper are done in spherical geometry. This introduces a small modification of the boundary
condition for the spheroidal part given in Appendix C. In order
to validate the above theoretical development we first perform a
test in the model 2 defined in the introduction. The source–receiver
configuration is the same as the one used in Section 1. The reference solution is computed in model 2 using mesh 2. The order 0
asymptotic solution is obtained by computing the seismogams in
model 1 using mesh 1. The results have already been shown in Section 1, Fig. 2, and are not accurate. On Fig. 7 is shown in dashed
line the asymptotic solution for the order 1 (left-hand plots) and
order 2 (right-hand plots) for the boundary condition and order 1
for the receiver and to be compared with the reference solution
(solid line). The source used is not in the shallow layer (161 km
deep), therefore, no correction is required for the source. This result
has to be compared with the two approximate solutions used in the
1144
Z component
order 1
Z component
order 2
T component
order 1
T component
order 2
0
2000
4000
6000
8000 0
2000
Time (s)
4000
6000
8000
Time (s)
Figure 7. Comparison between reference seismograms (solid line) obtained in model 2 and solutions obtained in model 1 with the asymptotic boundary
conditions at the order 1 (left-hand plots) and at the order 2 (right-hand plots) and with order 1 receiver correction. The geometrical configuration is the same
as for Fig. 2. Residuals are shown Fig. 8.
the reference solution and the solution obtained with the order 2
boundary conditions and the order 1 receiver solutions for the 20 km
(solid line) and the 10 km (dashed line) thick shallow layer. It can
first be observed that the residuals are larger for the vertical component than for the transverse component. For the vertical component,
the residual amplitude is roughly divided by 8 between the 20 and
10 km layer thickness cases, which is coherent with an ε3 asymptotic approximation. For the transverse component, some parts of
the signal have indeed a residual amplitude divided by 8 but some
others only by 4 or 2. This indicates that some parts of the residual
signal is dominated by the boundary condition effects (order 2) and
some other by the receiver correction (order 1) or even sphericity
effects (order 0 for the transverse component). The last two effects
are very small but still appear due to the good accuracy achieved for
the transverse component. For applications requiring high accuracy
convergence, it might be a good idea to go to the order 2 also for
the receiver correction and all sphercity terms. The effect of the receiver correction is shown in Fig. 9. In this case, it can be seen that
it affects slightly the amplitude for the fundamental surface wave
introduction: removing the shallow layer (Fig. 2), which is actually
the order 0 asymptotic solution, and using a mesh that doesn’t honour the physical discontinuity with an element boundary (Fig. 3). It
appears that the order 1 is accurate for the transverse component
but not for the vertical one. On the transverse component, some
small discrepancy with the reference solution can nevertheless be
observed for times corresponding to the Rayleigh waves. This is
due to the small transverse component of the spheroidal solution.
The order 2 corrects a large part of the error observed for the order
1. This is still not perfect but far much better than any of the approximate solutions tried in the introduction. The only differences
between the order 1 and 2 for the transverse component are due to
the small transverse component of the spheroidal solution. Indeed,
it appears that the order 1 for the toroidal solution is, in fact, the
order 2 accuracy (the order 2 terms of the toroidal asymptotic expansion for the boundary condition are all zero). To have an idea of
the convergence towards the reference solution as the thickness of
the shallow layer decreases, a similar test with a layer thickness of
10 km has been performed. In Fig. 8 is shown the residuals between
0.2
Normalized amplitude
Z component
0.0
-0.2
0.03
T component
0.00
-0.03
1000
2000
3000
4000
5000
6000
7000
8000
Time (s)
Figure 8. Residuals between the reference solution and the solution obtained with the order 2 boundary condition and the order 1 receiver solutions in model 2
with a 20 km (solid line) and a 10 km (dashed line) thick shallow layer. The amplitude of the traces is normalized with the maximum amplitude of the reference
solution.
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1000
2000
3000
4000
1145
5000
Time (s)
Figure 9. Effect of the order 1 local correction on the receiver. On the top traces is plotted the vertical component of the reference solution (thin solid line),
the solution with order 2 asymptotic boundary condition and no correction at the receiver (dotted line) and the residual (bold solid line). On the bottom traces
is plotted the same thing but this time the receiver correction is applied (dotted line).
R1
R2
Z component
T component
G1
0
1000
2000
3000
G2
4000
5000
6000
7000
8000
Time (s)
Figure 10. Same as Fig. 7 right-hand part, but this time, the source is 10 km depth (in the shallow layer) and an order 1 correction is applied for the source.
This figure has to be compared with the approximate solutions Figs 4 and 5.
but mainly the amplitude of some higher modes. In Fig. 10 is shown
the effect of the order 1 source correction if the source lies in the
shallow layer. It can be seen that this correction accurately corrects
the amplitude effects seen on Figs 4 and 5. Nevertheless, looking
carefully, it appears that the accuracy is not as good (especially for
the Rayleigh phase) as when the source is not in the shallow layer.
This indicates that the order 2 correction for the source might be
useful for higher accuracy.
Finally, we show a test in a more realistic Earth. We use PREMoc,
a modified PREM model with a thin oceanic crust but no ocean (see
Fig. 11). For this model, the Earth radius is 6366 km and the crustal
thickness is 6.5 km. In Fig. 12 are shown traces obtained with different numerical simulations. The reference solution is computed in
the PREMoc with a SEM mesh honouring all the model disconti
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nuities and especially the Moho. Then, the solution corresponding
to the order 0 of the asymptotic solution is shown. In that case the
asymptotic solution is just a regular computation with free boundary condition in the order 0 model. The order 0 model corresponds
to the elastic properties and density of the Ss operator for which
the crustal layer has been replaced by the continuity of the mantle
(see Fig. 11, right-hand graph). Note that here the order 0 model is
not constant in the shallow layer but is a degree 1 polynomial: the
prolongation of the PREMoc upper mantle. The traces obtained in
such a model have surface wave phases heavily shifted with respect
to the reference solution which shows how inaccurate this solution
is (Fig. 12). Next is computed the solution in PREMoc, but with a
SEM mesh that doesn’t honour the Moho. Instead the mesh interface that corresponds to the Moho is moved down by 20 km to a
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km/s, kg/m^3
8000
Vp
7000
10000
6000
Vp
Vs
rho
5000
5000
Vs
4000
rho
3000
0
0
1000 2000 3000 4000 5000 6000
2000
6350
6360
radius (km)
Figure 11. P and S wave velocities (Vp and Vs) and density (rho) as a function of the Earth radius of the PREMoc model, a modified PREM model used for
the last numerical experiment of this paper. It is a modified PREM in the sense that the Earth radius in 6366 km (6371 km for the original PREM), the Moho
depth is 6.5 km (24.4 km for the original PREM) and crustal properties correspond to an oceanic crust (obtained from CRUST2.0). On the left is plotted an
Earth scale view of the model. On the right is plotted a zoom of rho, Vp and Vs in the last kilometres of PREMoc (dashed lines) and of the order 0 model (solid
line) used for the asymptotic method. The last model correspond to the elastic properties and density of the Ss operator.
Z component
2400
2500
2600
2700
Time (s)
2800
2900
reference
order 0
not honored
order 2
T component
2200
2300
2400
2500
2600
Time (s)
Figure 12. Vertical (top traces) and transverse (bottom traces) components for the experiment in PREMoc (Fig. 11). The reference solution (solid lines) is
computed in PREMoc with a mesh honouring all the interfaces and especially the Moho. In dotted line is plotted the solution computed in the order 0 model,
that is a model for which the crust has been removed and replaced by the continuity of the mantle (see Fig. 11, right-hand plot). In dash–dotted line is plotted
the solution computed in PREMoc but with a mesh that is not honouring the Moho discontinuity (the mesh interface corresponding the Moho have been placed
at 20 km depth, which corresponds to the Moho depth of the real PREM). In dashed line is plotted the solution computed with the order 2 asymptotic boundary
condition and the order 1 receiver corrector. The corner frequency of the source is 1/65 Hz and is an irrelevant moment tensor. The time windows are focused
on the minor arc surface waves. The source depth is 25 km and the epicentral distance is 77◦ .
because of the time step, the computation of the asymptotic solution
is here seven times faster than that of the reference solution.
depth that corresponds to the Moho depth of the real PREM. This
represents the solution that is commonly used in global seismology.
It can be seen that this solution is more accurate than the previous
one, but the error is still very large. The phase shift is this time
overestimated where as in the example given Fig. 3, the phase shift
is underestimated. This shows how unpredictable is the error when
major model discontinuities are not honoured by a mesh interface.
Finally, the order 2 asymptotic solution is shown. The accuracy is
really good despite a small amplitude error. For this example, just
4 D I S C U S S I O N, C O N C LU S I O N A N D
PERSPECTIVES
We have presented a second order asymptotic method to compute
the effective behaviour of the wavefield in the presence of a thin
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heterogeneous shallow layer below the free surface. Using an order 2
matching asymptotic expansion, we have shown that, in the case of a
shallow layer thickness much smaller than the minimum wavelength
of the wavefield, the shallow layer elastic and density properties can
be removed and replaced by continuation of the elastic and density
properties of the medium just below the shallow layer and by a
particular DtN boundary condition. From the numerical method
point of view, the main interest of such a method is that the shallow
layer lower boundary doesn’t exist anymore and, therefore, doesn’t
need to be matched by a mesh interface. This releases the meshing
and explicit time marching time step problems. It has been shown,
that if the thickness is small enough with respect to the minimum
wavelength, the accuracy is excellent. The actual ‘small enough’
thickness required widely varies with the actual properties of the
elastic medium in the shallow layer and the wanted accuracy. In the
example given in this paper the ratio thickness with respect to the
horizontal wavelength is as small as 1/30, but is can raise up to 5
for smaller velocity contrasts.
The SEM implementation of the DtN boundary condition can
lead to an unstable Newmark time scheme if the thin layer is not
thin enough compared to the minimum wavelength that can accurately be sampled by the mesh. This can be a serious problem in
some situations. For example, if the thickness of the shallow layer
is smoothly varying laterally and becoming locally too thick so the
time scheme is unstable, the asymptotic solution cannot be used at
all. This problem can be solved using a spatial filtering of the displacement wavefield at the surface before using it in the boundary
condition DtN operator. The spacial filtering removes the unphysical high spatial frequencies of the wavefield and leads to a stable
scheme. Another potential solution in the introduction of auxiliary
variables following the work on non-reflecting absorbing boundaries
(Givoli 2004). These solutions will be explored in the near future.
Nevertheless, even if the scheme can be made stable in any case,
it doesn’t improve the fact that the accuracy that decreases rapidly
when the thickness of the layer is no longer small enough compared
the wavelength. This is the main limitation of this development: the
frequency range of accuracy is determined by the geometry (mainly
the thickness of the shallow layer) of the model. When the thin
layer assumption is broken by increasing the corner frequency of
the source or by increasing the thickness of the shallow layer, the
accuracy deteriorates and this development needs to be dropped.
Another obvious limitation is that this work only applies to small
scales directly located below the free surface and doesn’t help for
deeper small scale heterogeneities. An option to go around these
two problems is to move to volume homogenization (Capdeville &
Marigo 2007). In Capdeville & Marigo (2007), it is shown that, for a
given frequency band, an effective model and wave equation can be
found. In the layered model case, the order 2 effective medium is a
Backus filtering (Backus 1962) of the original model in the volume
together with a DtN boundary condition for the free surface similar
to the one obtained here. In that case, a different but valid solution is
found for each frequency band and for any depth of the small scale
heteogenities. The SEM implementation of Capdeville & Marigo
(2007)’s work will be the purpose of a future publication.
Applications of this work as it stands should be found in forward modelling as it allows to take into account complex shallow
structures. A good example of such an application is the accurate
implementation of crustal models at the global scale. Such a technique is of course not limited to the global scale and can be applied
for example at the scale of a basin to take accurately into account
a shallow structure like a sediment layer. A following application
is the crustal correction for imaging technique. Indeed, for classi
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1147
cal seismic imaging techniques, the crust is not inverted for, but
an a priori crustal model is used. By allowing to incorporate accurately the a priori crustal model in the forward modelling problem,
this work can be used to solve this classical problem. This work
can also be used for seismic imaging by allowing to invert for the
crust. Indeed, this works gives an integrated parametrization of these
complex shallow structures and of the local sources and receivers
interaction with the small scale of the Earth through a limited numbers of parameters (X 1L , X 1N , etc.) that can be inverted for more easily
than a thin vertical parametrization of the crust.
This work has been funded by the French ANR MUSE project under the BLANC program. Computations were performed on IPGP’s
cluster, IDRIS and CINES computers. Many thanks to Jean-Pierre
Vilotte, Jean-Paul Montagner and B. Romanowicz as well as Qinya
Liu and an anonymous reviewer for their help to improve the
manuscript. Many thanks to Dan Givoli and to Paco Sánchez-Sesma
for pointing out my variational crime.
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Givoli, D., Hagstrom, T. & Patlashenko, I., 2006. Finite element formulation
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Komatitsch, D. & Tromp, J., 2002. Spectral-element simulations of global
seismic wave propagation, part II: 3-D models, oceans, rotation, and gravity, Geophys. J. Int., 150, 303–318.
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A P P E N D I X A : S E C O N D O R D E R E X P R E S S I O N O F T H E B O U N D A RY C O N D I T I O N
For the toroidal case, one can check that the ε2 terms for the asymptotic boundary condition is 0. Therefore, expression (62) is also valid for
the order 2.
For the spheroidal case the matching condition (50) gives
1
(I − ε s X̂2k )s Ŷ k
(a) = 0,
(A1)
2,4
which can be written as
2
(0) Vk (a, ω)
TU k (a, ω) = −εω2 X ρ1 (0)Uk (a, ω) + ε2 kω2 X a2 (0) + k 3 X a1
(A2)
1
2
− ω2 X ρ1 (0) Vk (a, ω) + ε 2 kω2 X b2 (0) + k 3 X a1
Uk (a, ω),
TV k (a, ω) = ε k 2 X a1
(A3)
where all terms higher than ε2 have been truncated, and
ya
1
a0 (y ) − a0s (a) dy (y) = −
X a0
(A4)
y
1
(y) = −
X a1
ya
y
2
(y) = −
X a1
X a2 (y) =
ya
y
ya
y
(A5)
1
1
(y ) − X a1
(0) dy X a1
(A6)
1 1
ρ(y ) X a0
(y ) − X a0
(0) + 1 − a0s (a) X ρ1 (y ) − X ρ1 (0) dy (A7)
1 1
1
ρ s (a) X a0
(y ) − X a0
(0) + 1 − a0 (y ) X ρ1 (y ) − X ρ1 (0) dy + X a0
(0)X ρ1 (0),
(A8)
y
X b2 (y) =
ya
a1 (y ) − a1s (a) dy where for any function f ε , f (y) = f ε (εy + b). Furthermore, using that fact that
d
[X 1 (y) − X ρ1 (0)] = ρ(y) − ρ s (a),
dy ρ
and
d dy
(A9)
1
1
(y) − X a0
(0) = a0 (y) − a0s (a),
X a0
(A10)
it can be shown with some algebra that X a2 (0) = X b2 (0).
Using expressions (3) and (5), properties (18)–(20), the fact that
∇1 · Bkm = −kPkm · ẑ,
(A11)
∇1 (Pkm · ẑ) = kBkm ,
(A12)
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and the identity (23) we find from (A2) and (A3)
2
tzP SV = ε X ρ1 üz + ε 2 X a2 ∇1 · ü1 + ε 2 X a1
∇12 (∇1 · u1 )
(A13)
1
2
t1P SV = −ε X a1
∇1 (∇1 · u1 ) + ε X ρ1 ü1P SV − ε 2 X b2 ∇1 üz − ε 2 X a1
∇12 (∇1 uz ),
(A14)
t = Aε (u, ü),
(A15)
where the X coefficients are taken in y = 0 and the traction and displacement in (x, t), x belonging to the free surface. Using t = t1P SV +
tzP SV ẑ + t S H and u = u1P SV + uzP SV ẑ + u S H , we find, for all x belonging to the free surface and all time t
with
1
Aε (u, ü) = ε X ρ1 ü − X a1
∇1 (∇1 · u1 ) + X 1N ∇1 × ∇1 × u1
2 2
∇1 [(∇1 · u1 ) ẑ − ∇1 uz ] + X b2 [(∇1 · ü1 ) ẑ − ∇1 üz ] ,
+ε 2 X a1
(A16)
= X 2b , it
where the X coefficients are taken in y = 0. As written here, (A16) leads to a non-symmetric term in (81). Nevertheless, because
can be symmetrize. This can be seen on the spectral expressions (A2) and (A3) or directly on (A16) following the same procedure as Chaljub
(2000) for the gravity terms in the wave equation.
X a2
A P P E N D I X B : S P H E RO I DA L C O N T R I B U T I O N T O T H E S O U RC E C O R R E C T I O N
We develop here the order 1 spheroidal contribution to the correction to be applied to source if it is located in the shallow layer. Starting from
M : ∇u = M11 : 11 + M1z · (∇1 uz + ∂z u1 ) + Mzz uz ,
(B1)
and considering the following spheroidal displacement for any k and m,
uc = Ukc Pkm + Vkc Bkm ,
(B2)
we have, for any M,
Mzz ucz = ∂z Ukc Mzz Pkm · ẑ,
(B3)
1
M1z · ∇1 ucz + ∂z uc1 = ε TVc k M1z · Bkm ,
Le
(B4)
M11 : c11 = Vkc M11 : ∇1 Bkm .
(B5)
Using (54) and (55) for the spheroidal case, to the first order we find
1
(∂z Vk + kUk ) ,
Vkc = Vk − εL se X eL
(B6)
1
1
1
1
− ω2 X eρ
Vk ,
Ces ∂z Uk − ε k 2 X ea1
− Fes X ea0
TVc k = TV k − εk X ea0
(B7)
∂z Ukc =
X1
Ces
Feε − Fes
2 eρ
ε
1
∂
U
+
k
V
+
εω
Uk − εkae0
L se X eL
(∂z Vk + kUk ),
z
k
k
Ceε
Ceε
Ceε
where C εe = C ε (z e ), Ces = C s (z e ), F εe = F ε (z e ), Fes = F s (z e ),
ze − b
1
1
1
X ea0
,
= X a0
(0) − X a0
ε
1
1
1
= X a1
(0) − X a1
X ea1
(B8)
(B9)
ze − b
.
ε
(B10)
Using (20), (23), (A11), I : ∇1 Bkm = −kPkm · ẑ and combining with the toroidal solution, we finally have
M c : c = M0 : 1
1
L se Mc11 : (∂z 11 + ∇1 ∇1 uz ) + εae0 X eL
L se Mzzc ∇1 · ∂z u1 + ∇12 uz
− ε X eL
1
− ε X ea0
+ε
1
1
X eρ
X eρ
Ces 0
M1z · ∇1 ∂z uz + εω2 s M01z · u1 + εω2 s Mzz0 uz
s
Le
Le
Ce
1
1
− Fes X ea0
X ea1
X1
M01z · ∇1 (∇1 · u1 ) − ε eN
M0 · (∇1 × ∇1 × u1 ) ,
s
Le
L se 1z
(B11)
where
Mzz0 = Mzzc
C
Ces
Ceε
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Journal compilation (B12)
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Mrr0 = Mrrc + Mzzc
Fes − Feε
Ceε
0
c
Mφφ
= Mφφ
+ Mzzc
Mzr0 = Mzrc
0
c
= Mzφ
Mzφ
(B13)
Fes − Feε
Ceε
(B14)
L se
L εe
(B15)
L se
L εe
(B16)
Mr0φ = Mrcφ .
