True-amplitude Gaussian-beam migrationtinue waveﬁelds. Previous presentations of Gaussian-beam...

T

S

st

.©

GEOPHYSICS, VOL. 74, NO. 2 �MARCH-APRIL 2009�; P. S11–S23, 5 FIGS.10.1190/1.3052116

rue-amplitude Gaussian-beam migration

amuel H. Gray1 and Norman Bleistein2

sasp3Nrmbct

ecsabntmt

gaaptwtcuovg

ip

ed 8 Oc.Pheno

ABSTRACT

Gaussian-beam depth migration and related beam migra-tion methods can image multiple arrivals, so they provide anaccurate, flexible alternative to conventional single-arrivalKirchhoff migration. Also, they are not subject to the steep-dip limitations of many �so-called wave-equation� methodsthat use a one-way wave equation in depth to downward-con-tinue wavefields. Previous presentations of Gaussian-beammigration have emphasized its kinematic imaging capabili-ties without addressing its amplitude fidelity. We offer twotrue-amplitude versions of Gaussian-beam migration. Thefirst version combines aspects of the classic derivation ofprestack Gaussian-beam migration with recent results ontrue-amplitude wave-equation migration, yields an expres-sion involving a crosscorrelation imaging condition. To pro-vide amplitude-versus-angle �AVA� information, true-ampli-tude wave-equation migration requires postmigration map-ping from lateral distance �between image location andsource location� to subsurface opening angle. However,Gaussian-beam migration does not require postmigrationmapping to provideAVAdata. Instead, the amplitudes and di-rections of the Gaussian beams provide information that themigration can use to produce AVA gathers as part of themigration process. The second version of true-amplitudeGaussian-beam migration is an expression involving a de-convolution imaging condition, yielding amplitude-varia-tion-with-offset �AVO� information on migrated shot-do-main common-image gathers.

INTRODUCTION

Beginning with Hill’s �1990� paper on poststack migration, Gaus-ian-beam migration has been recognized as an accurate and versa-ile migration technique. Its reliance on ray tracing allows it to image

Manuscript received by the Editor 20 May 2008; revised manuscript receiv1CGGVeritas, Calgary,Alberta, Canada. Email: [email protected] School of Mines, Department of Geophysics, Center for Wave

mines.edu.2009 Society of Exploration Geophysicists.All rights reserved.

S11

Downloaded 10 Feb 2010 to 24.149.197.34. Redistribution subject to S

teep dips, and its decomposition of neighboring seismic traces intongular information allows it to propagate different angles to anyubsurface location independently, achieving multipathing. Hill’srestack method �2001� extends his work by presenting an efficientD migration, suitable for data acquired along a narrow azimuth.owack et al. �2003� and Gray �2005� remove the narrow azimuth

estriction by presenting variations suitable for �possibly wide-azi-uth� shot-record migration. All of this work establishes Gaussian-

eam migration as a powerful imaging technique, with accuracyomparable to wave-equation migration and flexibility comparableo Kirchhoff migration.

Albertin et al. �2004� present the first treatment of amplitude pres-rvation in Gaussian-beam migration, with an approach suitable forommon-offset, narrow-azimuth data.Aside from being limited to aingle acquisition geometry, the approach of Albertin et al. relies oncommon-offset Beylkin determinant that avoids caustic problemsy allowing time to be a complex quantity. �The Beylkin determi-ant is a Jacobian of transformation from image coordinates, whenhe image is formed, to surface coordinates, where the imaging sum-ation takes place�. This Jacobian is much more difficult to compute

han the Jacobian for common-shot migration.We present an alternative approach, based on a generalized strate-

y for true-amplitude migration �Bleistein et al., 2005�. �Here true-mplitude migration means to produce amplitudes on reflectors thatre proportional, up to a constant factor, to P-wave angle-dependentlane-wave reflection coefficients.�This strategy has been applied torue-amplitude wave-equation migration �Zhang et al., 2007�, ande follow the application closely. Our formulation also depends on

he Jacobian of a mapping from subsurface coordinates to surfaceoordinates; however, our Jacobian is much more straightforward tonderstand and compute than that of Albertin et al. �2004�. �On thether hand, optimizing the Gaussian-beam migration formulas in-olves high-frequency asymptotics, which complicate the final mi-ration weights.�Our Jacobian is also more general, in that it is not limited to imag-

ng data acquired using a particular geometry even though it was firstresented in the context of imaging complete wavefields. It applies,

tober 2008; published online 11 February 2009.

mena, Golden, Colorado, U.S.A. Email: [email protected]; norm@dix

EG license or copyright; see Terms of Use at http://segdl.org/

frct�ddct

maii

tgitwbmcbKdg�3

tbtcGiitit

titv�npdsvtt

oalc2c

ha

omrtacatAaatgpct

siotAa�ortltbtpp

T

s2�obc�itdtstwthtf

S12 Gray and Bleistein

or example, to both common-offset and common-shot data; its onlyeal requirement is that all input data �e.g., all offsets or all shots�ontaining information from a particular image location be includedo provide accurate amplitudes from that location. Zhang et al.2007� apply this Jacobian as a postmigration mapping from shot-omain common-image gathers �SDCIGs�, whose traces are in-exed by the lateral distance between image location and source lo-ation, to angle-domain common-image gathers �ADCIGs�, whoseraces are indexed by the opening angle at the imaged reflectors.

This approach to true-amplitude imaging can be applied to anyigration method — in particular, to Gaussian-beam migration. In

ddition, because the ray-based Gaussian beam contains the anglenformation as the imaging takes place, we can apply the mappingmplicitly during imaging, migrating directly intoADCIGs.

We present a derivation of true-amplitude Gaussian-beam migra-ion that follows those of Zhang et al. �2007� and Hill �2001�. Theeneral framework applies to any migration method: crosscorrelat-ng downward-continued wavefields from source and receiver loca-ions. We write the wavefields in terms of Green’s functions for theave equation, and we express the Green’s functions as Gaussian-eam expansions. Our derivation is in terms of wavefields, but aodification of the Green’s functions by Hill �2001� allows us to

onsider how migration acts on single traces �even after they haveeen slant stacked into neighboring beam center locations�, as inirchhoff migration. The derivation contains details of the steepest-escent calculation used by Hill �2001� to reduce the number of inte-rals, which correspond to loops in the computer implementation.We perform this analysis only for the 2D case; the correspondingD analysis is beyond the mathematical scope of this paper.�Zhang et al. �2007� show that the crosscorrelation imaging condi-

ion is appropriate for true-amplitude migration when it is followedy mapping from distance to angle. Here, we show that the same isrue for Gaussian-beam migration. In addition, we show that therosscorrelation imaging condition is appropriate for true-amplitudeaussian-beam migration when imaging directly into ADCIGs us-

ng ray information available to the migration, without the need tomage into intermediate SDCIGs. We also show the relationship be-ween the crosscorrelation imaging condition and the deconvolutionmaging condition usually used for true-amplitude Kirchhoff migra-ion.Avery simple example explains this relationship.

Because the action of Gaussian-beam migration on a single inputrace can be analyzed in a manner similar to Kirchhoff and general-zed Radon transform �GRT� migrations, we mention some connec-ions with those methods. As described by Hanitzsch �1997�, earlyersions of true-amplitude Kirchhoff migration relied on Bleistein’s1987� modification of Beylkin’s �1985� theory of imaging disconti-uities. Bleistein �1987� treats the imaging problem as an inverseroblem that integrates reflection data over acquisition surface coor-inates. He incorporates Beylkin’s �1985� geometric term into hisolution for the weight factor that makes the composition of the in-ersion operator with the forward modeling operator a formal identi-y, i.e., the inverse operator applied to reflection data yields a reflec-ivity model.

Many subsequent true-amplitude Kirchhoff migration results relyn this approach, which is limited to situations where the wavefieldsre well behaved �i.e., away from caustics�. Later generalizations al-ow for imaging in greater structural complexity by removing theaustic restriction and allowing for multiple arrivals �e.g., Xu et al.,001; Bleistein et al., 2005�. These generalizations are much moreomplicated than conventional single-arrival true-amplitude Kirch-


off migration; but, like our method, they provide greater accuracynd allow for imaging intoADCIGs.

