3D angle gathers from reverse time migration - CGG · 3D angle gathers from reverse time migration Sheng Xu1, Yu Zhang1, ... widely used tool for model building, ... For wide-azimuth

3D angle gathers from reverse time migration

Sheng Xu1, Yu Zhang1, and Bing Tang1

ABSTRACT

Common-image gathers are an important output of prestack

depth migration. They provide information needed for velocity

model building and amplitude and phase information for sub-

surface attribute interpretation. Conventionally, common-image

gathers are computed using Kirchhoff migration on common-

offset/azimuth data volumes. When geologic structures are

complex and strong contrasts exist in the velocity model, the

complicated wave behaviors will create migration artifacts in

the image gathers. As long as the gather output traces are

indexed by any surface attribute, such as source location, re-

ceiver location, or surface plane-wave direction, they suffer

from the migration artifacts caused by multiple raypaths.

These problems have been addressed in a significant amount

of work, resulting in common-image gathers computed in the

reflection angle domain, whose traces are indexed by the sub-

surface reflection angle and/or the subsurface azimuth angle.

Most of these efforts have concentrated on Kirchhoff and one-

way wave-equation migration methods. For reverse time

migration, subsurface angle gathers can be produced using the

same approach as that used for one-way wave-equation migra-

tion. However, these approaches need to be revisited when

producing high-quality subsurface angle gathers in three

dimensions (reflection angle/azimuth angle), especially for

wide-azimuth data. We have developed a method for obtaining

3D subsurface reflection angle/azimuth angle common-image

gathers specifically for the amplitude-preserved reverse time

migration. The method builds image gathers with a high-

dimensional convolution of wavefields in the wavenumber do-

main. We have found a windowed antileakage Fourier trans-

form method that leads to an efficient and practical

implementation. This approach has generated high-resolution

angle-domain gathers on synthetic 2.5D data and 3D wide-

azimuth real data.

INTRODUCTION

During the past three decades, progress in prestack depth

imaging has been considerable, both in theory and in practice.

The theoretical progress has provided better methods for extrap-

olating wavefields measured at the earth’s surface into the sub-

surface, and the practical progress has linked the migrations

more closely with velocity model building and interpretation.

In complex areas, imaging difficulties come from two major

components: prestack depth velocity model building and migra-

tion algorithms. Velocity model building estimates a velocity

model for the simulation of seismic wave propagation that takes

place during migration. This model forms the long-wavelength

(macro) part of the earth model, and migration provides the

short-wavelength (reflectivity) part. Seismic ray-based tomogra-

phy (Bishop et al., 1985; Liu and Bleistein, 1995; Billette and

Lambare, 1998; Zhou et al., 2003; Lambare, 2008) is the most

widely used tool for model building, but the nonlinearity and

uncertainty of the ray-based tomography algorithms expose to-

mography as a weak link in the imaging process (Nolte et al.,

2010). Another weak link is poor seismic illumination of regions

beneath the complex overburden, which makes adequate

imaging difficult or even impossible. Poor illumination is often

caused by inadequate seismic acquisition (Etgen, 2006), for

example, by conventional 3D narrow-azimuth streamer

acquisition when 3D wide-azimuth acquisition is needed (Etgen

et al., 2009).

Until recently, Kirchhoff migration has been the workhorse

method for prestack depth migration. This method has proved

successful through numerous examples when the velocity varia-

tions are mild (Thierry et al., 1999a; Operto et al., 2000). It has

also formed the basis from which “true-amplitude” migration

Manuscript received by the Editor 12 July 2010; revised manuscript received 28 October 2010; published online 10 March 2011.1CGGVeritas, Houston, Texas, U.S.A. E-mail: [email protected]; [email protected]; [email protected].

VC 2011 Society of Exploration Geophysicists. All rights reserved.

S77

GEOPHYSICS, VOL. 76, NO. 2 (MARCH-APRIL 2011); P. S77–S92, 14 FIGS.10.1190/1.3536527

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(Bleistein, 1987) has been extensively studied. This algorithm

migrates the input seismic data one trace at a time (Ettrich and

Gajewski, 1996) or one local group of traces at a time (Sun

et al., 2000) (with very similar operators applied to all traces in

the group); these processes imply that the cost of Kirchhoff

migration is proportional to the number of input traces. When

the number of input traces is relatively small within the migra-

tion aperture (as is usually the case with marine narrow-azimuth

surveys), Kirchhoff migration yields an efficient algorithm

(Thierry et al., 1999a), but when the number of input traces is

large within the migration aperture (as is usually the case with

marine wide-azimuth surveys), efficiency might be lost.

In addition, using ray tracing to approximate the Green’s

function of wave propagation can compromise the accuracy of

Kirchhoff migration, especially when the wavefield is compli-

cated. Its further approximation, choosing a single ray arrival of

the complicated wavefield at each image location guarantees a

noisy image in areas of many ray arrivals (Ettrich and Gajewski,

1996; Audebert et al., 1997; Thierry et al., 1999b;). Multiarrival

Kirchhoff migration algorithms overcome this problem, but they

tend to be complicated and relatively inefficient in three dimen-

sions (Operto et al., 2000; Xu and Lambare, 2004; Xu et al.,

2004).

Beam prestack depth migration methods (Hill, 2001; Gray,

2005) approximate the Green’s function with an expression that

allows multiple arrivals to be imaged without excessive algorith-

mic complication, and can be applied in a true-amplitude sense

(Gray and Bleistein, 2009). Like Kirchhoff migration, however,

the Green’s function approximation used by beam migration relies

on ray tracing and can become inaccurate if the migration velocity

model contains extremely strong variations (e.g., salt bodies) and

requires excessive smoothing. Still, the beam migration’s ability

to image complex structures and to control certain types of migra-

tion noise can usually ensure significantly better images in

complex areas than single-arrival Kirchhoff migration algorithms.

Whereas Kirchhoff and beam migrations use rays to approxi-

mate the Green’s functions for wave propagation, so-called

wave-equation migration algorithms use full waveform Green’s

functions that are numerically generated, for example, by finite

differences. The most computationally efficient algorithms for

doing this are collectively called one-way wave-equation migra-

tion (OWEM) (Claerbout and Doherty, 1972; Gazdag, 1978).

These algorithms decompose seismic wavefields inside the earth

into upgoing waves and downgoing waves under the assumption

of no interaction between these two wavefields; that is, no turn-

ing wave or vertical reflection generation during the synthesis of

wave propagation. For a very large and growing body of exam-

ples, OWEM has solved the problems of multiarrivals better

than single-arrival Kirchhoff migration.

For wide-azimuth seismic surveys, in which the number of input

traces is large compared with the migration aperture, OWEM tends

to gain efficiency relative to Kirchhoff migration. For such surveys,

efficient implementation of OWEM algorithms can be built either

for common-shot migration (Romero et al., 2000) or plane-wave

migration (Duquet et al., 2001; Zhang et al., 2005; Liu et al.,

2006; Soubaras, 2006). There are two major limitations of OWEM

algorithms: (1) turning waves are missing in the wave-propagation

simulation, which results in the poor imaging of the steep dip

events around 90�, and (2) the wave-propagation synthesis only

ensures the accuracy of the phase of the wavefield, whereas the

amplitudes of the wavefield are much less reliable and need further

correction (Zhang, 1993).

Implementation of the two-way wave equation in depth migra-

tion began many years ago in an algorithm called reverse time

migration (RTM) (Baysal et al., 1983; McMechan, 1983; Whit-

more, 1983); however, its use was limited due to its need for com-

puter power. With the increase in computer power, RTM has taken

off very rapidly over the last few years, and theoretical advantages

such as dip-unlimited accurate wave propagation and improved

amplitudes have provided significant imaging benefits in practice.

For example, in complex subsalt and salt flank areas, the numerical

Green’s function from the finite-difference solution of the two-way

wave equation has better amplitude behavior, so it is easier to

incorporate amplitude corrections into RTM than into OWEM.