(B17)
These last expressions give the order 0 effect of the shallow layer on the source. This result is the same than the one obtain by Capdeville &
Marigo (2007). As it has been done for the toroidal case, (B11) can be rewritten as
f(x, t) = F [ f (t)δ(x − xe )],
(B18)
where the operator F can be derived from (B11).
Spectral element implementation of a source in the shallow layer is similar as the implementation of the boundary condition. It nevertheless
requires to compute the surface gradient of a vector, which, in spherical geometry using the ‘cubed sphere’ coordinate system is not completely
trivial. Useful formula for this point can be found in Choblet (2005).
APPENDIX C: EFFECT OF SPHERICITY
For the sake of simplicity, this paper has been written for the flat geometry. Nevertheless, for large applications, including the examples given
in this paper, the sphericity is important. Working in spherical geometry leads to the same generic eq. (9) with similar matrices (10) and (11)
(see Takeuchi & Saito 1972, for details):
⎡
⎤
0
d/r
1/C
el /r
⎢ −ρω2 + a
−d/r
(−aγl + 2ρgr )/2
γl /r ⎥
⎢
⎥
(C1)
⎥
s Sl (r, ω) = ⎢
⎣
−γl /r
0
2/r
1/L ⎦
−ρω2 + bl
−2/r
(aγl + 2ρgr )/2 −el /r
with
2N
γl2
F2
− 2 ,
A
−
2
r
C
r
F
2F
,
el = γl .
d = 1−
C
C
#
where g is the norm of the gravity field at the radius r and γl = l(l + 1). In the toroidal case, we have:
2/r
1/L
,
t Sl (r, ω) =
−ρω2 + l N /r 2 −2/r
a=
4
r2
A−
F2
− N − ρgr ,
C
bl =
(C2)
with l = (l − 1)(l + 2) and l the angular degree.
√
For our frequency range of interest, l is large enough such that kr l + 12 l γl , where k is the horizontal wave number used all
along this paper. Assuming the asymptotic expansion is performed close to the free surface, we have k >> r1 . Simplifying (C1) and (C2), we
find the same expression than (10) and (11) but for two terms:
[s Sk ]23 = −[s Sk ]32 −
2k
(a1 − N ),
r∂ Ω
where r ∂
is the earth radius. This requires to add ε r
(C3)
2
∂
Ω
1
[X a1
(0) − X 1N (0)] (∇1 · u1 ẑ − ∇1 uz ) to expression (A16).
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Journal compilation Geophysical Journal International
Geophys. J. Int. (2010) 181, 897–910
doi: 10.1111/j.1365-246X.2010.04529.x
1-D non-periodic homogenization for the seismic wave equation
Yann Capdeville,1 Laurent Guillot1 and Jean-Jacques Marigo2
1 Équipe
de sismologie, Institut de Physique du Globe de Paris (UMR 7154), CNRS. E-mail: [email protected]
de Mécanique des solides (UMR 7649), École Polytechnique, Palaiseau, France
2 Laboratoire
Accepted 2010 January 18. Received 2010 January 18; in original form 2009 November 11
SUMMARY
When considering numerical acoustic or elastic wave propagation in media containing small
heterogeneities with respect to the minimum wavelength of the wavefield, being able to upscale
physical properties (or homogenize them) is valuable mainly for two reasons. First, replacing
the original discontinuous and very heterogeneous medium by a smooth and more simple one,
is a judicious alternative to the necessary fine and difficult meshing of the original medium
required by many wave equation solvers. Second, it helps to understand what properties of a
medium are really ‘seen’ by the wavefield propagating through, which is an important aspect
in an inverse problem approach. This paper is an attempt of a pedagogical introduction to nonperiodic homogenization in 1-D, allowing to find the effective wave equation and effective
physical properties, of the elastodynamics equation in a highly heterogeneous medium. It
can be extrapolated from 1-D to a higher space dimensions. This development can be seen
as an extension of the classical two-scale homogenization theory applied to the elastic wave
equation in periodic media, with this limitation that it does not hold beyond order 1 in the
asymptotic expansion involved in the classical theory.
In seismology or in seismic exploration, inhomogeneities of scale
much smaller than the minimum wavelength are a challenge for both
the forward problem and the inverse problem. This introduction is
focused on the forward problem.
In recent years, advances in numerical methods have allowed
to model full seismic waveforms in complex media. Among these
advances in numerical modelling, the introduction of the Spectral Element Method (SEM) (see, for example, Priolo et al. 1994;
Komatitsch & Vilotte 1998, for the first SEM applications to the
elastic wave equation and Chaljub et al. 2007 for a review) has
been particularly interesting. This method has the advantage to be
accurate for all kinds of waves and any type of media, as long as
a hexahedral mesh, on which the method most often relies, can be
designed for a partition of the space (note the SEM can be based on
tetrahedral meshes Komatitsch et al. 2001; Mercerat et al. 2006, but
at the price of lower efficiency). This method can be very efficient,
depending on the complexity of the mesh. Nevertheless, difficulties
arise when encountering some spatial patterns quite typical of the
Earth like 0th-order discontinuities in material properties.
Time signals in seismology, recorded at some receivers after the
Earth has been excited (e.g. by some quake), have a finite frequency
support [ f min , f max ]. This finiteness can be due to the instrument
response at the receiver, to the limited frequency band of the source,
to the attenuation in the medium, or to the bandpass filtering applied to the data by a seismologist. For the wave equation, the
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existence of a frequency cut-off f max of a wavefield propagating in
a medium implies the existence of a minimum wavelength for this
wavefield except in some special locations (close to a point source
for example). The knowledge of this maximum frequency—and of
the associated, minimum wavelength—allows to efficiently solve
the wave equation with numerical techniques like the SEM, in some
complex media, at a reasonable numerical cost. Indeed, when the
medium is smoothly heterogeneous and does not contain scales
smaller than the minimum wavelength of the wavefield, the mesh
design is mostly driven by the sampling of the wavefield and the
numerical cost is minimum. On the other hand, when the medium
contains heterogeneities at a small scale, the mesh design is driven
by the sampling of these heterogeneities, and this can lead to a very
high numerical cost. Another constraint on the mesh design is that
all physical discontinuities must be matched by an element interface.
If for technical reasons, a mesh honouring all physical interfaces
cannot be designed, the accuracy of the numerical solution is not
warranted and even worse, for special waves like interface waves,
the accuracy is difficult to be predicted. In realistic 3-D media, an
hexahedral mesh design is often impossible and requires simplifications in the model structure. Despite this trick, once the mesh is
designed, its complexity again involves a very small time step to
satisfy the stability condition of the explicit Newmark time scheme
used in most cases (Hughes 1987), leading to a very high numerical
cost. This time step problem can be avoided with unconditionally
stable time schemes (Seriani 1997, 1998), but at a price of a higher
complexity and numerical cost and such schemes are not widely
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GJI Seismology
Key words: Computational seismology; Wave scattering and diffraction; Wave propagation.
898
Y. Capdeville, L. Guillot and J.-J. Marigo
used so far. Let us mention that other methods based on tetrahedral meshes (whose design is much simpler than hexahedral ones
and can be automated) like the ADER scheme (Kaser & Dumbser
2006) or SEM based on tetrahedral meshes (Komatitsch et al. 2001;
Mercerat et al. 2006) exist, are promising, but so far less competitive
than the hexahedral version of SEM.
There is an alternative way to overcome these technical problems.
It is based on simple physical considerations, and intimately connected to the reason why we are able to model seismic data quite
simply in some situations. Indeed it is known (see, for example,
Backus 1962; Chapman 2004, or the work on highly heterogeneous
media of Zhang & LeVeque 1997; Fogarty & LeVeque 1999) that
heterogeneities whose size is much smaller than the minimum wavelength, only affect the wavefield in an effective way, and this is why
simple models can be very efficient to predict data in some cases. A
pertinent example is given in global seismology: very long period
data can be predicted with a good accuracy using a simple spherically symmetric model, despite the relatively well-known complex
structure in the crust at smaller scales. Being aware of this fact, and
rather than trying to mesh details much smaller than the minimum
wavelength of a wavefield, an appealing idea is to find a smooth,
effective model (and as we shall see, an effective wave equation)
that would lead to an accurate modelling of data, without resorting
to a very fine partition of the space. The issue is then the following:
given a known acoustic or elastic model, containing heterogeneities
at scales much smaller that the minimum wavelength of a wavefield
propagating through, may one find a smooth effective model and an
effective wave equation, that reproduce the full waveform observed
in the original medium? In other words, how may one upscale the
original medium to the scale of the wavefield?
In the static case, this kind of problem has been studied for a
long time and a large number of results have been obtained using the so-called homogenization theory applied to media showing
rapid and periodical variations of their physical properties. Since
the pioneering work of Auriault & Sanchez-Palencia (1977), numerous studies have been devoted either to the mathematical foundations of the homogenization theory in the static context (e.g.
Bensoussan et al. 1978; Murat & Tartar 1985; Allaire 1992), to
applications to the effective static behaviour of composite materials
(e.g. Dumontet 1986; Francfort & Murat 1986; Abdelmoula &
Marigo 2000; Haboussi et al. 2001a,b), to the application to the
heat diffusion (e.g. Marchenko & Khruslov 2005), to porous media (e.g. Hornung 1996), etc. In contrast, fewer studies have been
devoted to the theory and its applications in the general dynamical
context or to the non-periodic cases. However, one can for example
refer to Sanchez-Palencia (1980), Willis (1981), Auriault & Bonnet
(1985), Moskow & Vogelius (1997), Allaire & Conca (1998), Fish
et al. (2002), Fish & Chen (2004), Parnell & Abrahams (2006),
Milton & Willis (2007), Lurie (2009) or Allaire et al. (2009) for
the dynamical context, to Briane (1994), Nguetseng (2003) or to
Marchenko & Khruslov (2005) for the non-periodic case. Moczo
et al. (2002) have also used a kind of local homogenization to
take into account interfaces with the finite differences method. The
specific case of a long wave propagating in finely layered media
has been studied by Backus (1962) and the same results can be
extracted from the 0th-order term of the asymptotic expansion implied in homogenization theory. Higher order homogenization in the
non-periodic case has been studied by Capdeville & Marigo (2007)
and Capdeville & Marigo (2008) for wave propagation in stratified media, but the extension to media characterized by 3-D rapid
variation is not obvious from these works. Indeed, the non-periodic
homogenization strategy suggested in Capdeville & Marigo (2007)
is based on the knowledge of an explicit solution of the cell problem
(see main text for a definition of this concept), and such a solution
only exists for layered media. The challenge is therefore to present
a non-periodic homogenization that can be extended from the 1-D
to the 2-D/3-D case.
We first recall some general features of the homogenization theory in the context of 1-D periodic media, in a slightly different
manner as done by Fish et al. (2002). Then, we generalize these results, to 1-D non-periodic media in a way that can be extended form
1-D to 2-D/3-D. Numerical convergence tests of the asymptotic,
homogenized solution towards the reference one, are performed.
Our aim is to present in this paper, a clear introduction for the 1-D
case, of the techniques and hypotheses that will be presented later
for 2-D and 3-D wave propagation problems.
2 1-D PERIODIC CASE
We consider an infinite elastic bar of density ρ 0 and elastic modulus E 0 . In this first part, it is assumed that the bar properties are
-periodic, that is ρ 0 (x + ) = ρ 0 (x) and E 0 (x + ) = E 0 (x).
The bar is considered as infinite in order to avoid the treatment
of any boundary condition that normally would be necessary in
the following development. The boundary condition problem associated with homogenization has nevertheless been addressed by
Capdeville & Marigo (2007) and Capdeville & Marigo (2008) for
layered media, and will be the purpose of future works for a more
general case. An external force f = f (x, t) is applied to the bar
inducing a displacement field u(x, t) propagating along the x axis.
We assume that f (x, t) has a corner frequency f c which allows us
to assume that a minimum wavelength λm to the wavefield u exists.
The main assumption of this section is
ε=
1
λm
(1)
which means that the size of heterogeneities in the bar is much
smaller than the minimum wavelength of the propagating wavefield.
2.1 Set up of the homogenization problem
In this section, we give an intuitive construction of the homogenization problem. For a more precise and formal set up, one can
for example refers to Sanchez-Palencia (1980). A classical homogenization problem is built over a sequence of problems obtained by
varying the periodicity . For a fixed λm , one particular bar model of
periodicity is associated to a unique ε and therefore, the sequence
of problems can be indexed by the sequence of ε. The original problem, which has a given periodicity, let say 0 , corresponding to the
parameter ε0 = 0 /λm , is met only if ε = ε0 . To the sequence of
problems corresponds a sequence of elastic and density properties
named E ε and ρ ε (and we have E ε0 = E 0 and ρ ε0 = ρ 0 ). We
assume the external source does not depend on ε. Nevertheless, in
practice, the source if often represented as a point source which
can lead to some complications. This point will be discussed and
addressed at the end of this section. We assume here that f is smooth
both in space and time. For a given ε, the equation of motion and
constitutive relation in the bar are
ρ ε ∂tt u ε − ∂x σ ε = f
(2)
σ ε = E ε ∂x u ε ,
where u ε = u ε (x, t) is the displacement along x, σ ε = σ ε (x, t) is
the stress, ∂tt u ε the second derivative of u ε with respect to time and
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899
and
∂ y h = 0.
(11)
2.2 Resolution of the homogenization problem
In the following, the time dependence t is dropped to ease the
notations.
• Eqs (7) for i = −2 and (8) for i = −1 give
Figure 1. Sketch showing the infinite periodic bar and a zoom on one
periodic cell.
∂ x the partial derivative with respect to x. We assume zero initial
conditions at t = 0 and radiation conditions at the infinity.
To explicitly take small-scale heterogeneities into account when
solving the wave equation, a fast space variable is introduced (see
Fig. 1)
x
(3)
y= ,
ε
where y is called the microscopic variable and x is the macroscopic
variable. When ε → 0, any change in y induces a very small change
in x. This leads to the separation of scales: y and x are treated
as independent variables. This implies that partial derivatives with
respect to x become
∂
∂
1 ∂
→
+
.
(4)
∂x
∂x
ε ∂y
The solution to the wave eqs (2) is sought as an asymptotic expansion
in ε
εi u i (x, x/ε, t) =
εi u i (x, y, t) ,
u ε (x, t) =
i≥0
ε
σ (x, t) =
i≥0
ε σ (x, x/ε, t) =
i
i
i≥−1
(5)
εi σ i (x, y, t) ,
i≥−1
in which coefficients u and σ depend on both space variables
x and y and must be λm -periodic in y. This ansatz—the x and y
dependence of the solution—explicitly incorporates our intuition,
that the sought solution, depends on the wavefield at the large scale,
but also, locally, on the fast variations of elastic properties. Starting
the stress expansion at i = −1 is required by constitutive relation
between the stress and the displacement and the 1/ε in (4).
We introduce ρ and E
i
i
ρ(y) = ρ ε (εy) ,
E(y) = E ε (εy) ,
(6)
ρ∂tt u i − ∂x σ i − ∂ y σ i+1 = f δi,0 ,
(7)
σ i = E ∂x u i + ∂ y u i+1 ,
(8)
where δ i,0 takes for value 1 for i = 0 and 0 otherwise. These last
equations have to be solved for each i. Before going further, we
introduce the cell average, for any function h(x, y) λm -periodic in y
λm
1
h(x, y)dy .
(9)
h(x) =
λm 0
For any function h(x, y), λm -periodic in y, it can easily be shown
that
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σ −1 = E∂ y u 0 ,
which implies
∂ y E∂ y u 0 = 0.
(13)
0
Multiplying the last equation by u , integrating over the unit cell,
using an integration by part and taking account of the periodicity of
u 0 and E∂ y u 0 , we get
λm
0 2
∂ y u E dy = 0.
(14)
0
E(y) being a positive function, the unique solution to the above
equation is ∂ y u 0 = 0. We therefore have
u 0 = u 0 ,
(15)
σ −1 = 0.
(16)
Eq. (15) implies that the order 0 solution in displacement is independent on the fast variable y. This is an important result that
confirms the well known fact that the displacement field is poorly
sensitive to scales much smaller than its own scale.
• Eqs (7) for i = −1 and (8) for i = 0 give
∂ y σ 0 = 0,
(17)
σ 0 = E ∂ y u 1 + ∂x u 0 .
(18)
Eq. (17) implies that σ (x, y) = σ (x) and, with (18), that
∂ y E∂ y u 1 = −∂ y E ∂x u 0 .
0
0
(19)
Using the linearity of the last equation we can separate the variables
and look for a solution of the form
u 1 (x, y) = χ 1 (y)∂x u 0 (x) + u 1 (x),
(20)
where χ (y) is called the first-order periodic corrector. To enforce
the uniqueness of the solution, we impose χ 1 = 0. Introducing
(20) into (19), we obtain the equation of the so-called cell problem
(21)
∂y E 1 + ∂y χ 1 = 0 ,
1
the unit cell elastic modulus and density. E and ρ are independent of
ε and are λm -periodic. Introducing expansions (5) in eqs (2), using
(4) and identifying term by term in εi we obtain:
∂ y h = 0 ⇔ h(x, y) = h(x),
∂ y σ −1 = 0 ,
(10)
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χ 1 being λm -periodic and verifying χ 1 = 0. It is useful to note
that a general analytical solution to (21) exists and is
y
1
dy + b .
(22)
χ 1 (y) = −y + a
0 E(y )
The periodicity condition imposes
−1
1
a=
,
E
and b can be found using χ 1 = 0. We therefore have
−1
1
1
.
∂ y χ 1 (y) = −1 +
E
E(y)
(23)
(24)
900
E ρ∗ = E∂ y χ ρ .
Introducing (20) into (18), taking the cell average and using the fact
that we have shown that σ 0 does not depend upon y, we find the
order 0 constitutive relation,
σ 0 = E ∗ ∂x u 0 ,
(25)
where E ∗ is the order 0 homogenized elastic coefficient,
E ∗ = E 1 + ∂y χ 1 .
(26)
Using (24) in the last equation we have
−1
1
.
E∗ =
E
(39)
The periodicity condition on χ imposes ∂ y χ = −χ and therefore
E 1∗ = 0.
Finally using (20) and taking the average of eqs (7) for i = 1
gives the order 1 wave equation
2
2
1
ρ∂tt u 1 + ρχ 1 ∂x ∂tt u 0 − ∂x σ 1 = 0
σ 1 = E ∗ ∂x u 1 + E ρ∗ ∂tt u 0 .
(40)
It is shown in the Appendix that E ρ∗ = ρχ 1 and therefore, renaming σ̃ 1 = σ 1 − E ρ∗ ∂tt u 0 , the last equations can be simplified
to
(27)
• Eqs (7) for i = 0 and (8) for i = 1 give
ρ∂tt u 1 − ∂x σ̃ 1 = 0
ρ∂tt u 0 − ∂x σ 0 − ∂ y σ 1 = f,
(28)
σ 1 = E ∂ y u 2 + ∂x u 1 .