GRT migration takes a dual approach to Kirchhoff migration butbtains the same result. As described by Miller et al. �1987�, GRTigration reconstructs each subsurface scatterer by accumulating

eflection data from all isochrons that intersect at the scatterer loca-ion.At actual reflector locations, the live data that dominate the sumre from the isochron that passes tangent to the reflector surface. Theonceptual distinction of GRT migration lies in performing the im-ging operation in the subsurface, as opposed to Kirchhoff migra-ion’s summation of data expressed in terms of surface coordinates.lthough the isochrons are determined by the acquisition geometry

s well as the subsurface velocity structure, the quality of the imagend its amplitudes is determined by the geometry of the isochrons athe scatterer locations. As with Kirchhoff and Gaussian-beam mi-rations, a reflector is well imaged only if a portion of an isochronasses tangent to it. In Gaussian-beam migration, the isochrons areomposed of local portions, each determined by the real part of theraveltime surface for one of the localized Gaussian beams.

Several workers �e.g., Deng and McMechan, 2007� mentionhortcomings of true-amplitude migration, either from complexitiesn the migration velocity leading to uneven subsurface illuminationr from elastic or anelastic effects that lie outside acoustic inverseheory. Such statements apply equally to all migration methods.lso, shortcomings in processing before migration, or interpretation

fter migration, can compromise the fidelity of migrated amplitudes.In fact, the simpler of our two examples illustrates the effect of onef these problems — incomplete illumination resulting from finiteecording or migration aperture — on migrated amplitudes.� Al-hough these shortcomings imply limitations in our ability to ana-yze migrated amplitudes except in the simplest situations, we takehe position that migrated amplitudes should be as accurate as possi-le within the acoustic limits assumed by most migration methods. Ifhe cost of including accurate migration amplitudes is only a smallart of the total migration cost, the effort of including accurate am-litudes is warranted.

TRUE-AMPLITUDE GAUSSIAN-BEAMMIGRATION

he 2D formula — Crosscorrelation imaging condition

Prestack Gaussian-beam migration was developed first for marinetreamer-style common-offset, common-azimuth records �Hill,001; Albertin et al., 2004� and later for common-shot recordsNowack et al., 2003; Gray, 2005�. Because our presentation reliesn observations about true-amplitude wave-equation migration, weegin with recorded data in the form of an actual wavefield, i.e.,ommon-shot records. Although this approach differs from Hill’s2001� derivation of Gaussian-beam migration, most of the steps aredentical. In fact, if we consider the process that the method applieso any given input trace in a well-sampled survey, we find that the or-er of the input data is irrelevant; we can apply the method to inputraces gathered in any order with essentially identical migrated re-ults. If the input traces are well sampled spatially �receiver loca-ions and shot locations at regular inline x and crossline y intervalsith spacing that is fine enough to allow the unaliased slant stack of

he highest temporal frequencies present in the recorded data at theighest value of horizontal slowness� and if the subsurface illumina-ion at an image location from the downward-continued wavefieldsrom source and receiver locations is nonzero, then Gaussian-beam


mt

ta2impGaiK�ccml

wici

wdt�Cft

HpcvGms

cl

Camw

m

Thcbaf

ptilwmom

cc�owev

scIipastevmu

True-amplitude Gaussian-beam migration S13

igration will produce a migrated common-image gather �CIG� athat location that allows amplitude-versus-angle �AVA� analysis.

In a second departure from Hill’s �2001� derivation, we deriverue-amplitude Gaussian-beam migration in two dimensions onlynd present the 3D expression without derivation. For efficiency, theD and 3D derivations rely on a saddle-point �or steepest-descent�ntegral �Hill, 2001�, an asymptotic technique that generalizes theethod of stationary phase to complex exponential functions whose

hase is also complex. �This happens because the formulation ofaussian beams, a type of dynamic ray tracing, allows time t, which

ppears in the phase, to be a complex quantity: t � tRe � itIm, where� ��1. By contrast, the dynamic ray tracing used by standardirchhoff migration requires time along a raypath to remain real: ttRe.� The 2D derivation requires a saddle-point integral in a single

omplex variable, and the 3D derivation requires the integral in twoomplex variables. Because the theory for saddle-point integrals inore than one complex variable is very technical without shedding

ight on the problem at hand, we present it elsewhere.We begin with the expression for a single migrated 2D shot record

ith shot location xs � �xs,0�, receiver locations xr � �xr,0�, andmage locations x � �x,z�, using the imaging condition that is therosscorrelation of the downward-continued upgoing and downgo-ng wavefields �Zhang et al., 2007�:

I�x;xs� � �i� d� sgn��pU�x;xs;��pD*�x;xs;�� ,

�1�

here pU and pD are, respectively, the upgoing �recorded� and theowngoing �source� wavefields and * denotes complex conjuga-ion. With the postmigration mapping described by Zhang et al.2007�, this expression can be modified to convert SDCIGs intoAD-IGs. Still following Zhang et al. �2007�, we write asymptotic forms

or pU and pD*, also expressing them in terms of Green’s functions for

he 2D wave equation:

pU�x;xs;�� 2� dxrpU�xr;xs;��cos � r

Vr

�� i�AART,r

� exp�� i�� r�

��2� dxrpU�xr;xs;��

�cos � r

Vri�G*�x;xr;�� , �2�

pD*�x;xs;�� 2

cos � s

Vs

�� i�AART,s exp�� i�� s�

� 2cos � s

Vsi�G*�x;xs;�� . �3�

ere, Vs and Vr are surface velocities at the source and receiveroints; � s and � r are takeoff angles of the rays from source and re-eiver points to image location x; AART,s, AART,r, and � s, � r are real-alued amplitudes and traveltimes from asymptotic ray theory; and�x;x�;�� is the Green’s function for the wave equation. The di-ensionalities of pU and p

D* are different: in two dimensions, the

ource wavefield p* has dimensions of inverse length, but the re-
D

orded wavefield pU, the response to a 2D line source, is dimension-ess. Inserting expressions 2 and 3 into equation 1 results in

I�x;xs� ��4� d�i��cos � s

VsG*�x;xs;��

�� dxrpU�xr;xs;��cos � r

VrG*�x;xr;�� . �4�

The choice of Green’s function determines the migration method.hoosing a Green’s function of the form AART exp�i�� , where AART

nd � are both real, will lead to Kirchhoff migration. Choosing a nu-erical wavefield extrapolator based on a one-way wave equationill lead to a wave-equation migration.Instead of these choices, we write the Green’s function as a sum-ation of Gaussian beams �Hill, 1990, 2001�:

G�x;x�;�� i

4�� dpx

pzuGB�x;x�;��

�i

4�� dpx

pzAGB exp�i�TGB� . �5�

his expression is an integral of individual Gaussian beams uGB overorizontal slownesses px; pz � cos � /V is the vertical slowness. Inontrast to the amplitudes and times in equations 2 and 3, Gaussian-eam amplitude AGB and traveltime TGB �developed inAppendices And B� are complex quantities. From now on, we drop the subscriptrom AGB and TGB.