In addition to the ability to handle extreme velocity complexity,

many present-day RTM algorithms can handle anisotropy media

such as vertical transverse isotropy (VTI) and tilted transverse iso-

tropy (TTI) (Zhou et al., 2006a, 2006b; Duveneck et al., 2008;

Fletcher et al., 2008; Zhang and Zhang, 2008; Zhang and Zhang,

2009). On real data imaging examples, TTI RTM has given the

best images in a complex Gulf of Mexico wide-azimuth survey

(Huang et al., 2009), although the velocity models for TTI migra-

tion were simplified as structurally conformable transverse iso-

tropy (STI), which requires the symmetry axis to be consistent

with the normal vectors of reflectors (Audebert et al., 2006).

With the improved accuracy of RTM comes increased sensitivity

to the accuracy of the velocity model (Huang et al., 2008, 2009).

This sensitivity causes significant improvement in RTM images

when the velocity is accurate, but it also causes significant degrada-

tion of RTM images when the velocity is not accurate. For this rea-

son, migration velocity analysis is more important for RTM than

for other depth migration methods. The link between migration and

velocity model building is the set of common-image gathers (CIGs)

produced by the migration algorithms. A CIG is a set of images, all

at the same image location, with each image formed from different

subsets of input data. For example, a single common-offset/

common-azimuth data volume, a subset of the full acquired prestack

seismic data set, can be used to image the 3D earth; the collection

of images from all of the data subsets with different offset and azi-

muth forms the CIGs. The CIGs can contain traces all with differ-

ent offsets (with all the azimuthal information summed together), or

they can contain traces all with different offsets and azimuths.

The CIGs are commonly used for depth-domain amplitude-

variation-with-offset (AVO) analysis (Beydoun et al., 1993;

Tura et al., 1998; Dong and Ponton, 1999), and migration-based

velocity analysis (Jin and Madariaga, 1993; Symes, 1993; Liu

and Bleistein, 1995; Chauris et al., 2002; Zhou et al., 2003).

With a correct velocity model, all of the images at the same

image location should focus at the same depth, causing reflec-

tion events on the CIGs to appear flat. The flatness of seismic

events on CIGs is one of the most important criteria for validat-

ing the velocity model by focusing analysis. When events on

the CIGs are not flat, geophysicists improve their migration ve-

locity models by analyzing the curvature of the events, using

the analysis to guide a velocity update (Zhou et al., 2003). For

Kirchhoff migration, there is no significant additional cost to

compute common-offset CIGs (COCIGs). On the other hand,

migrating common-offset volumes by OWEM or RTM is

extremely expensive (Ehinger et al., 1996), so COCIGs are not

generally available for those migration methods.

S78 Xu et al.


The quality limitations of COCIGs are caused, in part, by the

underlying limitations of ray-based migration. More fundamen-

tally, COCIGs suffer from migration artifacts due to multiple

paths of wave propagation, whether or not the migration meth-

ods are capable of handling multiple paths of wavefield accu-

rately (Nolan and Symes, 1996; Xu et al., 2001), potentially

causing difficulties for velocity analysis and amplitude-versus-

reflection-angle (AVA) analysis. In fact, CIGs whose traces are

indexed by any attribute on the recording surface, such as the

source/receiver offset or surface incidence angle of the source

energy, are susceptible to such artifacts.

Xu et al. (2001) show that a necessary condition for artifact-

free CIGs is to parameterize the CIGs in a subsurface angle do-

main, such as in a reflection angle or opening angle. These

authors illustrate this fact in two dimensions using multiarrival

Kirchhoff migration on the Marmousi synthetic data set (Lailly

et al., 1991). Subsequent research has extended this work to ani-

sotropic media (Alerini and Ursin, 2009; Alkhalifah and Fomel,

2009) or three dimensions using CIGs in the reflection angle/

azimuth angle, and to 3D analysis in the multiple angle domain

(reflection angle, dip angle, azimuthal angle, and so on) (Aude-

bert et al., 2000; Koren et al., 2008).

Compared with multiarrival Kirchhoff and beam migrations,

OWEM and RTM appear to have limited capabilities for CIGs

indexed in the surface offset domain. For the subsurface angle

domain, Sava and Fomel (2003) propose an approach to output

local subsurface-offset CIGs from OWEM and then convert

them to subsurface (reflection) angle-domain CIGs (ADCIGs).

This approach is consistent with the work of de Bruin et al.

(1990). Converting local subsurface-offset CIGs into ADCIGs

has a simple form in the 2D isotropic case. First it requires the

migration imaging condition to be applied at a range of subsur-

face offsets, forming subsurface-offset CIGs, and next a 2D

Fourier transform to be applied to the local offset CIG. Then it

is necessary to map the transform wavenumber to the reflection

angle. The procedure is performed gather by gather locally, and

it is a reasonably efficient algorithm.

However, the gather conversion formula is very complex in

three dimensions (Fomel, 2004). It requires producing the local

subsurface-offset CIGs indexed by offset X and offset Y, and a

5D Fourier transform to the local subsurface-offset CIGs to the

wavenumber domain, followed by a high-dimensional mapping.

The definitions of the wavenumber directions and angles are

given in Appendix A and Figure 1, and the details of the 3D

angle conversion formula are given in Appendix B. For imple-

mentation, even producing local subsurface offset vector (X, Y)

CIGs from OWEM is very computationally intensive (Pell and

Clapp, 2007). The full implementation of this theory has to be

very challenging.

Sava and Fomel (2006) propose an alternative approach,

which is to produce time-delay CIGs and then convert them to

ADCIGs. The time-delay CIG has only one additional axis on

the images (delay time); its output is 4D (x, y, z, delay time).

However, for wide-azimuth seismic data, the expected 3D

ADCIGs are themselves 5D (x, y, z, and incident/reflection

angle, azimuth angle), so the time-shift imaging condition does

not provide sufficient information for a full conversion into 3D

ADCIGs when applied to wide-azimuth seismic data. As a

result, the final angle gathers generally lack azimuthal informa-

tion. Furthermore, the conversion from one data domain to

another suffers from sampling issues, possibly degrading the re-

solution of the final CIGs.

The above approaches to obtain ADCIGs for OWEM and

RTM are indirect, proceeding through an intermediate (subsur-

face offset or image time-lag) domain. It is also possible to com-

pute ADCIGs by directly decomposing the wavefields in the

subsurface into their local directional components (Soubaras,

2003; Wu et al., 2004). So far, this work has been performed on

2D or 3D isotropic migration algorithms, and it produces

ADCIGs only in the reflection angle (no azimuth). Furthermore,

this approach has been applied only to one-way wavefield propa-

gators, making it difficult to retain reliable amplitude information

(Zhang et al., 2003). Although there are other significant differ-

ences between the propagators for OWEM and RTM, there is no

essential strategic difference in the algorithm for CIG output.

Because of RTM’s superior accuracy, we adopt a direct angle-

gather-forming approach to RTM in this paper.

In summary, in complex media, RTM is the most accurate

method for the purposes of both structural and true-amplitude

imaging. For optimal wide-azimuth imaging, RTM needs to pro-

duce artifact-free CIGs for velocity updating and amplitude

analysis, and these CIGs should contain, not suppress, azimuthal

information. We illustrate such CIGs, indexed by reflection

Figure 1. Definition of angles. The wavenumber vectors ks and kr

give the propagation directions at image point x for the wavefieldsfrom the source location s and receiver location r, respectively.The directions can be represented as (hs, us), (hr, ur). The vectork is the migration vector; in the specular cases, it coincides withthe reflector normal vector, and can be represented as (hm, um).The h is the reflection opening angle, and u is the reflection azi-muth angle.

S79RTM 3D angle gathers


angle/azimuth angle, obtaining them by performing a wavefield

directional decomposition at locations in the subsurface. These

CIGs, called 3D angle-domain CIGs, can be produced in an iso-

tropic medium or in an STI anisotropic medium. Numerically,

we propose to transform the forward-propagated source wave-

fields and back-propagated receiver wavefields to the wavenum-

ber domain. The spatial wavenumber vectors thus provide the

phase direction of wave propagation.

