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Imaging conditions for prestack reverse-time migration

Sandip Chattopadhyay1and George A. McMechan1

ABSTRACT

Numerical implementations of six imaging conditions for

prestack reverse-time migration show widely differing abili-

ty to provide accurate, angle-dependent estimates of reﬂec-

tion coefﬁcients. Evaluation is in the context of a simple, one-

interface acoustic model. Only reﬂection coefﬁcients esti-

mated by normalization of a crosscorrelation image by

source illumination or by receiver-/source-waveﬁeld ampli-

tude ratio have the correct angle dependence, scale factor,

and sign and the required 共dimensionless兲units; thus, these

are the preferred imaging-condition algorithms. To obtain ac-

curate image amplitudes, source- and receiver-waveﬁeld ex-

trapolations must be able to accurately reconstruct their re-

spective waveﬁelds at the target reﬂector.

INTRODUCTION

Prestack migration that produces accurate amplitudes is a prereq-

uisite for inversion of amplitude variations with angle 共AVA 兲. The

migration algorithm must be dynamically and kinematically correct.

Correct amplitude information for AVA analysis can be obtained the-

oretically either from amplitude-preserved depth migration, after

correcting the data for geometric spreading 共Hanitzch, 1997兲,or

from a true-amplitude prestack depth migration of data processed to

preserve the recorded amplitudes 共Baina et al., 2002; Schleicher et

al., 2007b兲. Both procedures provide correct amplitude information

for AVA analysis only if they consider all propagation-related losses,

not just geometric spreading 共Deng and McMechan, 2007兲. An of-

ten-overlooked point is that the image condition must also be dy-

namically correct.

Beylkin 共1985兲and Bleistein 共1987兲propose obtaining accurate

amplitudes from prestack depth migration by using ray-based Kirch-

hoff migration, in which migration weights correct the dynamic

propagation 共geometric spreading兲losses and acquisition effects

共Thierry et al., 1999兲. Stationary phase analysis 共Bleistein, 1987;

Bleistein et al., 2001兲of the migrated image provides an estimation

of reﬂectivity as a function of incidence angle.

The requirement of imaging complex geologic structures has led

to wave-equation-based migration algorithms. Claerbout 共1971,

1985兲developed a one-way wave-equation-based depth migration

that was kinematically correct, i.e., designed to produce accurate

traveltimes, but whose geometric-spreading losses were inaccurate

共Zhang et al., 2003兲because of use of approximate pseudodifferen-

tial operators. Zhang et al. 共2005, 2007a兲introduced corrections to

one-way wave-equation extrapolations to improve geometric-

spreading behavior; these terms provide residual corrections to the

amplitude, rather than deﬁning the weights that give the full geomet-

ric spreading in Kirchhoff migration.

There is extensive literature about Kirchhoff migration in which

ray tracing is used to calculate migration weights that correct for

geometric spreading 共only兲共Schleicher et al., 1993; Tygel et al.,

1993; Hanitzsch, 1997; Vanelle and Gajewski, 2000; Vanelle et al.,

2006兲. These approximate weights and those derived by Zhang et al.

共2005, 2007a兲no longer are explicitly involved if the extrapolations

in the migration use the full two-way wave equation, which automat-

ically and accurately compensates for geometric spreading 共Haney

et al., 2005兲in either 2D 共cylindrical兲or 3D 共spherical兲, depending

on the extrapolator being used, and for propagation-related phase

shifts. This implies that reverse-time migration has an inherent am-

plitude-accuracy advantage over time-based methods and one-way

共depth-stepping兲wave-extrapolation methods.

Reverse-time migration 共McMechan, 1983; Whitmore, 1983;

Baysal et al., 1983兲promises better imaging of steep dips, compared

to Kirchhoff and one-way wave-equation-based algorithms. In

smooth media, reverse-time extrapolation is dynamically and kine-

matically correct, so it can be used to develop true-amplitude depth

migration if all propagation-related losses 共not just geometric

spreading兲are included in the extrapolator 共Deng and McMechan,

2007兲. No migration weights need to be calculated to correct for illu-

mination effects, as in traveltime-based Kirchhoff migrations

共Vanelle et al., 2006兲, because they are embedded and accurate in the

wave extrapolations.

Reverse-time migration is closely related to full-waveﬁeld inver-

sion. Mora 共1989兲shows that inversion can be thought of as an alter-

Manuscript received by the Editor 17 May 2007; revised manuscript received 13 October 2007; published online 2 May 2008.

1The University of Texasat Dallas, Center for Lithospheric Studies, Richardson, Texas, U.S.A. E-mail: sxc042200@utdallas.edu; mcmec@utdallas.edu.

© 2008 Society of Exploration Geophysicists. All rights reserved.

GEOPHYSICS, VOL. 73, NO. 3 共MAY-JUNE 2008兲; P. S81–S89, 10 FIGS.

10.1190/1.2903822

S81

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nating sequence of migration and model updates. Tarantola 共1984;

1986; 1988兲migrates residual waveﬁelds and calculates gradients of

seismic model parameters from waveﬁeld crosscorrelations, assum-

ing that all propagation effects were corrected for and that the images

are properly scaled. Thus, the concepts discussed in this paper also

apply to full-waveﬁeld inversion.

Most prestack depth-migration algorithms 共Kirchhoff, one-way

wave-equation, and reverse time兲do not inherently produce correct

amplitude information because they neglect attenuation and trans-

mission losses, and most do not use a correct imaging condition.

