Article

Imaging conditions for prestack reverse-time migration

Abstract and Figures

Numerical implementations of six imaging conditions for prestack reverse-time migration show widely differing ability to provide accurate, angle-dependent estimates of reflection coefficients. Evaluation is in the context of a simple, one-interface acoustic model. Only reflection coefficients estimated by normalization of a crosscorrelation image by source illumination or by receiver-/ source-wavefield amplitude ratio have the correct angle dependence, scale factor, and sign and the required (dimensionless) units; thus, these are the preferred imaging-condition algorithms. To obtain accurate image amplitudes, source- and receiver-wavefield extrapolations must be able to accurately reconstruct their respective wavefields at the target reflector.
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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 dimensionlessunits; 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 AVA . 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 Hanitzch, 1997,or
from a true-amplitude prestack depth migration of data processed to
preserve the recorded amplitudes Baina et al., 2002; Schleicher et
al., 2007b. Both procedures provide correct amplitude information
for AVA analysis only if they consider all propagation-related losses,
not just geometric spreading Deng and McMechan, 2007. An of-
ten-overlooked point is that the image condition must also be dy-
namically correct.
Beylkin 1985and Bleistein 1987propose obtaining accurate
amplitudes from prestack depth migration by using ray-based Kirch-
hoff migration, in which migration weights correct the dynamic
propagation geometric spreadinglosses and acquisition effects
Thierry et al., 1999. Stationary phase analysis Bleistein, 1987;
Bleistein et al., 2001of 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 1971,
1985developed 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
Zhang et al., 2003because of use of approximate pseudodifferen-
tial operators. Zhang et al. 2005, 2007aintroduced 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 only兲共Schleicher et al., 1993; Tygel et al.,
1993; Hanitzsch, 1997; Vanelle and Gajewski, 2000; Vanelle et al.,
2006. These approximate weights and those derived by Zhang et al.
2005, 2007ano 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 Haney
et al., 2005in either 2D cylindricalor 3D spherical, 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
depth-steppingwave-extrapolation methods.
Reverse-time migration McMechan, 1983; Whitmore, 1983;
Baysal et al., 1983promises 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 not just geometric
spreadingare included in the extrapolator Deng and McMechan,
2007. No migration weights need to be calculated to correct for illu-
mination effects, as in traveltime-based Kirchhoff migrations
Vanelle et al., 2006, because they are embedded and accurate in the
wave extrapolations.
Reverse-time migration is closely related to full-wavefield inver-
sion. Mora 1989shows 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 MAY-JUNE 2008; P. S81–S89, 10 FIGS.
10.1190/1.2903822
S81
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nating sequence of migration and model updates. Tarantola 1984;
1986; 1988migrates 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 Kirchhoff, one-way
wave-equation, and reverse timedo 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. 2007acompares 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 1971use the ratio of upgoing and downgo-
ing wavefields at temporal and spatial coincidence, which is the only
physically correct definition of the reflection coefficient Lumley,
1989. 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 1986and Chang and McMechan 1986use
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 1991use 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 1986use crosscorrelation of source and receiver
wavefields. Kaelin and Guitton 2006show 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: 1the model is scalar, uses con-
stant density, and contains one constant-velocity layer over a half-
space; 2source strength and directivity are known and repeatable
in modeling and migration; 3all source- and receiver-wavefield
propagation-related effects, i.e., reflections, transmissions, and at-
tenuations are included in the scalarwavefield extrapolator; 4the
data are primary reflections from the interface; and 5the geomet-
ric-spreading behavior is accurate.
These asumptions are consistent with those of Sava and Fomel
2006a,b. 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 Deng
and McMechan, 2007.
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 2.1 to 2.15 km/sis 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 Figure 1is 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 10 m both horizontally and vertically
and time increment 1msensure 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 Figure 2is extracted at 860 receiver loca-
tions along the top of the model in Figure 1for 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 1977.
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 Chang and McMechan, 1986. The time slices
of the seismograms are reversed for input to reverse-time extrapola-
tion Chang and McMechan, 1990兲共Figure 2.
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 McMechan, 1983.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
Larner et al., 1981. Incomplete reconstruction of the receiver
wavefield during reverse-time extrapolation is a consequence of the
finite recording aperture Chang and McMechan, 1994.