(29)
σ̃ 1 = E ∗ ∂x u 1 .
We stop here our resolution but we could go up to a higher order
(see Fish et al. 2002, for a 1-D periodic case up to the order 2).
Applying the cell average on (28), using the property (11), the fact
that u 0 and σ 0 do not depend on y and gathering the result with (25),
we find the order 0 wave equation
ρ ∗ ∂tt u 0 − ∂x σ 0 = f
2.3 Combining all orders together
Our aim is to solve for the homogenized wave equation using numerical methods like the Spectral Element Method (SEM). For such
a method it is convenient to combine all the orders together rather
than solving each order one after another. For that purpose, we solve
for û ε,1 solution of
(30)
σ 0 = E ∗ ∂x u 0 ,
(41)
where ρ ∗ = ρ is the effective density and E ∗ is defined by eq. (27).
This is the classical wave equation that can be solved using classical
techniques. Knowing that ρ ∗ and E ∗ are constant, solving the wave
equation for the order 0 homogenized medium is a much simpler
task than for the original medium and no numerical difficulty related
to the rapid variation of the properties of the bar arises. One of
the important results of homogenization theory is to show that u ε
‘converges’ to u 0 when ε tends towards 0 (the so-called convergence
theorem, see Sanchez-Palencia 1980).
Once u 0 is found, the first-order correction, χ 1 (x/ε) ∂ x u 0 (x), can
be computed. To obtain the complete order 1 solution u 1 using (20),
u 1 remains to be found. Subtracting (30) from (28) we have,
ρ∂tt û ε,1 − ∂x σ̂ ε,1 = f
(42)
σ̂ ε,1 = E ∗ ∂x û ε,1 .
(43)
One can check that
û ε,1 = u 0 + εu 1 + O(ε2 ) ,
(44)
σ̂ ε,1 = σ 0 + εσ̃ 1 + O(ε2 ) .
(45)
(46)
(47)
∂ y σ 1 = (ρ − ρ)∂tt u 0 ,
(31)
which, together with (29) and (20) gives
∂ y E∂ y u 2 = −∂ y E∂x u 1 + (ρ − ρ)∂tt u 0 ,
Furthermore, if we name
x ∂x û ε,1 û ε,1 = 1 + εχ 1
ε
we can show that
(32)
û ε,1 = u ε + O(ε2 ) .
(33)
Higher order terms in the asymptotic expansion can be added, as it
will be shown below for the partial order 2.
= −∂ y E ∂x u 1 − ∂ y (Eχ 1 )∂x x u 0 + (ρ − ρ)∂tt u 0 .
2.4 External point sources
u 2 (x,y) = χ 2 (y)∂x x u 0 (x) + χ 1 (y)∂x u 1 (x) + χ ρ (y)∂tt u 0 + u 2 (x),
In practice, the external source is often localized to an area much
smaller than the smallest wavelength λm allowing to consider it
ideally as a point source: f (x, t) = g(t)δ(x − x 0 ). Two potential
issues then arise:
(34)
where χ 2 and χ ρ are solutions of
∂y E χ 1 + ∂y χ 2 = 0 ,
∂ y E∂ y χ
ρ
(35)
= ρ − ρ ,
ρ
(i) In the vicinity of x 0 , there is no such a thing as a minimum
wavelength. The asymptotic development presented here is therefore only valid far away enough from x 0 .
(ii) A point source has a local interaction with the microscopic
structure that needs to be accounted for.
(36)
ρ
with χ and χ λm -periodic and where we impose χ = χ = 0
to ensure the uniqueness of the solutions. Introducing (34) into (29)
and taking the cell average, we find the order 1 constitutive relation
2
σ 1 = E ∗ ∂x u 1 + E 1∗ ∂x x u 0 + E ρ∗ ∂tt u 0
2
The first point is not really a problem because, for most realistic
cases, the point source assumption is not valid in the near field
anyway. One should nevertheless keep in mind that very close to
x 0 , which means closer than λm , the solution is not accurate but
not less than any standard numerical methods used to solve the
(37)
with
E 1∗ = E χ 1 + ∂ y χ 2
(38)
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homogenized solutions. Once the simulation is done, for a given
time step corresponding to t = 4.9 × 10−4 s, the complete order
1 solution can be computed with (46). We can also compute the
incomplete homogenized solution at the order 2
x x x ∂x + ε 2 χ 2
∂x x + ε 2 χ ρ
∂tt
û ε,3/2 = 1 + εχ 1
ε
ε
ε
(52)
×û ε,1 ,
12
10^3 kg/m^3, km/s
11
10
9
8
7
6
5
4
0
0.01
0.02
0.03
x (m)
0.04
0.05
Figure 2. Sample (5 cm) of the bar density (grey line, in 103 kg m−3 ) and
velocity (black line, in km s−1 ) for l 0 =6 mm.
wave equation. The second point is more important and can be
addressed the following way: the hypothetical point source is just
a macroscopic representation of a more complex physical process,
and what is relevant is to ensure the conservation of the energy
released at the source. Therefore we need to find an effective source
f̂ ε,1 that preserves the energy associated to the original force f up
to the wanted order (here 1). We therefore need
(u ε , f ) = (û ε,1 , f̂ ε,1 ) + O(ε2 ) ,
(48)
2
where (. , .) is the L inner product, and, for any function g and h is
(g, h) =
g(x)h(x)dx .
(49)
R
Using (47), (46) and an integration by part, we find
x f̂ ε,1 (x, t) = 1 − εχ 1
∂x f (x, t) .
ε
(50)
f̂ ε,1 needs to be used in (42) instead of f .
2.5 A numerical experiment for a periodic case
A numerical experiment in a bar of periodic properties shown in
Fig. 2 is performed. The periodicity of the structure is l 0 = 6 mm.
First, the cell problems (21), (35) and (36) with periodic boundary
conditions are solved with a finite element method based on the same
mesh and quadrature than the one that will be used to solve the wave
equation. This is not necessary, but for this simple 1-D case, it is
a convenient solution. This allows to compute E ∗ , the correctors
χ 1 , χ 2 and χ ρ as well as the external source term f̂ ε,1 . Then, the
homogenized wave equation,
ρ 0 ∂tt u − ∂x (E ∗ ∂x u) = f ,
ε,1
(51)
ε,1
where u = û and f = f̂ , is solved using the SEM (see
Capdeville 2000, for a complete description of the 1-D SEM).
A point source is located at x = 2 m. The time wavelet g(t)
is a Ricker with a central frequency of 50 kHz (which gives
a corner frequency of about 125 kHz) and a central time shift
t 0 = 6.4 × 10−5 s. In the far field, this wavelet gives a minimum
wavelength of about 4 cm; that corresponds to a wave propagation
with ε = 0.15. In practice the bar is of course not infinite, but its
length (5 m) and the time (4.9 × 10−4 s) at which is recorded the
displacement is such that the wave pulse does not reach the extremity of the bar. To be accurate, the reference solution is computed
with a SEM mesh matching all interfaces with an element boundary
(7440 elements for the 5 m bar). To make sure that only the effect of
homogenization is seen in the simulations, the mesh and time step
used to compute the reference solution are also used to compute the
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where û ε,3/2 is an incomplete order 2 solution and the ‘3/2’ is just a
notation to indicate that it is the order 1 plus second order correction
(‘1/2 order 2’). It is incomplete because u 2 has not been computed
and is missing in (52) to obtain û ε,2 . In other words, û ε,3/2 only
contains the order 2 periodic correction beyond the order 1 solution.
In Fig. 3 are shown the results of the simulation. On the upper
left plot (Fig. 3a) are shown the reference solution (bold grey line),
the order 0 solution (black line) and a solution obtained in the
bar with a E ∗ = E 0 (‘E average’, dashed line) for t = 4.9 ×
10−4 s as a function of x. As expected, the ‘E average’ solution
is not in phase with the reference solution and shows that this
‘natural’ filtering is not accurate. On the other hand, the order 0
homogenized solution is already in excellent agreement with the
reference solution. On Fig. 3(b) is shown the residual between the
order 0 homogenized solution and the reference solution û 0 (x, t) −
u ε (x, t). The error amplitude reaches 2 per cent and contains fast
variations. On Fig. 3(c) is shown the order 1 residual û 1 (x, t) −
u ε (x, t) (bold grey line) and the partial order 2 residual û ε,3/2 (x, t)−
u ε (x, t) (see eq. 52). It can be seen, comparing Figs 3(b) and (c),
that the order 1 periodic corrector removes most of the fast variation
present in the order 0 residual. The remaining fast variation residual
disappears with the partial order 2 residual. The smooth remaining
residual is due to the u 2 that is not computed. In order to check that
this smooth remaining residual is indeed an ε 2 residual, this residual
computed for ε = 0.15 is overlapped with a residual computed for
ε = 0.075 (which corresponds to l 0 = 3 mm) to which is applied
a factor 4 in amplitude. The fact that these two signals overlap is
consistent with a ε2 residual.
3 NON-PERIODIC CASE
We now give up the hypothesis of periodicity of E 0 and ρ 0 and consider more complex spectra for the size of the heterogeneities. As a
specific case, in the following, the bar properties in each cell are now
generated randomly around a constant mean value. Homogenization of random structures as studied by Papanicolaou & Varadhan
(1979) is not our purpose and the problem is considered as deterministic: the bar properties are completely known and unique for
each bar under study. An example of such bar is given in Fig. 4. We
still assume a minimum wavelength λm (or a maximum wavenumber km = 1/λm ) for the wavefield u, far enough from the source. It
is still reasonable to expect, to some sense, that heterogeneities in
the bar, whose size is much smaller than λm have a little influence
on the wavefield u and that an homogenization procedure can be
performed.
3.1 Preliminary: an intuitive solution
The first idea one can have is to consider the whole non-periodic
bar as a single periodic cell and apply results obtained in the previous section. The obtained
T effective medium has a constant density (ρ ∗ = limT →∞ 2T1 −T ρ 0 (x)dx) and elastic modulus ( E1∗ =
T
limT →∞ 2T1 −T E10 (x)dx) as shown in Fig. 4 in dashed line. Fig. 5,
902
1
0.02
a
b
reference
order 0
E average
0.01
0
0
-1
2
3
4
5
c
0.006
-0.01
4
0.006
0.004
0.004
0.002
0.002
0
0
-0.002
-0.002
-0.004
4.1
4.2
4.1
x (m)
4.2
d
-0.004
4.05
4.1
x (m)
4.15
4
Figure 3. (a) Grey line: displacement u ε (x, t) at t = 4.9 × 10−3 s computed in the reference model described Fig. 2. Black line: the order 0 homogenized
solution û 0 (x, t). Dashed line: solution computed in a model obtained by averaging the elastic properties (ρ 0 and E 0 ). (b) Order 0 residual, û 0 (x, t)−u ε (x, t).
(c) Grey line: order 1 residual, û ε,1 (x, t) − u ε (x, t). Black line: partial order 2 residual, û ε,3/2 − u ε (x, t) (see eq. 52) (d) Grey line: partial order 2 residual for
ε = 0.15. Black line: partial order 2 residual for ε = 0.075 with amplitude multiplied by 4.
into details, and given km , we can indeed distinguish three different
propagation regimes (see Aki & Richards 1980), as follows.
10
km/s
• Heterogeneities for which k 0 km (scale of the heterogeneities
much smaller than the wavelength), where the highly heterogeneous
medium can be considered as a homogeneous body with effective
elastic properties; this will be the domain of application of the
homogenization.
• Heterogeneities for which k 0 km (scale of the heterogeneities
much larger than the wavelength); the medium then can be considered as a smoothly-varying body.
• Heterogeneities for which k 0 ≈ km . The inhomogeneity scale
is comparable to wavelength; this is the domain where coda waves
do exist.
8
0
0.1
0.05
x (m)
The difficulty we are facing, is then to separate these scales correctly
in order to homogenize both elastic properties and the wave equation. Remember that this is not so obvious: in the periodic case, the
physical quantity to be manipulated, was {not} the elastic constant
— but its inverse.
To be able to separate wavenumber above k 0 from that one below k 0 , we introduce a mother filter wavelet w(x) (see Fig. 6). w
is normalized such that R w(x)dx = 1. When convolved with any
function, w acts as a low-pass spatial filter of corner spatial frequency 1. We define wk0 (x) = k0 w(xk0 ) the same but contracted
(if k 0 > 1) wavelet of corner spatial frequency k 0 . We still have
w (x)d x = 1. This allows to define a ‘filtering operator’, for
R k0
any function h(x)
h(x )wk0 (x − x )dx .
(53)
F k0 (h) (x) =
Figure 4. Black line: 10cm sample of the velocity c = E 0 /ρ 0 in a nonperiodic bar. Dashed line: ‘periodic’ homogenized
velocity if the whole
0
0
bar is considered has a single periodic cell (c =
E /ρ ). Grey line:
velocity obtained with the spatial filtering (c = F k0 (E 0 )/F k0 (ρ 0 )).
left-hand plot, shows a result obtained in such a medium compared to a reference solution computed in the original bar. The
direct arrival is acceptable, but the coda wave corresponding to
waves trapped in the heterogeneous medium can not be captured
with this simple constant homogenized medium. The waving-hand
explanation to that problem is that the wavefield interacts with heterogeneities of the medium whose wavenumbers are as high as
k 0 where k 0 = 1/λ0 is a wavenumber somewhat larger than the
maximum wavenumber of the wavefield, km . To be accurate, the effective medium should keep information up to k 0 . Without getting
R
k0
F (h) (x) is a smooth version of h where all wavenumbers larger
than k 0 have been muted to zero. If we apply this filter to 1/E 0 and
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1
1
reference
F(1/E)
reference
<1/E>
0
-1
903
0
0
0.1
time (ms)
0.2
-1
0
0.1
time (ms)
0.2
Figure 5. Examples of seismogram computed in bar models presented on Fig. 4. The distance between the source and the receiver is 1 m. The minimum
wavelength is 10 cm. Left-hand plot: grey line: reference solution computed in the original bar model; black line: solution obtained in the ‘periodic’ homogenized
model (ρ ∗ = ρ 0 , 1/E ∗ = 1/E 0 represented in dashed line Fig. 4). Right-hand plot: grey line: reference solution computed in the original bar model; black
line solution obtained by using 1/E ∗ (x) = F k0 (1/E 0 ) and ρ ∗ (x) = F k0 (ρ 0 ) as an effective medium (represented in grey line Fig. 4).
1
-2
0
x (m)
2
0
0
1
k (1/m)
Figure 6. Example of mother filter wavelet w(x) used in practice. On one hand, the cut-off spatial frequency is around 1 but is not sharp and on the other hand,
the spatial support can be considered as finite with a good approximation.
ρ 0 and use
−1
1
E ∗ (x) = F k0
(x)
E0
(54)
and ρ ∗ (x) = F k0 (ρ 0 )(x) we obtain the smooth medium (Fig. 4,
grey line) and the seismogram obtained in such a medium includes
the coda waves (Fig. 5, right-hand plot). This intuitive construction
of the effective medium seems convenient, but we now need to
obtain this result more formally. To do so, one need to construct
the sequence of models (E ε , ρ ε ) and fast parameters [E(y), ρ(y)]
from (E 0 , ρ 0 ) which is not as straightforward as for the periodic
case. Nevertheless, both are necessary to build the homogenization
asymptotic expansion. The idea here is to define and keep all the
bar properties with wavelength greater than λ0 and to homogenize
all wavelength λ smaller than λ0 using a spatial filtering similar to
(53).
3.2 Set up of the homogenization problem for
the non-periodic case
We first introduce the small parameter
ε=
λ
,
λm
(55)
where λ is a spatial length or a scale. For the periodic case, λ
would be 0 , the periodicity of the model. Because we are not in the
periodic case, another parameter is required
ε0 =
C
λ0
,
λm
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Journal compilation (56)
129
where λ0 is the user defined scale below which scales are considered
as small scale (microscopic) and above which scales are considered
as large scale (macroscopic). While ε is a formal parameter that
could be used to show a convergence theorem, ε 0 is a parameter
that indicates the degree of smoothness of the homogenized model
and accuracy of the homogenized solution (a small ε 0 corresponds
to a homogenized model with a lot of details and a precise solution; a large ε0 corresponds to a smooth homogenized model and
an imprecise solution). With that respect, the non-periodic case is
different from the periodic case. Indeed, for the periodic case no
parameter choice is left to the user, and the accuracy if fixed by the
frequency cut-off of the source and the geometry of the elastic structure. For the non-periodic case, we shall see that the introduction
of the second small parameters ε0 allows to specify the accuracy of
the solution independently from the geometry of the elastic model.
In the following we assume ε ≤ ε0 1. As for the periodic case,
we work at λm fixed. Therefore, a given spatial wavelength λ or
wavenumber k = 1/λ fully defines ε = λ/λm = 1/(kλm ). We define wm (y) = km w(ykm ). We assume that wm support in the space
domain is contained in [−α/km , + α/km ] and α is a positive number that depends upon the specific design of w (as wm has a finite
support in the frequency domain, it can not have one in the space
domain and α → ∞. Nevertheless, in practice, we assume that w
is designed in such a way that the support of wm can be considered
as finite is a good approximation and that a reasonably small α can
be found).
Let Yx = [x/ε0 − β/km , x/ε0 + β/km ] be a segment of R where
β is a positive number (much) larger than α. Yx is the sampling area
around x. We define T = {h(x, y) : R2 → R , 2βλm -periodic iny}
the set of functions defined in y on Y 0 and extended to R by
904
periodicity. We define the filtering operator, for any function h ∈ T
F (h) (x, y) =
h(x, y )wm (y − y )dy .
(57)
Unfortunately, the product of two functions of V is not in V unless
these two functions are periodic with the same periodicity smaller
than λm (and well chosen β as already mentioned).
In this section and the next one, we assume that we have been
able to define [ρ ε0 (x, y), E ε0 (x, y)] in T that set up a sequence of
parameters
x
,
ρ ε0 ,ε (x) ≡ ρ ε0 x,
(64)
ε
x
ε0
ε0 ,ε
,
x,
E (x) ≡ E
ε
R
F is a linear operator. For a perfectly sharp cut-off low-pass filter
in the wavenumber domain, we have, for any h
F [F (h)] = F (h) .
(58)
The last property is not exactly true in practice because, in order to
have a compact support for wm , we do not use a sharp cut-off in the
wavenumber domain. We nevertheless assume that (58) is true.
Finally let V, be the set of functions h(x, y) such that, for a given
x, the y part of h is periodic and contains only spatial frequencies
higher than km , plus a constant value in y
V = {h ∈ T /F (h) (x, y) = h(x)} ,
where
1
h(x) =
2βλm
βλm
−βλm
and that, with such a set of parameters, a solution to the problem
described below exists. This assumption is by far not obvious and the
construction of such a [ρ ε0 (x, y), E ε0 (x, y)] from [ρ 0 (x), E 0 (x)],
which is the critical point of this paper, is left for Section 3.4.
We look for the solution of the following wave equations
(59)
ρ ε0 ,ε ∂tt u ε0 ,ε − ∂x σ ε0 ,ε = f ,
h(x, y)dy ,
(60)
To solve this problem, the fast space variable y defined (3) is once
again used and, in the limit ε → 0 x and y are treated as independent
variables which implies the transformation (4).
