Seismic Image Waves

Transcription

Seismic Image Waves

Geophys. J . Int. (1996) 125,431-442
Seismic image waves
Peter Hubral,’ Martin Tyge12and Jorg Schleicher’?”
‘ Geophyrrc s institute, Karlsruhe Univerrrty, HertzstraJe 16,76187 Kurlsruhe, Germany
Department ofilpplred Mathematics, IMECCIUNICAMP, C P 6065,13081-970 Campinar, SP, Brazil
Accepted 1995 December 8. Received 1995 November 10; in original form 1995 July 20
Key words: seismic waves, seismic reflection imaging.
INTRODUCTION
The aim of this paper is to contribute to an interesting theory
proposed by Goldin (method of discontinuities in seismic
imaging; paper presented at the SvenFest, 1994 March 6-8,
Golden, CO) with close reference to the work of Fomel
(1994a, b). At the present stage, this theory is restricted
to simple objects, such as homogeneous acoustic media.
Nevertheless, it can be looked upon as a general approach in
order to formulate partial differential equations (which we call
image-wave equations) for various fundamental reflectionimaging problems. By introducing and making use of the
concept of a seismic image wave, a geometrically appealing
explanation for the general approach can be found.
An image wave is not a physical wave but behaves like one.
The concept is best understood with the help of Fig. 1. Fig. l(a)
depicts the familiar situation of a propagating (physical) body
wave. Three wavefronts of an elementary wave are shown at
three different instants of time. In Fig. l(b), another familiar
situation is shown; however, this is commonly not viewed
as an example of wave propagation. We see three different
(purely kinematically) migrated images of one seismic reflector,
obtained with three different (constant) migration velocities.
By comparison with the situation depicted in Fig. l(a), it is
not difficult to accept that this can be conceptually understood
*Now at: Department of Applied Mathematics, IMECC/UNICAMP,
CP 6065, 13081-970 Campinas, SP, Brazil.
0 1996 RAS
as a certain kind of ‘wave propagation’. In this case it is the
image of the seismic reflector that ‘propagates’; we can call
this phenomenon the ‘image wave’. In the same way as Fig. 1(a)
depicts a physical wavefront at three different instants of time,
Fig. l(b) can be said to depict an image wavefront at three
different ‘instants of migration velocity’.
Comparing Figs l(a) and (b), we see that they have much
in common. Their common features are stressed in Fig. l(c),
which is easily seen to be a generalization of the situations of
Figs l(a) and (b). Fig. l(c) shows a set of curves in the (a, b)
domain, where for each curve a parameter c is kept constant.
Identifying the parameter c with the time variable t and the
coordinates (a, h) with Cartesian space coordinates (x, z), we
immediately have the same situation as depicted in Fig. l(a).
O n the other hand, by identifying c with the migration velocity
v and the coordinates (a, b) with (x, z), we observe the same
situation as depicted in Fig. l(b). Not much imagination
is necessary to accept that Fig. l(c) serves to describe many
other seismic-reflection-imaging problems, some of which are
described in detail in this paper. For example, by identifying c
with v, but (a, b) with (x, t ) , we can interpret the same set of
curves as time-migrated reflector images for three different
migration velocities.
As an important example of seismic image waves, we
examine the depth- or time-migrated reflector image of a
primary reflection [e.g. a seismic horizon in a zero-offset (ZO)
or common-midpoint (CMP) stacked section] taken as a
function of the (constant) migration velocity. When the
43 1
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SUMMARY
The concept of a seismic image wave is introduced and explained with the aid of some
examples. Seismic reflector images in various domains (e.g. depth-migrated reflections
in the depth domain or common-offset reflections in the time domain) behave like
snapshots of elementary body waves. These ‘propagating’ images are thus referred to
as ‘image waves’. The propagation variable, however, is now not time as it is for
physical waves. It can be any other parameter involved in the seismic imaging process,
for example the migration velocity or the common source-receiver offset. In parallel
to the acoustic wave equation, which governs true elementary physical body waves,
partial differential equations (here called the image-wave eikonal equation and the
image-wave equation) can be derived that describe the propagation of image waves as
a function of the problem-specific propagation variable. The concept of image waves
is suited to solving a variety of different imaging problems. Image waves can be
propagated, for example by a finite-difference or spectral-method algorithm.
432
P. Hubral, M . Tygel and J. Schleicher
X
X
c
v
l
J
&
This describes the propagation of an acoustic pressure wavefield p ( x , z, r) in a medium with velocity v, assumed to be
constant throughout. A subscript x , z, or t denotes the partial
derivative of p ( x , z, t ) with respect to the corresponding variable. The kihematics of acoustic waves (i.e. the propagation of
wavefronts) are described by the eikonal equation
v=v 3
1
T;+ T I = -
c=c3
-
Figure 1. (a) Propagating wavefront at three different instants of time.
(b) Depth-migrated reflector images for three different migration
velocities. (c) Graph showing Cartesian coordinate axes a and b, which
define a suite of curves, where the parameter c is constant for each one.
migration velocity is changed continuously, the migrated
reflector image propagates in a similar manner to an elementary body wave. I n the case of a time migration, it propagates
in the time domain; in the case of a depth migration it
propagates in the depth domain. Different depth- or timemigrated images resulting from different migration velocities
represent different ‘snapshots’ of this image wave. The construction of a variety of migrated image-wave snapshots is often
desirable in order to study the influence of the migration
velocity o n the reflector images (e.g. in connection with a
velocity analysis o r in order to find the best possible images).
Another example of an image wave, which we will investigate
below, is the primary-reflection image in a common-offset
(CO) section of a subsurface reflector below a constant-velocity
overburden. This reflector image also propagates when changing the CO continuously. This latter type of propagation is
implicitly performed when changing a CO into a simulated
ZO seismic section. This change is often done either in one
step by a migration-to-zero offset (MZO) or in two steps by
applying the normal-moveout (NMO) followed by the dipmoveout (DMO) correction.