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 Chang and McMechan, 1986. 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 Loewenthal and Hu, 1991. 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 from wave-
field extrapolation, 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 which can be made arbitrarily
smalland multipathing in geometrically complicated models,
which does not occur in the test model used below because it has
constant velocity above a flattarget 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 Kaelin and Guitton, 2006. The problem is that the secondary
reflections overlie the direct waves and vice versaand that by defi-
nition direct waves from the source satisfy the image condition ev-
erywhere Chang and McMechan 1986. 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 spectralre-
flections lies between 1.5 and 3.0 km along the reflector Figure 3.
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 see Figure 1 for the
model and single-shot survey geometry. 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 aray
tracing and bfinite-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 from 1.5 to 3.0 km
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
1
V2cos
1V1cos
2
V2cos
1V1cos
2,1
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 cylindri-
calwaves, so the calculated reflection coefficients are more accu-
rate at larger propagation distances than at smaller distances Aki
and Richards, 1980, p. 200–208.
The critical angle for this model is about 77.62°. The reflection co-
efficients in Figure 4 and later figuresare 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 Figure 4.
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 2007
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. 1990, this involves slant
stacking. Rickett and Sava 2002estimate incident angles by calcu-
lating equivalent offset and depth wavenumbers after prestack
downward-continuationmigration, 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 2002
apply the radial trace method to reverse-time migration. Sava and
Fomel 2006ause 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 a zero-lag crosscorrelationof the receiver
wavefield by a constant-amplitude-spike fixed-timewavefront-
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. 1993make 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
single-unit-spikesource-wavefield constant-time trajectory, and
so has the same unit as the receiver wavefield e.g., pressure. 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
in spaceof the overlap associated with the wavefield crosscorrela-
tion is minimal and corresponds to the wavelength of the receiver
wavefield because the source wavefield is effectively a single-spike
image time trajectory in space at each time step. The upgoing and
downgoing receiver and sourcewavefields 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.
1992, Tygel et al. 1994, and Schleicher and Santos 2001ana-
lyzed in detail in the context of Kirchhoff migration. Tygel et al.
1994derived the migration pulse stretch factor as
2 cos
cos
/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° at horizontal position 1.5 kmand 60° at hori-
zontal position 3.0 kmin 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
Sx,z,tis propagated forward in time from the source location, and
the receiver wavefield Rx,z,tis propagated backward in time from
the receivers. The image is formed by multiplying a zero-lag cross-
correlationthe two wavefields at each time step Claerbout, 1971;
Kaelin and Guitton, 2006. 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 aand buse 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 0.0118. 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
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imagex,z
time
Sx,z,tRx,z,t,2
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 Claerbout, 1971; Kaelin and Guitton,
2006:
imagex,ztimeSx,z,tRx,z,t
timeS2x,z,t,3
which is equivalent to the deconvolution imaging condition using a
matched filter in the frequency domain Lee et al., 1991. The
source-normalized crosscorrelation image has the same dimension-
lessunit, scaling, and sign as the reflection coefficient. Normaliza-
tion by the square of the receiver illumination R2also is possible
Kaelin and Guitton, 2006, which gives correct dimensionless
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 not just a spike, 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 Tygel et al., 1994.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 regardless of input pulse
shape. 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 Figure 5agives the poorest
relative-amplitude-versus-angle behavior Figure 6aof all the im-
aging conditions considered. The source-normalized crosscorrela-
tion image Figure 5bgives a properly scaled, relatively accurate
amplitude-versus-angle curve Figure 6bup to 60°, and has cor-
rect dimensionlessunits.
Ratio of upgoing over downgoing wavefield amplitudes
These imaging conditions are based on Claerbout’s 1971imag-
ing principle: A reflector exists where the source downgoingand
receiver upgoingwavefields 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 at the maximum-amplitude trajectory
in Figure 3bor at the location of the maximum of the source-nor-
malized crosscorrelation at the maximum-amplitude trajectory in
Figure 5b. In both options, the reflection-coefficient amplitude is
imagex,z
Ux,z,t
Dx,z,t,4
where Ux,z,tis the upgoing receiverwavefield and Dx,z,tis
the downgoing sourcewavefield.