Equations 4 and 5 show the versatility of the Gaussian-beam ap-roach. Each term uGB represents a partial wavefield in a limited spa-ial region surrounding the beam’s central raypath. These beams arendependent of one another as they range through the subsurface, al-owing the overlapping beams to contribute multiple arrivals to theavefield at any subsurface location. This differs from Kirchhoffigration, whose Green’s functions can accommodate multipathing

nly with considerable effort. As mentioned, several workers haveade this extra effort.For Gaussian-beam migration, Hill �1990, 2001� chooses initial

onditions for the dynamic ray-tracing equations that ensure �1� lo-alized planar beams uGB at initial source and receiver locations and2� regular, nonsingular behavior of the individual beams through-ut the subsurface. These depend on sensible choices for parameters

0, the initial half-width of the Gaussian beams, and �r, the refer-nce frequency. �A typical value for �r is 2� �10 Hz, and a typicalalue for w0 is one wavelength at the reference frequency.�The first of these conditions allows us to think of decomposing the

ource and recorded wavefields pD and pU into local plane-waveomponents that match the initial directions of the Gaussian beams.n particular, it allows us to consider propagating these componentsnto the subsurface using individual beams uGB. If this can be accom-lished, then we can migrate the local plane-wave components fromset of surface locations that is much sparser than the original set ofource and receiver locations. Hill �1990, 2001� provides recipes forhis procedure, which is the heart of Gaussian-beam migration. Inssence, Gaussian-beam migration replaces the migration of indi-idual data traces over all directions into the subsurface by theigration of localized, directional components of the wavefield with

nique initial directions into the subsurface. Thus, at selected beam-


cgp

tnsaatatppsfsc

in�eofttetpn

wtpaua

ct

I

w

abfL

t

t

Tlmc

gpaett

flssetvt

�ss


enter locations, local slant stacks are performed and the data mi-rated over those initial directions using Gaussian beams as theropagators.To make this rigorous, we first describe the local slant stack, then

he partition of unity that permits each input trace to contribute to aumber of beam centers with weights that sum to unity. The locallant stack acts as a simple delay applied to an input trace; the delayccounts for the traveltime difference between a plane wave arrivingt the actual receiver location and the same plane wave arriving athe beam center location L � �L,0�. In the frequency domain, this isccomplished by applying the phase shift exp��i�px�xr � L�� tohe input trace whose receiver is located at �xr,0�. Hill �2001� incor-orates this phase shift as a modification to his Green’s function ex-ansion 5, and Gray �2005� suggests a modification to this phasehift when the elevations of L and xr are different. �Effectively, thisactor is used to approximate the wavefield resulting from a pointource at �x�,0� when the source of ray propagation is at a nearby lo-ation �L,0�.�

Because the Gaussian-beam wavefront is perfectly planar only atts initial location on the beam center and gradually accumulatesonzero curvature away from the beam center, approximating thenearly linear� delay by the linear slant stack incurs some kinematicrror; however, this error usually is negligible. Potentially more seri-us is the amplitude error incurred when the origin of the Green’sunction �the receiver location� is far from the initial point of the rayracing �the beam center�. However, Hill’s choice of partition of uni-y ensures that the traces far from a beam center are downweightedxponentially relative to traces close to a beam center, whose ampli-ude error is small. We use this approximation in our synthetic exam-les; its error is not enough to degrade the migrated amplitudes sig-ificantly.The partition of unity is given by Hill �1990� as

1 ��L

�2�w0

� �

�r

L

exp��

�r �xr � L�2

2w02 � , �6�

here �L is the spacing between beam-center locations.Asimple in-erpretation of this formula is as a discrete approximation, with ap-ropriate change of variable, to the formula for the infinite integral ofGaussian function: ��

� exp��x2/2�dx � �2� . Other partitions ofnity are possible, but equation 6 has benign effects on amplitudend it generalizes easily to three dimensions.

When we insert equations 5 �modified by the phase shift to ac-ount for the difference between receiver and beam-center loca-ions� and 6 into equation 4 and rearrange terms, we obtain

�x;xs� ��L�r

4�2�2�w0

� L� d�i�

cos � s

Vs� dpsx

pszA

s* exp�� i�T

s*�

�cos � L

VL� dpLx

pLzA

L* exp�� i�T

L*�Ds�L,pLx,�� ,

�7�

here


Ds�L,pLx,��

�r3/2� dxrpU�xr;xs;��

� exp��i�pLx�xr � L��

� exp��

�r �xr � L�2

2w02 � �8�

nd where AL and TL are Gaussian-beam amplitude and time fromeam center location L. Notation for receiver slowness changesrom pr to pL to emphasize that all receiver-side rays emanate from.Next, we transform from source and receiver slownesses ps and pL

o midpoint and offset slownesses pm and ph �Hill, 2001; Gray, 2005�,

pmx � pLx � psx, �9�

phx � pLx � psx,

o arrive at the migration formula:

I�x;xs� ��L�r

8�2�2�w0

� L� d�i� � � dpmxdphx

pszpLz

cos � s

Vs

cos � L

VLA

s*A

L* exp�� i��T

s* � T

L*��

�Ds�L,pLx,�� . �10�

his formula expresses a migrated record as a sum over beam-centerocations, where each beam center contributes partial images fromany directions. We note again that A and T are complex valued be-

ause of our use of Green’s functions composed of Gaussian beams.To reduce the computational load of equation 10, Hill �2001� sug-

ests evaluating the integral over phx with its steepest-descent ap-roximation. �If A and T were both real valued, we would instead usestationary-phase approximation. In the 2D case, in fact, the steep-st-descent approximation to the integral agrees formally with a sta-ionary-phase approximation, with the normally real quantities inhe stationary-phase formula replaced by complex quantities.�

In Hill’s �2001� common-offset derivation, local slant-stacked re-ection data D depend directly on pmx; but in the present common-hot formulation, D depends on pmx through pLx. In the steepest-de-cent calculation for fixed pmx, we find the value of phx that minimiz-s the imaginary part of the total traveltime T � Ts � TL. This sta-ionary value phx

0 is used with pmx in equation 9 to determine criticalalues psx

0 , pLx0 ; these are used to evaluate D at the time corresponding

o the total time along beams psx0 and pLx

0 . The critical values psx0 ,

pLx0 are also related to vertical slownesses psz

0 � cos � s/Vs, pLz0

cos � L/VL, canceling those factors appearing in equation 10. Wehow the details of the steepest-descent integral in Appendix A; in-erting its result into equation 10 leads to

I�x;xs� ��L�r

8�2w0L� d��i� � dpmx

�A

s*A

L* exp�� i�T*��T*��phx

0 �Ds�L,pLx

0 ,�� , �11�


warb

e

wsFwib

ettmnfupdecgpvmae

wtiwgecmw

T

ddv2d�

Ecnm

R

T

�mp

unceeo

Atnc�Ccti

Fwtfmcdoetttmg

p


here T* � Ts* � TL

*. In this equation, As* and Ts

* are evaluatedlong beam psx

0 ; and D, AL*, and TL

* are evaluated along beam pLx0 . The

eal parts of Ts* and TL

* identify traveltimes from source and receiveream-center locations to the image location.The second derivative, T*��phx

0 �, is assumed to be nonzero; it isvaluated inAppendix B as

T *��ph0� � ��rw0

2� 1

Vs2psz

2

Im�Qs�Qs

�1

VL2 pLz

2

Im�QL�QL

� ,

�12�

here Qs and QL are complex dynamic ray-tracing quantities de-cribed in Appendix B and Im�Q� denotes the imaginary part of Q.or either source or receiver, the vertical slowness pz � cos � /V,here V is velocity at source or receiver beam center location and �

s takeoff angle relative to the vertical of the raypath from source oream center to image location.The total error of our final migration formula is the combination of

rrors from �1� our approximation in equation 4 for the Green’s func-ions, �2� the weighted time shifts applied to associate each inputrace with a variety of beam centers �the slant stack�, �3� the approxi-ation in the steepest-descent integral, and �4� kinematic and dy-amic errors arising from insufficient ray illumination in the subsur-ace. Results from numerous numerical studies suggest that the fail-re of the Gaussian-beam approximation of equation 4 can be aroblem when the velocity structure is extraordinarily complex andetailed. On the other hand, our ability to estimate highly complicat-d migration velocities remains limited, and the advantages of effi-iency, flexibility, and steep-dip performance of Gaussian-beam mi-ration usually result in better-than-adequate imaging — even com-ared with wave-equation and reverse-time methods — in complexelocity structures that are not specified with perfect precision. Asentioned, the kinematic and dynamic errors from the slant-stack

pproximation appear to be negligible, even for land data with mod-rately varying surface elevation and near-surface velocity.

The approximation in the steepest-descent integral can be serioushen the magnitude of T ��phx

0 � is small. It is difficult to predict whenhis can happen, except to say that it will occur only when the behav-or of the Gaussian beams is pathological; this, in turn, is associatedith velocity complexities. In any case, even if Gaussian-beam mi-ration �or any migration� produces good images in the presence ofxtreme velocity variation, uneven subsurface illumination will pre-lude the analysis of migrated amplitudes. Uneven subsurface illu-ination is more likely to occur with Gaussian-beam migration,ith its reliance on ray tracing, than with wave-equation migration.

he 2D formula: Deconvolution imaging condition

Minor changes allow us to produce a migration formula using theeconvolution imaging condition. This imaging condition is writtenifferently from equation 1; in fact, we modify the standard decon-olution imaging condition used for wave-equation migration to theD true-amplitude migration expression derived by Keho and Bey-oun �1988�. This is equivalent to expression 9 of Zhang et al.2007�:

R�x;xs� �1

2�

cos � s

Vs� d�i�

pU�x;xs;��pD*�x;xs;��

pD�x;xs;��pD*�x;xs;��

.