For all kinds of seismic imaging problems, one can specify
certain image waves. For each of them, one can then formulate
a problem-specific image-wave equation. This describes the
image-wave propagation as a function of the problem-specific
propagation variable (which is the migration velocity or the
02
’
This comes from the above acoustic wave equation ( 1 ) after
inserting the ray-theoretical ansatz
(3)
Here f ( t ) is a high-frequency pulse that may slowly change its
shape along the wavefront t = T ( x , z). Thus, the problem of
propagating a given pressure wavefield [incorporating a set of
individual elementary body waves of type (3)] in (x, z) space
from the instant of time to to the instant of time t > to can be
expressed symbolically by
P ( X > z,
t o ) P ( X 3 z, t ) ’
(4)
In the seismic literature, a variety of techniques [e.g. finitedifference (FD) methods and spectral or integral techniques]
exists to solve the initial-value problem (4), which will not be
further discussed for that reason
The main purpose of this paper is to show how differential
equations of the form of the wave equation (1) and the eikonal
equation (2) can be designed for various seismic-reflectionimaging problems. In the next section we define some of these
problems and in the ensuing section we state and interpret the
solutions. All mathematical derivations can be found in the
Appendix.
-+
SEISMIC I M A G I N G PROBLEMS
In this section we consider various reflection-imaging problems,
for each of which an initial-value problem analogous to eq. (4)
can be formulated.
Depth remigration
The imaging process required in order to construct (from a
given depth-migrated seismic section, including a set of arbitrarily complicated subsurface reflector images) depth-migrated
seismic sections that would have resulted for a continuum of
(constant) migration velocities is subsequently referred to as
depth remigration. The term ‘remigration’ is chosen as a
0 1996 RAS, G J I 125, 431-442
v =
Y=V2
common offset in the above two cases). In connection with
each image-wave equation there exist also (1) an image-wave
eikonal equation (which propagates the image wavefronts) and
(2) Huygens image waves, i.e. (hypothetical) elementary pointsource image waves that are ‘excited at each point of a
propagating image wavefront. These two concepts are helpful
in formulating the image-wave equation for the specific imaging
problem of interest.
Image-wave equations can be derived using a general
approach. Before applying this approach to the examples
mentioned above, let us explain it by returning to the familiar
situation of a propagating (physical) body wave (Fig. la)
described by the homogeneous acoustic wave equation
Seismic image waves
A comparison with eq. (3) reveals that the function v = V ( x ,z)
can be considered as an ‘image-wave eikonal.’ Each elementary
image wave p(x, z, u ) in eq. ( 5 ) propagates when continuously
changing the migration velocity t), in the same way as the
physical elementary wave p(x, z, t ) in eq. (3) propagates when
continuously changing the time t. Snapshots of the image wave
( 5 ) in the (x, z ) domain at different instants of migration
velocity u thus describe the selected depth-migration reflector
image for the respective value of u. The problem of transforming
the snapshot of the total image wavefield p(x, z, uo), involving
many individual reflector images of type ( 5 ) , to the snapshot
of the total image wavefield p(x, z, u) can then, by analogy with
the initial-value problem (4), be symbolically expressed by
Here, u, is some arbitrary constant initial migration velocity.
As shown below, there exists for this particular imaging problem
a partial differential equation (the image-wave equation) for
the zero-offset or post-stack depth-migrated section p(x, z , u),
with u as the independent propagation variable.
Time remigration
Let us now assume in Fig. l(c) that all parameters are as in
the previous case, apart from the parameter b, which now
represents time, i.e. b = t. Let us consider again a ZO (or poststack) situation, so that a curve implicitly given by u = e ( x , t )
in the time domain describes the image wavefront of a selected
time-migrated reflector image
(7)
This reflector image again propagates like a body wave [now
in ( x , t ) space] if the constant migration velocity u changes
continuously. Note that although the migration velocity u may
be identical to the one used in depth migration, the shape of
the migrated reflector image will be different, and thus the
image-wave eikonal P(x, t ) in the time domain will be different
from V ( x ,z ) in the depth domain.
The image wavefield p ( x , t, u) consists of many individual
image waves of type (7) and the respective image-wave equation
0 1996 RAS, G J I 125, 431-442
describes the propagation
P(X,
t , u,)
-.+P ( X > t, 0)
(8)
from the instant of uo to the instant of u. In other words, given
(1) the (ZO or post-stack) time-migrated seismic section,
p(x, t, u,), i.e. the snapshot of the total image wavefield
(computed, say, for the wrong migration velocity u,) and ( 2 )
the particular image-wave equation, yet to be found, one can
derive any other time-migrated section, p ( x , t, u), from p(x, t, u,)
for a continuous range of migration velocities (for u > u ,
and/or u < u,).
Migration to zero offset (MZO)
This particular reflection-imaging problem combines the
familiar N M O and D M O corrections into a single step. Let
us consider Fig. l(c) with u = <, b = t and c = h, where h
signifies half the common offset and 5 the midpoints of the
shot-receiver pairs moved along the seismic line. Each curve
in Fig. 1(c) describes, then, the reflection traveltime of a selected
CO reflector image, resulting from the same depth reflector
but recorded with a different offset h. The recorded image
wavefield can be represented, as above, by
(9)
Each curve h = H(5, t ) is defined by a different constant value
of h. To perform the MZO imaging process implies solving
the following initial-value problem:
P ( 5 , t, h = h,)
--*
p ( 5 , t, h = 0 ) .
This can be understood as a propagation in (t,t ) space of the
total image wavefield p ( & t, h), i.e. the CO section incorporating
a set of (elementary) image waves of type (9), from the instant
at which h = h, to the instant at which h = 0. In other words,
for each value of h, the function p ( 5 , t , h) describes a snapshot
of the total image wavefield in the (5, t ) domain. Note that
this transformation depends on the (constant)medium velocity
u, assumed to be known.
Dip-moveout (DMO) correction
As indicated before, the M Z O image transformation ( 10) can
be subdivided into two familiar seismic imaging processes.