In the first option, the ratio is calculated at the imaging time tthat
corresponds to the maximum amplitude of the source wavefield at
location x,zLoewenthal and Hu, 1991. 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 auses the crosscorrelation imaging condition
and bis 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 spectral
reflections. The two panels are scaled to have the same maximum
amplitude; the unscaled ampitudes are accurate in b, but not in a.
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 aand bare 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 ais 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 bis not rescaled
because the source-normalized crosscorrelation corrects the image-
amplitude scale for the source strength. The source-normalized im-
age amplitude in b兲共0.0112is close to the theoretical reflection co-
efficient 0.0118. Compare with Figure 5.
Prestack migration imaging conditions S85
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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 dimensionlessre-
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 where source and re-
ceiver wavefields coincide in time and space. The corresponding
cross-reflector-image width Figure 7a and bis a single point, which
is equivalent to crosscorrelating only the maximum absolute ampli-
tudes or in the source and receiver wavefields. Defining the
imaging-condition equation 4 only in terms of the image time Fig-
ure 3bby 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 Figure 7a and b.
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. 2006in-
cluded a damping factor in the numerator. If spectra are being divid-
ed, smoothing the source spectrum effectively removes the zero di-
vide Guitton et al., 2007. Guitton et al. 2006also note that using
crosscorrelation equation 2is attractive because it avoids the zero-
divide problem entirely, but at the price of destroying the reflection-
amplitude information, as described above Lumley, 1989. 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 receiver/sourceamplitude 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 Figure 10a and cand at the maximum of the
source-normalized crosscorrelation Figure 10b and d. 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 compare Figures 8 and 10.
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 ais the receiver/source amplitude ratio at the ex-
citation image time defined by the maximum source-wavefield am-
plitude, and bis at the location of the maximum of the source-nor-
malized crosscorrelation. In each panel, the open rectangles from
1.5 to 3.0 kmindicate the extent of the reliable spectralreflec-
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 aand bare 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 aand b, the image amplitudes are not
rescaled; the amplitude-ratio imaging condition produces correct
amplitude scale. The image amplitudes at zero incident angle are a
0.0093 and b0.0110, respectively; the theoretical value is 0.0118.
Compare with Figure 7.
S86 Chattopadhyay and McMechan
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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 Lumley,
1989, 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 and ta-
peredto 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 such as common-source gatherswill 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 2004solve this dilemma by incorporating slowness, i.e.,
propagation-direction, information into the extrapolation, either ex-
plicitly see the prestack parsimonious migration of Hua and Mc-
Mechan, 2003or implicitly see the analysis of Claerbout’s 1985
survey-sinking algorithm by Stolk and de Hoop, 2006.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 2007show application in the context of
recovery of accurate angle-dependent reflection coefficients and
subsequentAVA inversionfrom 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 Sun et al., 2006at the
image time and location should give separate P and converted P-S
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 a5 and b2.
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 a and band 2 c and d
Figure 9, using the receiver/source amplitude-ratio imaging condi-
tions; imaging conditions for aand care evaluated at the time of
the maximum source-wavefield amplitude. Imaging conditions for
panels band dare evaluated at the location of the maximum of the
source-normalized crosscorrelation.