�13�


quations 2–6 are substituted into equation 13, and source and re-eiver slownesses ps, pr are transformed to midpoint and offset slow-esses pm, ph in the numerator exactly as in equation 9 to arrive at aigration formula:

�x;xs� � ��L�r

16�3�2�w0

cos � s

Vs

� L� d�

i�

pDpD* � � dpmxdphx

pszpLz

cos � s

Vs

�cos � L

VLA

s*A

L*exp��i�T*�Ds�L,pLx,�� . �14�

his formula is similar to equation 10 except for the presence ofpDp

D* in the denominator and an extra factor of ��1/2��cos � s/Vs�

��. We approximate the integral over ph in the numerator by theethod of steepest descent, with the same result as before, multi-

lied by psz � �1 � Vs2psx

2 /Vs, evaluated at the critical value of psx.The product pDp

D* also can be expressed as a product of integrals

sing equations 3 and 5; because the product has no phase, we do noteed the machinery of the steepest-descent method to evaluate it. Itan be evaluated numerically at each image location, arriving at anxpression for the total illumination from the source. Instead, usingquations 3, 5, and B-8, we approximate each integral by its leading-rder asymptotic expression, arriving at

R�x;xs� � ��L�r

8�2w0

cos � s

VsL� d��i� � dpmx

�A

s*A

L*�Ts��psx

0 ��exp��i�T*�

�As�2�T*��phx0 �

Ds�L,pLx0 ,�� .

�15�

s in equation 11, A and T are complex. When multiple arrivals fromhe source occur at an image location, the expression in the denomi-ator will be inaccurate and equation 15 will be a kinematically ac-urate imaging formula without amplitude fidelity. As Zhang et al.2007� point out in their equation 9, using equation 15 produces SD-IGs whose traces contain reflection coefficients �multiplied by aonstant factor�; as with the crosscorrelation imaging condition 11,he traces in each SDCIG are indexed by the lateral distance betweenmage location and source location.

Both of our derivations can be interpreted in two different ways.irst, we can view them as procedures for downward-continuingavefields followed by application of an imaging condition. This is

he classical approach to presenting wave-equation migration; inact, we rely on a recent advance in true-amplitude wave-equationigration theory �Zhang et al., 2007� to derive equation 11 from the

rosscorrelation imaging condition. Alternatively, we can view theerivations as procedures for migrating individual input traces ontoutput locations. For each input trace and a given beam center andmergence angle, we first perform a weighted delay that associateshe trace with the emergence angle at the beam center and then maphe trace into the subsurface using complex amplitude and travel-ime functions. This interpretation generalizes standard Kirchhoffigration to local slant-stack migration, variations of which have

ained popularity in recent years.As a consequence of the second interpretation, the order of the in-

ut traces is largely irrelevant. For example, the same derivations


cFgaspgsto

lictarafppti

T

scpspstcqfseidIttb

ldl

R

I

i�

dipid

asssm

M

ccbd�rgevG

srpilaAitWfdav


an be applied to common-offset input data volumes �Hill, 2001�.or each input trace and a given beam center and incident and emer-ence angles, the initial delay associates the trace with the incidentnd emergence angles at the source and receiver locations corre-ponding to the beam center �Hill, 2001, Figure 2b�.At the end of therocedure, which includes mapping the migrated data to opening an-les, each input trace will have been migrated in a true-amplitudeense. True-amplitude SDCIGs will be output when the deconvolu-ion imaging condition is applied; true-amplitude ADCIGs will beutput when the crosscorrelation imaging condition is applied.As an implementation detail, we note the amplitudes A in formu-

as 11 and 14 are complex. These normalized amplitudes are givenn Appendix B as �VQ0/V0Q, where V is local wavespeed, Q is aomplex dynamic ray-tracing quantity, and subscript zero refers tohe initial location of the ray. Although most of the other quantitiesre real or complex in some familiar fashion, complex-valued Q iselatively unfamiliar. In Gaussian-beam migration, it is real initiallynd gradually acquires a nonzero imaginary part as a ray propagatesrom the source or the receiver beam center into the subsurface. Thehases of Q from the source and receiver then combine with thehase of exp��i��T

s* � T

L*��/�T*��ph

0� to form a total phase shifthat needs to be applied to the local slant-stack data D while comput-ng the image.

he 3D true-amplitude migration formulas

Except for the presence of quantities in all three spatial dimen-ions, the derivations of 3D true-amplitude migration formulas pro-eed as for the 2D derivations. Hill �2001� and Zhang et al. �2007�rovide a template for these derivations.As mentioned, however, theteepest-descent integrals that make up the final step are more com-licated in the 3D case. Here, we present the final migration expres-ions for both crosscorrelation and deconvolution imaging condi-ions. As a technical condition, in three dimensions, P and Q are 2Domplex matrices that are diagonal initially �Hill, 2001�; the matrixuotient PQ�1 must remain diagonally dominant for the followingormulas to hold. We force this condition by assuming that secondpatial derivatives of velocity appearing in the coupled differentialquations for P and Q can be neglected. This is equivalent to assum-ng that the velocity is a piecewise linear function of the spatial coor-inates except for isolated negligible terms near velocity interfaces.f the velocity function is smoothed so that its relative variations onhe order of a wavelength are small, this is not a damaging assump-ion for kinematic ray behavior, although it can affect ray amplitudeseneath significant velocity boundaries, such as salt interfaces.Neglecting the second spatial derivatives of velocity leads to sca-

ar matrices P and Q, which facilitates the evaluation of the steepest-escent integrals. Then the 3D migration expressions for crosscorre-ation and deconvolution imaging conditions are, respectively,

I�x;xs� � ��3V�x��Lx�Ly�r

2

16�2w02

L� d� � � dpmxdpmy

�A

s*A

L* exp��i�T*�

�det�Tij*�

Ds�L,pLx0 ,pLy

0 ,�� , �16�

�x;xs� � ��3�Lx�Ly�r

2

16�2w2

cos � s

VsL� d� � � dpmxdpmy

0


�A

s*A

L*�det�Ts,ij

* ��exp��i�T*�

�As�2�det�Tij*�

Ds�L,pLx0 ,pLy

0 ,�� .

�17�

n equations 16 and 17,

Ds�L,pLx,pLy,��

�r3� � dxrdyrpU�xr;xs;��

� exp�i�pL · �xr � L��

� exp��

�r �xr � L�2

2w02 � �18�

s the 2D generalization of the weighted slant stack of equation 8, L�Lx,Ly,0�, xr � �xr,yr,0�, pL � �pLx,pLy,pLz�, and the sum is a

ouble sum over Lx and Ly. Furthermore, Tij* � �� 2T*„ph

0…/� phx� phy�

s the Hessian matrix of second derivatives of T* evaluated at criticaloint ph

0, replacing T*� in the steepest-descent equation 12, and Ts,ij*

s a similar Hessian for Ts*. Under our assumption that Qs and QL are

iagonal, the determinant of Tij* is given by

det�Tij*� � det��rw0

2

VspszIm�Qs�Qs

�1 ��rw0

2

VL pLzIm�QL�QL

�1� ,

�19�

nd the determinant of Ts,ij* is the same as equation 19 without the

econd term on the right-hand side. Complex matrices Qs and QL arecalar multiples of the 2�2 identity matrix. The scalars have theame values as solutions Qs and QL of the 2D dynamic ray equations,aking it easy to evaluate the determinant numerically.

apping surface offset to opening angles

With SDCIGs, it is easier to attach a physical meaning to the de-onvolution imaging condition than to the crosscorrelation imagingondition; understanding this meaning helps in mapping Gaussian-eam migrated data from distance to subsurface opening angle. Theeconvolution imaging condition forms the quotient of the upgoingreflected� wavefield with the downgoing �incident� wavefield. At aeflector location, the quotient is the reflection coefficient at the an-le of incidence, or half-opening angle, measured at a particular lat-ral offset from image point to source location. This heuristic obser-ation can be justified rigorously and used to help formADCIGs foraussian-beam migration.With Gaussian-beam migration, we know the locations of the

ource and receiver beam centers. If, at a reflector, we also know theay directions from those locations, we can assign the migrated am-litude to the opening angle. Gaussian-beam migration provides thisnformation as it migrates data: The subsurface ray angles at imageocations from source and receiver beam centers can be determinednd applied after migration to map offsets in SDCIGs to angles inDCIGs. This is different from �and more cumbersome than� apply-

ng ray directions during imaging to produceADCIGs without goinghrough the intermediate step of producing SDCIGs �Gray, 2007�.

here true-amplitude migration is valid, theseADCIGs are suitableor AVA analysis. In the absence of the mapping from offset to inci-ence angle, Gaussian-beam migration using the deconvolution im-ging condition produces SDCIGs that can be applied in amplitude-ersus-offset �AVO� studies in areas of mild structural complexity.