These are the N M O and DMO corrections. Applying the
NMO correction to a CO section p ( 5 , t, h = h,) involves performing the following time-domain stretching operation (which
cannot be conceived of as an image-wave propagation):
P(5, 4 h = ho)
-+
8(5,z,h = M .
Here, the time stretch
by
T~ = t 2 - 4h2/u2,
T
(11)
(i.e. the N M O correction) is defined
(12)
where u is the medium velocity. After this operation, the NMOcorrected CO section can be represented as consisting of
elementary image waves
To construct the desired simulated ZO section p(<, T, h = 0 ) =
p(<, t , h = 0) from this section, i.e. the DMO correction, we
generalization of the term ‘residual migration’ (Rothman, Levin
& Rocca 1985; Larner & Beasley 1987), which generally implies
construction of a new depth-migrated image directly from an
old one, by making a small change to the migration velocity.
A remigration as discussed here, however, allows not only for
one small change, but for a continuum of arbitrarily large
changes of the migration velocity.
Let us again consider Fig. 1( b), i.e. assume that in Fig. 1 (c)
we have u = x , b = z , and that c represents the migration
velocity u. The curves in Fig. l(c) describe, then, the locations
of one searched-for subsurface reflector image, implicitly given
by functions u = V ( x ,z), for three different values of u. These
reflector images result from a depth migration of the reflection
data acquired with a certain measurement configuration. We
choose here a ZO configuration that is also representative for
a CMP-stacked section. The signals attached to the migratedreflector location u = V ( x ,z ) can be interpreted as one particular elementary seismic image wave (i.e. a depth-migrated
reflector image) described by
433
434
P. Hubral, M . Tygel and J . Schleicher
have to solve the initial-value problem
P(5,7, h = ho) P(5, 7, h = 0 ) .
(14)
+
GENERAL APPROACH
In this section, we describe how the image-wave equations for
the above seismic imaging problems are derived by a procedure
hereinafter referred to as the general approach. We start by
using the ‘design’ of the acoustic wave equation (2), briefly
presented in the Introduction, as an example.
Let us outline the general approach with the help of formulas
(1) to ( 3 ) .It is well known that the wave equation (1) provides,
with the ray ansatz (3), the eikonal equation (2). Its solution
for a homogeneous medium with velocity u,
t = T ( x ,z)= to
+ V1 J ( x
-
-
xoy
+ (z
-z0y,
(15)
is nothing more than the kinematic description of a classical
elementary Huygens wave (here simply called the Huygens
wave). In (x,z) space, this is a circle with radius u ( t - to),
centred at ( x o , zo). In (x,t ) space (e.g. for z = O ) , it is a
hyperbola.
Let us assume that a continuous set of Huygens waves
originates at all points along a wavefront to = T(x,z). For this
purpose, assume in Fig. l(c) that a = x , b = z and c = t. Then,
according to Huygens principle, each wavefront t = T(x,z)for
a constant value t > to is nothing more than the envelope of
all elementary Huygens waves (15) placed into the wavefront
to= T ( x ,z). The procedure described above represents the
classical steps taken to arrive at the most fundamental results
of geometrical seismics (optics) from the wave equation (1).
This familiar procedure represents, on account of being physically justified, the forward process, or, metaphorically speaking,
water flowing from the tap into the tub. But can the chain of
steps described above be reversed, i.e. can eq. (1) be derived
from eq. (15), thus making the water flow from the tub into
the tap?
The answer is, yes, it can be done. The basic step necessary
in this respect is to find the eikonal equation (2) from formula
(15). This is achieved by substituting the partial derivatives of
the eikonal T(x, z),
T, =
VJfX
(x- xo)
x0y (z - zo)2
-
+
3
r,= u J ( x
( z - zo)
- x0y
+ (z-
zo)2
’
(16)
back into formula (15). Now that the eikonal equation (2) is
derived from the Huygens wave (15), we have to ask whether
we can also obtain the wave equation (1) from the eikonal
equation (2). This is indeed possible when certain assumptions
are made. It is easy to accept (Goldin 1990) that any partial
differential equation of the form
Depth remigration
To design the image-wave equation for this problem let us
look at Fig. 2. It shows a continuous subsurface reflector X o ,
shown as a string of diamonds at the instant of the (incorrect)
migration velocity uo. From each diamond an elementary
Huygens image wave emerges, which is plotted at the instant
of migration velocity u > uo. Admittedly, these Huygens image
waves no longer resemble true classical Huygens waves, as
they are not ‘surrounding’ their origins. What they have in
common with true Huygens waves, however, is that they
provide the subsurface reflector image C, i.e. the propagated
image wavefront a t the instant of v as their common envelope.
The envelope (bold curve in Fig. 2) will correspond to the
target reflector if u corresponds to the exact migration velocity.
The Huygens image waves are constructed as follows: points
P(xo, zo) on the reflector Zo are (kinematically) ZO demigrated
with the velocity uo. This provides a set of diffraction hyperbolas (also called Huygens curves) in the time domain of the
form
1
Pxx
+ P z z - SP,,+ F ( P n P z * P t > P> x , z,t ) = 0 ,
(17)
will give the same eikonal equation (2) as a result of ansatz
(3), irrespective of the particular form of the function F , which
depends on the listed arguments only. In other words, all wave
equations of type (17) share the same kinematic behaviour of
These curves have the ZO reflection-time curve (from which
the reflector X o was constructed) as their common envelope.
ZO migration of each diffraction hyperbola with the new
migration velocity u provides a Huygens image wave that
appears to have its origin at P(xo,zo) on the incorrectly
0 1996 RAS, G J I 125, 431-442
Wave propagation
wave propagation. The simplest of these equations is surely
the one with F = 0. So, why not accept eq. (17) without the
function F , i.e. choosing F = O? We will then have found an
equation that kinematically propagates one or many elementary waves of the form (3), say, from the instant to to the
instant t > t o , provided the initial propagation direction of all
waves is also specified. In mathematical terms, we can then
solve the initial-value problem (4) by involving the wave
equation (17) in its simplest form, i.e. without the lowerorder terms.