The proposed method is not computationally efficient because

the imaging condition becomes a high-dimensional spatial con-

volution with angle calculation in the inner loop. To reduce the

numerical cost, we propose to form the CIGs in a local window,

in a process equivalent to local plane-wave decomposition. Next

we propose to use the antileakage Fourier transform (ALFT) in

lieu of the fast Fourier transform (FFT) (Xu et al., 2005). The

ALFT algorithm helps to reduce the Gibbs phenomenon, which

causes severe accuracy problems when working in a small local

window (Xu et al., 2010). Furthermore, the ALFT ranks the

local plane waves according to their energy, which can be used

for limiting the number of the most energetic plane waves in

the convolution imaging condition instead of using the full set

of them; this scheme helps to reduce the computational cost

significantly. Finally, we show examples of angle-domain CIGs

on 2.5D synthetic data and on a Gulf of Mexico wide-azimuth

data set.

ASYMPTOTIC EXPRESSION FOR 3D

TRUE-AMPLITUDE MIGRATION WITH

ANGLE-GATHER OUTPUT

In this section, we review the formulation for 3D true-ampli-

tude migration with subsurface angle-gather output. This formu-

lation yields an asymptotic formula that we adapt in the next

section to apply it to RTM. For the acoustic wave equation, the

reflectivity R at subsurface location x depends on reflection angle

h and local subsurface azimuth angle u thus, R¼R(x, h, u). Our

angle conventions are shown in Figure 1. The wavenumber vec-

tors ks and kr are the wave-propagation-phase directions at

image location x for the wavefields from the source location s

and the receiver location r, respectively. The angle pairs (hs, us),

(hr, ur), (hm, um) are, respectively, the dip angles and azimuth

angles of vectors ks, kr, and their vector sum k. By Snell’s law,

the plane orthogonal to vector k is a potential reflector plane

(which coincides with an actual reflector when ks and kr form a

specular wavenumber pair). The angle relationships are defined

as (see details in Appendix A)

cos h ¼ k � kr

kj j krj j

cos u ¼ ks � krð Þ � nx � ks þ krð Þð Þks � krj j nx � ks þ krð Þj j

8>>><>>>:

; (1)

where the unit vector is defined as nx¼ (1, 0, 0). Our convention

of subsurface azimuth angle is similar to that of Sava and Fomel

(2005) and different from that of Bleistein et al. (2005), in

which the starting point of the azimuth angle is defined differ-

ently. In equation 1, the definition of angles is a general case,

which is applicable to isotropic, VTI, and TTI media.

The Kirchhoff approximation for seismic synthesis in forward

modeling states that, in the frequency domain, the recorded

wavefield dG(r, x, s) at location r due to a source at location s

is given by (Bleistein et al., 2005)

dGðr;x; sÞ ¼ð

dxRðx; h;uÞ� rG0ðr;x; xÞG�0ðx;x; sÞ�

�G0ðr;x; xÞrG�0ðx;x; sÞ�; (2)

where G0(x, x, s) is the frequency-domain Green’s function for

propagation in the background medium from location s to loca-

tion x; * denotes its complex conjugate; R(x, h, u)¼R(x, h, u)n,

with R(x, h, u) being angle-dependent reflectivity and n the vector

normal to the reflection plane; and r is the spatial gradient oper-

ator. The integral, over the full spatial volume, is the sum of

incremental reflectivity contributions from all subsurface locations

x. Asymptotically, the Green’s function can be expressed as

G0ðr;x; xÞ¼Aðr; xÞeixTðr;xÞ;

G0ðx;x; sÞ ¼ Aðx; sÞe�ixTðx;sÞ; (3)

and the gradient of the asymptotic Green’s function is given by

rG0ðx;x; sÞ � �ixrTðx; sÞAðx; sÞe�ixTðx;sÞ: (4)

In equations 3 and 4, T(x, y) and A(x, y) are the traveltime and

ray amplitude from the location y to x, respectively. As a result,

the reflected wavefield in equation 2 can be expressed as

dGðr;x; sÞ ¼ ixð

dxRðx;h;uÞn � rTðr;xÞð

þrTðx; sÞÞAðr;xÞAðx; sÞeixðTðr;xÞþTðx;sÞÞ

¼ ixð

dxRðx;h;uÞn � qðr;x; sÞAðr;x; sÞeixTðr;x;sÞ

¼ð

dxRðx;h;uÞKðr;x; s;xÞeixTðr;x;sÞ; (5)

where the Kirchhoff integral kernel Kðr; x; s;xÞ ¼ ixn � qðr; x; sÞAðr; x; sÞ contains a migration dip vector term q(r, x, s)¼rT(r, x)

þrT(x, s). In the isotropic case, the migration dip vector can also

be expressed as qðr; x; sÞ ¼ 2 cos hvðxÞ n, where h is the incident angle

and V(x) is the wave-propagation velocity at image location x.

This dip vector is combined with an amplitude product term

A(r, x, s)¼A(r, x)A(x, s) in the migration kernel. We note here

again that equation 5 is valid only with the high-frequency asymp-

totic approximation. The phase term is the summation of the trav-

eltimes of the wavefields at the image point from the source loca-

tion and the receiver location:

Tðr; x; sÞ ¼ Tðr; xÞ þ Tðx; sÞ: (6)

If there is an observed seismogram dGobs(r, x, s) at location r

due to a source at location s, this observed data should be con-

sistent with the synthesized reflection data dG(r, x, s). This con-

sistency requires that the background velocity model and the

reflectivity model both be correct. In the theory of true-ampli-

tude migration, the assumption is that the obtained background

velocity gives a reasonable estimate to the long-wavelength var-

iations of the earth model, so that the traveltime of wave propa-

gation could be well defined. The task of the inversion is to find

a good reflectivity model to achieve a fit between the observed

data dGobs(r, x, s) and the synthetic data dG(r, x, s). Using a

weighted least-squares inversion scheme (Jin et al., 1992), one

S80 Xu et al.


can build an objective function to measure the misfit between

the calculated data dG(r, x, s) and the acquired/observed data

dGobs(r, x, s):

CðRÞ ¼ 1

2

ðððdsdrdxQ dGobs � dGk k; (7)

where k�k denotes the L2-norm and Q¼Q(x, h, s, r) is the weight

for the least-squares inversion, to be determined. Minimizing the

misfit function 7 gives the least-squares solution for the reflectivity

model R(x, h,u); this solution is the so-called true- or preserved-

amplitude migration image, or migration/inversion image.

From equation 5, the synthetic data dG(r, x, s) depend line-

arly on the reflectivity model R(x, h, u), and equation 7 puts the

problem into a framework of linear inversion theory. Solving

equation 7 for reflectivity R(x, h, u) at specific angle h0 and u0,

the R(x, h0, u0) can be performed iteratively using the Gauss-

Newton method, whose first iteration solution (assuming a null

initial reflectivity guess) yields the angle-dependent reflectivity

model:

Rðx; h0;u0Þ ¼ �o2C

oR2

� ��1oC

oR: (8)

On the right-hand side of equation 8, the first term is the inverse of

the Hessian; the second term is the gradient of the objective func-

tion. The gradient of the objective function has the expression

oC

oRðx; h0;u0Þ ¼

ðððdsdrdxdðh� h0Þdðu� u0Þ

� QK�ðr; x; s;xÞe�ixTðr;x;sÞdGobs; (9)

where the two d functions (d(h� h0) and d(u�u0)) are angle

binning functions that restrict the integration to desired reflection

angle h0 and azimuth angle u0 (Xu et al., 2001). The Hessian

½o2CoR2� has the expression

o2C

oR2

� �¼ Hðx; x0; h0;u0Þ

¼ððð

dxdsdrd h� h0ð Þd u� u0ð ÞK�ðr; x; s;xÞ

�QKðr; x0; s;xÞe�ix Tðr;x;sÞ�Tðr;x0;sÞð Þ

�ðð

dhdud h� h0ð Þd u� u0ð Þð

dkK�ðr; x; s;xÞ

�QKðr; x0; s;xÞo x; s; rð Þo h;u; kð Þ

��e�ik�ðx�x0Þ; (10)

where x is the true focus location and x0 is the focus location

with the background velocity model. The second step of equa-

tion 10 is the result of changing integration variables from sur-

face coordinates (s, r) to subsurface angles (h, u); the Jacobian��o x;s;rð Þo h;u;kð Þ

�� is the result of this change of variables.