True-amplitude prestack depth migration requires correct imple-

mentation of the imaging condition. This paper explicitly compares

the most common imaging conditions to make clear which are viable

for recovering accurate AVA amplitudes and which are not. Schle-

icher et al. 共2007a兲compares a smaller subset of image conditions

and reaches similar conclusions to ours.

Six imaging conditions have been proposed for use with reverse-

time migration to improve the image obtained or reduce the compu-

tational cost. Claerbout 共1971兲use the ratio of upgoing and downgo-

ing waveﬁelds at temporal and spatial coincidence, which is the only

physically correct deﬁnition of the reﬂection coefﬁcient 共Lumley,

1989兲. By physically correct, we mean that the image amplitude is an

accurate approximation of the reﬂection coefﬁcient, both numerical-

ly and in being dimensionless.

Hu and McMechan 共1986兲and Chang and McMechan 共1986兲use

a hybrid method of ray tracing for the source extrapolation for the

excitation-time imaging condition with ﬁnite-difference receiver

waveﬁeld extrapolation in prestack reverse-time migration. Loe-

wenthal and Hu 共1991兲use a ﬁnite-difference source extrapolation

to calculate the excitation-time imaging condition on the basis of the

arrival time of the maximum-amplitude direct-wave energy. Whit-

more and Lines 共1986兲use crosscorrelation of source and receiver

waveﬁelds. Kaelin and Guitton 共2006兲show that normalization of

the crosscorrelated image by source illumination further improves

accuracy of the reﬂectivity information in the crosscorrelation im-

age.

In this study, we analyze three classes of imaging conditions and

compare the extracted image amplitudes with the analytic reﬂection

coefﬁcient for the model. The main objective is to analyze the reﬂec-

tion-coefﬁcient amplitudes obtained using different imaging condi-

tions in reverse-time migration to identify those that have acceptable

accuracy for the subsequent development of true-amplitude reverse-

time migrations. No previous study explicitly compares the estimat-

ed angle-dependent reﬂection coefﬁcients with each other or with

analytically computed values, as we do below.

ASSUMPTIONS AND GENERATION

OF TEST DATA

The prestack reverse-time migration used here has three parts:

forward-time extrapolation from the source, reverse-time extrapola-

tion of the receiver waveﬁeld, and application of the imaging condi-

tion at each time step.

To isolate the effect of imaging-condition choice from other prop-

agation-related effects and to extract accurate amplitude informa-

tion from the migrated image, we enforce the following conditions

on a simple synthetic test example: 共1兲the model is scalar, uses con-

stant density, and contains one constant-velocity layer over a half-

space; 共2兲source strength and directivity are known and repeatable

in modeling and migration; 共3兲all source- and receiver-waveﬁeld

propagation-related effects, i.e., reﬂections, transmissions, and at-

tenuations are included in the 共scalar兲waveﬁeld extrapolator; 共4兲the

data are primary reﬂections from the interface; and 共5兲the geomet-

ric-spreading behavior is accurate.

These asumptions are consistent with those of Sava and Fomel

共2006a,b兲. The imaging condition is independent of the method used

to reconstruct the source and receiver waveﬁelds at each image

point, so the results are equally applicable to Kirchhoff or wave-

equation-based methods, and no generality is lost by using a con-

stant-velocity model for illustration. Also, for the ﬂat reﬂector used

below, the recovered angle-dependent reﬂection coefﬁcients along

the reﬂector are the same as those that are a function of angle at any

single point on the reﬂector. In general, each subsurface point is illu-

minated with a different incident angle by each source, so that angle-

dependent reﬂectivity is obtained for each subsurface point 共Deng

and McMechan, 2007兲.

Using a nonattenuating model that contains a single reﬂector with

constant velocity above the reﬂector inherently excludes the effects

of transmission and wave-attenuation losses during waveﬁeld ex-

trapolation. Thus, the calculated reﬂection coefﬁcient is not contam-

inated by these effects and can be evaluated by comparing it directly

to the theoretical values. The compressional velocity contrast in the

model 共2.1 to 2.15 km/s兲is kept small to increase the critical angle

to a large offset.

Forward modeling and data preprocessing

A 2D scalar eighth-order ﬁnite-difference extrapolator is used to

generate a synthetic common-source data gather. The model’s hori-

zontal extent 共Figure 1兲is 8.6 km and its vertical extent is 1.3 km.

The reﬂector depth is 0.8 km from the top of the computational grid.

The spatial grid increment 共10 m both horizontally and vertically兲

and time increment 共1ms兲ensure stability and reduce grid disper-

sion. The source is explosive; the source time function is a Ricker

wavelet with a dominant frequency of 11 Hz. A common-source

synthetic seismogram 共Figure 2兲is extracted at 860 receiver loca-

tions 共along the top of the model in Figure 1兲for 4501 ﬁnite-differ-

ence time steps during source-waveﬁeld extrapolation. Absorbing

boundaries are applied at all four edges of the grid, using the A2 al-

gorithm of Clayton and Engquist 共1977兲.

The synthetic data are preprocessed before migration by muting

the direct wave and tapering the data in time and space to reduce ap-

erture-edge artifacts 共Chang and McMechan, 1986兲. The time slices

of the seismograms are reversed for input to reverse-time extrapola-

tion 共Chang and McMechan, 1990兲共Figure 2兲.