�ttpw

wftwweeserl

wdtdwnaiib

pcctlimcdbopt

mTsetcteeacui

cuba

celfFprpFt

Fsflt

a

Ffzdt


In the wave-equation migration formulation of Zhang et al.2007�, the crosscorrelation imaging condition 1 provides SDCIGs;he migrated amplitude in a particular offset bin is directly propor-ional to the product of angle-dependent reflection coefficient multi-lied by the angular width of the offset bin. In particular, the angularidth of a 2D offset bin is

�� 8�AART,s2 cos�� s�x

Vs� , �20�

here AART,s is the asymptotic ray-theoretic amplitude along the rayrom the source point to the image point, � s is the takeoff angle rela-ive to the vertical of that ray at the source point, �x is the spatialidth of the offset bin �maximum offset minus minimum offsetithin the bin�, and Vs is the wavespeed at the source point. Fromquation 20, near-offset bins �small � s� of a given size contain a larg-r range of opening angles than far-offset bins �large � s� of the sameize. In other words, the crosscorrelation imaging condition produc-s angle-dependent reflection coefficients normalized by d� s/dx oreflection coefficient densities. Equation 20 provides the map fromateral distance to opening angle.

Zhang et al. �2007� point out the difficulty of this procedure forave-equation migration when rays, not directly related to theownward continuation, are used for the mapping.As an alternative,hey propose a modified imaging condition that produces CIGs in-exed by subsurface offset; these can be transformed into ADCIGsithout relying on ray information. Obviously, this extra work is noteeded for Gaussian-beam migration, where the ray information isvailable and well behaved when the conditions for true-amplitudemaging are met. Also, as mentioned, the two-step process of form-ng SDCIGs and transforming them into ADCIGs is relatively cum-ersome.When SDCIGs are formed, deconvolution imaging condition 15

roduces reflection coefficients as a function of lateral distance, butrosscorrelation imaging condition 11 produces reflection coeffi-ient density. Reflection coefficients are more useful than a densityhat needs to be mapped into reflection coefficients, so the deconvo-ution imaging condition is theoretically preferable when migratingnto SDCIGs. �As we show, however, the situation reverses when weigrate directly into ADCIGs: there, the crosscorrelation imaging

ondition produces reflection coefficients.� On the practical side, theeconvolution imaging condition contains a quotient that needs toe regularized, but the crosscorrelation imaging condition containsnly a product.Although the regularization can be straightforward toerform, it might compromise the accuracy of the migrated ampli-udes, especially in areas where the source illumination is weak.

An extra issue that arises in three dimensions is absent in two di-ensions: the presence of a second angle, the subsurface azimuth.ypically, this angle is ignored, with all azimuths summing into eachurface or subsurface offset or angle bin. If a reflector is illuminatedqually at all azimuths, then summing all azimuths together scaleshe migrated amplitudes equally for each opening angle, with nohange to the migrated amplitudes except for a single scale factor. Ifhis is not the case, summing all azimuths together scales the migrat-d amplitudes differently at different opening angles, introducing anrror into the AVA analysis. In Gaussian-beam migration, the samengle information used to identify the opening angle at an image lo-ation is also available to identify the subsurface azimuth angle. Fullse of this information will result inADCIGs indexed by both open-ng angle and subsurface azimuth angle, with considerable extra


omputer memory and disk needed to hold the larger gathers.As ourse of wide-azimuth data increases, so will our use of CIGs indexedy two quantities — either x- and y-offset or opening and azimuthngles.

EXAMPLES

We present two examples: a constant-velocity synthetic and aomplex structural synthetic. The simplicity of the constant-velocityxample allows us to demonstrate the nature of the mapping fromateral distance to subsurface angle.This example models reflectionsrom density contrasts in a medium of constant velocity 2000 m/s.our horizontal reflectors with identical reflection coefficients arelaced at depths of 1000, 2000, 3000, and 4000 m. A single shotecord, with a recording aperture of 7000 m on either side of the shotoint, is migrated; the results of true-amplitude GBM are shown inigures 1–3. Half-opening angles were limited to 60° in the migra-

ion.In Figure 1, the crosscorrelation imaging condition is used, and in

igure 2 the deconvolution imaging condition is used. Figure 1ahows a decay in amplitudes with depth, similar to the horizontal-re-ector example of Zhang et al. �2007�. Because of the lateral transla-

ional symmetry of the problem for this simple geometry, the migrat-

)

b)

1000

2000

3000

4000

5000

Depth(m)

Distance (m)-5000 0 5000

igure 1. �a� Migrated image �crosscorrelation imaging condition�rom a single shot record in a constant-velocity medium with hori-ontal reflectors caused by identical density contrasts. Amplitudesecay visibly from top to bottom. �b� Normalized amplitudes alonghe reflectors in �a�.


eAeepttdd

ttftgspo

dv

wtd

vdciaisatauwvwtio

FiNff

Fdzass


d record is equivalent to an SDCIG. To convert the gather into anDCIG, we must perform one of the mappings proposed by Zhang

t al. �2007�. In this constant-velocity, horizontal reflector case,quation 20 provides an analytic relative decay function for the am-litudes, vertically and laterally. In two dimensions, squared ampli-ude AART,s

2 is proportional to 1/�x2 � z2 and cos � s � z/�x2 � z2, sohe crosscorrelation imaging condition produces a constant angle-ependent reflection coefficient scaled by a factor proportional to theecay function z/�x2 � z2�.Figure 1b shows the migrated amplitudes along the reflectors;

hese amplitudes are the approximations, generated by the migra-ion, to the decay function. Figure 1b also illustrates the exact decayunction for the shallowest reflector, showing good agreement withhe migrated amplitudes. Migration aperture truncation artifacts be-in to interfere with the migrated amplitudes at distances corre-ponding to half-opening angles approaching 60°, and migrated am-litudes decay rapidly beyond those distances, nullifying any benefitf amplitude preservation.In Figure 2 �deconvolution imaging condition�, the goal is to pro-

uce amplitudes proportional to reflection coefficients, without anyertically and laterally varying scaling. This goal is accomplished

a)

b)

igure 2. �a� Same as Figure 1a except that the deconvolution imag-ng condition is applied.Amplitudes are similar for all reflectors. �b�ormalized amplitudes along the reflectors in �a�. Amplitude arti-

acts near the maximum offsets are migration aperture truncation ef-ects.


ith the same fidelity as in Figure 1, with aperture truncation effectshat are more visible because of the relatively larger amplitudes pro-uced at greater distances.As mentioned, when the crosscorrelation imaging condition pro-

ides SDCIGs, the migrated amplitude in a particular distance bin isirectly proportional to the product of angle-dependent reflectionoefficient multiplied by angular width of the distance bin. The samemaging condition can be used with subsurface angle informationvailable in Gaussian-beam migration to produce ADCIGs directlyn the migration. This is done by �1� summing the ray angles from theource and beam center at each image location to find the openingngle for migrating a sample onto an image location and �2� placinghe migrated amplitude into the bin corresponding to that openingngle. Then the amplitude in an angle bin is proportional to the prod-ct of angle-dependent reflection coefficient multiplied by angularidth of the angle bin. In other words, the laterally and verticallyarying normalization of equation 20 becomes proportional to theidth of the angle bins. Because the angle bins are all the same size,

he normalization becomes a constant, so the crosscorrelation imag-ng condition is appropriate for migrating directly in the subsurfacepening angle domain. �For the same reason, the deconvolution im-

a)

b)