The approximation in equation 10 is valid when the high-fre-

quency asymptotic approximation x(T(r, x, s)�T(r, x0, s))� k �(x� x0) holds (Beylkin, 1985). When the background velocity

model is close enough to the true velocity model, the Green’s

function on the x0 and x are very close within the band of seis-

mic frequencies. Therefore, in the first-order approximation,

under a high-frequency asymptotic approximation, the amplitude

kernel could be approximated as (Beylkin, 1985)

Kðr; x; s;xÞ � Kðr; x0; s;xÞ: (11)

Then, as long as the least-squares weight satisfies

Q � �1

2pð Þ3K2ðr; x0; s;xÞo h;u; kð Þo x; s; rð Þ

��; (12)

the right-hand side of equation 10 can be made asymptotically

equivalent to the Dirac delta function (Jin et al., 1992):

dðx� x0Þ ¼1

2pð Þ3ð

dke�ik�ðx�x0Þ: (13)

In this case, the Hessian becomes the diagonal matrix:

Hðx; x0; h0;u0Þ ¼ dðx� x0Þ: (14)

The Jacobian in equation 10 includes the Beylkin determinant

(Beylkin, 1985). It provides the connection between local direc-

tion vectors at an image location and the actual acquisition ge-

ometry. For wide-azimuth acquisition, in cases of isotropic

media, it can be developed in detail as (Appendix C)

o h;u; kð Þo x; s; rð Þ

�� ¼ x2 qj j3 4pð Þ4V2 xð ÞA2 x; sð ÞA2 x; rð Þps

zprz

4 cos h sin h;

(15)

where V(x) is the velocity at image point x and psz is the vertical

component of slowness vector of the wave propagation at surface.

With the help of equation 15, the weight Q of the least-squares

inversion becomes

Q � 16pV xð Þpsz pr

z

sin h: (16)

Note that this weight function does not depend directly on the

wavefield amplitude. This very regular production comes from

the wide-azimuth acquisition geometry. In general, though, the

weight depends, through the Beylkin determinant, on the devia-

tion of the acquisition geometry from the ideal wide-azimuth,

wide-offset case. Wide-azimuth acquisition (in principle) covers

the earth’s surface completely with an areal array of receivers to

record energy from each shot and an areal array of shots to send

the energy to be recorded at each receiver. As a result of this,

the weight (equation 16) has lost the dependence on wavefield

amplitude that is used to compensate for insufficient acquisition,

so it is different from the common-offset migration weight of

Bleistein (1987).

Substituting equations 9, 14, and 16 into equation 8 provides

the following asymptotic form for 3D wide-azimuth true-ampli-

tude migration with output ADCIGs:

Rðx; h0;u0Þ

¼ððð

d h� h0ð Þd u� u0ð Þ

� dsdrdx32pix cos hA s; x; rð Þps

z prze�ixTðr;x;sÞdGobs

sin h;

(17)



which gives 3D true-amplitude Kirchhoff migration for ADCIGs

in both the reflection angle and the azimuth angle. However, the

above inversion formula has a singularity at h¼ 0�, which is dif-

ferent from its 2D inversion counterpart (Zhang et al., 2007a).

To understand why this happens, one can imagine that the

reflection and azimuth angles distribute on a hemisphere as lati-

tude and longitude, respectively. The expression h¼ 90� gives

the equator, with all the azimuths equally distributed on it;

h¼ 0� is the north pole where the azimuth angle is ambiguously

defined. The 1=sin h term in the above imaging condition is

used to compensate for the diminishing area when the reflection

angle tends to 0�. However, this causes difficulty in obtaining

accurate amplitudes around the 0� reflection angle.

Note that the term 1=sin h in expression 17 is different from

the one in Bleistein et al. (2005). It is a necessary term of Jaco-

bian for the translation to the azimuth angle (polar system). The

reason for this difference is due to our direct binning function

d(u�u0), whereas the binning function d(sin h(u – u0)) was

used in expression 38 of Bleistein et al. (2005).

3D TRUE-AMPLITUDE MIGRATION

FOR REVERSE TIME MIGRATION

The true-amplitude-migration expression 17 provides a recipe

for producing 3D ADCIGs in the reflection angle and the azi-

muth angle. However, it is an asymptotic expression, and it can

be applied directly only to migration algorithms that calculate

amplitude terms explicitly and rely on specified initial direc-

tions, that is, to Kirchhoff and beam migrations. To apply the

theory to RTM, it is necessary to translate expression 17 into a

true-amplitude RTM formula. The resulting RTM formula will

still be asymptotic in terms of amplitude, even though the

underlying wave-propagation engine is not. To simplify the deri-

vation, we consider isotropic RTM (the anisotropic case can be

treated similarly, with extra complications in notation). We also

consider decomposing the full migration in expression 17 into

migration shot by shot; if we ignore multiple reflections, the

RTM migration operator is linear to input data dGobs, and the

final wide-azimuth output is the sum of contributions from all of

the shot experiments.

To produce a true-amplitude formula for RTM, we need to

replace the asymptotic expressions for the Green’s function (am-

plitude and traveltime terms) with their exact counterparts

(Zhang et al., 2007b). We do this by rewriting migration expres-

sion 17 as

�Rðx;h0;u0Þ ¼ 16pððð

vðxÞsin h

d h� h0ð Þd u� u0ð ÞpF pB dsdx:

(18)

Here �Rðx; h0;u0Þ ¼ Rðx; h0;u0Þ= qj j ¼ Rðx; h0;u0ÞvðxÞ=2 cos h0

is introduced (Bleistein, 1987) to maintain consistency with the

output definition in Zhang et al. (2007b). The integral over

source locations ensures that all recorded seismic traces contrib-

ute to the final image and to the angle gathers. The wavefields

pF and pB are given by

pF ¼ G� x; sð Þpsz; pB ¼

ðdrG� x; rð Þpr

zdGobs: (19)

The forward propagation of the source signature pF satisfies

1

V2

o2

ot2�r2

� �pFðx; tÞ ¼ 0;

pFðx; y; z ¼ 0; tÞ ¼ dðx� sÞÐt�1

f ðt0Þdt0:

8>><>>: (20)

In equation 20, the time integration of the source wavelet f (t)incorporates the term ps

z into pF (Zhang et al., 2007a). The back

propagation of the recorded data pB satisfies the equation:

1

V2

o2

ot2�r2

� �pBðx; tÞ ¼ 0;

pBðx; y; z ¼ 0; tÞ ¼ dGobsðx; y; s; tÞ:(21)

In equations 20 and 21, V¼V(x) is the velocity, d(x�s) f(t) is

the source signature, and r2 is the Laplacian operator.

Consistent with the theory presented by Zhang et al. (2007a),

the final integral 18 over shot locations s is the sum of all the

contributions for all shots. For a single shot, the contribution to

the final 3D angle CIGs is

�Rsðx; h0;u0Þ ¼ 16pð

VðxÞsin h

d h� h0ð Þd u� u0ð ÞpF pB dx:

(22)Of the two migration imaging conditions, crosscorrelation (multipli-

cation of wavefields) and deconvolution (division of wavefields),

proposed by Claerbout (1971) and Zhang et al. (2007b), shows that

the crosscorrelation imaging condition is appropriate for 3D

ADCIGs. Equation 22, and its ray-theoretical counterpart 17, both

have the form of a crosscorrelation imaging condition, which is

easy to apply and numerically stable.

In expression 22, the forward-propagated wavefields and

backward-propagated data are in a general form, applicable to

any wave-equation-based migration, OWEM or RTM. The

migration formula has a simple form that is independent on the

velocity model, isotropic or STI anisotropic.