Source

Receivers

m

m

m

Figure 1. Two-dimensional model and single-shot survey geometry

used to generate the test data.

S82 Chattopadhyay and McMechan

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Reverse-time extrapolation

A 2D scalar eighth-order ﬁnite-difference extrapolator is used for

reverse-time extrapolation of the receiver waveﬁeld. During re-

verse-time extrapolation, at each ﬁnite-difference time step, a con-

stant-time slice from the recorded data at that time is inserted into the

grid at the corresponding receiver locations 共McMechan, 1983兲.In

reverse-time extrapolation, the grid spacing is equal to the trace

spacing and the time step of the reverse-time extrapolation is equal

to the time increment of the data. Interpolation might be required

共Larner et al., 1981兲. Incomplete reconstruction of the receiver

waveﬁeld during reverse-time extrapolation is a consequence of the

ﬁnite recording aperture 共Chang and McMechan, 1994兲.

IMAGING CONDITIONS

In this section, we consider two instances each of three implemen-

tations of prestack imaging conditions: excitation-time imaging

conditions computed by ray tracing and forward ﬁnite-difference

modeling; nonnormalized and source-normalized crosscorrelations;

and ratios of the receiver and source waveﬁelds calculated at the ex-

citation time deﬁned by the maximum amplitude of the source wave-

ﬁeld and at the locations of maximum amplitude of the source-nor-

malized crosscorrelation.

Excitation-time imaging conditions

For a common-source gather, the excitation-time imaging condi-

tion for prestack migration is the one-way traveltime from the source

to each point in the image grid 共Chang and McMechan, 1986兲. This

traveltime can be calculated by ray tracing from the source point into

the model, using interpolation between rays to estimate the arrival

time at each grid point, or doing a ﬁnite-difference extrapolation

from the source and detecting when the maximum amplitude occurs

at each grid point 共Loewenthal and Hu, 1991兲. The traveltimes these

two methods produce are not the same. The arrival time from ray

tracing will be less than that of the maximum amplitude 共from wave-

ﬁeld extrapolation兲, but wavelet processing to zero phase so that the

maximum amplitude coincides with the reﬂection time makes them

comparable.

Other sources of time-difference include numerical dispersion in

the ﬁnite-difference calculation 共which can be made arbitrarily

small兲and multipathing in geometrically complicated models,

which does not occur in the test model used below because it has

constant velocity above a 共ﬂat兲target reﬂector. The excitation-time

imaging condition is applied at each ﬁnite-difference time step dur-

ing reverse-time receiver-waveﬁeld extrapolation by extracting the

amplitude at all points that satisfy the imaging condition at that time.

Source- and receiver-waveﬁeld extrapolations may be done

through a smoothed model to avoid production of additional reﬂec-

tions and losses during extrapolation. Using a nonsmoothed velocity

model to maximize kinematic accuracy is valid for ray-based extrap-

olations, but in two-way wave-equation extrapolations, the ampli-

tude artifacts in a nonsmoothed model can be large and difﬁcult to re-

duce 共Kaelin and Guitton, 2006兲. The problem is that the secondary

reﬂections overlie the direct waves 共and vice versa兲and that by deﬁ-

nition direct waves from the source satisfy the image condition ev-

erywhere 共Chang and McMechan 1986兲. In the examples below, we

use a constant velocity equal to that of the layer above the reﬂector so

that we can evaluate the imaging condition with minimal secondary

artifacts.

Figure 3a and b shows the images produced by prestack migration

of the data in Figure 2, using the excitation-time imaging condition

computed by ray tracing and by ﬁnite-difference source extrapola-

tion, respectively. The horizontal extent of the reliable 共spectral兲re-

ﬂections lies between 1.5 and 3.0 km along the reﬂector 共Figure 3兲.

Beyond 3.0 km, the shape of the migration-impulse response causes

the drift to shallower depths. Figure 4a and b compares image ampli-

tudes at the respective image times with the theoretical predictions.

The theoretical reﬂection coefﬁcient is

Position (km)

iTme

()

s

Figure 2. Time-reversed-input seismograms used for all the reverse-

time migrations in the examples that follow 共see Figure 1 for the

model and single-shot survey geometry兲. The data contain only pri-

mary reﬂections and are tapered in time and space directions to re-

duce aperture-edge effects. For clarity, only every ﬁfteenth trace is

plotted.

0.0 1.0 2.0 3.0 4.0 5.

0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

h

t

pe

D)

mk(

a

)

b

)

0.0 1.0 2.0 3.0 4.0 5.0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

htp

eD)

m

k(

Figure 3. Reverse-time-migrated images obtained using excitation-

time imaging conditions. Excitation time is calculated using 共a兲ray

tracing and 共b兲ﬁnite-difference source-waveﬁeld extrapolation. The

horizontal position of the source is at 1.5 km and the depth of the re-

ﬂector is 0.8 km. The solid horizontal lines 共from 1.5 to ⬃3.0 km兲

are the part of the reﬂector that corresponds to the reliable spectral

reﬂections; these have incident angles from 0° to ⬃60°. The two

panels are scaled to have the same maximum amplitudes.

Prestack migration imaging conditions S83

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R共

1兲⳱

V2cos共

1兲ⳮV1cos共

2兲

V2cos共

1兲ⳭV1cos共

2兲,共1兲

where

1is the angle of incidence,

2is the transmission angle, and

V1and V2are the P-wave velocities in media 1 and 2, respectively.