Note, however, that an image-wave equation ‘derived’ in
this way will be, in principle, incorrect with respect to the
propagation of amplitudes. To design a dynamically correct
image-wave equation for a certain problem, one must also
consider the corresponding transport equation. In this respect,
the function F plays a relevant role. It is only accidental that
for the true wave equation, F = O is also the dynamically
correct choice. As we consider only kinematic properties of
image-wave propagation, we have no criterion to decide
whether this may also be the case for any of the image-wave
equations derived below.
The above design strategy of the wave equation has shown
that reversing the classical chain of steps from eq. (1) via eq. (2)
to eq. (15) enables one to ‘derive’ the wave equation (1) from
the Huygens wave equation (15). It would appear that this
inversion can be regarded only as a rough and ready method
of arriving at the wave equation. However, if we regard the
wave equation (1) as an image-wave equation, i.e. if we demand
from it no more than the propagation of image waves with
desirable kinematic properties (ignoring their dynamic characteristics), we are not doing anything illegal. With this understanding, the design procedure outlined above can now be
applied to the aforementioned imaging problems.
Seismic image waves
435
Offset-0 m,vo=3000m/s,v-3500m/s
0
I
Q
---
I
I
I
I
Wrongly migrated reflector C,
Depth remigration Huygens image waves
-200
-600
-400
-200
0
400
200
600
Distance [m]
Figure 2. An incorrectly migrated subsurface reflector constructed for the migration velocity uo is indicated by a string of diamonds. From each
diamond an exploding Huygens image wave emerges, which is shown at the instant u > uo. The envelope of all Huygens image waves forms the
migrated target reflector (bold curve) constructed for the migration velocity u.
migrated reflector image Zo. In fact, if v is continuously
increased, such that v > vo, or continuously decreased, such
that v < v,, the elementary Huygens image waves will propagate away from each point on the (frozen) reflector image C,.
They behave just like the classical true Huygens waves, which
would propagate away from a fixed true (frozen) wavefront
to = T(x,z ) and form an advanced wavefront t = T(x,z) as
their envelope for t > to. The Huygens image wave for depth
remigration that emanates from point P ( x o ,zo) on Zo is given
by
z=-Jz;+-i-(x-x,)2
V
vi
for v # vo (see Appendix). It results from (kinematically) ZO
migrating each point on the diffraction hyperbola (18) from
the (x, t ) domain into the (x, z ) domain with the migration
velocity v. This provides a set of lower half-circles in the (x, z)
domain with formula (19) as the common envelope.
We now replace the migration velocity v in eq. (19) by the
image-wave eikonal V(x, z) and take partial derivatives of this
modified equation with respect to x and z (see Appendix).
Substituting these partial derivatives again into eq. (19) and
performing an appropriate elimination, we obtain the imagewave eikonal equation
V
v:+ v;--v,=o.
Z
( 20)
The same image-wave eikonal equation obviously results from
0 1996 RAS, GJI 125, 431-442
V
PXX
+ P z z + ZP”.
=0
(21)
with the ansatz (5). With the help of this equation, the initialvalue problem (6) can now be solved. For this purpose, the
initial value p(x, z, vo) should be obtained beforehand by a
depth migration with any algorithm (e.g. the algorithm of Stolt
1978) for the incorrect migration velocity vo.
Time remigration
v -v;
VO
the image-wave equation
A point p ( x o ,to) in a time-migrated section p ( x , t , v), which
was constructed from a ZO (or CMP-stack) section for the
migration velocity v, relates to a point P ( x o , zo) in the corresponding depth-migrated section p ( x , z, v) by the following
depth-to-time stretch relationship:
(XO,
2z*/vot
--* (XO,
to)‘
(22)
This allows the transformation of the depth-remigration
Huygens image wave (19) from the (x, z) domain into the ( x , t )
domain, thus providing the time-remigration Huygens image
wave
4(x - xo)2
t = Jt;+
v;
- u2
In the same manner as before, from this Huygens image wave
(23) (see Appendix) one can derive the time-remigration
-600
436
to < t , < t. It also describes an imploding Huygens wave for
point P(x, z ) on t l = T ( x ,z) in the (x,,, z,,) space and in the
time range t , > t > to. Thus a derivation of the eikonal equation
image-wave eikonal equation
t
vef
-
4
=0 ,
which describes the kinematics of image waves of type (7). The
image-wave equation that propagates time-migrated reflector
images of the form po(x, t ) f [ u - %(x, t ) ] directly in the (x, t )
domain, as a result of continuously changing the (constant)
migration velocity v, is (Fomel 1994a)
4p,,
+ vtp,,
=0.
(25)
Exploding versus imploding Huygens image waves
All of the above remigration problems can be solved (see
Appendix) by formulating the respective remigration Huygens
image waves. These can be looked upon as generalizations of
classical ‘exploding Huygens waves’, which we simply called
Huygens waves above. In fact, the remigration problems could
also be solved with ‘imploding Huygens image waves’, as well
as for other elementary wavefronts. Let us define a n imploding
Huygens wave in connection with the propagation of the true
wavefront t = T(x, z ) controlled by the eikonal equation (2) in
the range to < t < t l . The classical design strategy to propagate
a wavefront is to place exploding Huygens waves of the type
(15) into all points P(x,,, zo) of the frozen wavefront to = T(x,z )
and construct the new propagated wavefront t = T(x,z) as the
envelope of all exploding Huygens waves (15) defined for
t > to. However, the wavefront propagation in the indicated
range could also be described by placing the origins of so-called
imploding Huygens waves into all points P ( x , z ) of the
advanced (frozen) wavefront t , = T ( x ,z). An imploding
Huygens wave is then defined by formula (15) for t < to.