COMMON-IMAGE GATHERS IN THE

REFLECTION AND AZIMUTH

ANGLE DOMAINS

To create ADCIGs from RTM, equation 18 needs the subsur-

face wavefields PF and PB from equations 20 and 21. Once

these wavefields have been computed, we determine the (local)

wave-propagation directions ks and kr from them at each image

location; then equation 1 provides the reflection angle h and

azimuth angle u. The imaging condition in the frequency/wave-

number domain is derived in Appendix D:

�Rs x; h0;u0ð Þ ¼ 16pVðxÞsin h0

Xx

Xk

Xkr�ksð Þ

d h� h0ð Þd u� u0ð Þ

� pFðks;xÞpBðkr;xÞeik�x: (23)

This equation expresses the reflectivity for each opening/azimuth

angle as a sum of wavefield contributions over a range of dip/azi-

muth angles that all obey the law of reflection. (Contributions from

dip/azimuth angles that do not agree with those of an actual reflec-

tor tend to cancel out by the integral summation.) To improve effi-

ciency, we rewrite this expression in the wavenumber domain:

Isðk; h0;u0Þ ¼Xx

Xðkr�ksÞ

dðh� h0Þdðu� u0Þ

� pFðks;xÞpBðkr;xÞ; (24)

S82 Xu et al.


and then we can express equation 23 as

�Rs x; h0;u0ð Þ ¼ 16pVðxÞsin h0

Xk

Is k; h0;u0ð Þeik�x: (25)

As shown in Appendix D, the summation over frequency in

equation 24 provides an imaging condition appropriate for

RTM. Equation 25 is to compute the angle-dependent reflectiv-

ity by a 3D spatial inverse Fourier transform. The following

steps summarize the whole procedure:

1) Calculate the wavefields PF and PB using an RTM wave-

propagation engine. The wavefield from the source location is

a forward propagation (in time) of a boundary source condi-

tion; the wavefield from the receiver locations is a backward

propagation (in time) of the prefiltered acquired seismic data.

Store all wavefields on a local disk.

2) Apply a 4D spatial/temporal Fourier transform to the source

and receiver wavefields (forward in time for the source wave-

field, backward in time for the receiver wavefield) to express

the wavefields in the frequency/wavenumber domain.

3) Apply the imaging condition 24 to the wavefields and decom-

pose the image by reflection angle h and azimuth angle u,

according to equation 1.

4) Apply a 3D inverse spatial Fourier transform on Is(k, h0, u0),

to obtain the 3D ADCIG �Rs x; h0;u0ð Þ using formulation 25.

As outlined here, the proposed algorithm is quite computa-

tionally intensive because the imaging condition 24 is a convo-

lution over the 3D wave vectors from both the source side and

the receiver side. Inside the convolution loop is a binning pro-

cess with azimuth angle and reflection angle binning. This bin-

ning requires computing the reflection and azimuth angles h0

and u0 for each ks, kr pair in the inner loop. With the outer

loop of frequency, the total number of computational loops for

the full-azimuth imaging is seven for a common-shot migration.

The computational effort for wave propagation is the same

for this angle-gather procedure as for conventional RTM. How-

ever, the additional 4D Fourier transforms and the angle decom-

position in the wavenumber domain significantly increase the

computational cost because the imaging condition is a multidi-

mensional convolution. For a common-shot RTM performing

wide-azimuth imaging in the Gulf of Mexico, the migration

apertures are typically 20 km in both inline and crossline direc-

tions with an inline sample rate of 25 m and a crossline sample

rate of 40 m, and the maximum depth is typically 15 km with a

depth sample rate of 15 m. The total number of wavenumbers

for a single subsurface wavefield can reach 4� 108 (800� 500

� 1000), so the convolution of two wavefields can reach

1.6� 1017 floating-point operations. Furthermore, we have an

additional outer loop over approximately 102 frequencies. The

total number of floating-point operations is so large that the

algorithm, as described so far, is well beyond the capability of

present-day supercomputers.

To reduce the cost, we first use the dispersion relationships

on the wave vectors to constrain the decomposition and convo-

lution. For homogeneous isotropic velocity models, the norms of

all slowness vectors are a fixed constant value, allowing the

reduction of seven loops to five. For heterogeneous and/or ani-

sotropic velocity models, the norms of the slowness vectors are

not anymore a fixed value when the velocity varies in the local

window, but they can be constrained to range between minimum

and maximum values, determined by maximum and minimum

values of phase velocity. In cases with a wide range of veloc-

ities (such as marine environments over salt), the computational

cost reduction due to this constraint is much less dramatic than

that of cases with a small range of velocities.

Next we decompose the wavefields to image in overlapping

local spatial windows. In this decomposition, the local window

sizes typically have an equal size in three spatial dimensions.

We typically use a window size of a few wavelengths at domi-

nant frequency with 50% overlap in each spatial dimension.

Using local windows reduces the cost of the 3D convolution by

several orders of magnitude because it reduces the total number

of wavenumbers. Using local windows also allows more efficient

use of dispersion relationships because the local range of slow-

ness vector norms is usually much smaller than the global range.

However, two issues arise from the local window scheme.

First, if the window size is very small, the small number of spa-

tial samples in the input window will result in a small number

of wavenumbers in the Fourier domain. The sparse representa-

tion of wave-propagation directions caused by the small number

of wavenumbers will cause poor resolution in the angle domain.

For example, 32 wavenumbers in the x-direction will result in

an angle resolution greater than 6�. Second, as the window size

becomes smaller, the effects of window boundary truncation

become larger, so that Gibbs artifacts can contaminate the wave-

number components and degrade the migration results. Xu et al.

(2010) investigated these two phenomena and propose a wave-

number-domain oversampling scheme inside an antileakage Fou-

rier transform (ALFT) (Xu et al., 2005), which provides good

angular resolution and reduces the boundary Gibbs artifacts by

ALFT. In the examples below, we use an oversampling ratio of

eight in all three spatial dimensions. The oversampling scheme

increases the number of wavenumbers dramatically. After over-

sampling, the wavenumbers for each wavefield in a local win-

dow might reach as many as 224, reintroducing the possibility of

a huge convolution cost.

Use of the ALFT allows us to control the cost of the convolu-

tion. The oversampling scheme in the ALFT favors the use of a

small local window size over the preservation of all wavenum-

ber information from the wavefields, and the small window size

tends to give a better approximation of plane waves to the

wavefields by an oversampling scheme inside the ALFT algo-

rithm (Xu et al., 2010). Furthermore, the ALFT can be used to

reveal the most energetic wavenumbers. In applications, a rela-

tively small number of wavenumbers will contribute the most

energy of the wavefield in a local window. Thus, a few ener-

getic wavenumbers from both source and receiver wavefields

will contribute to the convolution.

Here, we use the simple function y¼ 0.1x (Figure 2a) to illus-

trate the problem and ALFT solution. Let us start from a regular

sampling with the sampling locations x [ [0,…, 63]. The sampled

data contain 64 samples with sampling rate 1 (Figure 2a).

Because the input data are sampled on a regular grid, the Fou-

rier components are orthogonal to each other and no frequency

leakage occurs. A straightforward ALFT cannot improve the

reconstruction in this ideal case. However, with the wavenumber

sampling rate of FFT theory, Figure 2b shows the continuous

reconstruction of Fourier components computed by FFT. The

ringing artifacts appearing at the two edges are the famous



Gibbs phenomenon. This example tells us that truncating the

sampling range introduces noise near the boundary, especially

for those wavefields whose wavenumbers are not at the exact

wavenumber sampling location of FFT theory.

The solution we adopt is to oversample the data in the wave-

number domain because in the ALFT the wavenumber could be

an arbitrary value. In a small window-size application, the num-

ber input trace for ALFT is much smaller than for large window-

size applications, especially in the case of numerous dimensions.

This is the point that dramatically saves the computational cost of

ALFT. However, after oversampling the wavenumber, the number

of wavenumbers might be much bigger than when using FFT

theory. It could even reach the same number of wavenumbers as

when using large window-size applications. Thus it results in

high computational cost for the wavenumber-domain convolution.

Fortunately, with smaller window sizes, the seismic events are

closer to linear events; the “spiky wavenumbers” subset of the

total wavenumbers could represent well the input data.

We hereby illustrate this concept by a 1D example. Figure 3

shows the spectra comparison — the red line is the spectrum com-

puted by FFT and the green line is computed by ALFT. In the

example of Figure 3, an interesting phenomenon shows up in the

area of the first nonzero wavenumber. The ALFT spectrum shows

only one value with large energy. Figure 4a shows the recon-

structed continuous function (blue line). Compared to the exact

solution (red line), it has very small errors, and the biggest mis-

matches are in the two boundary areas (see Figure 4b).