The solution to the wave eqs (65) is again sought as an asymptotic
expansion in ε, but this time we look for u ε0 ,i and σ ε0 ,i in V
is still the y average of h(x, y) over the periodic cell. In other words,
V is the set of functions that present only fast variations plus a
constant value in y. An example of a function h in V is given in
Fig. 7. For any periodic function g with a periodicity smaller than
λm , choosing β such that an integer number of periodicity fits in Y 0 ,
we have F (g) = g and therefore g belongs to V. This implies that
the periodic case is a particular case of the following development.
As the periodicity has been kept, we still have
∀h ∈ V, ∂ y h = 0 ,
(65)
σ ε0 ,ε = E ε0 ,ε ∂x u ε0 ,ε .
u ε0 ,ε (x, t) =
(61)
σ ε0 ,ε (x, t) =
and
∞
εi u ε0 ,i (x, x/ε, t) =
i=0
∞
εi σ ε0 ,i (x, x/ε, t) =
i=−1
∀h ∈ V, ∂ y h = 0 ⇔ h(x, y) = h(x) .
∞
εi u ε0 ,i (x, y, t) ,
i=0
∞
(66)
εi σ ε0 ,i (x, y, t) .
i=−1
ε0 ,i
ε0 ,i
Note that imposing u
and σ
in V is a strong condition that
mainly means that only slow variations must appear in x and only
fast in y. Introducing expansions (66) in the wave eqs (65) and using
(4) we obtain
(62)
It can be shown that, for any function h ∈ V, its derivative is also in
V, and that
y
h(x, y )dy ∈ V .
(63)
∀h ∈ V with h = 0 ⇒ g(x, y) ≡
ρ ε0 ∂tt u ε0 ,i − ∂x σ ε0 ,i − ∂ y σ ε0 ,i+1 = f δi,0 ,
0
a
(67)
b
0
0.5
c
0
50
100
50
k (1/m)
100
d
0
0.5
0
y (m)
Figure 7. Example of a function h(x, y) ∈ T but not in V (graph a) and h(x, y) ∈ V (graph c) for a km = 16 m−1 plotted for a given x as a function of y and
their respective power spectra (graphs b and d) for positive wavenumber (k). It can be seen that, for h(x, y) ∈ V the power spectrum is 0 is the range [0 m−1 ,
16 m−1 ]. Both functions are periodic with a periodicity of 0.5 m.
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σ ε0 ,i = E ε0 ∂x u ε0 ,i + ∂ y u ε0 ,i+1 ,
(68)
which need to be solved for each i ≥ −2 and i ≥ −1, respectively.
905
We can once again use the linearity of the last equation and we can
separate the variables and look for a solution of the form
u 2 (x, y) = χ ε0 ,2 (x, y)∂x x u 0 (x) + χ ε0 ,2x (x, y)∂x u 0 (x)
+ χ ε0 ,1 (x, y)∂x u 1 (x) + χ ε0 ,ρ (x, y)∂tt u 0 + u 2 (x) ,
(81)
We follow the same procedure as for the periodic case. We work at ε 0
fixed and, to ease the notations, the ε 0 superscript is only kept for the
bar properties and correctors, but dropped for u ε0 ,i , σ ε0 ,i . Because
the y periodicity is kept in V, the resolution of the homogenized
equations is almost the same as in the periodic case.
• As for the periodic case, eqs (67) for i = −2 and (68) for
i = −1 gives σ −1 = 0 and u 0 = u 0 .
• Eqs (67) for i = −1 and (68) for i = 0 implies σ 0 = σ 0 and
∂ y E ε0 ∂ y u 1 = −∂ y E ε0 ∂x u 0 .
(69)
u 1 (x, y) = χ ε0 ,1 (x, y)∂x u 0 (x) + u 1 (x) .
(70)
As u 1 ∈ V and u 0 = u 0 , χ ε0 ,1 must lie in V and satisfies
∂ y E ε0 1 + ∂ y χ ε0 ,1 = 0 ,
(71)
with periodic boundary conditions. We impose χ ε0 ,1 (x) = 0. A
solution in V to the last equation exists only if E ε0 have been
correctly build, that is, using the general solution (22), 1/E ε0 must
lie in V. If this condition is met, χ ε0 ,1 (x, y) is in V and
1 −1
1
.
(72)
(x) ε
∂ y χ ε0 ,1 (x, y) = −1 +
E ε0
E 0 (x, y)
As for the periodic case, we find the order 0 constitutive relation
σ 0 (x) = E ε0 ∗ (x)∂x u 0 (x) ,
(73)
E ε0 ∗ (x) = E ε0 1 + ∂ y χ ε0 ,1 (x) ,
=
1
E ε0
−1
(x) .
(74)
(75)
ρ ∂tt u 0 − ∂x σ 0 − ∂ y σ 1 = f ,
(76)
σ 1 = E ε0 ∂ y u 2 + ∂x u 1 .
(77)
To be able to obtain σ 1 in V, (76) implies that ρ ε0 must lie in V.
Taking the average on (76) together with (73) allows to find the
order 0 wave equation
ρ ε0 ∂tt u 0 − ∂x σ 0 = f
(78)
σ 0 = E ε0 ∗ ∂x u 0 .
(79)
Subtracting (78) from (76), together with (77) gives
∂ y E ε0 ∂ y u 2 = −∂ y E ε0 ∂x u 1 − ∂ y E ε0 χ ε0 ,1 ∂x x u 0
−∂ y E ε0 ∂x χ ε0 ,1 ∂x u 0 + (ρ ε0 − ρ ε0 )∂tt u 0 .
(80)
2010 The Authors, GJI, 181, 897–910
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Journal compilation ε0 ,ρ
(82)
∂ y E ε0 ∂x χ ε0 ,1 + ∂ y χ ε0 ,2x = 0 ,
(83)
∂ y E ε0 ∂ y χ ε0 ,ρ = ρ ε0 − ρ ε0 ,
ε0 ,2
ε0 ,2x
ε0 ,ρ
(84)
ε0 ,2
131
ε0 ,2x
ε0 ,ρ
and χ
in V and χ = χ
= χ =
with χ , χ
0. An important point here is to check that it exists a solution to
(82) in V. The general solution of (82) is
y
y
1
(x,
y
)dy
−
χ ε0 ,1 (x, y )dy + b .
χ ε0 ,2 (x, y) = a
ε0
0 E
0
(85)
The periodic condition
and χ ε0 ,1 = 0 give a = 0. We therefore
y
have χ ε0 ,2 = − 0 χ ε0 ,1 (x, y )dy + b and, thanks to (63), χ ε0 ,2
is indeed in V. Similarly, it can be shown that χ ε0 ,2x is in V. On
the other hand, in general, χ ε0 ,ρ is not in V as the product of two
functions in V is not in V. Nevertheless, in the periodic case with a
periodicity smaller than 1/k 0 , the product of two functions are still
periodic with the same periodicity and therefore belongs to V. In
that case, χ ε0 ,ρ is indeed in V .
Introducing (81) into (77) and taking the average, we find the
order 1 constitutive relation
σ 1 = E ε0 ∗ ∂x u 1 + E ε0 ,1x∗ ∂x u 0 + E ε0 ,1∗ ∂x x u 0 + E ε0 ,ρ∗ ∂tt u 0 (86)
with
E ε0 ,1∗ = E ε0 χ ε0 ,1 + ∂ y χ ε0 ,2 ,
(87)
= E ε0 ∂x χ ε0 ,1 + ∂ y χ ε0 ,2x ,
(88)
E ε0 ,ρ∗ = E ε0 ∂ y χ ε0 ,ρ .
(89)
ε0 ,2
• Eqs (67) for i = 0 and (68) for i = 1 give
ε0
ε0 ,2x
and χ
are solutions of
where χ , χ
ε ε ,1
ε
,2
∂y E 0 χ 0 + ∂y χ 0
= 0,
E ε0 ,1x∗
with
C
ε0 ,2
ε0 ,2
in V and χ = 0 impose
As we have already seen, χ
∂ y χ ε0 ,2 = −χ ε0 ,1 and therefore E ε0 ,1∗ = 0. Similarly, we have
E ε0 ,1x∗ = 0.
Finally, using (70) and taking the average of eqs (67) for i = 1
gives the order 1 wave equation
ρ ε0 ∂tt u 1 + ρ ε0 χ ε0 ,1 ∂x ∂tt u 0 − ∂x σ 1 = 0
σ 1 = E ε0 ∗ ∂x u 1 + E ρ,ε0 ∗ ∂tt u 0 .
(90)
In general, ρ ε0 χ ε0 ,1 is not in V, but we still have E ρ,ε0 ∗ = ρ ε0 χ ε0 ,1 and similarly to the periodic case, the last equation can be rewritten
as
(ρ ε0 + ∂x E ρ,ε0 ∗ )∂tt u 1 − ∂x σ̃ 1 = 0
σ̃ 1 = E ε0 ∗ ∂x u 1 ,
(91)
where σ̃ 1 = σ 1 − E ρ,ε0 ∗ ∂tt u 0 .
As we have seen earlier, in general ρ ε0 χ ε0 ,1 and χ ε0 ,ρ are not in V
which means that, in general, the whole non-periodic development
presented is only valid for the order 0 and the first order corrector.
It is valid for higher order only if ρ ε0 has no fast variation or for
periodic variations. Nevertheless, in practice , χ ε0 ,ρ and ρ ε0 χ ε0 ,1 are
very close to be in V and the whole development can be used as we
will see in the non-periodic example.
906
As for the periodic case, the different orders can be combined as
shown in Section 2.3.
but it can be considered as a 1-D case). The direct solution explained
above to build E ε0 is therefore not available for higher dimensions
(it still is for ρ ε0 ). We propose here a procedure that gives a similar
result as the explicit construction without the knowledge that the
construction should to be done on 1/E 0 . The main interest of the
procedure is it can be generalized to a higher space dimension. It
is based on the work of Papanicolaou & Varadhan (1979) on the
homogenization for random media. They suggest to work with the
gradient of correctors rather than to work on corrector directly. If
we name
3.4 Construction of Eε0 and ρ ε0
We present here two ways of building E ε0 and ρ ε0 in T with the
following constraints obtained in the previous sections:
(i) ρ ε0 and χ ε0 must lie in V (see eqs 76 and 72).
(ii) ρ ε0 and E ε0 must be positive functions.
(iii) ρ ε0 (x, x/ε0 ) = ρ 0 (x) and E ε0 (x, x/ε0 ) = E 0 (x).
The first constraint is necessary to obtain solutions in V at least
up to the order 1.
(98)
H ε0 (x, y) = E ε0 (x, y)G ε0 (x, y) ,
(99)
a solution to our problem in V up to the order 1 is found if we can
build E ε0 (x, y) such that (H ε0 , G ε0 ) ∈ V and G ε0 = 1.
To do so, we propose the following procedure:
3.4.1 Direct construction
The 1-D case is interesting because it gives an explicit formula for
χ ε0 , and implies that 1/E ε0 should be in V (constraint (i)) so that a
solution to the non-periodic homogenized problem exists. Thanks
to this explicit constraint, we can propose, for a given x and any y
∈ Yx ,
ρ ε0 (x, y) = F k0 (ρ 0 )(x) + [ρ 0 − F k0 (ρ 0 )](ε0 y) ,
G ε0 = ∂ y χ ε0 ,1 + 1 ,
(i) Build a start E sε0 defined as, for a given x and for any y ∈
Yx , E sε0 (x, y) = E 0 (ε0 y) and then extended to R in y by periodicity
(E sε0 is therefore in T ). Then solve (71) with periodic boundary
conditions on Yx to find χsε0 ,1 (x, y).
(ii) Compute G εs 0 = ∂ y χsε0 ,1 + 1, then Hsε0 (x, y) =
E sε0 (x, y)G εs 0 (x, y) and finally
(92)
−1
1
1
1
k0
(x)
+
(ε
E ε0 (x, y) = F k0
−
F
y)
,
0
E0
E0
E0
G ε0 (x, y) =
ε
1
ε
G s 0 − F G εs 0 (x, y) + 1 ,
F G s 0 (x, x/ε0 )
(93)
and then extended to R in y by periodicity. ρ ε0 and E ε0 are by
construction in T . Thanks to the fact that, for any h
F k0 F k0 (h) = F k0 (h) ,
(94)
H ε0 (x, y) =
it can be checked that ρ ε0 and 1/E ε0 are in V (which is not the case
of E ε0 ). Note that this is not completely true in practice because the
filter w has not a sharp cut-off, which implies that (94) is not fully
accurate. We consider this side effect as being negligible. We define
x
,
(95)
ρ ε0 ,ε (x) = ρ ε0 x,
ε
E ε0 ,ε (x) = E ε0 x,
x
ε
,
F
×
(100)
1
ε G s 0 (x, x/ε0 )
Hsε0 − F Hsε0 (x, y) + F Hsε0 (x, x/ε0 ) .
(101)
At this stage, we have (H ε0 , G ε0 ) ∈ V 2 and G ε0 = 1.
(iii) From (99) and (74), we have
E ε0 (x, y) =
E
(96)
∗,ε0
Hsε0
ε (x, y) ,
G s0
(102)
F Hsε0
(x) = H (x) = ε0 (x, x/ε0 ) .
F Gs
ε0
(103)
(iv) Once E ε0 (x, y) is known, follow the whole homogenization
procedure to find the different correctors can be pursued.
and one can check that ρ ε0 ,ε0 = ρ 0 and E ε0 ,ε0 = E 0 .
For most standard applications, ρ ε0 and E ε0 are positive functions
for any filter wavelet w. Nevertheless, for some extreme cases (e.g. a
single discontinuity with several orders of magnitude of elastic
modulus contrast), some filter wavelet w designs could lead to a
negative E ε0 . In such an extreme case, on should make sure that the
design of w allows ρ ε0 and E ε0 to be positive functions.
Finally, we can check that
1
1
1
k0
=
=
F
(x) ,
(97)
E ε0 ,∗
E ε0
E0
Once again, we insist on the fact that the main interest of this
procedure is that obtaining an explicit solution to the cell problem
is not required and it can be extended to 2-D or 3-D.
Remarks
• In practical cases, the bar is finite and Yx can be chosen to enclose the whole bar. In that case, the dependence to the macroscopic
location x in χsε0 ,1 , G εs 0 , Hsε0 and E sε0 disappears.
• The step (i) of the implicit construction procedure involves
to solve (71) with periodic boundary conditions on Yx . This step
implies the use of a finite element solver on a single large domain
(if Y 0 is set as the whole bar) or on a set of smaller domains (Yx )
and this implies that a mesh, or a set of meshes, of the elastic
properties in the Yx domain must be designed. Therefore, even if
the meshing problem for the elastic wave propagation in order 0
homogenized model is much simpler for the than for the original
model, the problem is still not mesh free. Fine meshes still must be
which is the intuitive effective elastic modulus (54) guessed in
Section 3.1.
3.4.2 Implicit construction
For higher dimensions than 1-D, a cell problem, similar to (71),
arises (see, for example Sanchez-Palencia 1980). Unfortunately, in
general, there is no explicit solution to this cell problem leading to an
analytical solution equivalent to (72) (there is one for layered media,
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designed to solve the homogenization problem. Nevertheless, these
meshes can be based on tetrahedra even if the wave equation solver
is based on hexahedra. Moreover, as the homogenization problem is
time independent, the consequences of very small or badly shaped
elements on the computing time are limited.
(i) Taking Yx as R (β infinite), the first step allows to find
T
∂ y χsε0 ,1 (y) = EC0 (ε0 y) − 1 where C = (limT →∞ 2T1 −T E10 (x)dx)−1 .
ε0
ε0
(ii) H and G are straight forward to compute from step (i).
We have
1
1
1
k0
−
F
(104)
(ε0 y) + 1,
G ε0 (x, y) = k 1 (x)
0
0
0
E
E
F
E0
where the fact that, for any h, F (h) (x/ε0 ) = F (h) (x). We also
find H ε0 (x, y) = [F k0 ( E10 )(x)]−1 .
(iii) The third step allows to find
−1
1
1
1
k0
k0
(ε
(x)
−
F
y)
+
F
E ε0 (x, y) =
0
E0
E0
E0
10
9
km/s
We can check that this procedure gives a correct result on our
1-D case.
907
8
7
0
0.1
0.05
x (m)
k0
(105)
E
∗,ε0
(x) = F
k0
1
E0
−1
(x)
(106)
3.5 Convergence of the asymptotic solution with ε 0
As for this non-periodic case, we have built a classical periodic
homogenization scheme, the convergence theorem is still valid: u ε0 ,ε
converges in the appropriate sense to the leading order asymptotic
term u ε0 ,i=0 as ε tends towards 0 (see Section 2.1). In the periodic
case, one particular ε (ε = 0 /λm ) corresponds to the ‘real’ case
we wish to approximate. In the non-periodic case, it is ε = ε 0 ,
but this is true for any ε0 . Let us name u r e f the reference solution
obtained in the ‘real model’ (ρ 0 , E 0 ). Thanks to the condition (iii)
of Section 3.4, we have, for all ε0 , u r e f = u ε0 ,ε0 . Therefore, still
using the classical convergence theorem, we know that the leading
order asymptotic term u ε0 ,0 will converge towards u r e f as ε0 tends
towards 0. Furthermore, using (47), we also have
û ε0 ,1 = u r e f + O ε02 .
(107)
3.6 A numerical experiment for a non-periodic case
As mentioned above, the only case that can be considered practically
is ε = ε0 . A consequence of this is, in order to check the convergence
with ε, the only way is to vary ε0 through the filter wk0 used to
separate the scales.
As for the periodic case, we perform a numerical experiment using SEM. We generate a bar model composed of slices of 0.64 mm
thick in which the properties E and ρ are constant and determined
randomly. A sample of the E 0 values are shown on Fig. 8. We
first build E ε0 (x, y) and ρ ε0 (x, y) using one of the two methods
described in Section 3.4 (they both give the same result). Two examples of homogenized E ∗,ε0 can be seen on Fig. 8. Then the cells
problem (71), (82), (83) and (84) are solved for each x of the SEM
mesh using the same finite elements method than the one used for
2010 The Authors, GJI, 181, 897–910
C 2010 RAS
Journal compilation the periodic case. This part is more time consuming than for the periodic case because these equations have to be solved on many large
Yx segments or on a single global Y 0 segment. The wave equation
ρ̃∂tt u − ∂x (E ε0 ,∗ ∂x u) = f ,
ε0
which are the desired results.
C
Figure 8. Sample (10 cm) of the bar velocity (black line, km s−1 ) and two
examples of order 0 homogenized velocities (bold grey line for ε 0 = 0.125
and bold dashed grey line for ε 0 = 0.25). The density has a similar but
uncorrelated pattern.
133
ρ,ε0 ∗
(108)
ε0 ,1
ε0 ,1
, u = û and f = f̂ , is then
where ρ̃ = ρ + ε0 ∂x E
solved using the SEM.
The source is the same as for the periodic case as well as the set
up of the SEM. In the case presented here, ε 0 0.125. Once the
simulation is done, for a given time step corresponding to t = 4.9 ×
10−4 s, the complete order 1 solution can be computed with (46).
We can also compute the incomplete order 2 solution
x
x
∂x + ε02 χ ε0 ,2x x,
∂x
û ε0 ,3/2 = 1 + ε0 χ ε0 ,1 x,
ε0
ε0
x
x
∂x x + χ ε0 ,ρ x,
∂tt
û ε0 ,1 .