In fact, imploding Huygens waves are simply constructed
from exploding ones due to an inherent duality. Both are in
fact described by one and the same formula. Eq. (15) describes
an exploding Huygens wave for point P ( x o , z o ) on the
wavefront to = T(x,z ) in the (x, z ) space and in the time range
Let ( and t denote the coordinates of a point P ( ( , t ) in the
C O section. To solve the M Z O problem, let us first consider
the inverse problem. This is the simulation of the C O section
p ( ( , t, h = h,) from the Z O section p ( 5 , t , h = 0). For brevity,
we will refer to this problem as a migration to CO (MCO).
An exploding M C O Huygens image wave is identical to an
imploding M Z O Huygens image wave. This latter wave, in
0 1996 RAS, G J I 125, 431-442
Note that formula (25) differs from the corresponding equation
derived by Fomel (1994a) by a factor 4 in the first term. This
is because Fomel used V = v/2 instead of v in his work.
Given a time-migrated section, p ( x , t, uo) say, for the
(incorrect) migration velocity vo, all other time-migrated sections, p ( x , t, v), can be obtained by applying the image-wave
equation (25) in order to solve the initial-value problem (8).
One may expect that, by analogy to the depth remigration
problem, a previously migrated image (obtained with a certain
migration velocity uo) is necessary as an initial value. It is,
however, interesting to observe that this is not the case. The
ZO or CMP-stacked section can be taken to represent the
time-migrated image wavefield for zero migration velocity,
p ( x , t, uo = 0) (Chun & Jacewitz 1981 ). Hence, this section can
serve directly as the initial value in the remigration imagewave equation (25). The initial-value problem ( 8 ) is then to
transform the initial image wavefield p ( x , t , u,, = 0) directly into
the desired image wavefield p ( x , r , u ) . In fact, by a suitable
implementation (e.g. in the form of an FD scheme), p(x, t, v)
will not only be provided for one single value of v, but also
for the whole range of migration velocities from u,, = 0 up to
a certain limiting value. In this way, a n optimal migration
velocity u can be found. Application of the stretch relationship
(22) to p ( x , t, u ) provides the depth-migrated image p ( x , z, v) in
a second step.
(2) with the help of an imploding Huygens wave would
formally be achieved by taking the partial derivatives of the
eikonal T ( x ,z ) in eq. (15) with respect to x,, and z, not with
respect to x and z.
What has been said with respect to the duality of the true
Huygens wave (15) also applies to any Huygens image wave.
For instance, the depth-remigration Huygens image wave ( 19)
propagates in the variables (x, z, v) away from point P(xo,z,,),
i.e. it is exploding. For a point P ( x , z), it is imploding in the
variables (x,,, yo, v). Imploding (image) Huygens waves can,
therefore, be just as useful in arriving at the (image-)wave
equation as exploding ones.
In this connection, we wish to emphasize one interesting
aspect involving imploding and exploding Huygens waves.
This aspect involves their role in the relationship between a
forward and an inverse seismic-imaging problem. Let us refer
to all imaging problems discussed above (which can be looked
upon as transforming an input image into an output image)
as direct imagiug problems. Then, solving the respective inverse
problem means transforming the output image back into the
input image. It is easily observed that the roles that exploding
and imploding Huygens (image) waves assume in a forward
problem are exchanged when addressing the inverse problem.
In other words, exploding Huygens waves turn into imploding
ones, and vice versa.
Imploding and exploding Huygens image waves are very
characteristic functions of the specific seismic-reflectionimaging problem considered. In fact, in previous works (Hubral,
Schleicher & Tygel 1996; Tygel, Schleicher & Hubral 1996) the
importance of the duality that exists between imploding and
exploding Huygens image waves has already been emphasised,
without, however, referring to them as such, and without
recognizing their potential for deriving image-wave equations.
In the cited works (where similar problems to those presented
here were solved in a 3-D laterally inhomogeneous medium
with the help of integral operators), the snapshot of an
exploding Huygens image wave was called an outplanat and
the snapshot of a n imploding Huygens image wave, a n inplanat.
Inplanats, then, describe the stacking curves in problem-specific
Kirchhoff-type integral operators, while outplanats describe
the corresponding smear-stack curves.
In the MZO problem, considered in the next section, we
will make use of the duality of Huygens image waves for the
development of the image-wave eikonal equations. To solve
the M Z O problem, we will use imploding Huygens image
waves. However, we find it easier to construct exploding
Huygens image waves for the respective inverse problem. From
the discussion above, both types of Huygens image waves
are identical.
Seismic image waves
Substituting the image wavefront h = H ( & t ) into eq. (26),
taking partial derivatives with respect to [ and t, and performing the elimination procedure (see Appendix) leads to the
MZO image-wave eikonal equation
(
tH l + - ui4H :
)-(
t 2 + -24):
Ht-tHH:=O,
which, by ansatz (9), also results from the MZO-wave equation
(Fomel 1994b)
This, in turn, can be employed to solve the initial-value
problem (10).
DMO correction
Suppose that an NMO correction has been applied to the CO
section p(x, t, h = h,) according to formula (11). What needs
to be done to obtain the ZO section p(x, t , h = 0) from the
NMO-corrected CO section p(x, z, h = h,) is called the DMO
correction. The exploding Huygens image wave for point
P(C,, to) of the inverse DMO problem is
z=t,
J
1--
(5
This is obtained by applying the NMO stretching operation
( 1 2 ) to each point P(5, t ) of the exploding MCO (or imploding
MZO) Huygens image wave ( 2 6 ) . The (velocity-independent)
NMO image-wave eikonai equation for the image-wave eikonal
h = &([, z) is then found from eq. ( 2 9 ) as (see Appendix)
HH? -H
+ zH, = 0 .
(30)
This image-wave eikonal equation corresponds to the DMO
image-wave equation (see also Fomel 1994b)
hp”<<- hp“hh- Zp”hr = 0 .
which is used to solve the initial-value problem (14).