Equation 1 is a plane-wave approximation; the data are 2D 共cylindri-

cal兲waves, so the calculated reﬂection coefﬁcients are more accu-

rate at larger propagation distances than at smaller distances 共Aki

and Richards, 1980, p. 200–208兲.

The critical angle for this model is about 77.62°. The reﬂection co-

efﬁcients in Figure 4 共and later ﬁgures兲are displayed as functions of

incident angle. The two images in Figure 3 are similar, including

their impulse-response artifacts. The image amplitudes depend only

on the source waveﬁeld; they have an indeterminant scale factor that

is proportional to the source amplitude 共Figure 4兲.

Incident angles can be calculated in a variety of ways. For simple

models, ray tracing might sufﬁce, but for complicated models, espe-

cially in which multipathing occurs, angles must be computed local-

ly from the waveﬁelds. For example, Deng and McMechan 共2007兲

use constant-time trajectories extracted from the propagating source

waveﬁeld along with a migrated image to estimate incident angles in

the space domain. As in de Bruin et al. 共1990兲, this involves slant

stacking. Rickett and Sava 共2002兲estimate incident angles by calcu-

lating equivalent offset and depth wavenumbers after prestack

共downward-continuation兲migration, using radial trace transforms

of the output prestack images so no particular image condition or ex-

trapolation method is implied or required. Biondi and Shan 共2002兲

apply the radial trace method to reverse-time migration. Sava and

Fomel 共2006a兲use multiple image-time lags to estimate reﬂection

angles. For the simple examples in this paper, we obtain accurate an-

gles analytically.

Application of the excitation-time imaging condition is equiva-

lent to multiplication 共a zero-lag crosscorrelation兲of the receiver

waveﬁeld by a constant-amplitude-spike 共ﬁxed-time兲wavefront-

trajectory source waveﬁeld at each time step. This is equivalent to

source-waveﬁeld illumination of constant amplitude at all grid

points. Thus, the relative amplitude gradient in the image is similar

to the theoretical reﬂection coefﬁcient only where the actual source

amplitude is nearly constant, which is at near-normal incidence. In

Figure 4a and b, this occurs at incidence angles less than about 20°.

Hanitzch et al. 共1993兲make a similar observation for Kirchhoff mi-

gration and suggested a similar maximum angle of about 26°. The

actual image amplitude at zero incidence angle is given in each re-

ﬂection-coefﬁcient plot for each image-amplitude curve, and the im-

age-amplitude curves are scaled so that they have the correct ampli-

tude at zero incidence angle. In real data, this factor would be un-

known, requiring calibration.

The units of the image amplitudes that the excitation-time imag-

ing conditions produce are not physically correct. The image at each

time step is the product of the receiver-waveﬁeld amplitude and the

共single-unit-spike兲source-waveﬁeld constant-time trajectory, and

so has the same unit as the receiver waveﬁeld 共e.g., pressure兲. How-

ever, a physically correct reﬂection coefﬁcient is dimensionless.

The cross-reﬂector widths of the images in Figure 3a and b depend

on the angle of overlap between the source and receiver waveﬁelds.

This can be understood by noting that each incidence angle corre-

sponds to an apparent wavenumber. At zero incidence angle, the

source and receiver waveﬁelds propagate vertically and the length

共in space兲of the overlap associated with the waveﬁeld crosscorrela-

tion is minimal and corresponds to the wavelength of the receiver

waveﬁeld 共because the source waveﬁeld is effectively a single-spike

image time trajectory in space at each time step兲. The upgoing and

downgoing 共receiver and source兲waveﬁelds pass through each other

in a space of one wavelength in the direction corresponding to the

wavenumber vector. At nonzero incidence, the source and receiver

waveﬁelds have different apparent wavenumbers in any given direc-

tion. Apparent wavelengths always are greater than actual wave-

lengths, so the migrated image necessarily broadens. This is the

waveﬁeld equivalent of the pulse broadening that Hanitzsch et al.

共1992兲, Tygel et al. 共1994兲, and Schleicher and Santos 共2001兲ana-

lyzed in detail in the context of Kirchhoff migration. Tygel et al.

共1994兲derived the migration pulse stretch factor as

共2 cos

cos

兲/v, where

is the incident angle,

is the reﬂector

dip, and vis the local velocity. For the ﬂat reﬂector considered here,

cos

⳱1. The expected ratio of cross-reﬂector pulse widths at inci-

dence angles of 0° 共at horizontal position 1.5 km兲and 60° 共at hori-

zontal position 3.0 km兲in Figure 3a and b therefore is approximately

cos 0° /cos 60° ⳱2.

Crosscorrelation imaging conditions

For the crosscorrelation imaging condition, the source and receiv-

er waveﬁelds are independently propagated using the same scalar,

two-way, ﬁnite-difference extrapolator. The source waveﬁeld

S共x,z,t兲is propagated forward in time from the source location, and

the receiver waveﬁeld R共x,z,t兲is propagated backward in time from

the receivers. The image is formed by multiplying 共a zero-lag cross-

correlation兲the two waveﬁelds at each time step 共Claerbout, 1971;

Kaelin and Guitton, 2006兲. For a single common-source gather,

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

a) b)

Incident an

g

le

(

°

)

Scaled image amplitude

Scaled image amplitude

Reflection coefficient

Reflection coefficient

Incident an

g

le

(

°

)

0.00

0.05

0.10

0.15

Reflection coefficient and scaled image amplitude

0.00

0.05

0.10

0.15

Reflection coefficient and scaled image amplitude

Figure 4. Image amplitudes and theoretical reﬂection coefﬁcients as

a function of incident angle for excitation-time imaging conditions.