+χ ε0 ,2 x,
ε0
ε0
(109)
On Fig. 9 are shown the results of the simulation. On the upper left
plot (Fig. 9a) are shown the reference solution (bold grey line), the
order 0 solution (black line) and a solution obtained in bar with an
effective E ∗ = F k0 (E 0 ) (‘E average’, dashed line) for t = 4.9 ×
10−4 s as a function of x. Note the strong coda wave trapped in the
random model on the left of the ballistic pulse which was not at all
present in the periodic case. As expected, the ‘E average’ solution is
not in phase with the reference solution and shows that this ‘natural’
filtering is not accurate. On the other hand, the order 0 homogenized
solution is already in excellent agreement with the reference solution. On Fig. 9(b) is shown the residual between the order 0 homogenized solution and the reference solution û ε0 ,0 (x, t) − u ref (x, t). The
error amplitude reaches 1 per cent and contains fast variations. On
Fig. 9(c) is shown the order 1 residual û ε0 ,1 (x, t) − u r e f (x, t) (bold
grey line) and the partial order 2 residual û ε0 ,3/2 (x, t) − u r e f (x, t)
(see eq. 109). It can be seen comparing Figs 9(b) and (c) that the
order 1 periodic corrector removes most of the fast variation present
in the order 0 residual. The remaining fast variation residual disappears with the partial order 2 residual. The smooth remaining
residual is due to the u 2 that is not computed. In order to check
that this smooth remaining residual is indeed an ε 2 0 residual, the
same residual, computed for ε0 = 0.125 is compared to the partial order 2 residual computed for ε0 = 0.0625 (multiplying its
amplitude by 4) and for ε0 = 0.25 (dividing its amplitude by 4).
908
1
b
a
0.01
ref
order 0
E average
0.005
0
0
-0.005
-1
2
3
4
3
4
3.5
eps=0.0625 x 4
eps=0.125
eps=0.25 /4
0.002
0.001
0
0
-0.001
3.5
4
-0.002
x (m)
2
3
x (m)
4
Figure 9. (a) Grey line: displacement u r e f (x, t) at t = 4.9 × 10−3 s computed in the reference model described Fig. 2. Black line: the order 0 homogenized
solution û 0 (x, t). Dashed line: solution computed in a model obtained by averaging the elastic properties [F k0 (ρ 0 ) and F k0 (E 0 )]. (b) Order 0 residual,
û 0 (x, t) − u r e f (x, t). (c) Grey line: order 1 residual, û ε0 ,1 (x, t) − u r e f (x, t). Black line: partial order 2 residual, û ε0 ,3/2 − u r e f (x, t) (see eq. 109). (d) Black
line: partial order 2 residual for ε 0 = 0.125 mm. Grey line: partial order 2 residual for ε 0 = 0.0625 mm with amplitude multiplied by 4. Dashed line: partial
order 2 residual for ε 0 = 0.25 mm with amplitude divided by 4.
It can be seen that these three signals overlap but not completely.
This is consistent with a ε2 0 residual but it shows that the approximations made, mainly on the fact that support of the filters wk0 has
been truncated to make their support finite, has some effect on the
convergence rate.
filter to discontinuities. Because the spatial cut-off k 0 is chosen
larger than the maximum wavenumber km , the spatial oscillations of
the medium may be faster than the maximum ones of the wavefield.
For the SEM point of view, this must be taken into account and
the classical rule used to sample the wavefield (e.g. 2 minimum
wavelength per degree 8 elements) for piecewise content medium
does not fully apply. The solution is just to increase the number
of elements per wavelength, but this unfortunately has a numerical
cost. The optimum sampling of the wavefield in such a case remains
to be studied.
A critical aspect of this work is that the methodology exposed
for the 1-D case can be extended to higher dimensions because
it is not based on the knowledge of an analytical solution of the
cell problem as it was the case in Capdeville & Marigo (2007). An
important perspective of this work is then to extend it to 2- and 3-D.
This should allow to solve many meshes difficulties that arise when
using the SEM for wave propagation simulation in complex 3-D
media.
Finally, we underline the fact that the results obtained may have
important applications in inverse problems for the Earth structure:
first it is shown how to define a multiscale parametrization depending on the wavefield properties; second, physical quantities
that should be inverted for (here 1/E), directly appear in the set
of equations involved in the homogenization procedure. Moreover,
the periodic correctors should allow to make inferences about local
effect at source and receiver locations.
A patent (Capdeville 2009) has been filed on the non-periodic
homogenization process by the ‘Centre national de la recherche scientifique’ (CNRS) (this is by no mean a restriction to any academic
research on the subject).
4 C O N C LU S I O N S A N D P E R S P E C T I V E S
We have presented an extension of the classical two scale homogenization from 1-D periodic media to 1-D non-periodic media. This
extension does not hold beyond the order 1, but for the periodic
case. This non-periodic homogenization procedure is based on the
introduction of a spatial filter, that allows to separate scales in the
heterogeneity pattern of the medium. In the wavenumber domain,
the location of the filter cut-off k 0 that separates slow and fast
variations in the physical properties, is left at the discretion of the
user, but must be chosen such that k 0 is larger than the maximum
wavenumber of the wavefield km to obtain an accurate result. For the
leading order of this asymptotic theory, the wavefield computed in
the homogenized medium converge towards the reference solution
as ε0 = km /k 0 . The level of accuracy can therefore be chosen by
selecting the appropriate k 0 .
In contrast to the periodic case, the general solution presented
is computationally intensive. Indeed, the non-periodic homogenization requires to solve the so-called cell problem (eq. 71) over the full
model (in one time or in a multitude of smaller problems) and not on
a single cell as in the periodic case; and this can be very challenging. Nevertheless, this cell problem is time-independent, and has to
be solved only once for the whole medium (using a classical finite
element method); and simulating wave propagation with SEM, in
the smooth, homogenized medium is much less time-consuming,
than doing the same in the rough, initial one.
Another issue is that the effective medium resulting from the
homogenization procedure at the leading order, is often oscillating
in space. These oscillations result from the application of the spatial
The authors thank Leonid Pankratov and Jean-Pierre Vilotte for
their reference suggestions, Dimitri Komatitsch and an anonymous
134
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2010 The Authors, GJI, 181, 897–910
C 2010 RAS
reviewer for their constructive remarks. This work was supported
by the ANR MUSE under the blanc programme.
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APPENDIX A:
We show here that E ρ∗ = ρχ 1 . We start with
∂ y [E∂ y χ ρ ] = ρ − ρ.
(A1)
910
Multiplying the last equation by χ 1 , taking the cell average and
using the fact that χ 1 = 0, we find
1 χ ∂ y E∂ y χ ρ = χ 1 ρ .
(A2)
Using an integration by part, we have
− ∂ y χ 1 E∂ y χ ρ = χ 1 ρ .
(A3)
Using (24) we find
E∂ y χ ρ = χ 1 ρ ,
(A4)
which, using the definition E ρ∗ = E∂ y χ ρ , is the wanted result.
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Journal compilation Geophys. J. Int. (2010) 000, 000–000
2-D non-periodic homogenization to upscale elastic media
for P-SV waves
Yann CAPDEVILLE1 , Laurent GUILLOT1 , Jean-Jacques MARIGO2
1
Équipe de sismologie, Institut de Physique du Globe de Paris (UMR 7154), CNRS. email: [email protected]
2
Laboratoire de Mécanique des solides (UMR 7649), École Polytechnique
keywords: Wave propagation, Theoretical Seismology, Computational seismology, Seismic anisotropy,
wave scattering and diffraction, numerical solutions.
27 May 2010
SUMMARY
The purpose of this article is to give an upscaling tool valid for the wave equation in
general elastic media. The present paper is focused on P-SV wave propagation in 2-D,
but the methodology can be extended without any theoretical difficulty to the general 3-D
case. No assumption on the heterogeneity spectrum is made and the medium can show
rapid variations of its elastic properties in all spatial directions. The method used is based
on the two-scale homogenization expansion, but extended to the non-periodic case. The
scale separation is made using a spatial low-pass filter. The ratio of the filter wavelength
cutoff and the minimum wavelength of the propagating wavefield defines a parameter ε0
with which the wavefield propagating in the homogenized medium converges to the reference wavefield. In the general case, this non-periodic extension of the homogenization
technique is only valid up to the leading order and for the so-called first order corrector.
We apply this non-periodic homogenization procedure to two kinds of heterogeneous media: a randomly generated, highly heterogeneous medium and the Marmousi2 geological
model. The method is tested with the Spectral Element Method as a solver to the wave
equation. Comparing computations in the homogenized media with those obtained in the
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2
Y. CAPDEVILLE
original ones shows that convergence with ε0 is even better than expected. The effects of
the leading order correction to the source and first correction at the receivers’ location are
shown.
1 INTRODUCTION
Seismic waves are widely used to study or image the Earth interior at all scales. In the seismological
or seismic exploration fields, one current challenge is to understand and to account for the effect
of heterogeneities much smaller than the minimal wavelength of a wavefield propagating through
complex media. Indeed, the earth structure is often highly heterogeneous, at least at the crust and
smaller scales. Nevertheless, it is well known that, in some cases at least, one can obtain relatively
accurate ground displacement predictions when using simple propagation media, even if the real ones
show a high complexity in the spatial distribution of their elastic properties at smaller scale than
the minimum propagating wavelength. For example, very long period surface waves at the global
Earth scale can be modeled with a reasonable accuracy using simple spherically symmetric elastic
models, and yet, the crust is highly heterogeneous at small scales. What happens is that waves naturally
“upscale” (or, equivalently, “homogenize” or “see an effective medium of”) the real medium. Being
able to understand in what sense a wave is upscaling a real medium is important for both the imaging
techniques (the inverse problem) and for waveform modeling (the forward problem). From the seismic
imaging (inversion) perspective, it is indeed of importance to understand in what sense the wavefield
upscales the real medium to be able to interpret the imaging results. For the forward problem, small
scale heterogeneities are a difficulty for all numerical wave equation solvers. Replacing the original
discontinuous and heterogeneous medium by a smooth and simpler one, is an attractive alternative to
the fine and difficult meshing of the original medium, required by many wave equation solvers, that
usually leads to a high computational cost.
In the geophysical community, taking into account small scales is referred to as finding the “effective medium” of a complex medium. In the seismic community it is referred to as “to upscale” a
medium. In solid mechanics, this procedure is referred to as “to homogenize the medium”. In geophysics, a theoretical effort on effective medium has been going on since the sixties when Hashin &
Shtrikman (1963) or Hill (1965) defined upper and lower bounds for the effective elastic properties of
heterogeneous assemblages. Other and more recent contributions to this topic are described in Mainprice et al. (2000). For wave propagation in the seismic exploration context, an important contribution
was that of Backus (1962) who showed how to compute effective properties for a wave propagating in
finely layered media. This work is still widely used within the seismic community. Since then, work
138
2-D non-periodic homogenization, PSV case
3
has focused on obtaining a more general upscaling theory (see, for example, Grechka (2003), Gold
et al. (2000) or Tiwary et al. (2009) for a review of some upscaling methods used in the exploration
industry). In mechanics, the method used is the so-called two-scale homogenization. The latter is unfortunately often restricted to periodic media (for applications of the homogenization to the dynamic
case, one may refer to Sanchez-Palencia (1980), Willis (1981), Auriault & Bonnet (1985), Moskow &
Vogelius (1997), Allaire & Conca (1998), Fish & Chen (2004), Lurie (2009) or Allaire et al. (2009))
or dedicated to the formal mathematical foundations of the non-periodic case (e.g. Nguetseng (2003),
Marchenko & Khruslov (2005)). When considering a layered medium, it is possible to extend the
two-scale homogenization method to the non-periodic case (Capdeville & Marigo, 2007) and it can be
shown that the order 0 homogenization (the homogenization theory relies on an asymptotic expansion)
gives the same result as the Backus (1962) averaging technique. For higher-dimensional problem, the
two-scale homogenization solution is well know and it is has been applied to the elastic wave equation
(e.g. Fish & Chen 2004). Nevertheless, in practice, it is still limited to the periodic case. The challenge
of our work is therefore to extend the two-scale homogenization theory to the non-periodic case for a
spatial dimension higher than 1, for P-SV waves. The reader is encouraged to read the introductions
to this topic given by Capdeville et al. (2010) for a 1-D wave propagation, and by Guillot et al. (2010)
in the case of an anti-plane elastic motion in 2-D.
The wave equation solver used here is the Spectral Element Method (SEM) (see, for example,
Priolo et al. (1994) and Komatitsch & Vilotte (1998) for the first SEM applications to the wave equation and Chaljub et al. (2007) for a review). This method has the advantage of being accurate for all
type of waves and all types of media, as long as a hexahedral mesh, on which most of this method
implementations rely, can be designed for a partition of the space. This method can be very efficient,
depending on the complexity of the mesh. Nevertheless, difficulties arise when encountering some
spatial patterns typical of the Earth like a discontinuity of material properties. In 3-D realistic media,
the hexahedral mesh design is often impossible.
We first introduce some concepts of spatial filtering and study wave propagation in two distinct
elastic media for which computing a reference solution with SEM is a possible but difficult and timeconsuming alternative. We apply two naive upscaling solutions and show they are not accurate. We
then develop the non-periodic homogenization for the P-SV wave propagation in 2-D. We then show
with examples that the method is accurate and generates wavefields that converge rapidly towards the
reference ones (computed in the original, non-homogenized medium, with SEM).
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Y. CAPDEVILLE
2 PRELIMINARIES
In this preliminary section, we introduce some spatial filtering notions, we define an elastic model
and suggest two trivial upscaling processes. Finally, we give examples of wave propagation in two
complex models and compare the results to the ones computed in the corresponding trivially upscaled
models.
2.1 Spatial filtering
For any function h, we define its 2-D-Fourier transform as
Z
h(x)eik·x dx ,
h̄(k) =
R2
(1)
where x = t (x1 , x2 ) is the position vector, k = t (k1 , k2 ) is the wave-number vector and t the transpose
operator. Let λ = 1/|k| be the wavelength associated with a wave-number vector k. Our development
requires to separate low from high wave-numbers of a given distribution h̄(k) around a given wavenumber k0 . For that purpose, we introduce a low-pass space filter operator which, for any function h,
is defined as:
F k0 (h) (x) =
Z
R2
h(x′ )wk0 (x − x′ )dx′ ,
(2)
where wk0 is a wavelet, such that

 1 for |k| ≤ k ;
0
w̄k0 (k) =
 0 for |k| > k0 .
(3)
F k0 ◦ F k0 = F k0 ,
(4)
An important property of F k0 is
where ◦ is the function composition. In practice, in order to have a wavelet wk0 for which a compact
support is a good approximation, we do not use such a sharp cutoff but a smooth transition form 1 to 0
around k0 . In that case, the property (4) is only approximated. An example of such a wavelet is shown
in Fig. 1 and its design is detailed in appendix A.
2.2 Elastic models
In the following, we consider that an “elastic model” in which we wish to propagate waves is fully
defined by the spatial distributions of its density ρ(x) and elastic tensor
c(x) = {cijkl (x)}, (i, j, k, l) ∈ {1, 2} .
(5)
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5
|w̄k0 |(k)
wk0 (x)
0.0015
0.0005
x2
20 k
m
22 k
m
kx2 (1/m)
0.0010
18 k
m
x1
22 k
m
0.0000
0.0000
18 k
m
20 k
m
0.0005
0.0010
0.0015
kx1 (1/m)
0
1
Figure 1. Wavelet example on the left (centered on t (20km, 20km)) and its power spectrum for positive
wavenumbers on the right. The wavelet power spectrum is 1 for |k| < 6×10−3 m−1 , 0 for |k| > 10×10−3 m−1
and values in-between are given by a cosine-taper (see appendix A).
The elastic tensor is positive-definite and satisfies the following symmetries:
cijkl = cjikl = cijlk = cklij ,
(6)
reducing the maximum number of independent parameters necessary to characterize c to 6. If the
model is isotropic, there are only two independent parameters. Therefore, in the isotropic case, knowing the P and S wave velocities and the density, or the two Lamé elastic parameters and the density,
is enough to characterize c and is therefore enough to fully define an elastic model.
2.3 Naive upscaling techniques based on spatial filtering
Assuming the existence of a minimum wavelength λm for a given wavefield propagating in a given
elastic medium (ρ, c), it is known by seismologists that this wavefield is in most cases insensitive
to scales much smaller than λm . As mentioned in the introduction, a typical case of this phenomenon
occurs for the crust which, despite its known complexity at small scales, can be modeled at long period
with a reasonable accuracy using a simple spherically symmetric model. If an original medium (ρ, c)
has spatial variations on scales much smaller than λm , there are at least two naive ways to upscale
this model based on the spatial filter F k0 , where k0 is a user-defined wavenumber, preferably (much)
larger than 1/λm . This wavenumber cutoff k0 allows to define the parameter
ε0 =
λ0
,
λm
(7)
where λ0 = 1/k0 , and the two naive upscaling procedures are the following ones:
• The “elastic filtering” upscaling. It is based on low-pass spatial filtering of the density and of the
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elastic tensor. The effective model is therefore (ρ∗,ε0 , c∗,ε0 ) = F k0 (ρ) , F k0 (c) (in this paper, the ∗
superscript is used to point out any effective property).
• The “velocity filtering” upscaling. It is based on low-pass spatial filtering of the density and of
the elastic wave velocities. The model is computed from the effective density ρ∗,ε0 = F k0 (ρ) and
velocities Vp∗,ε0 = F k0 (Vp ) and Vs∗,ε0 = F k0 (Vs ).
At this point, a problem already appears with this low-pass filtering idea: filtering velocities or elastic
parameters does not produce the same effective media for high velocities contrasts (it would in a
medium with only weak velocity contrasts), therefore which one should be chosen (if any)? In the
following subsection these two upscaling procedures are nevertheless tested on two elastic model
examples.
2.4 Two elastic models and naive upscaling examples
In this section, we study the propagation of waves in two distinct elastic media, both of them containing heterogeneities whose size is much smaller than the minimum wavelength of the wavefield.
As mentioned in the introduction, the method used to compute the reference solution and the solutions in the upscaled medium is the SEM. The mesh used to compute this reference solution matches
all physical discontinuities making possible high precision but one that comes with a high numerical
cost. Such a simulation is only made possible by the 2-D geometry and is prohibitive in 3-D. We test
here three different solutions to avoid the thin meshing of the original medium and the resulting high
numerical cost:
(i) one based on velocity filtering upscaling;
(ii) one based on elastic filtering upscaling;
(iii) one based on a sparser mesh than the one imposed by physical interfaces but good enough to
sample the wavefield. In that case, the physical discontinuities of the model are not matched by any
element boundary.
Solutions (i) and (ii) are defined in the previous subsection and solution (iii) is sometimes used when
the mesh design is too difficult. For example, Komatitsch & Tromp (2002) proceeded in this way to
avoid the difficult meshing of a complex Earth’s crust model.