(31)
Offset- 1OOOm, Velocity- 3500 m/s
500
I
I
I
1
1
I
400
600
400
w
300
Q
E
Y
!i
F
200
100
--- Imploding MZO Huygens image waves
Q
0
-600
Zero-offset reflection-time c u m
400
-200
0
200
Distance [m]
Figure 3. A reflection-timecurve in a zero-offset section is indicated by a string of diamonds. An imploding Huygens image wave, which is shown
at the instant h > h, = 0, pertains to each diamond. The envelope of all Huygens image waves forms the common-offset reflection-timecurve (bold
curve) with offset h.
0 1996 RAS, GJI 125,431-442
turn, serves as a point of departure to design the searched-for
MZO image-wave eikonal and image-wave equation. To
visualize imploding Huygens image waves with the help of an
example, we have constructed Fig. 3. This shows a continuous
ZO (h = 0) reflection-time curve, To, as a string of diamonds.
For each point on this string, exploding MCO (i.e. imploding
MZO) Huygens image waves are displayed (dashed curves) at
the instant h = h,. They provide the CO reflection-time curve
(bold curve) as their common envelope.
The exploding MCO Huygens image wave for a point
P(5,, to) on the ZO reflection-time curve, h = H ( l , t o ) = 0, is
constructed as follows. First, migrate point P(5,, t o ) into the
(x, z ) domain with the ZO configuration. This provides a lower
half-circle with centre at ({,,O) and radius vt,/2 for point
P(co, to). Then, demigrate this isochron with the CO configuration, defined by the value h, by treating it as a reflector.
This provides the exploding MCO Huygens image wave (see
Appendix) in the variables l and t at the instant of h,
437
438
C O N C L U D I N G REMARKS
+
ACKNOWLEDGMENTS
The authors thank Sergey Goldin for making his presentation
at the SvenFest, which was the main source of inspiration for
this work, available to them, and for many fruitful discussions
with respect to the subject. We also thank Makky S. Jaya, and
F. Liptow for useful discussions. The research has been supported in part by the Commission of the European Union in
the framework of the JOULE 11 programme, by the Deutsche
Forschungsgemeinschaft (DFG, Germany), by the National
Council of Technology and Development (CNPq, Brazil), and
the Silo Paulo State Research Foundation (FAPESP, Brazil).
Jorg Schleicher gratefully acknowledges additional support by
the Alexander von Humboldt foundation in the framework of
the Feodor Lynen programme. The responsibility for the
content remains with the authors. This is Karlsruhe University,
Geophysical Institute, Publication No. 677.
REFERENCES
Bleistein, N., 1990. Born DMO revisited, 60th Ann. Int. Mtg., SOC.
Expl. Geophys., Expanded Abstracts, 1366-1 369.
Chun, J.H. & Jacewitz, C.A., 1981. Fundamentals of frequency domain
migration, Geophysics, 46, 717-733.
Fomel, S.B., 1994a. Method of velocity continuation in the problem
of seismic time migration, Russian Geol. Geophys., 35, 100-111.
Fomel, S.B., 1994b. Kinematically equivalent differential operators for
offset continuation of seismic sections, Russian Geol. Geophys.,
35, 146-160.
Fomel, S.B., 1994c. Amplitude preserving offset continuation in
I: The offset continuation equation, Stanford
theory-Part
Exploration Project, Rept 84, 1-18.
Goldin, S.V., 1990. A geometrical approach to seismic processing: The
method of discontinuities, Stanford Exploration Project, Rept 67,
171-209.
Hubral, P., Schleicher, J. & Tygel, M., 1996. A unified approach to
3-D seismic reflection imaging-Part I: Basic concepts, Geophysics,
in press.
Lamer, K. & Beasley, C., 1987. Cascaded migration: Improving the
accuracy of finite-difference migration, Geophysics, 52, 618-643.
Rothrnan, D.H., Levin, S.A. & Rocca, F., 1985. Residual migration:
Applications and limitations, Geophysics, 50, 110-126.
Stolt, R.H., 1978. Migration by Fourier transform, Geophysics, 43,
23-48.
Tygel, M. & Hubral, P., 1989. Constant velocity migration in the
various guises of plane-wave theory, Suru. Geophys., 10, 331-348.
0 1996 RAS, GJ1 125, 431-442
In this paper, we have provided a conceptual foundation for
deriving differential equations for seismic-reflection-imaging
problems. Some examples of such differential equations
describing time remigration and offset continuation have
already been discussed by Fomel ( 1994a, b). Differential
equations of this form can be solved numerically, in a fully
analogous way to the wave equation, by means of integral,
spectral, o r F D techniques.
By introducing the concept of a seismic image wave, together
with the corresponding image-wave eikonal equation, imagewave equation, and exploding and imploding Huygens image
waves, we have provided a rigorous design strategy, by which
such differential equations, termed image-wave equations, can
be derived for a variety of imaging problems. In order to
explain the newly introduced concepts, we related them to
classical ones, connected with the familiar acoustic-wave equation. In a very natural and systematic way, we were able to
employ the new strategy to design very useful new partial
differential equations for a variety of seismic-reflection-imaging
problems. We believe that with the help of our geometrically
motivated concepts, all the formal mathematical considerations, as found in Fomel (1994a, b), are now much easier to
understand.
As the derivations in this paper are based on kinematic
consideration only, no criterion is provided with respect to the
dynamic qualities of the image-wave equations obtained. Note,
however, that Fomel (1994~)proves that the M Z O imagewave equation (28) is also dynamically correct, i.e. it leads to
an amplitude-preserving MZO, in the sense of Bleistein (1990)
and Tygel et al. (1996). The way in which a n amplitudepreserving time-migration (by which reflection coefficients can
be estimated) can be achieved with the help of eq.(25) is
discussed in Fomel ( 1994a); the topic of amplitude-preserving
imaging has not been addressed in this work.
It has been shown that the point of departure to derive
problem-specific image-wave equations involves an analytical
formulation of the problem-specific Huygens image wavefronts.