In complex areas, the spiky events in the wavenumber domain

are generally more than one. In practice, we might need only to

take from 50 to 100 highest energy wavenumbers at a local win-

dow for a fixed time slice instead of 224 wavenumbers for convo-

lution. This scheme can reduce the convolution cost significantly,

and it reduces the cost of applying the imaging condition

expressed by equation 24. In practice, the computational effort of

spatial windowing, applying the ALFT, and performing the full-

angle imaging significantly exceeds the effort of a conventional

RTM (from five to 10 times more computationally expensive),

but remains feasible with the supercomputers of today (e.g., PC

clusters).

NUMERICAL EXPERIMENTS

To validate the ADCIG algorithm, we first test our 3D RTM

in a constant-velocity medium. We use a model consisting of

five flat reflectors with a depth spacing of 1.0 km. We synthe-

sized a wide-azimuth common-shot data set by simulating a sin-

gle wide-azimuth shot record. The receiver locations are distrib-

uted on a 100� 100 m grid with a maximum offset of 5.0 km

in both X and Y directions. The data were synthesized with

Figure 3. The spectra on a grid of wavenumbers with a resam-pling ratio of 16. The red line is the spectrum computed by dis-crete Fourier transform (DFT); the green line is the spectrumcomputed by ALFT. (a) The full spectrum; (b) the zoom-in overan area of major difference.

Figure 2. Regularly sampled function y¼ 0.1x and its Fourierreconstruction. (a) The function y¼ 0.1x on a regular grid with asampling rate of 1 on the support x [ [0, …, 63]. (b) Reconstructedcontinuous function using Fourier components computed by FFT;the Gibbs effect is very clear.

S84 Xu et al.


acquisition of a single shot located at surface location (5.0 km,

5.0 km), which is the center point of the receiver array (Figure

5). Five tiles of synthetic seismic data are shown in Figure 6.

Common-shot RTM was performed to output 3D angle CIGs

with eight azimuth angles. Two image locations (3.0 km, 3.0

km) and (3.75 km, 5.0 km) were selected to output full-azimuth

ADCIGs. In Figure 7, the two image locations and three com-

mon azimuth angle CIGs are plotted. Because this is a single

common-shot RTM, the contributions to the angle gathers are

one point for each location within the correct azimuth. These

are specular arrivals identified at the correct reflection angles by

our procedure. Our approach of computing 3D ADCIGs illus-

trates a superior resolution; the specular energy is well concen-

trated to the theoretical angle position, and this is well resolved

in both the reflection angle and the azimuth angle.

Figure 5. Acquisition geometry of a constant-velocity (2.0 km/s)synthetic example. The blue diamonds are a sparse representationof the receiver grid (plots 1 from every 5 in both X and Y direc-tions). The star indicates the shot location. The orange lines indi-cate the receiver locations corresponding to the display in Figure 3.

Figure 6. Seismic data at receiver locations along the orange linesof Figure 2. There is one display per line, sampled at the Y loca-tion of 1.0 to 5.0 km with an interval of 1.0 km.

Figure 7. Two full-azimuth ADCIGs created by RTM. The topshows the CIG in angles; the bottom shows the CIG locations.

Figure 4. The reconstructed continuous function by ALFT. (a)The full function; (b) the zoom-in over an area of large difference(the red line represents the exact solution).



Next we simulated a complete wide-azimuth acquisition by

laterally translating our single-shot record onto a regular grid of

shot locations, spaced on a regular 100� 100 m grid over the

whole survey. In Figure 8, we show the stack of all contribu-

tions from this complete wide-azimuth acquisition. The termina-

tions of angle gathers at the different depths are consistent with

the theoretical solution, with good amplitude preservation.

For complex structural imaging, ADCIG algorithms for

OWEM or RTM have so far considered 2D or 3D reflection

angles only (Sava and Fomel, 2003, 2006; Zhang et al., 2007b)

without considering reflection azimuth in three dimensions. Here

we first provide an example of reflection/azimuth angle-domain

CIGs from RTM using the Sigsbee synthetic data set. The origi-

nal model is 2D; we generalized it to a 2.5D model by padding

the model in the y-direction. To maintain valid 3D amplitudes,

we used a high-order finite-difference modeling scheme to gen-

erate 3D synthetic data with zero-azimuth (single-streamer) ac-

quisition. For this restricted 3D data set, we expect 3D ADCIGs

to have significant energy with full reflection angles only at

zero azimuth.

Figure 9 shows the stacked image of the central line using 3D

RTM. We selected 18 locations (represented by 18 green lines

in Figure 9) to output CIGs. The 2D ADCIGs converted from

subsurface offset gathers (Sava and Fomel, 2003) are shown in

Figure 10. The maximum subsurface offset is 5000 feet with a

75-feet offset increment. The ADCIGs are sampled every 2�.These gathers display a smearing effect, also observed by Sava

and Fomel (2003); this effect is due to sampling issues in the

gather conversion from space to wavenumber and then from

wavenumber to angle. The zero-azimuth 3D ADCIGs are shown

in Figure 11. The 3D angle CIGs exhibit a better S/N and higher

resolution with sharper event terminations at far angles than the

2D results from the conventional approach.

Finally, a true wide-azimuth test was performed from a deep-

water survey in the Garden Banks area of the Gulf of Mexico.

The seismic data were acquired with a maximum inline offset

of 8.0 km and crossline offset of 4.0 km. Figure 12 shows a

TTI RTM inline stacked section in a complex area: a minibasin

between two complex salt bodies with a rough top of salt geo-

metries and overhangs. The subsalt events are well imaged.

Two TTI RTMs were performed. First a narrow-azimuth sub-

set of the wide azimuth data was migrated. Figure 13 shows a

single 3D ADCIG from this migration. The reflection angle

range is from 0� to 40� with a sampling rate of 2�. Six subsur-

face azimuth sectors were used with a sampling rate of 30�. The

contributions of azimuth angles to the CIGs were also plotted

above the 3D ADCIG. For the narrow-azimuth subset of the

data, the energy in the ADCIG is concentrated at zero azimuth,

with some leakage to the nearby azimuth angles at shallow

depths. Second, the full wide-azimuth data set was migrated.

Figure 8. Zero-azimuth ADCIGs in the synthetic example.

Figure 9. Stack images of the Sigsbee 2.5D synthetic example.The green lines indicate the locations of the CIG plotted inFigures 7 and 8.

Figure 10. Two-dimensional ADCIGs built according to theapproach of Sava and Fomel (2003).

Figure 11. Zero-azimuth ADCIGs built according to our proposed3D ADCIG approach.

S86 Xu et al.


Figure 14 shows the 3D angle CIG, to be compared with Figure

13. The energy distribution is more uniformly spread over azi-

muth angles, except at great depths around azimuth angle 90�,which has less energy at far reflection angles because the maxi-

mum crossline offset (90� surface azimuth) is only one-half the

maximum inline offset (0� surface azimuth).

CONCLUSIONS

We have derived the theory of true-amplitude RTM with

ADCIG output, and we have applied it to generate 3D subsur-

face reflection/azimuth ADCIGs. Our method for generating the

ADCIGs is based on the decomposition of wavefields in the

subsurface into directional components. Because a full decom-

position into all possible directional components is prohibitively

inefficient, our decomposition selects those components associ-

ated with the dominant energy. Use of an antileakage Fourier

transform facilitates the selection of the dominant wavefield

components. Results from a synthetic data set demonstrate that

our approach produces higher quality angle gathers, with better

angular resolution, than angle gathers from conventional 2D

schemes. The Gulf of Mexico example demonstrates that the

approach is stable for real data applications.

We expect further applications of 3D ADCIGs in velocity

updating, via wave-equation tomography, and AVA analysis.

More generally, we consider our decomposition method to be a

promising tool for analyzing, and possibly controlling, the

behavior of wavefields in the subsurface.