2.4.1 First example: square random model
The first model is a randomly generated 2-D elastic medium. It consists of a 30×30 km2 square matrix
of 300 × 300 elements of constant elastic properties surrounded by a 5 km thick strip of constant
elastic properties corresponding to P and S wave velocities of 5 kms−1 and 3.2 kms−1 respectively
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7
Figure 2. Square random model. Density, Vp and Vs are presented.
t=8 s
40 km
40 km
30 km
30 km
C
20 km
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
A
B
20 km
D
10 km
10 km
0 km
0 km
0 km
10 km
20 km
30 km
40 km
Figure 3. Configuration of the experiment for the random square model. Two source locations A and B are used
(marked with green squares). Pink diamonds labeled from 1 to 40 are receiver locations and the line “CD” is a
line of receivers with a 50 m vertical sampling rate. The thin black line square corresponds to the boundary of
the random elastic properties area. The plotted field is a kinetic energy snapshot at t=8 s for an explosion located
in A with a Ricker wavelet in time of central frequency 1.5 Hz.
and a density of 3000 kg m−3 (see Fig. 2). In each element of the matrix, the constant elastic properties
and density are generated independently and randomly with a uniform distribution within ±50% of
the outer strip elastic values and density.
The geometrical configuration of the experiment is given in Fig. 3. We compute the wave propagation induced by an explosion with a Ricker wavelet (i.e. second derivative of a Gaussian function)
time function with a central frequency of 1.5 Hz (corresponding roughly to a corner frequency of
3.6 Hz). Ignoring the fluctuations of wave velocities in the inner square and far away enough from the
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source, we can estimate the minimum wavelength λm of the wavefield generated by the explosion to
be roughly equal to 800 m. To obtain the promised accuracy of the SEM, we must generate a mesh
based on square elements that honors all physical discontinuities of the model. In this case, the geometry is so simple that the mesh generation is trivial. Nevertheless this simple matrix geometry imposes
small 100×100 m2 elements to honor the interfaces with element boundaries. Knowing that a degree 4
spectral element (a tensorial product of degree 4 polynomial basis) can roughly handle one wavelength
per element, the mesh is oversampling the wavefield by a factor 8 in each direction, leading to a factor
512 in numerical cost (a factor 8 in each direction and a factor 8 in time to match the Newmark time
marching scheme stability condition). For this simple 2-D case, this factor 512 can readily be handled
and this allows us to compute a reference solution. Nevertheless, one can imagine that for a 3-D case,
meshing the original model can quickly be out of reach for a reasonable computing power and the
temptation would be high to either use a mesh that does not honor the physical interfaces or to simplify the model. We therefore test here the three simple solutions (i), (ii) and (iii) mentioned above.
For the solutions (i) and (ii), we use λ0 = 267m which implies a ε0 = 0.3. For the solution (iii), we
simply use a mesh with 142 × 142 elements to mesh the matrix instead of 300 × 300 elements used to
compute the reference solution. Using this sparser mesh, we are still oversampling the wavefield (by
a factor 4 in each direction) but none of the physical interfaces is matched by any element boundary.
We first generate a reference solution using the SEM mesh matching all interfaces. A snapshot of the
kinetic energy of the wavefield generated by the source A is plotted in Fig. 3 for t = 8 s. In Fig. 4, we
pick a representative receiver (receiver 22) and compare waveforms obtained for the three solutions (i),
(ii) and (iii) to the reference solution. It clearly appears that none of them provide a good solution, at
least for standard SEM accuracy. It appears that low-pass filtered solutions (i) and (ii) have first arrival
propagating faster than in the original medium. The coda is also faster and the time delay increases
with time. It is interesting to note that this time shift observed for the first arrival is consistent with
the “velocity shift” observed when comparing time arrivals of waves propagating in random media to
time arrivals computed with the corresponding average velocity (Shapiro et al., 1996). Solution (iii),
despite being also slightly too fast, provides a better solution for the first arrival. For coda, amplitude
errors and phase time shifts can clearly be observed. Another interesting situation is shown in Fig. 5
for the same explosion as for the previous case, but located in B (see Fig. 3) at the center of the random
area. On the kinetic energy snapshot for t = 5 s (Fig. 5, left graph), an incomplete energy ring can
be observed after the main P front. In Fig. 5, right graph, this phase is clearly seen on the reference
solution (black line) around t = 8 s on the vertical component (x2 ) of receiver 38, outside of the
random area. This phase is a ballistic S wave which is not normally generated by an explosion located
in a simple medium (as can be seen for source A in Fig. 4). This is a S wave generated by a strong P
144
x1 component
9
x2 component
0.2
1
(i) velocity average
0
0
-1
1
-0.2
0.2
(ii) elastic average
0
0
-1
1
-0.2
0.2
(iii) sparser mesh
0
0
-1
5
10
time (s)
-0.2
15
5
10
time (s)
15
Figure 4. x1 (horizontal, left column) and x2 (vertical, right column) components of the velocity recorded at
receiver 22 for the source A (see Fig. 3). On each graph, the reference solution is plotted in black. In red,
the solution obtained using solution (i) (top line of graphs), (ii) (middle line of graphs) and (iii) (bottom line
of graphs). The traces amplitude is normalized to the maximum amplitude of all traces for this receiver. The
amplitude axis range is [-1,1] for the horizontal component and [-0.2,0.2] for the vertical component.
to S wave conversion on an interface located very close to the source. All the solutions proposed in
this section fail to reproduce this effect (see Fig. 5, where only the elastic filtering upscaling solution
is represented (red line)).
2.4.2 Second example: the Marmousi2 model
Our second example is derived from the Marmousi2 elastic model (Martin, 2004; Martin et al., 2006),
which is itself derived from the famous Marmousi acoustic model designed by the Institut Français
du Pétrole (Versteeg, 1994). It is a 2-D geological (a section) model based upon the real geological
setting from North Quenguela in the Quanza basin of Angola. The section is primarily composed of
shale units with some sand and salt layers and a complex faulted area in the center of the section.
From a technical point of view, 199 horizon lines are provided and each of them correspond to the top
of a layer. When recombined together, it is possible to generate 435 closed objects from the horizons
to which constant or depth gradient elastic properties and density can be assigned. The density, P
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Y. CAPDEVILLE
0 km
t=5 s
x1 xomponent
10 km
20 km
30 km
40 km
40 km
40 km
0.5
0
30 km
30 km
-0.5
utut
20 km
utut
-1
x2 component
20 km
0.6
0.4
10 km
0.2
10 km
0
-0.2
0 km
0 km
0 km
10 km
20 km
30 km
-0.4
40 km
5
10
time (s)
15
Figure 5. Left graph: kinetic energy snapshot at t=5 s for an explosion located at the center of the random square
(red triangle). Right graph: horizontal (x1 , top graph) and vertical (x2 , bottom graph) components of the velocity
recorded at receiver 38 (blue triangle). The reference solution is plotted in black. The solution computed with
the upscaled model obtained using the elastic filtering is plotted in red.
and S wave velocities are plotted in Fig. 6. For the original Marmousi and Marmousi2 models the
top layer is a water layer corresponding to the ocean. We replace this layer by an elastic layer with
the same P wave velocity but a non-zero S wave velocity. The reason for this modification is to
avoid the occurrence of a solid-fluid interface and the associated boundary layer from the point of
view of homogenization which we shall present below. This case is similar to the one encountered
close to a free surface (see for example Capdeville & Marigo 2008) and will be addressed in future
works. We wish to pursue the same experiment as for the previous example for an explosion located
at x0 = t (8 km, −100 m) (see Fig. 8) with a Ricker time function of 6 Hz central frequency (15 Hz
of corner frequency) and to do so we once again need a reference solution. Compared to the previous
example, the hexahedral element mesh design if far from being trivial and leads to a complex mesh
geometry and a high numerical cost. Because of the 2-D configuration, some free software can help
in its design; once the necessary closed objects are generated from the horizon lines, which is the
difficult part here, we use “gmsh” (Geuzaine & Remacle, 2009), an open source mesh generator, to
complete the mesh. A sample of this latter is shown in Fig. 7. Due to the large number of layers and
some being very thin (less than a meter thick), the computation is heavy: it took seven days to compute
the reference solution using 64 CPU of a recent PC cluster. This reference solution can be computed
for this 2-D example, but it would be impossible for a similar but 3-D model. The mesh would be
impossible to design and even if one manages to do so, the numerical cost would be out of reach
146
0.0 km
0.0 km
2.5 km
5.0 km
2.5 km
5.0 km
2.5 km
5.0 km
7.5 km
10.0 km
12.5 km
15.0 km
12.5 km
15.0 km
12.5 km
15.0 km
11
-0.5 km
-1.0 km
-1.5 km
-2.0 km
-2.5 km
-3.0 km
-3.5 km
1.6
1.8
2.0
2.2
2.4
2.6
Density (tonne/m^3)
0.0 km
0.0 km
7.5 km
10.0 km
-0.5 km
-1.0 km
-1.5 km
-2.0 km
-2.5 km
-3.0 km
-3.5 km
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Vp (km/s)
0.0 km
0.0 km
7.5 km
10.0 km
-0.5 km
-1.0 km
-1.5 km
-2.0 km
-2.5 km
-3.0 km
-3.5 km
0.5
1.0
1.5
2.0
2.5
3.0
Vs (km/s)
Figure 6. Marmousi2 model. Density, Vp and Vs are presented. Grey lines correspond to physical interfaces.
for a reasonable size cluster. Once again we test the three solutions (i), (ii) and (iii) proposed at the
beginning of this section. For these three solutions, we use a simple regular mesh with a conforming
de-refinement with depth to take advantage of the vertical velocity gradient. With such a mesh, the
numerical cost is of course much cheaper and it took about one hour, still with 64 CPU, to compute
each of these three solutions. It is worth noting that, for such a model, because of the vertical velocity
gradient, the minimum wavelength increases with depth (from λm = 25 m at the top of the model to
λm = 170 m at the bottom). Therefore the spatial filtering we suggested previously for solutions (i)
and (ii) may not be well adapted, and for such a case, a variable filtering with depth based on wavelet
expansion would certainly be more appropriate. We nevertheless use the F k0 filtering operator with
λ0 = 50 m (which implies ε0 = 2 at the top of the model and ε0 = 0.3 at the bottom). The filtering
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Y. CAPDEVILLE
7500 m
-1000 m
8000 m
8500 m
9000 m
-1000 m
-1500 m
-1500 m
-2000 m
-2000 m
-2500 m
7500 m
8000 m
8500 m
-2500 m
9000 m
Figure 7. Sample of the spectral element mesh (black lines) used here. All physical discontinuities (grey lines)
are matched by a mesh interface. The background color is the S velocity with the same color code as for Fig. 6.
t=1.4 s
7000 m
8000 m
9000 m
0m
0m
-1000 m
-1000 m
-2000 m
-2000 m
-3000 m
-3000 m
7000 m
8000 m
9000 m
Figure 8. Kinetic energy snapshot at t = 1.4 s in the marmousi2 model for an explosion located at x0 =
t
(8 km, −100 m) (red diamond). The blue diamond is the receiver location used in Fig. 9 and Fig. 20.
is then too harsh at the top of the model, but, because the velocity contrasts are relatively weak there,
we hope it is good enough (and we will see that the homogenization procedure with the same spatial
filtering parameters produces good results). The results of the computations for the three solutions
are shown in Fig. 9 for the receiver location shown in Fig. 8. This location is chosen near a physical
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13
interface of strong velocity contrast, where the 2-D effects are expected to be important. Even if this
example is less spectacular than the previous one, it appears that the first arrival is faster for solutions
(i) and (ii) than for the reference solution and that larger differences can be observed in the coda. The
results for the solution (iii) are of better quality but some apparent misfits remain. Nevertheless, the
three solutions give a better result for the Marmousi model than for the square random model. The
main reasons are that the propagation distance compared to the minimum wavelength is shorter in the
Marmousi model, and that the power spectrum of the elastic properties decreases faster with the wave
number k in the Marmousi2 model than in the square random model. Actually, for the Marmousi2
model, the three solutions can provide a good result just by decreasing ε0 for solutions (i) and (ii), or
by using an even finer mesh for solution (iii). Nevertheless, computing these solutions is expensive,
and even in that case, depending on the model spectrum, and on the type of waves studied, there is no
guarantee that these solutions will converge to the reference solution. For surface waves for example,
or for interface waves in general, none of these solutions would provide an accurate result (Capdeville
& Marigo, 2008).
3 THEORETICAL DEVELOPMENT
3.1 Notations
Let us first define some notations that will be used in this section. For any 4th-order tensor A and
second order tensor b, we note
[A : b]ij = Aijkl bkl ,
(8)
where the sum over repeated subscripts is assumed. For any 4th-order tensors A and B, we note
[A : B]ijkl = Aijmn Bmnkl .
(9)
We will use the following compact notation for partial derivatives with respect to any variable x of a
given function g:
∂x g ≡
∂g
.
∂x
(10)
Finally, we will sometimes use the classical notation for time partial derivative: for any u
u̇ ≡
∂u
.
∂t
(11)
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x1 component
x2 component
0.6
(i) velocity average
0
0
-0.6
-1
0.6
(ii) elastic average
0
0
-0.6
-1
0.6
(iii) sparser mesh
0
0
-1
-0.6
2
3
time (s)
2
3
time (s)
Figure 9. Horizontal (x1 , left column) and vertical (x2 , right column) component of the velocity recorded at
receiver shown in Fig. 8. For all graphs, the reference solution is plotted in black. In red is plotted the solution
obtained using solution (i) (top line of graphs), (ii) (middle line of graphs) and (iii) (bottom line of graphs).
The traces amplitude is normalized to the maximum amplitude of all traces for this receiver. The amplitude axis
range is [-1,0.9] for the horizontal component and [-0.6,0.7] for the vertical component.
3.2 Problem set up
We consider an infinite elastic plane characterized by its density, ρ0 (x), and elastic tensor, c0 (x), distributions. The plane is considered as infinite in order to avoid the treatment of any boundary condition
that normally would be necessary in the following development. The boundary condition problem associated with homogenization has nevertheless been addressed by Capdeville & Marigo (2007) and
Capdeville & Marigo (2008) for layered media, and will be the purpose of future works for a more
general case. No assumption on the spatial variability of ρ0 (x) and c0 (x) is made, which implies that
they can vary at any scale and in any direction. The plane is submitted to an external source force
f = f (x, t) and we wish to study the displacement u(x, t) = t (u1 , u2 )(x, t) associated with the wave
propagating in the plane. We assume that f (x, t) has a corner frequency fc which allows to assume
that, in the far field, a minimum wavelength λm to the wavefield u exists. The displacement u is driven
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15
by the wave equation,
ρ0 ∂tt u − ∇ · σ = f ,
(12)
associated with the following constitutive relation between the stress σ and the strain ǫ(u) = 21 (∇u+
t ∇u)
tensors:
σ = c0 : ǫ(u) .
(13)
The initial conditions at t = 0 are assumed to be zero and radiation boundary conditions at the infinity
are assumed (actually modeled using the Perfectly Matched Layers version of Festa et al. 2005).
3.3 Homogenization problem set up
To solve the so-called two-scale homogenization problems, a small parameter ε is classically introduced :
λ
,
λm
ε=
(14)
where λ is a spatial wavelength or a scale. For a periodic medium, λ would be a characteristic length
of the periodicity of the model. In the non-periodic case, another parameter is required
ε0 =
λ0
,
λm
(15)
where λ0 is the user-defined scale below which a wavelength is considered as belonging to the small
scale (microscopic) domain. Reciprocally, a wavelength larger than λ0 is considered as belonging to
the large scale (macroscopic) domain. The parameter λ0 is user-defined, but it makes sense to assume
that the wavefield does interact with heterogeneities whose scales are smaller than λm . Therefore,
choosing an ε0 << 1, which means considering as microscopic, heterogeneities whose size is much
smaller than the minimum wavelength, is probably a good guess.
In order to explicitly take microscopic scale heterogeneities into account, a fast space variable is
introduced:
y=
x
.
ε
(16)
y is the microscopic variable and x is the macroscopic one. When ε → 0, any change in y induces a
very small change in x. This leads to the separation of scales: y and x are treated as independent
variables. This hypothesis implies that partial derivatives with respect to x become:
1
∇x → ∇x + ∇y ,
ε
(17)
where ∇x = t (∂x1 , ∂x2 ) and ∇y = t (∂y1 , ∂y2 ).
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We define the wavelet wm (y) = wkm (y) where wkm is the low-pass filter wavelet defined in (3) or
in appendix A and km = 1/λm . We assume that the support of wm in the space domain is contained
in [−αλm , +αλm ]2 where α is a positive number that depends upon the specific design of w (see
appendix A for details).
Let Y0 = [−βλm , βλm ]2 be a square of R2 where β is a positive number larger than α and Yx the
same square but translated by a vector x/ε0 . We define T = {h(x, y) : R4 → R , Y0 -periodic in y}
the set of functions defined in y on Y0 and periodically extended to R2 . We define the filtering oper-
ator, for any function h ∈ T :
Z
h(x, y′ )wm (y − y′ )dy′ .
F (h) (x, y) =
(18)
R2
Finally let V be the set of functions h(x, y) such that, for a given x, the y part of h is periodic and
contains only spatial frequencies higher than km , plus a constant value in y:
V = {h ∈ T /F (h) (x, y) = hhi (x)} ,
(19)
where
hhi (x) =
1
|Y0 |
Z
h(x, y)dy ,
(20)
Y0
is the y average of h(x, y) over the periodic cell. It can be easily shown that, for any function or tensor
h in T and for any i,
h∂yi hi = 0 .
(21)
In this section and the next one, we proceed in the same way as in Capdeville et al. (2010) and
Guillot et al. (2010). We first assume that we have been able to define (ρε0 (x, y), cε0 (x, y)) in T with
the conditions
ρε0 (x, x/ε0 ) = ρ0 (x)
(22)
cε0 (x, x/ε0 ) = c0 (x)
that set up a sequence of models indexed by ε
x
ρε0 ,ε (x) ≡ ρε0 (x, ) ,
ε
(23)
x
ε0 ,ε
ε0
c (x) ≡ c (x, ) ,
ε
and that, with such a set of parameters, a solution to the problem described below exists. This assumption is by far not obvious and the construction of such a (ρε0 (x, y), cε0 (x, y)) from (ρ0 (x), c0 (x)),
which is the critical point of this article, is left for section 3.5.
Of course, there is only one real elastic model (ρ0 , c0 ), but the homogenization relies on a series
of models indexed by the small parameter ε. In the periodic case, the construction of a ε-indexed
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17
series of elastic models is simply obtained by changing the periodicity length of the original elastic
model. This series is formal and it is only used to build the asymptotic expansion of the solution
and to establish the convergence theorem (e.g. Sanchez-Palencia (1980)). For practical applications,
only the real periodicity length is used. In the non-periodic case, things are more difficult because
there is no periodicity to vary to build a series of models. In the periodic case, we first build a εindexed series of elastic models and then define the cell elastic model that depends on y but not on
ε (e.g. Capdeville et al. (2010)). In the non-periodic case, the opposite is done and the cell elastic
model (ρε0 (x, y), cε0 (x, y)) with a user-defined ε0 is first built and then the formal ε-indexed series
of elastic models with (23). The ε-indexed series of elastic models defined in (23) is once again formal
and only the cell elastic model (ρε0 (x, y), cε0 (x, y)) is needed for practical applications.
We work at ε0 fixed and, to ease the already heavy notations, we drop the ε0 superscript that should
appears on stress and displacement solutions and expansion coefficients. One should nevertheless keep
in mind that the solutions of the problem depend on ε0 and one should read, for example, uε0 ,ε when
uε appears.