The formulation of a Huygens image wave requires no more
than chaining a purely kinematic migration and demigration
of a point in either the (x, t ) or (x, z ) domain. In this two-step
sequence, different measurement configurations o r velocities
(or possibly even some anisotropy parameter) can be considered in each step. In this way, a problem-specific propagation variable can be defined that describes the propagation
of image waves. Of course, the specification of a Huygens
image wave may not always lead to a n operationally simple
image-wave eikonal or image-wave equation. In such cases,
one has to find valid approximations that, in turn, will yield
simple approximate image-wave equations, the latter being
suitable only within certain parameter ranges.
There are, of course, other important concepts (like rays,
ray amplitudes, caustics, etc.) related to image-wave equations.
These concepts can also be considered to be generalizations of
familiar ones connected with the wave equation (1) and eikonal
equation (2). For instance, there exist harmonic plane image
waves that, if substituted into the image-wave equations, lead
t o an image-wave dispersion relation. These would be a
generalization of the dispersion relation, which for the case of
the wave equation ( 1) is k; k; = w2/v2,where k, and k, are
the components of the wave-number vector and w is the
circular frequency. Starting from harmonic planar image waves,
one can formulate solutions for the above initial-value problems in the spectral domain (Fomel 1994a) by following, for
example, a similar strategy to that described in Tygel & Hubral
(1989) in connection with a time-to-depth migration based on
the classical wave equation (1).
How to generalize the concept of image waves to inhomogeneous media remains an unsolved problem. At first sight,
the restriction t o constant velocity seems intrinsic to the
concept. However, at least in media consisting of constantvelocity layers, an application of the remigration image-wave
equations seems possible using a layer-stripping approach.
Moreover, and most intriguingly, even a reflector image below
an inhomogeneous overburden behaves formally like a wavefront if the overburden is continuously changed, even though
we do not know how to construct an appropriate image-wave
equation that describes this propagation at present.
Seismic image waves
The envelope of this ensemble is the desired exploding Huygens
image wave at the instant of migration velocity v. The condition
for the envelope curve is
Tygel, M., Schleicher, 3. & Hubral, P., 1996. A unified approach to
3-D seismic reflection imaging-Part IT: Theory, Geophysics,in press.
A P P E N D I X A: DERIVATION OF THE
IMAGE-WAVE EIKONAL EQUATIONS
aF
-=0,
a5
This Appendix is devoted to the derivation of all image-wave
eikonal equations dealt with in this paper, as derived from
Huygens image waves. For the remigration problem we use an
exploding Huygens image wave, and for the MZO problem
we use an imploding Huygens image wave.
which, after some algebra, yields the stationary
5=
In order to get a better understanding of this function, we
have constructed snapshots of this exploding Huygens image
wave for point P(x,,, zo) in the ( x , z ) domain for different
velocities u in Fig. Al. It is now our aim to find the imagewave eikonal eq. (4) for the depth-remigration image-wave
eikonal u = V ( x , z ) . For that purpose, we substitute this
expression into eq. fA6) and differentiate both sides of eq. (A6f
with respect to x and z. Setting, for convenience, the simplifying
notation, M ( x , z ) = V ( x ,z)/vo, we find
Then, we proceed to depth migrate this curve using the same
ZO configuration, but a different constant migration velocity,
v, i.e. v # uo. To accomplish this task, we initially construct the
isochrons for each point, M = M ( [ , T), on the Huygens curve
(A1). These isochrons are the lower half-circles
O=M,{[z6+=
r=2V JW.
+
Squaring and equating expressions (Al) and (A2) yields the
following ensemble of lower half-circle isochons, parametrized
by 5:
+ z2
x - (dvo)2x,,
1 - (v/vo)’ ’
-(V/V~)’[(X,, -
5)’
+ z;] = 0 .
(x - x,,)’
1
l’’
(1-M’)’
(x - Xo)’
-
(A3)
I’
x [kml
0.5
1
/
/
1.5
2
2.5
3
3.5
\
Figure A l . Snapshots of Huygens image waves lor depth remigration. The source point is at x = 2 km, z = 2 km. The original migration velocity
was uo = 2.5 km. Curves with increasing maxima at x = 2 km are Huygens image waves for decreasing velocities (u = 2.4 km s-’ to u = 1.8 km s - l
in steps of 100 m s-I), and curves with decreasing minima are Huygens image waves for increasing velocities (u = 2.6 km s-’ to u = 3.2 km s-’ in
steps of 100 m s-l). All displayed Huygens image waves are exploding ones for a remigration from u,, to u or imploding ones for a remigration
from u to uo.
0 1996 RAS, GJI 125, 431-442
Let us consider a point Mo(xo,z o )in the (x, z)domain. Locating
the coincident source-and-receiver pairs of a ZO configuration
o n the x-axis at x = 5 we demigrate the point Ma with the
constant migration velocity v,, to find the so-called Huygens
curve t = T(5)in the (5, t ) domain,
5 ) = (x-5)’
5 value
which, if substituted back into eq. (As), yields the exploding
Huygens image wave
Depth remigration
F(x, Z ,
439
P. Hubral, M . Tygel and J. Schleicher
440
( 1 ) specify a point P(t0,to) in the ZO section (i.e. the
input space);
(2) construct its ZO depth-migrated image;
(3) CO demigrate each point on this image with the CO
configuration into the (5, t) space (i.e. the output space) and
find the resulting envelope.
and
1 =M ,
{ [z: +
+
( x - Xo)Z
1
(1 - M2)’
ZO depth-migration of point Po(<,, to) yields the image (i.e.
the half circle)
Since from eq. (A6),
1
t = - (’4’12
which, if substituted into eq. (AS) yields, after a little algebra,
the eikonal equation for M ( x , z),
Returning to V ( x , z)= v o M ( x , z ) , we find the desired imagewave eikonal equation (20).
Time remigration
4(x - xo)2
u:- vz ’
Replacing the velocity u in eq. (A12) by the image-wave eikonal
P(x, t), and taking, as before, partial derivatives with respect
to the variables x and t, we find,
[
t;+
V;-
P2
and
[
- ‘iZ
4(x
P2
4(x - xo)2
1 = &C t;
1
11
4 ( x - X0)Z
0;-
+ 4 v;( x - xPZo y
-
- Xo)Z
x-XO
uo -
-
v;
- &2 ’
u24(x - x o y
(u: - P Z ) Z
.