ACKNOWLEDGMENTS

We sincerely thank Sam Gray for his enormous help with this

manuscript. We thank Gilles Lambare and Robert Soubaras for

their constructive suggestions and discussion. We thank Chu-ong

Ting and his team for their help with the numerical examples. We

thank Yan Huang and Bing Bai for their test of Gulf of Mexico real

data. We thank Jerry Young and Scott Shonbeck for their support

of this work. We thank three anonymous reviewers for reviewing

this manuscript and giving constructive suggestions.Figure 13. Three-dimensional ADCIG from a narrow-azimuthexperiment in the Garden Banks area of the Gulf of Mexico.

Figure 14. Three-dimensional ADCIG from a wide-azimuthexperiment in the Garden Banks area of the Gulf of Mexico.

Figure 12. Stack image of one inline in a real data example in theGarden Banks area of the Gulf of Mexico. The yellow line indi-cates the location of the 3D ADCIGs plotted in Figures 13 and 14.



APPENDIX A

LOCAL COORDINATE SYSTEM AND ANGLE

DEFINITIONS

At a reflection point, given the wave-propagation direction vec-

tors ks and kr (Figure 1), the local coordinates can be described

using three orthogonal vectors:

k ¼ ks þ kr

l ¼ k� nxð Þ � k

m ¼ k� l

8<: ; (A-1)

where nx¼ (1, 0, 0) is a unit vector. The basis vectors of this local

coordinate system are the normalized unit vectors:

n1¼nðlÞ¼ l

lj j ; n2¼nðmÞ¼ m

mj j ; n3¼ nðkÞ¼ k

kj j :

(A-2)

We define the reflection angle h and the azimuth angle u in this

local coordinate system, as shown in Figure 1, with the following

formulas:

cos h ¼ k � kr

kj j krj j

cos u ¼ ks � krð Þ � nx � ks þ krð Þð Þks � krj j nx � ks þ krð Þj j

8>>><>>>:

: (A-3)

For tomography, in the isotropic or STI cases (the general TI case

can be done in similar analog), we have knowledge of h, u, and

n3, and we need to reconstruct the two directions n(ks) and n(kr).

We first construct an auxiliary vector,

a ¼ tan h n1 cos uþ n2 sin uð Þ; (A-4)

then we have two vectors:

v1 ¼ n3 þ a; v2 ¼ n3 � a: (A-5)

Then the required directions are

nðksÞ ¼ nðv1Þ; nðkrÞ ¼ nðv2Þ: (A-6)

APPENDIX B

CONVERSION FORMULA FROM

OFFSET-DOMAIN TO ANGLE-DOMAIN CIG

Consider an isotropic 2D medium; the wavenumber vectors ks

from the source wavefield and kr from the receiver wavefield can

be expressed with components as

ks ¼ ðksx; k

szÞ; kr ¼ ðkr

x; krzÞ: (B-1)

The direction of the wavenumber vector indicates the wavefield-

propagation-phase direction for the source wavefield and the re-

ceiver wavefield, respectively. For wave-equation migration, one

might work in the CMP and offset domain, where the wavenum-

bers can be expressed in terms of offset h, midpoint x, and depth

wavenumbers kh, kcmp, kz. The relationships with ks and kr are

defined by the following transformation:

kh ¼ðkr

x� ksxÞ

2; kcmp ¼

ðkrxþ ks

xÞ2

; kz ¼ðkr

z þ kszÞ

2: (B-2)

Furthermore, at image point x, the components of the wavenum-

bers from the source side and receiver side that are perpendicular

to the migration dip vector are exactly preserved as guaranteed by

Snell’s law.

For a single mode in an isotropic velocity model, the vector

lengths of ks and kr are the same:

ksj j ¼ krj j; (B-3)

where j�j denotes the modulus of a vector. Next we rewrite equa-

tion B-3 in component form:

ðksxÞ

2 þ ðkszÞ

2 ¼ ðkrxÞ

2 þ ðkrzÞ

2: (B-4)

Then, moving ðkrzÞ

2to the left side and ðks

xÞ2

to the right side, we

obtain

ðkrz � ks

zÞðkrz þ ks

zÞ ¼ ðkrx � ks

xÞðkrx þ ks

xÞ: (B-5)

Using the transformation relation B-2, we rewrite equation B-5,

for kz= 0, as

ðkrz � ks

zÞ ¼2kcmpkh

kz: (B-6)

To compute the reflection angle h, we have

tan h ¼ jkr � ksjjkr þ ksj

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkr

x � ksxÞ

2 þ ðkrz � ks

zÞ2

ðkrx þ ks

xÞ2 þ ðkr

z þ kszÞ

2

s: (B-7)

Finally, substituting equations B-2 and B-6 into equation B7, we

obtain

tan h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2

h þ ðkrz � ks

zÞ2

4k2cmp þ 4k2

z

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2

h þ ð2kcmpkh

kzÞ2

4k2cmp þ 4k2

z

vuut ¼ kh

kz

��;

(B-8)

which is an expression similar to formulas obtained by de Bruin

et al. (1990) and Sava and Fomel (2003).

In three dimensions, wavenumbers are expressed as

ks ¼ ðksx; k

sy; k

szÞ; kr ¼ ðkr

x; kry; k

rzÞ: (B-9)

Three-dimensional wave-equation migration works with wave-

numbers khx, khy, kcmpx, kcmpy, kz, whose relationships with 3D

wavenumbers ks and kr are

khx ¼ðkr

x � ksxÞ

2; khy ¼

ðkry � ks

yÞ2

;

kcmpx ¼ðkr

x þ ksxÞ

2; kcmpy ¼

ðkry þ ks

yÞ2

;

kz ¼ðkr

z þ kszÞ

2:

(B-10)

The isotropic additional condition (velocity invariance) in compo-

nent form is

ðksxÞ

2 þ ðksyÞ

2 þ ðkszÞ

2 ¼ ðkrxÞ

2 þ ðksyÞ

2 þ ðkrzÞ

2: (B-11)

Anisotropic cases involve a more complex dispersion relationship,

and the formulation becomes significantly more complicated. As

in two dimensions, moving the z-component to the left side, and

S88 Xu et al.


the x-, y-components to the right side, yields

ðkrz � ks

zÞðkrz þ ks

zÞ ¼ ðkrx � ks

xÞðkrx þ ks

xÞþ ðkr

y � ksyÞðkr

x þ ksyÞ; (B-12)

which, for kz= 0, becomes

ðkrz � ks

zÞ ¼2ðkcmpxkhx þ kcmpykhyÞ

kz: (B-13)

To compute the reflection angle h in three dimensions, we have

tan h ¼ jkr � ksjjkr þ ksj

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkr

x � ksxÞ

2 þ ðkry � ks

yÞ2 þ ðkr

z � kszÞ

2

ðkrx þ ks

xÞ2 þ ðkr

y þ ksyÞ

2 þ ðkrz þ ks

zÞ2

vuut ;

(B-14)

Finally, substituting equations B-10 and B-13 into equation B-14,

we obtain

tan h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2

hx þ 4k2hy þ ðkr

z � kszÞ

2

4k2cmpx þ 4k2

cmpy þ 4k2z

vuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2

hx þ 4k2hy þ ð

2ðkcmpxkhxþkcmpykhyÞkz

Þ2

4k2cmpx þ 4k2

cmpy þ 4k2z

vuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

hxk2z þ k2

hyk2z þ ðkcmpxkhx þ kcmpykhyÞ2

k2z ðk2

cmpx þ k2cmpy þ k2

z Þ

vuut : (B-15)

This is consistent with Fomel (2004), but it does not resemble the

2D formulas of de Bruin et al. (1990) or Sava and Fomel (2003),

even with the approximation of null khy (which might be valid for

marine narrow-azimuth data),

kky � 0; kh � khx: (B-16)

tanh�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

hxk2z þ k2

cmpxk2hx

k2z ðk2

cmpx þ k2cmpyþ k2

z Þ

s¼ kh

kz

��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2z þ k2

cmpx

k2cmpxþ k2

cmpy þ k2z

s:

(B-17)

The 2D formula applies in a 3D case only if kcmpy: 0, which

implies that the reflectors have no dip in the crossline direction,

and that the medium is identical to 2.5 dimensions.