We look for the solutions of the following wave equation and constitutive relation
ρε0 ,ε ∂tt uε − ∇ · σ ε = f ,
(24)
σ ε = cε0 ,ε : ǫ(uε ) ,
where ǫ(uε ) = 21 (∇uε + t ∇uε ). The initial conditions at t = 0 are assumed to be zero and radiation
boundary conditions at the infinity are assumed. To solve this problem, the fast space variable y,
defined by (16), is used. In the limit ε → 0, x and y are treated as independent variables, implying the
transformation (17), or similarly, with strain operators:
1
ǫ(u) → ǫx (u) + ǫy (u) ,
ε
(25)
where ǫx (u) = 21 (∇x u + t ∇x u) and ǫy (u) = 21 (∇y u + t ∇y u).
The solution to the wave equations (24) is then sought as an asymptotic expansion in ε with ui and σ i
in V:
ε
u (x, t) =
ε
σ (x, t) =
∞
X
i i
ε u (x, x/ε, t) =
i=0
∞
X
i=−1
i
i
∞
X
εi ui (x, y, t) ,
i=0
∞
X
ε σ (x, x/ε, t) =
(26)
i
i
ε σ (x, y, t) .
i=−1
where the superscript i in a power on ε but not on ui and σ i . Note that the condition for ui and σ i to
be in V is a strong condition which basically means that only slow variations in x and fast variations
in y are allowed. It is the equivalent to the y periodic condition in the periodic case. Introducing the
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Y. CAPDEVILLE
expansions (26) in the wave equations (24) and using (25) we obtain:
ρε0 ∂tt ui − ∇x · σ i − ∇y · σ i+1 = f δi,0 ,
σ i = cε0 : ǫx ui + ǫy ui+1 .
(27)
(28)
The external source is build as independent of y. For a point source, this is not a valid construction (see
Capdeville et al. 2010) and this is taken into account in section 3.4.3. To solve this homogenization
problem up to the order i0 , (27) and (28) need to be solved for each i, up to i0 . This is the purpose of
the next section.
3.4.1 Order 0 solution and first order corrector
The resolution of the system (27,28) is classical and can be found with more details in, for example,
Sanchez-Palencia (1980) or in Guillot et al. (2010).
Solving (27) and (28), for i = −2 and i = −1 respectively allows to show that σ −1 = 0 and
that u0 = u0 . The last equality implies that u0 does not depend upon the fast variable y. This is an
important result that is intuitively well known: to the order 0 the displacement field does not contain
any fast variation (that is, is insensitive to small scale heterogeneities).
Equations (27) for i = −1 and (28) for i = 0 give
∇y · σ 0 = 0 ,
σ 0 = cε0 : ǫx
(29)
u0 + ǫy u1 .
(30)
The two last equations lead to
∇y · cε0 : ǫx u0 = −∇y · cε0 : ǫy u1 .
(31)
Thanks to the linearity of the last equation, we look for a solution as
u1i (x, y) = χεi 0 ,kl (x, y)ǫ0x,kl (x) + u1i (x) .
(32)
where ǫ0x = ǫx u0 and χε0 is the so-called first-order corrector, a 3rd order tensor. Introducing (32)
in (31) we find that χε0 is solution in V of
ε0
= 0,
∂yi Hijkl
(33)
with
ε0
0
0
,
= cεijmn
Gεmnkl
Hijkl
1
0
Gεijkl
=
δik δjl + δjk δil + ∂yi χεj 0 ,kl + ∂yj χεi 0 ,kl ,
2
154
(34)
(35)
19
where, to enforce the uniqueness of the solution, hχε0 i = 0 is imposed. Taking the cell average of (30),
using (32), property (21) and the fact that u0 does not depend upon y, we find the order 0 constitutive
relation:
0
σ = c∗,ε0 : ǫx u0 ,
(36)
where the effective elastic tensor is
c∗,ε0 (x) = hHε0 i (x) .
(37)
Equations (27) for i = 0 gives
ρε0 ∂tt u0 − ∇x · σ 0 − ∇y · σ 1 = f .
(38)
Taking the cell average of the last equation, using property (21) and taking into account the fact that
f has been build as independent of y, we find
ρ∗,ε0 ∂tt u0 − ∇ · σ 0 = f ,
(39)
where ρ∗,ε0 = hρε0 i. The last equation together with the order 0 constitutive relation (36) are the
order 0 effective wave equations. (36) and (39) form a classical elastic wave equation for the effective
elastic model (ρ∗,ε0 , c∗,ε0 ).
At this stage, solving the effective equations (36) and (39), u0 and the average stress σ 0 can be
found. To obtain the complete order 0 stress tensor, σ 0 needs to be computed using
σ 0 (x, y) = Hε0 (x, y) : ǫx u0 (x) .
(40)
In this paper, we stop our development to the order 0 and first order correction, which means we
do not solve for u1 . For the 1-D case, u1 is always equal to zero (see Capdeville et al. 2010),
but for higher-dimensional problems like the one we tackle here, u1 is not equal to zero in general
(see Guillot et al. (2010)). Nevertheless we will notice in the examples that it might be small, in some
cases at least.
Finally, note that the physical interpretation of the effective elastic tensor formula (37) is not
obvious. It can be interpreted as the average of the elastic tensor, plus a correction made of the average
of the elementary stresses associated with the displacements χε0 ,kl . This interpretation can be linked to
a heuristic approach to obtain an effective elastic tensor by computing the average stresses and strains
associated with a set of elementary static problems and finding the average tensor linking them. This
approach is known as the “average method”, and was developed by Suquet (1982). This idea has been
used in the dynamical case by Grechka (2003), but for a set of elementary problems based on a set of
boundary conditions applied to the unit cell instead of a set of external forces.
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Y. CAPDEVILLE
3.4.2 Practical resolution
Practically, to solve the homogenized equations, presented in the previous section, with classical wave
equation solver like SEM, different orders are combined together (Fish & Chen, 2004; Capdeville &
Marigo, 2007; Capdeville & Marigo, 2008; Capdeville et al., 2010):
ε,i û
(x) = u0 (x) + ε u1 (x) + ... + εi ui (x) ,
ε,i (x) = σ 0 (x) + ε σ 1 (x) + ... + εi σ i (x) ,
σ̂
(41)
(42)
where σ̂ ε,i and ûε,i are solutions of an order i combined effective equation. Knowing ûε,i , ûε,i
can be found using a high corrector operator that we will not explicit here and it can be shown that
uε (x) = ûε,i (x) + O(εi+1 ) .
(43)
In the present article, because we stop the expansion at the order 0, ûε,0 and σ̂ ε,0 are simply u0
and σ 0 and the combined effective equation is simply the equations (36) and (39). At the order 0,
the solutions ûε,0 and σ̂ ε,0 are
ûε,0 (x) = u0 (x) ,
σ̂ ε,0 (x) = Hε0 (x, x/ε) : ǫx
(44)
u0 (x) .
(45)
Applying the first order corrector to ûε,0 (x), we can obtain a partial order 1 solution
ûε,1/2 (x) = u0 (x) + χε0 (x, x/ε) : ǫx u0 (x) ,
(46)
where the 1/2 superscript means “partial order 1”. To obtain a complete order 1 solution, u1 should
be computed, which we will not do here. Because, it is only a partial order 1 solution and, in general,
uε (x) 6= ûε,1/2 (x) + O(ε2 ) ,
(47)
on the contrary of the 1-D case (in the 1-D case, u1 can be shown to be 0, see Capdeville et al.
(2010)), unless u1 is small, which appears to be the case at least for the random square example
presented in this paper.
Finally, the only ε that is of practical interest is ε = ε0 as, thanks to (22), it is the only case
for which uε is equal to the solution of the original problem uref . Note that, for all ε0 , we have
uref = uε0 . Using the above development and keeping in mind that u0 depends on ε0 (u0 stands for
uε0 ,0 ) that we therefore have uref (x) = u0 (x) + O(ε0 ).
3.4.3 External source term
We have shown in a previous work (Capdeville et al., 2010) that, for an external point source, the
original force or the moment tensor should be corrected. As in this article we stop the asymptotic
156
21
expansion at the order 0, nothing needs to be done for a vector force, which is not the case for a
moment tensor. For a moment tensor located in x0 , the external force is
f (x, t) = g(t)M · ∇δ(x − x0 )
(48)
where g(t) is the source time wavelet and M the symmetric moment tensor. As shown by Capdeville
et al. (2010), to ensure the conservation of the energy released by the source in the original model, we
need to find a moment tensor Mε0 ,ε,0 such that
(uε , f ) =
ûε,0 , f ε0 ,ε,0 + O(ε) ,
(49)
where ( . , . ) is the L2 inner product and
f ε0 ,ε,0 (x, t) = g(t)Mε0 ,ε,0 · ∇δ(x − x0 ) .
(50)
Using an integration by parts and the symmetry of the moment tensor, (49) becomes
M : ǫ (uε ) |x0 = Mε0 ,ε,0 : ǫx
ûε,0
|x0 + O(ε) .
(51)
Using (25) and (32), one finally finds, at the order 0
Mε0 ,ε,0 = Gε0 (x0 , x0 /ε) : M .
(52)
3.5 Construction of ρε0 (x, y) and cε0 (x, y)
The next (and essential) step, is to build ρε0 and cε0 (x, y) such that u0 , u1 and σ 0 are in V.
It can be seen from (32) and (40) that u0 , u1 and σ 0 are in V if cε0 (x, y) can be build such that
χε0 and Hε0 are in V. Note that if this is the case, Gε0 is also in V (gradients of function in V are also
in V). Therefore, we seek for ρε0 (x, y) and cε0 (x, y) such that
(i) ρε0 , Hε0 and χε0 ,kl are in V;
(ii) ρε0 and cε0 must be positive definite;
(iii) ρε0 (x, x/ε0 ) = ρ0 (x) and cε0 (x, x/ε0 ) = c0 (x).
The construction of ρε0 (x, y) is trivial. To do so, we introduce an initial ρε0 ,s (x, y) = ρ0 (ε0 y) defined
on R2 × Yx and then periodically extended to R2 in y. ρε0 ,s depends on x because the cell domain
used in y, Yx , depends on x. If the Y0 cell is chosen as a the whole domain, then this x dependence
disappears. We can then define
ρε0 (x, y) = F (ρε0 ,s ) (x, x/ε0 ) + (ρε0 ,s − F (ρε0 ,s ))(x, y) .
(53)
We indeed have ρε0 ,s in T and ρε0 (x, x/ε0 ) = ρ0 (x). Because F is a low pass filter we have
hρε0 ,s − F (ρε0 ,s )i = 0 .
(54)
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22
Y. CAPDEVILLE
We therefore have hρε0 i = F(ρε0 ,s ) and thanks to (4) we have F (ρε0 ) = hρε0 i. ρε0 is therefore indeed
in V. ρε0 is a positive function with a well chosen wavelet wm . Moreover, with such a definition, we
have,
ρ∗,ε0 = hρε0 i = F(ρε0 ,s )
(55)
For cε0 , the process is not trivial and we follow the procedure described by Capdeville et al. (2010)
and Guillot et al. (2010) which is inspired by the homogenization procedure for random media (Papanicolaou & Varadhan, 1979). The main idea is to search for two intermediate fields Gε0 and Hε0 in
V such that Gε0 can be written as
Gε0 =
and
1
∇y χε0 + t ∇y χε0 + I4 ,
2
(56)
Hε0 = cε0 : Gε0 ,
(57)
∇y · Hε0 = 0 ,
(58)
hGε0 i = I4 ,
(59)
4
=
where [∇y χε0 ]ijkl = ∂yi χεj 0 ,kl , [t ∇y χε0 ]ijkl = ∂yj χεi 0 ,kl and Iijkl
1
2
(δik δjl + δjk δil ).
To do so, we propose the following procedure:
• Step 1: build a start cε0 ,s defined as cε0 ,s (x, y) = c0 (ε0 y) for y in Yx and then periodically
extended to R2 . Then solve (33) with periodic boundary conditions in Yx to find χε0 ,s (x, y).
• Step 2: compute
Gε0 ,s (x, y) =
1
∇y χε0 ,s + t ∇y χε0 ,s + I4 ,
2
(60)
Hε0 ,s (x, y) = cε0 ,s (x, y) : Gε0 ,s (x, y) .
(61)
F (Gε0 ,s ) being symmetric and, for well chosen wavelet wm , positive definite, it can be inverted. This
allows to build, for any y ∈ Yx ,
Gε0 (x, y) = [(Gε0 ,s − F (Gε0 ,s )) (x, y)] : [F (Gε0 ,s ) (x, x/ε0 )]−1 + I4 ,
(62)
Hε0 (x, y) = [(Hε0 ,s − F (Hε0 ,s )) (x, y) + F (Hε0 ,s ) (x, x/ε0 )] : [F (Gε0 ,s ) (x, x/ε0 )]−1 .
(63)
Gε0 and Hε0 are periodically extended in y from Yx to R2 .
• Step 3: From (57) we can build
cε0 (x, y) = Hε0 : (Gε0 )−1 (x, y) .
(64)
Using (62) and (63) in (64), it can be seen that the tensor to be inverted in the above equation, is in
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23
fact (Gε0 ,s − F (Gε0 ,s )) (x, y) + F (Gε0 ,s ) (x, x/ε0 ). The latter is symmetric and positive definite
for well chosen wavelet wm , meaning it can be inverted and that (64) can be computed. It can be also
noted that
c∗,ε0 (x) = hHε0 i (x) = F (Hε0 ,s ) : F (Gε0 ,s )−1 (x, x/ε0 ) .
(65)
• Step 4: once cε0 (x, y) is known, the whole classical homogenization procedure can be pursued.
Remark: in practical cases, the domain is finite and Yx can be chosen to enclose the whole domain.
In that case, the dependence to the macroscopic location x in χε0 ,s , Gε0 ,s , Hε0 ,s and cε0 ,s disappears.
Following these steps, we indeed have by construction cε0 (x, x/ε0 ) = c0 (x) and cε0 is positive
definite for a well chosen wavelet wm . It is also important to check that, at the end of the procedure,
χε0 is indeed in V (Hε0 is in V by construction). At step 2, we have, by construction, (Hε0 , Gε0 ) ∈ V
and hGε0 i = I4 . Gε0 can be written under the from (56) if, and only if, ∇y × Gε0 = 0. Knowing that
for any h, ∇y × F (h) = F (∇y × h), and that, ∇y × Gε0 ,s = 0, we indeed have ∇y × Gε0 = 0.
It therefore exists a corrector χε0 such that (56) can be written. Furthermore, knowing that for any h
and g such h = ∇y g, h ∈ V with hhi = 0 implies that g lies in V, we indeed have χε0 in V. At
this stage we have found Hε0 and Gε0 , unique solutions to our problem, and we know it exists a χε0
in V satisfying (56). We ensure the uniqueness of χε0 by enforcing hχε0 i = 0. To find χε0 , we can
either solve (56), or find cε0 with (64) and solve again (33). We have chosen the last alternative. An
illustration of the process is sketched in Fig. 10. It can be seen that the power spectrum component
of the corrector which is represented (lower right graph) is equal to zero for |k|/ε0 < 6 × 10−3 m−1
which implies it belongs to V.
Finally, if we follow the physical interpretation of the effective elastic tensor formula (37) given
at the end of section 3.4.1, the above steps can be interpreted as follows: we first compute the stresses
and stains associated with a set of elementary static problems for a fine elastic model for which all
heterogeneity scales of the original model are considered as small scales (this is step one). Then, the
scales are separated on these stresses and stains using the low-pass filter (this is step 2). The two scale
elastic tensor is finally built as the relation between the scale separated stresses and stains obtained at
step 2.
One can notice that the symmetry of the effective elastic tensor does not appear to be obvious
from (65). In the periodic case and for layered media, (65) analytically gives a symmetric elastic
tensor (Guillot et al., 2010). Nevertheless, we are not able to prove it in the general case for the time
being. Let us define the skewness of the effective elastic tensor as
max c∗ − t c∗
d(x) =
(x) ,
max(c∗ )
159
(66)
24
Y. CAPDEVILLE
0.0015
100
ky2 (1/m)
0.0010
χε10,s,11
0
0.0005
-100
15000
20000
0.0000
0.0000
25000
ε0 y1 (m)
0.0005
0.0010
0.0015
ky1 (1/m)
0
1
0.0015
0.4
0 ,s
Gε1111
ky2 (1/m)
0.0010
0
0.0005
-0.4
15000
20000
0.0000
0.0000
25000
ε0 y1 (m)
0.0005
0.0010
0.0015
ky1 (1/m)
0
1
0.0015
0.4
ky2 (1/m)
0.0010
0
Gε1111
0
0.0005
-0.4
15000
20000
0.0000
0.0000
25000
ε0 y1 (m)
0.0005
0.0010
0.0015
ky1 (1/m)
0
1
0.0015
40
χε10,11
ky2 (1/m)
0.0010
0
0.0005
-40
15000
20000
25000
0.0000
0.0000
0.0005
0.0010
0.0015
ky1 (1/m)
ε0 y1 (m)
0
1
Figure 10. Illustration of the construction of the correctors in V. On the left column are plotted χ1ε0 ,s,11 (y1 , y2 ),
0 ,s
0
(x0 , y1 , y2 ) and χ1ε0 ,11 (x0 , y1 , y2 ) as a function of ε0 y1 for ε0 y2 = 20 km and
Gε1111
(y1 , y2 ), Gε1111
0 ,s
0
|(x0 , ky ) and
x0 = t (20 km, 20 km). On the right column are plotted |χ̄1ε0 ,s,11 |(ky ), |Ḡε1111
|(ky ), |Ḡε1111
|χ̄1ε0 ,11 |(x0 , ky ) at x0 = t (20 km, 20 km) and for positive wavenumbers. The actual ε0 corresponds to the
wavelet shown in Fig. 1.
160
25
where the max operator applies to the tensor components. In practice, a slight skewness of the effective
elastic tensor can be observed for the examples studied in this paper: typically d takes values below
10−3 with some localized peaks attaining 10−2 . Using the same algorithm on periodic or layered
media, we get values of the order of 10−5 . At this point, we do not know if the effective tensor
indeed has a slight skewness for general media or if this is just an accuracy issue. This important point
deserves to be studied in a future work.
4 VALIDATION TESTS
In order to validate our development, we apply the homogenization procedure to the two model examples studied in subsections 2.4.1 and 2.4.2. To do so we need to solve the cell problem (33) on the
whole domain with periodic boundary conditions (we choose Yx as the whole domain). Note that one
could rather choose to solve the cell problem on multiple smaller domains. This solution is not necessary here but might be interesting in 3-D or for large domains in 2-D. We use a relatively high order
finite element method based on a triangular mesh to solve the weak (or variational) form of the cell
problem equations. The finite element interpolation is based on the Fekete points (Pasquetti & Rapetti,
2004; Mercerat et al., 2006) and we employ a high order integration quadrature (Rathod et al., 2004).
In the following two examples, the polynomial expansion used over each element corresponds to a
degree 5 polynomial order on elements’ edge.