Figure 3(a)

1000

2000

3000

4000

5000

Depth(m)

Angle (°)0 80

igure 3. �a� Migrated CIG, with the crosscorrelation imaging con-ition applied in the opening-angle domain. Angle bins range fromero to 80°.Amplitudes are similar for all reflectors. �b� Normalizedmplitudes along the reflectors in �a�.Amplitudes have not been pre-erved as accurately as in Figures 1 and 2 because the binning of off-ets into angles is inaccurate, especially at shallow depths.


aocSt

g1d

rstTnabtc

1iehtata

hcActeGntmntt

gGggmaiomcgwcp2iapnvqoo

Fdt

Fbm


ging condition produces ADGICs whose traces have the inversef the decay function implied by equation 20. Thus, using the cross-orrelation imaging condition produces a decay at far offsets in theDCIGs as in Figure 1, and using the deconvolution imaging condi-

ion produces an unwanted extra gain inADCIGs at large angles.�We illustrate the crosscorrelation imaging condition on angle

athers in Figure 3. Instead of mapping the migrated record of Figurefrom distance to angle as Zhang et al. �2007� do, we buildADCIGsirectly by migrating a range of �translationally invariant� shot

igure 4. Stacked Gaussian-beam migrated image of the Sigsbee2aata set using the true-amplitude crosscorrelation imaging condi-ion.

igure 5. ADCIGs at five locations from true-amplitude Gaussian-eam migration. Several diffraction and reflection events arearked as in Zhang et al. �2007�.


ecords, using the crosscorrelation imaging condition. Figure 3hows an ADCIG at a single location. Disappointingly, the ampli-ude behavior is worse than for the SDCIG amplitudes of Figure 1.his problem is caused by the granularity of the opening-angle bin-ing. Specifically, the small number of migration grid points in eachngle bin at the smallest depths causes the angular width of the offsetins in equation 20 to be calculated with significant jitter. Far fromhe source, where the angle bins contain a large number of image lo-ations, this jitter decreases.

Although the aperture truncation effects slightly visible in Figureb and very visible in Figure 2b are present in Figure 3b, their effects dominated by the inaccuracy of the numerical approximation toquation 20. As this example shows, these truncation effects can beard to detect �especially when the crosscorrelation imaging condi-ion is used�, and they are not always considered when amplitudenalysis on CIGs is performed. However, as Figure 2b shows, aper-ure truncation effects may cause significant errors on amplitudenalysis.

Our second example uses the Sigsbee2a model data set �Paffen-olz, 2001�. We produced the migrated stack of Figure 4 by using therosscorrelation imaging condition to migrate the input traces intoDCIGs, followed by stack. We show some ADCIGs in Figure 5,

orresponding to those shown by Zhang et al. �2007�. Qualitatively,hese gathers resemble those obtained by true-amplitude wave-quation migration, but there are differences in amplitude details.enerally, the Gaussian-beam migration amplitudes are weaker be-eath salt than the wave-equation migration amplitudes. Because ofhe complexity of the model, it is impossible to determine whichethod produces more correct amplitudes, but the generally lower

oise level in the wave-equation migrated gathers and stack suggesthat the relative amplitudes produced by true-amplitude wave-equa-ion migration are better.

CONCLUSIONS

We have derived true-amplitude versions of Gaussian-beam mi-ration. Our derivation combines two migration methods �classicalaussian-beam migration and true-amplitude wave-equation mi-ration�, applying a commonly used criterion for true-amplitude mi-ration. This application produces expressions 11 and 15 in two di-ensions and expressions 16 and 17 in three dimensions for true-

mplitude migration with crosscorrelation and deconvolution imag-ng conditions. Our final expressions are similar to published resultsn Gaussian-beam migration, with modified migration weights. Theodifications arise from a detailed analysis of a steepest-descent

alculation used to approximate a computationally intensive inte-ral; the modified weights prove useful in migration. Migrationeights obtained without using the results of the steepest-descent

alculation produce migrated amplitudes that are similar to thoseroduced here, modified by a slowly varying scale factor. That is, theD terms involving T*��ph

0� in equation 12 and the 3D terms involv-ng det�T

ij*� in equation 19 vary slowly in the far field of the source

nd receiver beam-center locations, where the high-frequency ap-roximation of the steepest-descent calculation is valid. Therefore,eglecting these terms produces an amplitude error that is slowlyarying and can be difficult to measure. This illustrates that high-fre-uency asymptotic methods such as steepest descent produce first-rder effects on the kinematic behavior of wavefields but only sec-nd-order effects on the dynamic behavior.


oiTcCti

ta

totveotesetw

mtta

Ip

ahA

TefHdatr

g

Ttcs

LfAbc

ciwestdetmt

tfbtoreot

w

wr

trt


We also have provided physical interpretations of the amplitudesn migrated traces using both imaging conditions when migratingnto CIGs indexed by surface distance or using subsurface angle.he deconvolution imaging condition produces reflection coeffi-ients �multiplied by a constant factor� when migrating into SD-IGs, and the crosscorrelation imaging condition produces reflec-

ion coefficients �multiplied by a constant factor� when migratingntoADCIGs. Our examples illustrate these observations.

ACKNOWLEDGMENTS

We thank Ross Hill for several extremely interesting conversa-ions on this topic. We also thank Uwe Albertin, Yu Zhang, and annonymous reviewer for very constructive comments.

APPENDIX A

METHOD OF STEEPEST DESCENT FORASYMPTOTIC EVALUATION OF CERTAIN

COMPLEX INTEGRALS

In this appendix, we describe the asymptotic expansion of an in-egral with a complex exponent by the method of steepest descent. Inur 2D Gaussian-beam migration application, the integral is charac-erized by a single complex-valued function of a real variable. In-olving only a single complex variable, the application is a 1D steep-st-descent integral. In 3D Gaussian-beam migration, the integralver x- and y-offset ray parameters leads to a double integral overwo complex-valued functions of two real variables, or a 2D steep-st-descent integral. Although there is no general 2D method ofteepest descent �in two complex variables�, it is still possible tovaluate this integral asymptotically. Doing so, however, is beyondhe mathematical scope of this paper; we derive only the 1D formula,hich allows us to approximate equation 10 by equation 11.

The 1D method of steepest descent is a standard tool in classicalathematical physics. We begin by thinking of the original integra-

ion over variable phx along the real axis as a more general contour in-egral in the complex phx plane. That is, we consider the integral onn interval along the real axis, generically denoted I, as

I�� dzf�z�exp�� z�� . �A-1�

n our application, we must identify complex variable z � phx, com-lex amplitude

f �cos � s

Vs

cos � L

VL

As*A

L*

pszpLz

Ds,

nd complex time � � iT* to use the asymptotic formula derivedere. However, the derivation is simplest if we start with the form-1.

We seek an asymptotic expansion of this integral for large �.echnically, we should use a dimensionless large parameter — forxample, normalizing � by some reference frequency �differentrom the reference frequency used in Gaussian-beam migration�.owever, such a dimensionless parameter will be proportional to �,efined as high frequency, so that we can proceed with this form,voiding unnecessary notational clutter, knowing that the scale fac-ors attached to � will produce the desired dimensionless large pa-ameter.


We proceed by assuming there is a point zs on the interval of inte-ration for which

d� �zs�dz

� 0, � � �d2� �zs�

dz2 � 0. �A-2�

hese conditions echo the conditions of a simple stationary point inhe method of stationary phase. However, the interpretation in theomplex plane is somewhat different. In particular, for �z � zs�mall,

� �z� � � �zs� ��z � zs�2

2� �. �A-3�

et u�x,y� � Re�� z� � � �zs�� be the real part of the complex dif-erence in � . Because of the quadratic approximation in equation-3, it is fairly easy to show that the surface u�x,y� is locally hyper-olic — visually, a saddle. Hence, the point zs is interchangeablyalled a stationary point or a saddle point.