APPENDIX C

TRANSFORMING FROM SURFACE

COORDINATES TO SUBSURFACE ANGLES

To simplify the derivation, we start from the isotropic case. We denote

the elementary vectors on the unit illumination sphere, corresponding to

the directions of the wavenumber vectors ks, kr, and k as Xs, Xr, and

Xm, respectively. Thus, we have the following relationships:

Xm ¼ðjksjXs þ jkrjXrÞjjksjXs þ jkrjXrj

;

cos 2h ¼ Xs � Xr;

cos u ¼ ðXs � XrÞ � ðnx � ðXs þ XrÞÞjXs � Xrjjnx � ðXs þ XrÞj

:

8>>>>><>>>>>:

(C-1)

Here the condition of incident angle equal to reflection angle was

applied. This assumption is valid under the cases of an isotropic

velocity model, or an STI anisotropic model, in which the sym-

metrical axis is orthogonal to the reflection plane. As a conse-

quence, the wavenumber vectors for the incident wavefield and

reflected wavefield have the same norm.

We decompose the Jacobian into two parts:

oðh;u; kÞoðx; s; rÞ

�� ¼ oðh;u; kÞ

oðx;Xs;XrÞ

�� oðx;Xs;XrÞ

oðx; s; rÞ

��: (C-2)

The first determinant on the right-hand side is a local coordinate

transform. The second one is the link between the elementary vec-

tors on the illumination sphere and the source and receiver loca-

tions; this determinant is separable in shot and receiver:

oðx;Xs;XrÞoðx; s; rÞ

�� ¼ oðXs;XrÞ

oðs; rÞ

�� ¼ oðXsÞ

oðsÞ

�� oðXrÞ

oðrÞ

��: (C-3)

We recall the asymptotic form of the 3D acoustic Green’s

function,

Gðx;x; sÞ ¼ Aðx; sÞe�ixTðx;sÞ ¼ 1

4p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

VðxÞoXs

pszos

��

se�ixTðx;sÞ;

(C-4)

and then apply equation C-4 to both shot and receiver sides,

obtaining the following equations:

oXs

os

�� ¼ ð4pÞ2VðxÞA2ðx; sÞps

z;

oXr

or

�� ¼ ð4pÞ2VðxÞA2ðx; rÞpr

z : (C-5)

Now we go back to the first Jacobian on the right-hand side of

equation C-2, where we need the following relations:

Xs!ðhs;usÞ : dXs! sinhsdhsdus;oðhs;usÞ

oXs

��¼ 1

sinhs

;

Xr!ðhr;urÞ : dXr! sinhrdhrdur;oðhr;urÞ

oXr

��¼ 1

sinhr

:

(C-6)

Because

k ¼ xq ¼ xjqjðsin hm cos um; sin hm sin um; cos hmÞ;(C-7)

we can directly compute the Jacobian:

ok

oðhm;um;xÞ

��

¼xjqjcoshm cosum xjqjcoshm sinum �xjqj sinhm

�xjqj sinhm sinum xjqj sinhm cosum 0

jqj sinhm cosum jqj sinhm sinum jqjcoshm

��

¼x2jqj3 sinhm: (C-8)



Using equations C-6 and C-8, we have

oðh;u; kÞoðx;Xs;XrÞ

��

¼ oðhs;usÞoðXsÞ

�� oðhr;urÞ

oðXrÞ

�� oðkÞoðhm;um;xÞ

�� oðh;u; hm;umÞoðhs;us; hr;urÞ

��

¼ x2jqj3 sin hm

sin hs sin hr

oðh;u; hm;umÞoðhs;us; hr;urÞ

��: (C-9)

Deriving the Jacobian j oðh;u;hm ;umÞoðhs;us ;hr;urÞ

j is the most difficult part of this

appendix. To achieve it, we write the following four independent

equations:

2coshm cosh¼ðcoshsþ coshrÞ2coshsinhm cosum¼ sinhs cosusþ sinhr cosur

2coshsinhm sinum¼ðsinhs sinusþ sinhr sinurÞ

2cosusinhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� sin2 hm cos2 umÞ

q¼ sinhs cosus� sinhr cosur

8>>>>>>><>>>>>>>:

(C-10)

In equation C-10, the first three equations are the component

form of k¼ ksþkr; the fourth equation comes from the third

equation of C-1. To simplify our discussion, we define

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� sin2 hm cos2 umÞ

qand take the derivatives with respect

to all the variables in equation C-10, and write the result in ma-

trix form:

L11 L12 L13 L14

L21 L22 L23 L24

L31 L32 L33 L34

L41 L42 L43 L44

0BBBB@

1CCCCA

oh

ou

ohm

oum

0BBBB@

1CCCCA

¼

R11 R12 R13 R14

R21 R22 R23 R24

R31 R32 R33 R34

R41 R42 R43 R44

0BBBB@

1CCCCA

ohs

ous

ohr

our

0BBBB@

1CCCCA; (C-11)

which gives the expression of the Jacobian


�� ¼ jRjjLj : (C-12)

The determinant of the left matrix jLj ¼ j(Lij)j has the following

expression:

jLj ¼

�2 cos hm sin h 0 �2 sin hm cos h 0

�2 sin h sin hm cos um 0 2 cos h cos hm cos um �2 cos h sin hm sin um

�2 sin h sin hm sin um 0 2 cos h cos hm sin um 2 cos h sin hm cos um

2r cos h cos u �2r sin u sin h � sin 2hm cos2 um cos u sin hr

sin2 hm sin 2um cos u sin hr

��

��¼ 16r cos2 h sin2 h sin hm sin u

;

and the determinant of the right matrix jRj ¼ j(Rij)j is

jRj¼

�sinhs 0 �sinhr 0

coshscosus�sinhssinus coshrcosur �sinhrsinur

coshssinus sinhscosus coshrsinur sinhrcosur

coshscosus�sinhssinus�coshrcosur sinhrsinur

��

��¼4rcoshsinhssinhrsinusinh: (C-13)

Substituting the expressions of jLj and jRj into equation C-12, we

obtain


�� ¼ sin hs sin hr

4 cos h sin h sin hm: (C-14)

With the help of equations C-14 and C-9, we get the final formula-

tion of the Jacobian:

oðh;/; kÞoðx; s; rÞ

�� ¼ x2jqj3ð4pÞ4V2ðxÞA2ðx; sÞA2ðx; rÞps

zprz

4 cos h sin h:

(C-15)

APPENDIX D

IMAGING CONDITION IN THE

WAVENUMBER DOMAIN

Given subsurface wavefields pF(xs, ts) from the source and

pB(xr, tr) from the receivers, the contribution to the subsurface

reflectivity from one shot, using the crosscorrelation imaging con-

dition, is

Isðx;h0;u0Þ ¼ dðh� h0Þdðu�u0ÞpFðxs; tsÞpBðxr; tsÞjxs¼xr¼xts¼tr

:

(D-1)

If we note that pF(ks, x) is the 3D spatial forward Fourier trans-

form and 1D temporal forward Fourier transform of pF(xs, ts), we

can express the subsurface wavefield from the source as

pFðxs; tsÞ ¼Xx

Xks

pFðks;xÞeiks�xs eix�ts : (D-2)

Likewise, if we note that pB(kr, x) is the 3D spatial forward Fou-

rier transform and 1D temporal inverse Fourier transform of

pB(xr, tr), we can express the subsurface wavefield from the

receivers as

pBðxr; trÞ ¼Xx

Xkr

pBðkr;xÞeikr�xr e�ix�tr : (D-3)

Compared with equation D-2, the negative sign in the Fourier

transform in equation D-3 is due to the back propagation.

Substituting equations D-2 and D-3 into equation D-1, and

applying the ts¼ tr and xs¼ xr¼ x condition, we write the imag-

ing condition in terms of source and receiver wave vectors:

Isðx; h0;u0Þ ¼Xx

Xks

Xkr


� pFðks;xÞeiks�xpBðkr;xÞeikr�x

Then the relations

k ¼ ðks þ krÞ; kh ¼ ðkr � ksÞ: (D-4)

allow us to rewrite the imaging condition in terms of midpoint

and offset wave vectors:

Isðx; h0;u0Þ ¼Xx

Xk

Xkh


� pFðks;xÞpBðkr;xÞeik�x: (D-5)

S90 Xu et al.


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