Potential-Field Inversion for Topographic Sources with Uncorrelated

Index
Table of contents
Potential-Field Inversion for Topographic
Sources with Uncorrelated Top and Bottom
F. Caratori Tontini, L. Cocchi, C. Carmisciano
Istituto Nazionale di Geofisica e Vulcanologia, Sede di Portovenere, ITALY
Summary
Inversion of large-scale potential-field anomalies usually proceeds in the Fourier domain, where a
large amount of data can be properly addressed. The commonly adopted geometry is based on a layer
of constant thickness, i.e. with a bottom level at a fixed distance from the top level. We propose a
method that overcomes this limiting geometry by inverting in the usual iterating scheme for top and
bottom levels of any shape. Randomly generated synthetic models will be explored both for gravity
and magnetic data, and finally the good performance of this method will be tested by the real isostatic
residual anomaly of the Tyrrhenian Sea in Italy. The final result is a density model that allows the
investigation of the distribution of the oceanic crust in this region, which is still a point under
discussion.
Introduction
Large-scale potential-field anomalies are usually inverted in the 2-D limit, being the horizontal
extension of the source effectively greater than the depth extension. Linear inversion methods by
discrete cells are particularly ineffective dealing with a large amount of data, since the dimensions of
the associated matrices are of the same order of the squared number of data. Increasing with the
amount of data, usually the inversion is performed in the Fourier domain. Since the pioneering work of
Parker (1972), different methods have been developed. Particularly Parker and Huestis (1974) have
provided the iterative framework for the inverse problem. Oldenburg (1974) has modified this method
in order to determine the geometry of the density interface from the gravity anomaly. This approach
has been successfully used also in order to compute the field at a constant level by using an equivalent
source distribution (Hansen and Miyazaki, 1984; Pilkington and Urquhart, 1990). More recently
Hussenhoeder et al. (1995) have inverted magnetic data directly also in the case of uneven tracks.
However, dealing with the inversion finalized at determining the density or magnetization distribution,
many of these methods are characterized by assuming a constant thickness for the layer. Maybe in
some geological settings this could be a straightforward assumption, for example in the analysis of
marine magnetic anomalies (Macdonald et al., 1983). In the general case however the constant
thickness of the layer may be a limiting assumption and thus the density or susceptibility obtained
under this hypothesis may be significantly different. We have modified thus the method and its
theoretical framework in order to deal with any kind of top and bottom of the layer. This has been
obtained by a simple mathematical modification of the method by Parker and Huestis (1974).
Methodology
Parker (1972), has demonstrated that the total-intensity magnetic anomaly ∆T of a layer confined
between horizontal surfaces zt and z b, characterized by a magnetization distribution M, can be written
in the Fourier domain as
+∞ (− k )
µ
ℑ[∆T ]= 0 θ mθ f exp(k z 0 )∑
ℑ M (z tn − z bn ),
2
n!
n =1
n
[
]
EGM 2007 International Workshop
Innovation in EM, Grav and Mag Methods:a new Perspective for Exploration
Capri, Italy, April 15 – 18, 2007
[1]
where θm and θ f are phase factors depending on the directions of magnetization and ambient field and
ℑ denotes the Fourier operator. Equation [1], and its equivalent gravitational expression which we do
not report here for conciseness, can be used to evaluate anomalies generated by topographic sources or
to calculate isostatic residual gravity anomalies, and moreover they can be introduced in the inverse
models by an iterative procedure.
In the case of a layer of constant thickness t in fact, Parker and Huestis (1974) have demonstrated that
by isolating the n = 1 term of the summation of Equation [1], we obtain a self-consistent equation
+∞ (− k )
2ℑ[∆T ]
ℑ[M ]=
−∑
ℑ Mz tn ,
µ 0θ mθ f exp(k z 0 )(1 − exp(− k t )) n= 2 n!
n
[ ]
[2]
which can be iteratively solved by making an initial guess for M and then solving for a revised
magnetization distribution until some numerical convergence is reached. We propose a straightforward
modification of this approach in order to deal with any kind of top and bottom surfaces. It is simply
starting from Equation [1] that we can isolate again the n = 1 term of the summation as follows
+∞ (− k )
− 2ℑ[∆T ]
ℑ[M (z t − z b )]=
−∑
µ 0θ mθ f k exp(k z 0 ) n= 2 n!
n −1
[ (
)]
ℑ M z tn − z bn .
(
[3]
)
We can note at this level that since zt = zb is a root of the polynomial ztn − zbn , it is also clear that (zt –
zb) is a divisor of this polynomial.
This means also that the term ztn − zbn can be factorized as (zt – zb) Pn-1 (zt – zb), where P n (zt – zb) is
the complete polynomial of order n:
(
)
Pn (x, y ) = x n + x n −1 y + L + xy n −1 + y n ,
[4]
so that defining a weighted magnetization Mw=M(zt – zb), we obtain a new self-consistent equation
+∞ (− k )
− 2ℑ[∆T ]
ℑ[M w ]=
−∑
µ 0θ mθ f k exp(k z 0 ) n = 2 n!
n −1
ℑ[M w Pn−1 (z t , z b )],
[5]
which can be solved iteratively for Mw until some convergence is reached, and the true magnetization
M is thus recovered by dividing for (zt – zb ). We report here also the gravitational equivalent
expression:
+ ∞ (− k )
− ℑ[g z ]
ℑ[ρ w ]=
−∑
2πγ exp(k z 0 ) n= 2
n!
n −1
ℑ[ρ w Pn −1 (z t , z b )] ,
[6]
with similar notations. In Figure 1 we show a synthetic test highlighting the good performances of the
method. The top, bottom and magnetization/density distributions are obtained by synthetic fractal
surfaces.
EGM 2007 International Workshop
Innovation in EM, Grav and Mag Methods:a new Perspective for Exploration
Capri, Italy, April 15 – 18, 2007
Figure 1. Results from a synthetic test.
Conclusions
As a conclusion we show the result of the method obtained inverting the isostatic residual gravity
anomaly of the Tyrrhenian Sea in Italy, evaluated by the GEOSAT and ERS-1 satellite data of
Sandwell and Smith (1997), to define the distribution of oceanic crust in this region. It is well-known
in fact in previous literature that the oceanic crust emplacement took place in the Southern portion of
the Tyrrhenian Sea, where the roll-back movement has recorded the highest velocities. Since the
change of rheologic characteristics from continental to oceanic crust may be considerable in terms of
crustal density, a reconstruction of the density distribution of the Tyrrhenian Sea can be used to define
the oceanic portion of the basin, which is characterized by the highest density values as happens
within the bold lines of Figure 2.
EGM 2007 International Workshop
Innovation in EM, Grav and Mag Methods:a new Perspective for Exploration
Capri, Italy, April 15 – 18, 2007
Figure 2. The Tyrrhenian Sea gravity model (a) bathymetry; (b) bottom estimated by an Airy type model; (c) isostatic
residual anomaly; (d) density distribution. The bold line in subplot (d) encloses the supposed blocks of oceanic crust.
References
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EGM 2007 International Workshop
Innovation in EM, Grav and Mag Methods:a new Perspective for Exploration
Capri, Italy, April 15 – 18, 2007