4.1 First example: square random model
We first apply the non-periodic homogenization procedure to the random square model described
in section 2.4.1. In figure Fig. 11 are shown sections in Vs (left plot) and in the total anisotropy
(right plot) computed from the order 0 homogenized coefficients ρ∗,ε0 and c∗,ε0 for ε0 = 0.3. At any
∗,ε0
∗,ε0
|}/max{ciso
}, where the max
given location x, the total anisotropy is defined as max{|c∗,ε0 − ciso
∗,ε0
operator applies to the tensor components and ciso
is the closest isotropic elastic to c∗,ε0 (in the sens
of, for example, Browaeys & Chevrot 2004). The homogenized quantities show relatively rapid spatial
variations, but these are smoother than for the original medium. The apparent anisotropy is significant
with average values around 1.5%. In Fig. 12 is shown a comparison of the order 0 homogenized
solution to the filtered wave velocities solution (alternative (i) of section 2.4.1) for source A and
receiver 22. In the left column plots, we compare the x1 component of the order 0 homogenized
velocity (û˙ 0 , in red line) to the reference solution (black line) as a function of ε0 (from 2.4 to 0.3).
1
On the right column is presented the same comparison but for the filtered wave velocities solution. It
appears that, when both upscaling processes are used with a large ε0 (that is, too much smoothing with
161
26
Y. CAPDEVILLE
0.04
5000
4000
m/s
0.02
3000
0
2000
5000 6000 7000 8000 9000 10000
x1(m)
5000 6000 7000 8000 9000 10000
x1 (m)
Figure 11. Left graph: 1-D section of Vs (Black line) at x2 = 32 km as a function of x1 for the original “square”
p ε0 ,∗
/ρε0 ,∗ ( Red line) at x2 = 32 km as a function of x1 for the
model (see Fig. 2); 1-D section of Vs = c2222
homogenized model. Right graph: 1-D section of the total anisotropy at x2 = 32 km for the original model
(black line) and for the order 0 homogenized model (red line). The total anisotropy is computed, at a given
∗,ε0
0
location x, as max{|c∗,ε0 − c∗,ε
iso |}/max{ciso }, where the max operator applies to the tensor components
0
∗,ε0
. This function is a non-linear function which explains the
and c∗,ε
iso is the closest isotropic elastic tensor to c
rapid oscillations observed on the total anisotropy (red line, right graph). Individual components of the effective
elastic tensor show similar slow oscillations to the one observed on the left graph.
respect to λmin ), the coda of the direct wave disappears. Nevertheless, the ballistic P wave has a correct
time arrival for the homogenized solution, whereas this is not the case for the filtered wave velocities
solution. When ε0 decreases, that is when more and more details are incorporated in the upscaled
model, the coda wave appears. Nevertheless, once again, the phase is correctly predicted only for the
homogenized solution and it seems that the filtered velocities solution have a poor convergence with
ε0 . To look more closely at the convergence issue, we define the error Ei (u̇) of a solution in velocity
u̇ at a given receiver i
qR
tmax
(u̇ − u̇ref )2 (xi , t)dt
0
Ei (u̇) = qR
,
tmax
ref )2 (x , t)dt
(
u̇
i
0
(67)
where uref is the reference solution and tmax is here 20 s. We define the combined error from receiver
5 to receiver 35 (see Fig. 3) as
35
E c (u̇) =
1 X
Ei (u̇) .
31
(68)
i=5
In Fig. 13 is shown the error as defined above for a wave propagation computed for source A (see
Fig. 3) as a function of ε0 . It appears that the error for the filtered wave velocity model solution has
162
order 0 homogenization
27
velocity average
1
ε0= 2.4
0
Normalized amplitude
-1
1
ε0= 1.2
0
-1
1
ε0= 0.6
0
-1
1
ε0= 0.3
0
-1
10
time (s)
5
15
10
time (s)
5
15
Figure 12. x1 component velocity traces computed for the source A at receiver 22 for the reference solution
(black line), for the order 0 homogenized solution (û˙ 01 , left column, red line) and for the velocity filtering
upscaled model (right column, red line) for ε0 = 2.4, 1.2, 0.6 and 0.3.
error
1
0.1
order 0 homogenization
order 0 homogenization + corrector
velocity average
0.01
2.4
1.2
0.6
0.3
0.15
ε0
Figure 13. Combined error as defined by equation (68) as a function of ε0 for an explosion located in A (see
Fig. 3) for the solution computed in the velocity filtering upscaled model (blue line), for the order 0 homogenized
solution (û˙ 0 , in dashed black line) and for the order 0 homogenized plus first order correction (û˙ 1/2 as defined
by (46), in red line).
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28
Y. CAPDEVILLE
0.02
error
0.015
0.01
order 0
order 0 + corrector
0.005
0
5
10
15
20
x1 (km)
25
30
35
Figure 14. Error for the order 0 homogenized solution, Ei (û˙ 0 ), (black line) and for the order 0 homogenized
solution plus first order corrector, Ei (û˙ 1/2 ), (red line) for receivers 5 to 35 (see Fig. 3) plotted as a function of
their location along the x1 axis and for ε0 = 0.15.
a poor convergence with ε0 . Furthermore, as it could already be seen in Fig. 12, this error is much
larger than the one obtained for the homogenized solution. For the order 0 homogenized solution,
the error E c (û˙ 0 ) decreases first slowly for large ε0 . This can be understood in Fig. 12, left column:
the coda is fully constructed only for ε0 ≤ 0.6. Once the coda is fully constructed, the convergence
is unexpectedly fast (in between ε20 and ε30 ) whereas we expect a convergence in ε0 only. This fact
certainly implies that, at least for this specific example, higher order terms of the asymptotic expansion
are small with respect to the leading term. This is confirmed by the introduction of the first order
correction in the calculation of the error E c (û˙ 1/2 ): its effect can be observed only for the smallest ε0
values. For small ε0 , we expect a convergence of the leading term as ε0 , rather than as ε20 . The effect
of the first order correction can nevertheless clearly be seen by improving the fit for small values of ε0 .
This can also be seen in Fig. 14 where the error for the order 0 homogenized solution, Ei (û˙ 0 ), and for
the order 0 homogenized solution supplemented by the first order correction, Ei (û˙ 1/2 ), for receivers
5 to 35 are plotted as a function of their location along the x1 axis and for ε0 = 0.15. It appears that,
when adding the first order correction to the leading term of the expansion, the error is, as expected,
always minimized. An interesting observation is that the error determined for the sole leading term
varies more rapidly with x1 than when the first order correction is taken into account. This is expected
since the fast scale (y) dependence of the first order correction implies variations of the wavefield at
the microscopic scale. Note that this error as a function of x1 is largely under-sampled in Fig. 14 as
we only have one receiver every 1km compared to the 100m long of the edge of a random element. To
investigate more closely the first order correction effect, in Fig. 15 is plotted the first order correction
û˙ 1/2 − û˙ 0 along the line CD (see Fig. 3) for t = 5.5 s, and compared to u̇ref − û˙ 0 . It can be seen that
164
x1 component
29
x2 component
0.04
relative amplitude
0.006
0
0
-0.006
18
19
20
x2 (km)
21
22
-0.04
18
19
20
x2 (km)
21
22
Figure 15. Cut along the line CD (see Fig. 3) for u̇ref − û˙ 0 (black line) and for û˙ 1/2 − û˙ 0 (red line) at t = 5.5s.
On the left graph is plotted the x1 component normalized by the maximum of u̇1,ref and on right graph the x2
component normalized by the maximum of u̇2,ref .
the fast oscillations are the same for both curves. The remaining differences are due to un-computed
higher order asymptotic terms.
Finally, in Fig. 16 is shown the leading order moment tensor correction (52) effect for the source B.
It can be seen that the moment tensor correction and the order 0 homogenized model allow to correctly
reproduce the observed strong S wave with the correct time arrivals as well as the full waveform.
In the above study, the random model was generated such that the density and the Lamé parameters were uncorrelated. Other tests were realized using other kinds of correlations between parameters
and they all give similar results. We nevertheless show here the result when only the density varies
randomly, the P and S waves velocities being kept constant in the whole domain. This case is interesting because it is known to be difficult case for another upscaling method developed by Gold et al.
(2000). For our approach, such a case presents no specific difficulty as it can be seen in Fig. 17.
4.2 Marmousi2 model example
The same homogenization procedure is applied to the Marmousi2 model described in section 2.4.2.
The spatial filter is the same as the one used in section 2.4.2, which, due to the change in velocities with
depth (and then of the minimum wavelengths), implies an evolution of the values of the ε0 parameter
from 2 at the top of the model to 0.3 at the bottom. This is a strong limitation of our filtering technique
which does not allows to obtain a roughly constant value for ε0 throughout the whole domain. This
is an aspect that should be investigated in a future work and a filtering technique allowing to adapt
locally the cutoff of the filter is probably an interesting lead to follow. In Fig. 18 are plotted the S
165
30
Y. CAPDEVILLE
x1 xomponent
0.5
0
-0.5
-1
x2 component
0.6
0.4
0.2
0
-0.2
-0.4
10
time (s)
5
15
Figure 16. Velocity traces recorder at receiver 38 for source B. The reference solution (black line) is compared
to the elastic filtering upscaling solution (green line) and to the order 0 homogenized solution with moment
tensor correction (52) (red line).
5
6
7
time (s)
8
9
Figure 17. x1 velocity component recorded at receiver 22 and source A (see Fig. 3) computed using SEM in
a model with randomly generated density variations but with constant P and S velocities (reference solution,
black line), in the corresponding order 0 homogenized medium with ε0 = 0.6 (red line) and in velocity averaged
model still with ε0 = 0.6 (green line). Note that for the velocity averaged model, only the density is low-pass
filtered with ε0 = 0.6 as the wave velocities remain constant.
166
0.0 km
0.0 km
2.5 km
5.0 km
7.5 km
10.0 km
12.5 km
15.0 km
12.5 km
15 km
31
-0.5 km
-1.0 km
-1.5 km
-2.0 km
-2.5 km
-3.0 km
-3.5 km
0.5
1.0
1.5
2.0
2.5
3.0
Vs (km/s)
0 km
0.0 km
2.5 km
5 km
7.5 km
10 km
-0.5 km
-1.0 km
-1.5 km
-2.0 km
-2.5 km
-3.0 km
-3.5 km
0
5
10
15
20
25
30
Anisotropy (%)
Figure 18. Order 0 Marmousi2 homogenized model for λ0 = 50 m. Top graph: S velocity (
graph: total anisotropy as defined Fig. 11.
p
c∗2222 /ρ∗ ) Bottom
velocity and the total anisotropy of the order 0 homogenized model. This smooth model allows to use
a trivially simple mesh compared to the original mesh presented in Fig. 7. A sample of this mesh,
with the homogenized S wave velocity in background, is presented in Fig. 19. As already mentioned
in section 2.4.2, the simulations with such a simple mesh are much faster and it took only one hour
to compute the homogenized solution compared to the seven days required to obtain the reference
solution using the same computing power. Traces recorded at the receiver location shown in Fig. 8 are
shown in Fig. 20. The traces obtained using the order 0 homogenized medium are more accurate than
the velocity filtering solution based on the same spatial filter. The fact the results are not as spectacular
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Y. CAPDEVILLE
7500 m
-1000 m
8000 m
8500 m
9000 m
-1000 m
-1500 m
-1500 m
-2000 m
-2000 m
-2500 m
7500 m
8000 m
8500 m
-2500 m
9000 m
Figure 19. Sample of the spectral element mesh (black lines) used to solve the wave equation with the order 0
homogenized Marmousi2 model. The background color is the corresponding order 0 homogenized S velocity
with the same color code as for Fig. 18.
here compared to the square random model example are mainly due to the heterogeneity spectrum of
the Marmoursi2 model which roughly decreases as 1/k (k being the wavenumber of heterogeneities),
while it is almost flat in the case of the random square model. Unfortunately, we can not pursue the
same convergence analysis as it was done for the random square model example, mainly because of the
presence of absorbing boundary conditions. Indeed, the Perfectly Matched Layers we are using (Festa
& Vilotte, 2005) are not adapted to take anisotropy into account. Therefore, the anisotropy created by
the homogenization at the domain boundaries is an issue that prevents to lead a precise convergence
analysis as the one done for the random square example. Nevertheless, the results are good enough
to show the interest of the procedure in such a case. The fact that the naive upscaling technique gives
a relatively good result for the Marmousi2 elastic model indicates that a careful study needs to be
carried out to determine in which case the non-periodic homogenization should be used face to a
naive filtering technique. For the Marmousi2 model, the answer is not obvious, even if one needs a
low precision waveform modeling. Indeed, we have not tracked for more difficult waves than direct
waves for this test. For example, for interfaces waves propagating along strong and heterogeneous
discontinuities of the model, the use of non-periodic homogenization is expected to be more critical
than for transmitted waves.
168
traces
residuals
x2 (vertical) component
x2 (vertical component)
33
0.05
0.5
0
0
-0.05
-0.5
x1 (horizontal) component
x1 (horizontal) component
0.4
1
0.2
0
0
-0.2
-1
1.5
2
2.5
3
time (s)
1.5
2
2.5
3
time (s)
Figure 20. Left column graphs: velocity traces recorded at receiver 48. The reference solution (black line) is
compared to the velocity filtering upscaled model (green line) and to the order 0 homogenized solution (red
line). Both vertical (top graph) and horizontal (bottom graph) components are shown. Right column graphs:
residuals between the reference solution and the order 0 homogenized solution (red line) and between the reference solution and the velocity filtering upscaled model (green line) for both vertical (top graph) and horizontal
components (bottom graph).
5 CONCLUSIONS AND PERSPECTIVES
We have presented a two-scale homogenization procedure which can be applied to the upscaling process in non-periodic media. The critical point of this procedure is the practical construction of the fast
(microscopic) part of the density and elastic tensor cε0 (x, y) implied in well-known classical homogenization procedures (in periodic media). Once this is done, the homogenization expansion is similar
to the one of classical two-scale periodic homogenization. In the general case, it is not possible to go
beyond the calculation of the leading order of the expansion, and that of the first order corrector. This
nevertheless allows to find an effective medium to any general elastic medium with fast variations in
all spatial directions. It also allows to retrieve the leading order corrector to a moment tensor source
type as well as the first order correction at a receiver location, and then to take into account local structure effects. The study of two examples in this article, the random model, as well as the Marmousi2
geological model, demonstrates the efficiency and accuracy of the method. This is an important step
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forward since the results of Backus (1962), which are applicable to non-periodic but layered media,
and compared to the classical two-scale homogenization theory, which is applicable to media showing
fast variations in their physical properties in higher spatial dimensions, but only in the periodic case.
As already mentioned when studying wave propagation throughout the Marmousi2 model, the
low-pass filter used in the homogenization procedure has a the spatially-constant wavenumber content, which may not be appropriate when applied to media where strong variations in the heterogeneity spectrum arise. Other kinds of filtering, like ones involving wavelets for instance, may be more
pertinent - and this will be the topic of a future work. The issue of boundary conditions in an homogenization procedure has not been treated in this article. It will be important to tackle this problem in
a future work as it is known that the boundary conditions are important for surface waves and that
the subsurface structures strongly influence waveforms (Capdeville & Marigo, 2007; Capdeville &
Marigo, 2008).
The practical extension to 3-D is obviously a priority. It should not be a problem as the theoretical
difficulties that were faced when going from 1-D to 2-D (see Capdeville et al. (2010)) are not present
when going from 2-D to 3-D.
The range of applications of such a development seems wide. One of then is the waveform
modeling in complex media: for a given medium being able to upscale its properties to the wanted
scale (knowing the corner frequency of the source) and to use the leading order effective medium
(ρε0 ,∗ , cε0 ,∗ ) in the favorite wave equation solver of a user, like finite differences or the Spectral Element Method, is an important alternative to the classical complex, and often impossible, meshing of
the original medium. Note that, if the difficulty of the meshing for the forward problem and its consequences on the numerical cost can be avoided when using a homogenized model, the design of a mesh
(or of multiple small meshes) for the homogenized problem itself can not be avoided. Nevertheless,
the design of this mesh can be based on tetrahedron elements (even if the wave equation solver is
based on a hexahedral mesh) and the mesh sampling is independent on the frequency cutoff of the
seismic sources that will be used. Moreover, this or these meshes will be used only once for a given
elastic model. For such a type of applications, a careful study of the type of elastic models, type of
waves, etc, for which the homogenization is required with respect to more naive upscaling remains to
be done.
Another application is related to the study of the time arrival of the ballistic phase, in seismic
exploration or geophysical imaging. It is known that this time arrival is only sensitive to a smooth
version of the real medium. A natural question is therefore: is this smooth medium the elastic model
(ρε0 ,∗ , cε0 ,∗ ) for a large ε0 ? Fig. 12 seems to suggest it, but this should be studied with care as it is
probably not the case.
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35
Using our homogenization procedure for applications to the inverse problems is also in sight.
A major and well-known result of our work is that microscopically (with respect to the wavefield)
isotropic media, are macroscopically fully anisotropic, and this should be taken into account in tomographic studies for instance. Moreover, when inverting full waveforms, it may also not be a good
idea to track for interfaces as they are homogenized (that means, smoothed) by the wavefield anyway.
Finally, let us notice that this development gives the opportunity to build a multi-scale parametrization
for the elastic properties and a well posed parametrization to take into account local effects on sources
and receivers, of the inverse problem.
Some applications to other fields but with similar equations, like the stress loading of a complex
geological structure, could also be considered.
A patent (Capdeville, 2009) has been filed on the non-periodic homogenization process by the
"Centre national de la recherche scientifique” (CNRS) (this is by no mean a restriction to any academic
research on the subject).
6 ACKNOWLEDGMENTS
The authors would like to thank Gaetano Festa for kindly providing his 2-D Spectral Element program
that was used and modified for this article. The comments and suggestions of two anonymous reviewers greatly helped to improve the manuscript. The computations were performed using the Institut
du développement et des ressources en informatique Scientifique (IDRIS)’s and Institut de Physique
du Globe de Paris (IPGP)’s clusters. This work was funded by the Agence National de la Recherche
(ANR) MUSE project under the blanc program.
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APPENDIX A:
Spatial low-pass filter design. To be able to separate the scales around λ0 , we introduce a mother filter
wavelet w(x) such that its power spectrum is



1
for k ≤ a ;

 |k|−a
1
w̄(k) =
for |k| ∈]a, b[ ; ,
2 1 + cos π b−a



 0
for |k| ≥ b .
(A1)
where k = |k|, a and b are two real around 1 defining the tapper transition from 1 to 0 of the low pass
filter. The space wavelet in the space domain is obtained with a Hankel transform:
Z ∞
w̄(k)J0 (k|x|)kdk ,
w(x) =
0
where J0 is the Bessel function of the first kind of order 0. Note that we have
R
R2
(A2)
w(x)dx = 1. We
define wk0 (x) = k0 w(xk0 ) the same but contracted (if k0 > 1) wavelet of corner spatial frequency
R
k0 . We still have R2 wk0 (x)dx = 1. If a = b = 1, the low pass filter has a perfectly sharp cutoff for
k = k0 . In that case the drawback is the space support of wk0 is infinite and cannot be truncated with
a good accuracy. A solution is to chose a smaller than 1 and b larger than 1 knowing that the largest
|b − a| is, the best a compact support for wk0 is an accurate approximation. An example of such a
wavelet is shown in Fig. 1.
175

Université Paris VII – Denis Diderot Institut de Physique du Globe de

Transcription

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