Panels 共a兲and 共b兲use image times computed by ray tracing and by ﬁ-

nite-difference source-waveﬁeld extrapolation, respectively. The

image amplitudes are scaled to have the same reﬂection coefﬁcient

at zero angle as the theoretical one 共0.0118兲. The numbers at the left

end of the image amplitude curves are the actual zero-angle values

and have arbitrarily large errors that depend on the source strength.

Compare with Figure 3.

S84 Chattopadhyay and McMechan

image共x,z兲⳱兺

time

S共x,z,t兲R共x,z,t兲,共2兲

where xand zare horizontal and depth coordinates, respectively, and

tis time. The image unit is amplitude squared; thus, the image mag-

nitude has arbitrary scaling that depends on the source strength and

so has no physical interpretation as a reﬂection coefﬁcient.

The crosscorrelation image can be normalized by the square of the

source illumination strength 共Claerbout, 1971; Kaelin and Guitton,

2006兲:

image共x,z兲⳱兺timeS共x,z,t兲R共x,z,t兲

兺timeS2共x,z,t兲,共3兲

which is equivalent to the deconvolution imaging condition using a

matched ﬁlter in the frequency domain 共Lee et al., 1991兲. The

source-normalized crosscorrelation image has the same 共dimension-

less兲unit, scaling, and sign as the reﬂection coefﬁcient. Normaliza-

tion by the square of the receiver illumination 共R2兲also is possible

共Kaelin and Guitton, 2006兲, which gives correct 共dimensionless兲

units, but not correct reﬂection amplitudes.

An obvious difference between the crosscorrelation images in

Figure 5 and the excitation-time images in Figure 3 is that the cross-

reﬂector image widths in Figure 5 are approximately double those in

Figure 3. According to the explanation in the “Excitation-time imag-

ing conditions” section, this is expected. The source waveﬁeld here

has a full wavelet 共not just a spike兲, so that the crosscorrelation of a

wavefront in the source and receiver waveﬁelds has a cross-reﬂector

width of approximately double the apparent wavelength of each and

is a function of incident angle 共Tygel et al., 1994兲.The shapes of the

migrated wavelets in Figure 5 are closer to zero phase than are those

in Figure 3, for the same reason that correlated vibroseis traces are

zero phase. If the reﬂection does not change a pulse shape, such that

the incident and reﬂected pulses are very similar, their crosscorrela-

tion is necessarily close to zero phase 共regardless of input pulse

shape兲. Crosscorrelation changes the wavelet shape.

Figure 6 compares relative image amplitudes with theoretical re-

ﬂection coefﬁcient as a function of incident angle. The nonnormal-

ized crosscorrelation image amplitude 共Figure 5a兲gives the poorest

relative-amplitude-versus-angle behavior 共Figure 6a兲of all the im-

aging conditions considered. The source-normalized crosscorrela-

tion image 共Figure 5b兲gives a properly scaled, relatively accurate

amplitude-versus-angle curve 共Figure 6b兲up to ⬃60°, and has cor-

rect 共dimensionless兲units.

Ratio of upgoing over downgoing waveﬁeld amplitudes

These imaging conditions are based on Claerbout’s 共1971兲imag-

ing principle: A reﬂector exists where the source 共downgoing兲and

receiver 共upgoing兲waveﬁelds coincide in time and space. The re-

ﬂectivity strength depends on both the source and the receiver wave-

ﬁelds at the image time and location. The ratio of upgoing to down-

going wave amplitudes can be computed either at the location of the

maximum amplitude in the waveﬁeld that results from the excita-

tion-time imaging condition 共at the maximum-amplitude trajectory

in Figure 3b兲or at the location of the maximum of the source-nor-

malized crosscorrelation 共at the maximum-amplitude trajectory in

Figure 5b兲. In both options, the reﬂection-coefﬁcient amplitude is

image共x,z兲⳱

U共x,z,t兲

D共x,z,t兲,共4兲

where U共x,z,t兲is the upgoing 共receiver兲waveﬁeld and D共x,z,t兲is

the downgoing 共source兲waveﬁeld.

In the ﬁrst option, the ratio is calculated at the imaging time tthat

corresponds to the maximum amplitude of the source waveﬁeld at

location 共x,z兲共Loewenthal and Hu, 1991兲. This procedure is like the

0.0 1.0 2.0 3.0 4.0 5.

0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ht

pe

D)

mk(

a) b)

0.0 1.0 2.0 3.0 4.0 5.0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

htpeD

)

m

k

(

Figure 5. Reverse-time-migrated images for crosscorrelation imag-

ing conditions. Panel 共a兲uses the crosscorrelation imaging condition

and 共b兲is the source-normalized crosscorrelation. The horizontal

position of the source is 1.5 km. In each panel, the solid horizontal

line from 1.5 to 3.0 km indicates the extent of the reliable 共spectral兲

reﬂections. The two panels are scaled to have the same maximum

amplitude; the unscaled ampitudes are accurate in 共b兲, but not in 共a兲.