C‘
Squaring of eq. (A15) and further substitution of the result
into eq. (A14) yields the time-remigration image-wave eikonal
equation
fi
v: - 4
(‘418)
where A , = z2+ ( x - 5 h)’. A cascaded migration and demigration is realized by insertion of (A17) into (A18). This leads
to
1
t =-(BY
u
+Byz),
(A191
where B , is given by
’>O
;(
+(to-5
T h)’+
2(~-50)(50-
5 T h).
(A20)
The envelope of all functions (A19) for all points on the half
circle (A17) is given by atlax = 0. From this equality, we can
determine and eliminate the parameter x in eq. (AIO). We find
Inserting expression (A21) into (A19) finally provides the
exploding MCO Huygens image wave in the (t,t ) space, which,
at the instant h, is given by
4 (x - x o )
-
-tK
2
+ A !p),
(0; - t Z ) 2
The combination of the above three equations immediately
yields the intermediate result
2=,-
V
B+ = -
Substitution of z = vt/2 and zo = u0t0/2 into expression (A6)
leads to the exploding time-remigration Huygens image wave
o = VV-
- ( x - toy.
CO demigration of a point P(x, z ) on this image gives
x-XO
z M,
~ - M ~ - M ~ M , ’
t = Jt;+
J(
=
=0 .
(A161
In agreement with the discussion above, formula (A22)
describes an exploding MCO Huygens image wave in ( 5 , t )
space, when going from h = O to h = h,. It is equal to the
imploding MZO Huygens image wave, which we require in
(5, t) space, when going from h = h, to h = 0. In Fig. A2 we
have constructed exploding MCO (imploding MZO) Huygens
image waves for a point P(C0, to) at different instants of h.
The next step in our aim to derive the MZO image-wave
eikonal equation (11) consists of inserting the image-wave
front h = H ( 5 , t) into eq. (A22) and taking derivatives with
respect to 5 and t . Prior to doing this, we introduce, for
convenience, in eq. (A22), the notations
B = hZ - ( 5 -to)’,
and
(A23b)
To arrive at the image-wave equation for the MZO problem,
we have to proceed as follows. First, we construct the imploding
MZO Huygens image wave (i.e. the exploding MCO Huygens
image wave) by the following steps:
(A23a)
from which we observe that
(A23c)
0 1996 RAS, G J l 125, 431-442
z
we have from eqs (AS) and (A7),
Seismic image waves
441
2.3
t
[SI
7
20
--
10
-
5
2.05
[kTi
h=lOOm
i
1.4
1.6
1.8
2
5
2.2
2.4
2.6
[kml
Figure A2. Snapshots of imploding MZO Huygens image waves. The source point is at 5 = 2 km, t = 2 s. The Huygens image waves pertain to
offsets or h = 100 m to h = 700 m in steps of 100 m. In the upper right corner, the region near the apices is enlarged.
1.95
1.4
1.6
2.2
2
1.8
5
2.05
2.4
2.6
[kml
Figure A3. Snapshots of imploding DMO Huygens image waves. The source point is at 5 = 2 km, t = 2 s. The Huygens image waves pertain to
offsets of h = 100 m to h = 700 m in steps of 100 m. In the upper right corner, the region near the apices is enlarged. Note the difference between
this and Fig. A2, indicating the effect of the NMO correction.
We again insert h = H ( [ , t ) into the above formulae and take
the partial derivatives of eq. ( A 2 2 ) with respect to 4 and t. We
find
2
2H 1 uto 1
l=-HA--U '
v A(2)SHHt'
0 1996 RAS, G J I 125,431-442
and
2
2H 1 vt,
1
2H 1 vt,
0 = - H A--- -HHI+--(~)
v
v A ( 2 ) B 2
v A
*
1
~(t-t,,).
1.95
442
From eq. (A24) we obtain, upon the use of eqs (A23),
H, =
section
z = t o J 1 - (to
- 5)2/h2,
1
t
H
- -[ t 2 - ( ~ H / v ) ~ ]
H
Bt
(A311
where z is the NMO corrected traveltime defined by
so that
B=
t(l H , - H )
Correspondingly, from eq. (A25),we obtain
H<=
-(2H/U)2)(5- 4 0 )
t
H
- - -[ t 2 - (2H/u)2]
H Bt
-(tZ
(A281
Substituting B from eq. (A27) into formula (A28),we arrive at
the following expression for 5 - to:
4-50=-
-H~H,
tH,-H.
and
where W, 1s given according to the above by
We finally use eq. (A23a)together with eqs (A27)and (A29)to
arrive a t the M Z O image-wave eikonal equation
H;-
4
H; +
( t2
+ ( 2H/u)2)H , = 1 .
Ht
The corresponding MZO image-wave equation (27) results
from this.
Eq. (A33)can be readily inverted for 4 - To; we find
_5 - _5 0_ _- 1
B
H,‘
Using
t = to/WN,from
eq. (A34) we obtain
DMO correction
To solve the DMO problem, we again use the imploding M Z O
Huygens image wave given in eq. (A22).However, as the initial
condition for a DMO correction is represented by a fixed
point P N with coordinates (5, z), we have in the NMO corrected
where we have made use of eq. (A36).By comparing eqs (A35)
and (A37), we immediately obtain the DMO image-wave
eikonal equation
-tH,
= AH; - A .
(A381
0 1996 RAS, G J I 125, 431-442
where t is the CO traveltime, as before. Eq. (A31) describes
imploding DMO Huygens image waves which are depicted in
Fig. A3. Inserting h = A(<,z) into formula (A31) and partially
differentiating the resulting equation with respect to ( and z
leads to
H , H ~ ( ~ ~ - ( ~ H / ~ ) ~ )

Seismic Image Waves

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