The Cauchy integral theorem tells us that the path of integrationan be deformed through the saddle point in a direction that is fortu-tous for asymptotic approximation of the integral, i.e., on a pathhere �u rises up to the saddle point and then falls again on the oth-

r side. If we think of those two half-paths as directed away from theaddle point for the moment, they are the paths of steepest descent ofhe real part of the exponent, hence the term method of steepestescent. Along those paths, the exponential has the formxp��as2/2�, with a real and s a new real variable of integration. Inhis case, the integral can be evaluated asymptotically by the Laplaceethod �Bleistein, 1984�, which is the analog for real exponents of

he method of stationary phase for imaginary exponents.In fact, we can identify those two directions of steepest descent at

he saddle point by exploiting the approximation of the function dif-erence in equation A-3. The minus sign in the exponent is alreadyuilt into the structure of the integrand in equation A-1, so the direc-ion we seek can be characterized by the complex angle or argumentf the approximation in equation A-3. This direction maximizes theate of decay of the exponential. This happens if the difference inquation A-3 is purely real so that its angle is equal to some multiplef 2� . Because the angle of the product of two complex numbers ishe sum of their angles,

arg�� 2 arg�z � zs� � 0,2� , . . . �A-4�

hich is equivalent to

arg�z � zs� � �1

2arg�� ,

arg�z � zs� � �1

2arg�� , . . . , �A-5�

here arg means angle of a complex number relative to the positiveeal axis.

These are the only two unique choices. We expect that the direc-ion of choice will be a rotation of the contour of integration — theeal line, positively oriented — through an acute angle. Thus, fromhe two unique choices of direction, we choose

arg�z � zs� � �1

2Arg�� , �A-6�


wtpsis

Tpcccac

�

e

sQfr

Ttd�

Wastoit

It

WGrt

Be

Q

W

As

abwrmt

t

IosA

siircclctw


ithArg denoting the angle that picks out this principal argument ofhe second derivative making an acute angle with the direction of theositive x-axis. Given this information, we can apply the method ofteepest descent �Bleistein 1984, his equation 7.3.11� to the integraln equation A-1 to obtain the contribution to the asymptotic expan-ion of I from the saddle point zs as follows:

I�� 2�

�� f�zs�exp�� zs� �

Arg�� 2

�� 2�

�� f�zs�exp�� zs��, � 0. �A-7�

hat is, with the form of the original integral in equation A-1, thehase shift in the exponent provides just the right factor to yield aomplex square root of � �, using as the principal argument thehoice that makes an acute angle with the direction of the originalontour of integration. If � is negative, the formula will contain theppropriate phase shift to transform this expansion into its complexonjugate.

Equation A-7, with phx substituted for z, �cos � s/Vs��cos � L/VL��A

s*A

L*/pszpLz�Ds substituted for f , and iT* substituted for � , yields

quation 11 as an approximate evaluation of equation 10.

APPENDIX B

STEEPEST-DESCENT APPROXIMATIONAPPLIED TO GAUSSIAN BEAMS

In this appendix, we derive the expression in equation 12 for theecond derivative T ��ph

0� in terms of the complex amplitudes Qs and

L of the two Gaussian-beam expansions appearing in the imagingormula for I�x;xs�, equation 10. We need to start from an explicitepresentation of a single Gaussian beam uGB as used in equation 5:

A ��V�s�Q�0�V�0�Q�s�

,

T � T�s,n� � � �s� �1

2PQ�1n2. �B-1�

he quantities Q and P are determined along the central ray as solu-ions of a system of dynamic ray equations, with ray-centered coor-inates s �measured along the central ray from its initial location at s

0� and n �measured perpendicular to the central ray at points s�.ithin a particular Gaussian beam, complex amplitude and time A

nd T are functions of position; several Gaussian beams from a givenource or receiver location might strike a particular subsurface loca-ion, so A and T are also functions of slowness px. This dependencef A and T on px interests us here. The kinematic ray equations andnitial conditions �at s � 0� for position x, time � , and slowness vec-or p � �� are

dx

ds� V�s�p,

dp

ds� � � 1

V�s��,

d�

ds�

1

V�s�,


x�0� � x�, p�0� �1

V�0��sin � ,cos � �, � �0� � 0.

�B-2�

n addition, quantities PGB and QGB used in Gaussian-beam migra-ion satisfy the dynamic equations and initial conditions

dQGB

ds� V�s�PGB�s�,

dPGB

ds� �

1

V2�s�� 2V

�n2 QGB�s� ,

QGB�0� ��rw0

2

V�0� ,, PGB�0� �

i

V�0�. �B-3�

e distinguish between the complex functions QGB and PGB foraussian beams and the corresponding real solutions of asymptotic

ay theory, QART and PART.The latter satisfy the same system of equa-ions B-2, but the initial conditions change as follows:

QART�0� � 0, PART�0� �1

V�0�. �B-4�

y comparing these initial conditions with those for QGB and PGB inquation B-3, we conclude that

ART�s� � Im�QGB�s��, PART � Im�PGB�s�� . �B-5�

e need this relationship in the following.

symptotic expansion of the Gaussian-beam integral for aingle Green’s function

In equation 5, the final integral expresses the Green’s function asn integral of contributions from a set of individual beams indexedy horizontal slowness px, with each term uGB representing a partialavefield in a limited spatial region surrounding the beam’s central

aypath. We begin our analysis by approximating this integral by theethod of steepest descent. We later apply that analysis to an evalua-

ion of T*��phx0 � in equation 12.

To approximate the Green’s function �expression 5�, we followhe discussion at the beginning ofAppendix A, setting

� �px� � � iT�px� . �B-6�

n addition to its dependence on slowness px, the function � dependsn location x. However, this dependence does not concern us here,o all indicated derivatives of � and T* are with respect to px. As inppendix A, the saddle point is determined by setting

d�

dpx� �i

dT

dpx� 0. �B-7�

For homogeneous media, one of the central rays passes throughubsurface location x; the saddle point occurs at this central ray. Thiss true because the imaginary part of T is zero for this central ray; thats, the imaginary part of T is a minimum and, correspondingly, theeal part of T is a maximum for this critical slowness value, which weall px

0. This is equally true for heterogeneous media, except in twoases: �1� when no central ray from source or receiver beam centerocation x� passes close to x �a shadow zone� and �2� when, in the dis-rete sampling of central rays used in the sum approximating the in-egral, one or more central rays from x� passes near the location xithout actually passing through it. Where an actual saddle point oc-


coi

TT

bvwtst�e

cscpotmiiitsmniltbs

t

iIBta

G

WtT

waoie

dGfoGsf

MG

pserHtswc

w

fpom

phBosta�stgtsp


urs, we can apply the steepest-descent formula of Appendix A tobtain the leading order asymptotic expansion of the Gaussian-beamntegral:

G�x;x�;�� A exp�� px

0��

2pz�2�� px

0��

A exp�i�T�px0��

2pz��2� i�T��px

0�

A ��V�s�Q�0�V�0�Q�s�

. �B-8�

his asymptotic approximation is valid only if T ��px0� is nonzero;

��px0� will be zero when x is at a caustic of the family of central rays.

Equation B-8 approximates the superposition of multiple contri-utions to the wavefield by a single term; this seems to negate the ad-antage of the Gaussian-beam expansion of the Green’s function,hich is multiple contributions to the wavefield from a range of ini-

ial directions. In our downward continuation of wavefields from theource and receiver locations, we use the approximation in a mannerhat incorporates most of the partial contributions. Thus, as Hill2001� shows, Gaussian-beam migration retains most of the arrivals,ven though it relies on the steepest-descent approximation.

As mentioned, saddle points do not exist in a shadow zone. In thisase, no contribution to the wavefield �and therefore no steepest-de-cent calculation� is performed because either the source or beam-enter location does not see the subsurface location. True saddleoints also do not exist when one or more central rays from a sourcer receiver location pass near the subsurface location x but none ac-ually pass through it, as in Hill’s �2001� Figure 2. In this case, weay still seek the dominant contribution to the integral and include it

n the wavefield at x.As in the steepest descent calculation, the dom-nant contribution comes from the central ray whose imaginary times a minimum. If this value of imaginary time is close to zero, thenhe central ray is very close to a stationary value and the steepest-de-cent calculation can be performed with negligible error. If the mini-um value of imaginary time is not close to zero, the central ray is

ot close to a stationary value and the steepest descent calculation isn error. However, the large value of imaginary time will supply aarge exponential decay to the wavefield contribution from this cen-ral ray, and both the contribution to the wavefield and the error wille negligible. In such quasi-saddle point cases, we can perform theteepest-descent integral as if the saddle point were a true one.