Prestack migration imaging conditions S87
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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. 2007bdiscuss 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 Fornberg, 1998. In Figure 7, typi-
cal depth errors within the spectral-reflection zone 1.5 to 3.0 km
horizontal positionare a maximum of 1 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 blackened
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 source-de-
pendentscaling. 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 because it corre-
sponds to the maximum amplitude of both the source and receiver
wavefieldsand the best image resolution from using only the peak
wavefield values. 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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Prestack migration imaging conditions S89
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Reverse-time migration (RTM) is used for subsurface imaging to handle complex velocity models including steeply dipping interfaces and dramatic lateral variations and promises better imaging results compared to traditional migration method such as Kirchhoff migration algorithm. RTM has been increasingly used in seismic surveys for hydrocarbon resource explorations. Based on the similarity of kinematics and dynamics between electromagnetic wave and elastic wave, we develop pre-stack RTM method and apply it to process ground penetrating radar (GPR) data. Finite-difference time domain (FDTD) numerical method is used to simulate the electromagnetic wave propagation including forward and backward extrapolations, the cross-correlation imaging condition is used to obtain the final image. In order to provide a velocity model with relatively higher accuracy as the initial velocity model for RTM, we apply a full waveform inversion (FWI) in time domain to estimate the subsurface velocity structure based on reflection radar data. For testing the effectiveness of the algorithm, we have constructed a complex geological model, common-offset radar data and common-shot profile (CSP) radar reflection data are synthesized. All data are migrated with traditional Kirchhoff migration method and pre-stack RTM method separately, the migration results from pre-stack RTM show better coincidence with the true model. Furthermore, we have performed a physical experiment in a sandbox where a polyvinyl chloride (PVC) box is buried in the sand, the obtained common-offset radar data and common-shot radar data are migrated by using Kirchhoff migration method and pre-stack RTM algorithm separately, the pre-stack RTM result shows that RTM algorithm could get better imaging results.
... However, there are two crucial problems in the RTM algorithm: (a) migration velocity model or relative permittivity (RP) model, which is usually not prior information and depends on the experts' decision. When the velocity or RP estimation is not appropriate, the accuracy of the RTM imaging will be seriously affected (Chattopadhyay and McMechan, 2008;Dai et al., 2012); (b) the artifacts remain in the RTM imaging, which is composed of noise interference, arc-shaped clutter, multiple waves, and crosstalk. ...
... The RTM is based on the fundamental principle of time reversal invariance: the recorded electromagnetic wavefields as the new source, inputting them in reverse time from the receiver position and calculating the backward wavefields (Fink, 1992;Zhu et al., 2016). RTM can be divided into three steps (Chattopadhyay and McMechan, 2008): (a) calculate and store the forward wavefields; (b) calculate and store the backward wavefields; (c) apply the imaging conditions. According to the fundamental theory of electromagnetic wave propagation (Taflove and Brodwin, 1975), the kinematics and dynamics of the GPR waves follow Maxwell equations, and the TM mode waves in the time domain can be expressed as: ...
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... where (x, y, z) defines the spatial coordinates of the imaging point, T max is the maximum recording time, and S and R represent the source and receiver wave fields, respectively (Chattopadhyay and McMechan, 2008). Both the receiver and source wave fields are independently propagated with the same scalar, two-way FD extrapolator. ...
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... As of now, RTM is the most accurate and robust depth migration method among all these methods (Baysal et al., 1983;McMechan, 1983) because it does not make any approximation to the wave equation, and makes full use of the kinematics and dynamics information of the wave equation. It can deal with all wave types (reflection, refraction and diffraction) and has no imaging inclination limit (Chattopadhyay and McMechan, 2008). The amplitude characteristics of the imaging results are maintained well . ...
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True-amplitude Kirchhoff migration (TAKM) is an important tool in seismic-reflection imaging. In addition to a structural image, it leads to reflectivity maps of the subsurface. TAKM is carried out in terms of a weighted diffraction stack where the weight functions are computed with dynamic ray tracing (DRT) in addition to the diffraction traveltimes. DRT, however, is time-consuming and imposes restrictions on the velocity models, which are not always acceptable. An alternative approach to TAKM is proposed in which the weight functions are directly determined from the diffraction traveltimes. Because other methods exist for the generation of traveltimes, this approach is not limited by the requirements for DRT. Applications to a complex synthetic model and real data demonstrate that the image quality and accuracy of the reconstructed amplitudes are equivalent to those obtained from TAKM with DRT-generated weight functions.
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In this work, we study the resolving power of seismic migration as a function of source-receiver offset. We quantify horizontal resolution by means of the region around the migrated reflection point that is influenced by the migrated elementary wave. To obtain an estimate for the mentioned zone of horizontal in uence after migration, we investigate the migration output at a chosen depth point in the vicinity of the specular reflection point, i.e., when the output point is moved along the reflector. The width of the spatial resolution resulting from migration of the reflection event is compared with the resolution predicted from theoretical ray-theory formulas for various data sets with different offsets. We find that the region of influence increases linearly with offset. In other words, the resolution power of seismic depth migration decreases when migrating data from larger offsets.