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

a) b

)

Scaled image amplitude

Reflection coefficient

Reflection coefficient

Incident angle (°) Incident angle (°)

0.00

0.05

0.10

0.15

Reflection coefficient and image amplitude

0.00

0.05

0.10

0.15

Reflection coefficient and image amplitude

Figure 6. Image amplitudes and theoretical reﬂection coefﬁcients as

a function of incident angle for crosscorrelation imaging conditions.

Panels 共a兲and 共b兲are nonnormalized and source-normalized cross-

correlation imaging conditions, respectively. The theoretical reﬂec-

tion coefﬁcient at zero incident angle is 0.0118. The image ampli-

tude in 共a兲is rescaled to match the reﬂection coefﬁcient at zero inci-

dent angle; the nonscaled value is 2.8393, but is arbitrary, depending

on the source amplitude. The image amplitude in 共b兲is not rescaled

because the source-normalized crosscorrelation corrects the image-

amplitude scale for the source strength. The source-normalized im-

age amplitude in 共b兲共0.0112兲is close to the theoretical reﬂection co-

efﬁcient 共0.0118兲. Compare with Figure 5.

Prestack migration imaging conditions S85

one used to compute Figures 3b and 4b, but it computes the ampli-

tude ratio instead of simply extracting the receiver waveﬁeld ampli-

tude. In the second option, the ratio is calculated along the maxi-

mum-amplitude trajectory of the source-normalized crosscorrela-

tion. It is equivalent to a source deconvolution using a matched ﬁlter,

so it scales the image amplitude to the correct 共dimensionless兲re-

ﬂection coefﬁcient and produces a single spike as output.

In both options, the ratio equation 4 is stable because it is per-

formed only at the image time, where the source-waveﬁeld ampli-

tude Dis locally maximum, by deﬁnition; only the ratio’s maximum

values correspond to the reﬂection coefﬁcients 共where source and re-

ceiver waveﬁelds coincide in time and space兲. The corresponding

cross-reﬂector-image width 共Figure 7a and b兲is a single point, which

is equivalent to crosscorrelating only the maximum absolute ampli-

tudes 共⫹or ⫺兲in the source and receiver waveﬁelds. Deﬁning the

imaging-condition equation 4 only in terms of the image time 共Fig-

ure 3b兲by deﬁnition coincides with the maximum of the source-

waveﬁeld amplitude, but does not guarantee that the maximum am-

plitude in the receiver waveﬁeld will be used. This can be forced by

evaluating the ratio at the spatial locations of the maximum of the

source-normalized crosscorrelation image for each reﬂector. This

gives a cross-reﬂector image width of one point 共Figure 7a and b兲.

Although we use only the peak points in the U/Dreﬂection-coef-

ﬁcient calculation, this procedure is not unstable or sensitive to

noise. The amplitudes involved are postextrapolation and so have

the beneﬁt of the smoothing and trace mixing that is inherent in ex-

trapolation. The maximum-amplitude samples are the most salient

of their associated wavelets; the noisy data example in the next sec-

tion clearly shows this effect for even a single common-source mi-

gration. For multiple sources, the AVA at each reﬂector point would

be computed and ﬁtted across sources, so the amplitude trend will

tend to be stable even if there are errors in individual observations.

The main advantages of using only the peak amplitudes are that

these corespond to the highest signal-to-noise ratio and that the zero-

divide question is moot.

To avoid division by zero in equation 4, Guitton et al. 共2006兲in-

cluded a damping factor in the numerator. If spectra are being divid-

ed, smoothing the source spectrum effectively removes the zero di-

vide 共Guitton et al., 2007兲. Guitton et al. 共2006兲also note that using

crosscorrelation 共equation 2兲is attractive because it avoids the zero-

divide problem entirely, but at the price of destroying the reﬂection-

amplitude information, as described above 共Lumley, 1989兲. Nonnor-

malized crosscorrelation images cannot be used for subsequent

quantitativeAVAanalysis.

In Figure 8a and b, both images automatically are scaled to the

neighborhood of the correct theoretical value by virtue of the ratio,

and the image amplitudes are slightly different from the theoretical

one at near-normal incidence angles. The location of the maximum

source-normalized crosscorrelation inherently corresponds to the

receiver and source waveﬁelds, both having local maximum ampli-

tudes. The corresponding 共receiver/source兲amplitude ratio provides

the correct reﬂection-coefﬁcient calculation. Thus, in Figure 8b, the

image amplitude curve is closer to the theoretical curve than it is in

Figure 8a, especially for near-zero incidence angles.

As ﬁnal examples, consider the effect of noise on the reﬂection-

coefﬁcient estimates that use the up/down versions of the imaging

condition. Figure 9 shows the same data used in the previous exam-

ples, with two different levels of Gaussian noise. The S/N is deﬁned

as the peak signal divided by the root-mean-square noise. Two S/N

considered are 5 and 2. Figure 10 shows the reﬂection-coefﬁcient es-

timates at the excitation image time deﬁned by the maximum source-

waveﬁeld amplitude 共Figure 10a and c兲and at the maximum of the

source-normalized crosscorrelation 共Figure 10b and d兲. As the noise

increases, the variance of the estimated reﬂection coefﬁcient in-

creases, but the average reﬂection coefﬁcients remain similar to

those estimated from the noise-free data 共compare Figures 8 and 10兲.