At a true saddle point identified by the critical slowness value px0,

he complex traveltime becomes real:

T�px0� � � �s� , �B-9�

n agreement with the traveltime obtained by asymptotic ray theory.n fact, the asymptotic expansion of the Green’s function in equation-8 should agree, at least to leading order, with the expansion ob-

ained by asymptotic ray theory. The leading order term of thatsymptotic expansion is

�x;x�;�� exp�i�� s� � i��

4�sgn��

2�2� ��QART�s�V�s�

. �B-10�

hen we compare this asymptotic expression for the Green’s func-ion with the one in equation B-8, we obtain an indirect evaluation of

�p0�, namely,
� x

T ��px0� �

QART�s�QGB�s�

QGB�0�pz

2V�0��

Im�QGB�s��QGB�s�

QGB�0�pz

2V�0�

�Im�QGB�s��

QGB�s��rw0

2

pz2V2�0�

, �B-11�

here pz � �1 � V2�0�px2/V�0�. The first equality in equation B-11

rises from equating the amplitudes and solving for T ��px0�. The sec-

nd equality exploits the derived relationship between QART and QGB

n equation B-5. The final equality uses the prescribed initial data ofquation B-3.

Substituting the final value of T ��px0� from equation B-11 into the

enominator of equation B-8 for the asymptotic expansion of thereen’s function provides the right asymptotic expansion away

rom caustics of central rays. A convenient cancellation of QGB�s�ccurs, leading to the same asymptotic leading term as is stated forART in equation B-10. This is not our main objective here, however,

o we do not carry out the details. We need identity B-11 for T ��px0�

ollowing.

ultiple Gaussian-beam integrations for the product ofreen’s functions

The imaging condition of equation 10 involves a product of com-lex conjugates of Green’s functions. Thus, creating an image of theubsurface requires two additional integrations to obtain the migrat-d amplitude at an image location, compared to the computationalequirements of a standard Kirchhoff true-amplitude migration.ere, following an approach proposed by Hill �2001�, we carry out

he asymptotic expansion of the integral over phx by the method ofteepest descent. In addition to producing Hill’s complex traveltime,e also produce the amplitude correction from the steepest-descent

alculation.Let us define

J�x;xs� �� dphxAs*A

L* exp��i�T*�phx��Ds�L,prx,��

�B-12�

ith T*�phx� � Ts* � T

L*. Here, J is the innermost integral �for fixed

pmx� in equation 10, and we have neglected the dependence of theunctions on all variables except the variable of integration phx. Inarticular, we have neglected the dependence on the other variablef integration pmx. The asymptotic expansion that we derive hereust be applied for each pmx.We propose to apply the general formula of equation A-7 for ap-

roximating the integral J in equation B-12, with � � iT* �phx�.Asappened for the calculation of the asymptotic Green’s function-11, there may be true simple saddle points �for which the methodf steepest descent is completely valid�, quasi-saddle points, orhadow zones. If x is in a shadow zone of either xs or L, no integra-ion will take place and beam center L will not contribute to the im-ge at x. If one or more central rays from both xs and L pass near xwith no central ray pairs actually striking x�, there will be a quasi-addle point. Then we can use the same argument as above to showhat the error in using the steepest-descent method to evaluate inte-ral J is negligible compared with the error in the method itself.As inhe evaluation of the Green’s function, we are justified in using theteepest-descent method for both true saddle points and quasi-saddleoints. Doing this yields a critical value p0 for which
hx

Rtd

�d

Eerp

raottbp

A

B

B

—

B

D

G

—

H

H—

K

M

N

P

X

Z


J�x;xs� �� 2�

��T* ��phx0 �

As*A

L*Ds�L,prx,��

� exp�� i�T *�phx0 � � i

�

4sgn�� .

�B-13�

ecall that pmx is fixed during the evaluation of integral J.As a result,he critical value phx

0 depends on pmx, and critical values of psx and prx

epend on pmx and phx0 .

Finally, from equation B-12, we can write T* ��phx0 � � T* ��psx�

T* ��prx�. Using equations 9 and B-11 to evaluate the individualerivatives, we obtain

T *��phx0 � � ��rw0

2� 1

Vs2psz

2

Im�Qs�Qs

�1

Vr2prz

2

Im�QL�QL

� .

�B-14�

quation B-14 evaluates T*� in the asymptotic expression for J inquation B-12. We then substitute this expression into equation 11 toeduce the double integral over pmx and phx into a single integral overmx, which is equation 12.To complete the discussion, we must mention that the incident or

eflected wavefield might have a caustic of central rays at some im-ge locations. Even though the regular behavior of the critical pair�s�f Gaussian beams will produce nonzero amplitudes at such loca-ions, the interpretation of the peak amplitude at those image loca-ions in terms of the geometric optics reflection coefficient will note valid. Briefly, such locations will produce nonsimple saddle
oints, violating the validity of the steepest-descent analysis.

REFERENCES

lbertin, U., D. Yingst, P. Kitchenside, and V. Tcheverda, 2004, True-ampli-tude beam migration: 74thAnnual International Meeting, SEG, ExpandedAbstracts, 398–401.

eylkin, G., 1985, Imaging of discontinuities in the inverse scattering prob-lem by inversion of a causal generalized Radon transform: Journal ofMathematical Physics, 26, 99–108.

leistein, N., 1984, Mathematical methods for wave phenomena: AcademicPress Inc.—–, 1987, On the imaging of reflectors in the earth: Geophysics, 52,931–942

leistein, N., Y. Zhang, S. Xu, S. H. Gray, and G. Zhang, 2005, Migration/in-version: Think image point coordinates, process in acquisition surface co-ordinates: Inverse Problems, 21, 1715–1744.

eng, F., and G. McMechan, 2007, True-amplitude prestack depth migra-tion: Geophysics, 72, no.3, S155–S166.

ray, S. H., 2005, Gaussian beam migration of common-shot records: Geo-physics, 70, no.4, S71–S77.—–, 2007, Angle gathers for Gaussian beam migration: 69th Annual Con-ference and Exhibition, EAGE, ExtendedAbstracts.

anitzsch, C., 1997, Comparison of weights in prestack amplitude-preserv-ing Kirchhoff depth migration: Geophysics, 62, 1812–1816.

ill, N. R., 1990, Gaussian beam migration: Geophysics, 55, 1416–1428.—–, 2001, Prestack Gaussian-beam depth migration: Geophysics, 66,1240–1250.

eho, T. H., and W. B. Beydoun, 1988, Paraxial ray Kirchhoff migration:Geophysics, 53, 1540–1546.iller, D., M. Oristaglio, and G. Beylkin, 1987, Anew slant on seismic imag-ing: Migration and integral geometry: Geophysics, 52, 943–964.

owack, R. L., M. K. Sen, and P. L. Stoffa, 2003, Gaussian beam migrationfor sparse common-shot and common-receiver data: 73rdAnnual Interna-tional Meeting, SEG, ExpandedAbstracts, 1114–1117.

affenholz, J., 2001, Sigsbee2 synthetic data set: Image quality as function ofmigration algorithm and velocity model error: 71st Annual InternationalMeeting, SEG, Workshop W-5.

u, S., H. Chauris, G. Lambaré, and M. Noble, 2001, Common-angle migra-tion:Astrategy for imaging complex media: Geophysics, 66, 1877–1894.

hang, Y., S. Xu, N. Bleistein, and G. Zhang, 2007, True-amplitude, angle-domain, common-image gathers from one-way wave-equation migra-
tions: Geophysics, 72, no.1, S49–S58.

True-amplitude Gaussian-beam migrationtinue waveﬁelds. Previous presentations of Gaussian-beam...

Documents

Transcript of True-amplitude Gaussian-beam migrationtinue waveﬁelds. Previous presentations of Gaussian-beam...