The effect of noise is somewhat mitigated by the smoothing that is

inherent in the trace mixing that occurs during extrapolation of the

receiver waveﬁeld. The source waveﬁeld is noise-free by deﬁnition.

0.0 1.0 2.0 3.0 4.0 5.

0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

htp

e

D)

m

k(

a

)

b

)

0.0 1.0 2.0 3.0 4.0 5.0

Position (km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ht

p

eD

)

mk

(

Figure 7. Reverse-time-migrated images obtained using imaging

conditions deﬁned by the amplitude ratio of the receiver and source

waveﬁelds. Panel 共a兲is the receiver/source amplitude ratio at the ex-

citation image time deﬁned by the maximum source-waveﬁeld am-

plitude, and 共b兲is at the location of the maximum of the source-nor-

malized crosscorrelation. In each panel, the open rectangles 共from

1.5 to 3.0 km兲indicate the extent of the reliable 共spectral兲reﬂec-

tions. The horizontal position of the source is at 1.5 km.

0 10203040506070

0.00

0.05

0.15

0 1020304050607

0

a) b)

0.10

0.0110

Incident angle (°) Incident angle (°)

Reflection coefficient and image amplitude

0.00

0.05

0.15

0.10

Reflection coefficient and image amplitude

Figure 8. Image amplitudes and theoretical reﬂection coefﬁcients as

a function of incident angle for amplitude-ratio imaging conditions.

Panels 共a兲and 共b兲are receiver/source amplitude ratios at the excita-

tion-image time deﬁned by the maximum source-waveﬁeld ampli-

tude and at the location of the maximum source-normalized cross-

correlation, respectively. In 共a兲and 共b兲, the image amplitudes are not

rescaled; the amplitude-ratio imaging condition produces correct

amplitude scale. The image amplitudes at zero incident angle are 共a兲

0.0093 and 共b兲0.0110, respectively; the theoretical value is 0.0118.

Compare with Figure 7.

S86 Chattopadhyay and McMechan

DISCUSSION

We evaluate six imaging conditions used in prestack reverse-time

depth migration. They are of three types—excitation-time, crosscor-

relation, and receiver-/source-waveﬁeld amplitude ratio—but all

can be seen as speciﬁc instances of crosscorrelation. The image am-

plitude and cross-reﬂector width are affected by the type of imaging

condition used. The excitation-time imaging condition contains in-

formation from only the receiver waveﬁeld, and the corresponding

image amplitude has the same scaling and unit as the receiver wave-

ﬁeld amplitude. The crosscorrelation imaging condition provides an

image amplitude that is the product of source and receiver wave-

ﬁelds and has the unit of amplitude squared. Only the source-nor-

malized crosscorrelation imaging condition and the amplitude-ratio

imaging condition produce image amplitudes that have the same

unit and scaling as the reﬂection coefﬁcient, and both give similarly

accurate estimated reﬂection coefﬁcients.

The cross-reﬂector width of the migrated image depends on the

correlation length of the source and receiver waveﬁelds. The corre-

lation length is a function of the overlap between the source and re-

ceiver waveﬁelds. The excitation-time imaging condition correlates

the receiver waveﬁeld with a spike source waveﬁeld 共Lumley,

1989兲, so that the cross-reﬂector resolution is better than that of the

crosscorrelation imaging condition. Source normalization of the

crosscorrelated image corrects the amplitude scale. However, the

resolution is same as that of the crosscorrelated image. The ampli-

tude ratio of the receiver and source waveﬁelds is equivalent to the

crosscorrelation of two spike functions, and the correlation length is

a single point, so that the resolution is better than for excitation-time

or crosscorrelation imaging conditions.

The present study uses a constant velocity for migration for a

model with a single layer over a half-space. The constant-velocity

model allows accurate calculation of traveltime to the reﬂector and

eliminates multipathing and secondary reﬂections during reverse-

time extrapolation. The data aperture must be large enough 共and ta-

pered兲to minimize edge artifacts over the angles of interest during

receiver waveﬁeld propagation. Where lateral gradients are strong

enough that multipathing occurs, migrations based on single data

subsets 共such as common-source gathers兲will contain images corre-

sponding to multiple source-to-reciever paths for the same reﬂection

time; only one of these images is a real reﬂector position. Stolk and

Symes 共2004兲solve this dilemma by incorporating slowness, i.e.,

propagation-direction, information into the extrapolation, either ex-

plicitly 共see the prestack parsimonious migration of Hua and Mc-

Mechan, 2003兲or implicitly 共see the analysis of Claerbout’s 1985

survey-sinking algorithm by Stolk and de Hoop, 2006兲.Inthe

present context, these issues are associated with the waveﬁeld ex-

trapolations—not how, but where the image condition is applied—

but they do need to be considered in a complete implementation.

Deng and McMechan 共2007兲show application in the context of

recovery of accurate angle-dependent reﬂection coefﬁcients 共and

subsequentAVA inversion兲from prestack reverse-time migration for

a more-complicated, multilayered model. This is straightforward to

do in wave-equation-based reverse-time migration and suggests a

path to truly “true-amplitude” migration that has not been consid-

ered in the Kirchhoff literature; hence, the reverse-time context for

this paper.

For application to ﬁeld data, geometric spreading must be 3D, so

extrapolation must be in 3D, and the recording aperture in both in-

line and crossline directions must be sufﬁcient, so that aperture-edge

effects do not contaminate the amplitudes in the region of interest.

This implies a requirement of wide-swath recording. The imaging

conditions are the same in 2D and 3D. The main consideration for

ﬁeld data is that the source wavelet used for generation of the source

waveﬁeld must be accurately estimated. To give correct scaling of

the reﬂection coefﬁcients, the source amplitude and spatial directivi-

ty must be calibrated to be internally consistent with the scaling of

the recorded waveﬁeld. This can be done using the direct wave or a

reﬂection from a known reﬂector, such as at a water bottom.A source

scaling error will embed a corresponding scale error in the reﬂection

coefﬁcients for each common-source gather.

Application to elastic data also is possible by substituting an elas-

tic extrapolator. Separating P- and S-waves 共Sun et al., 2006兲at the

image time and location should give separate P and converted 共P-S兲

reﬂection coefﬁcients. These applications are beyond the scope of

this paper.

a

)

b

)

Figure 9. The same synthetic data as in Figure 2, but with S/N of pan-

els 共a兲5 and 共b兲2.

0.00

0.05

0.10

0.15

0 10203040506070

0.00

0.05

0.10

0.15

0 1020304050607

0

Incident angle (°) Incident angle (°)

0 10203040506070 0 1020304050607

0

Incident angle (°) Incident angle (°)

Re

f

lection coe

ff

icient and

image amplitude

Re

f

lection coe

ff

icient and

image amplitude

0.00

0.05

0.10

0.15

0.00

0.05

0.10

0.15

Reflection coefficient and

image amplitude

Reflection coefficient and

image amplitude

a) b

)

c) d)

Figure 10. Image amplitudes and theoretical reﬂection coefﬁcients

as a function of incident angle for S/N of 5 共a and b兲and 2 共c and d兲

共Figure 9兲, using the receiver/source amplitude-ratio imaging condi-

tions; imaging conditions for 共a兲and 共c兲are evaluated at the time of

the maximum source-waveﬁeld amplitude. Imaging conditions for

panels 共b兲and 共d兲are evaluated at the location of the maximum of the

source-normalized crosscorrelation.

Prestack migration imaging conditions S87

There are some numerical issues that inﬂuence details of the mi-

grated images that are not discussed above, but should be considered

in implementation. Zhang et al. 共2007b兲discuss some of these.

Shifts in apparent image position can be caused by grid dispersion in

ﬁnite-difference extrapolations, by a time shift in the input data, by

the ﬁnite discretization of the image grid, by nonzero phase wave-

lets, and by interference such as grid-edge artifacts. Grid dispersion

increases with propagation-path length, but can be eliminated by us-

ing pseudospectral extrapolators 共Fornberg, 1998兲. In Figure 7, typi-

cal depth errors within the spectral-reﬂection zone 共1.5 to 3.0 km

horizontal position兲are a maximum of Ⳳ1 depth sample increment.

Beyond 3.0 km, the shallow image depths are attributed primarily to

the upward curvature of the migration impulse responses.

Crosscorrelation changes the wavelet shape. Deconvolution in

up/down division theoretically removes the source wavelet. Wavelet

estimation is important because the maximum amplitude of a cross-

correlation does not necessarily concide with the maximum ampli-

tude of either the source or the receiver waveﬁeld if the two wavelets

are not similar. Slight differences in image position do not signiﬁ-

cantly affect the estimated reﬂection angles, so a usable AVAmay be

obtained even if the reﬂector position has some uncertainty. In the

examples above, the wavelet time reference was the ﬁrst 共blackened兲

peak.

CONCLUSIONS

All three classes of imaging condition are based on the correlation

of source and receiver waveﬁelds. What type of imaging condition is

used greatly affects image amplitude, physical validity, and resolu-

tion. The crosscorrelation and excitation-time imaging conditions

produce images that have low resolution, image amplitudes that are

different from the reﬂection coefﬁcient, and arbitrary 共source-de-

pendent兲scaling. The source-normalized crosscorrelation image

amplitude represents the reﬂectivity of the model and has the correct

scaling and sign. The amplitude-ratio imaging conditions provide

the best resolution. The ratio of the receiver/source amplitudes at the

image time might not accurately represent the reﬂectivity of the

model; the receiver waveﬁeld might not have the peak amplitude at

the image time that is computed from the source waveﬁeld alone,

and thus it might not correspond to the peak amplitude ratio of the re-

ceiver and source waveﬁelds. The amplitude ratio of the receiver and

source waveﬁelds at the location of the maximum of the source-nor-

malized crosscorrelation gives the correct scale 共because it corre-

sponds to the maximum amplitude of both the source and receiver

waveﬁelds兲and the best image resolution 共from using only the peak

waveﬁeld values兲. Proper implementation of the imaging condition

is a necessary step for the future development of amplitude-preserv-

ing prestack depth migration. This study is one step in that process.

ACKNOWLEDGMENTS

The research leading to this paper was supported by the sponsors

of the UT-Dallas Geophysical Consortium; the Texas Advanced Re-

search Program, under grant 009741-0006-2006; the Petroleum

Research Fund of the American Chemical Society, under grant

47347-AC8; and a teaching assistantship from the Department of

Geosciences at the University of Texas at Dallas. We thank Feng

Deng for the 2D eighth-order scalar forward-modeling code and for

useful suggestions. This paper is contribution number 1125 from the

Department of Geosciences at the University of Texas at Dallas.

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