GEOPHYSICS, VOL. 73, NO. 5 共SEPTEMBER-OCTOBER 2008兲; P. VE135–VE144, 16 FIGS., 1 TABLE.
10.1190/1.2952623
3D prestack plane-wave, full-waveform inversion
Denes Vigh1 and E. William Starr1
ABSTRACT
Prestack depth migration has been used for decades to derive velocity distributions in depth. Numerous tools and
methodologies have been developed to reach this goal. Exploration in geologically more complex areas exceeds the
abilities of existing methods. New data-acquisition and dataprocessing methods are required to answer these new challenges effectively. The recently introduced wide-azimuth
data acquisition method offers better illumination and noise
attenuation as well as an opportunity to more accurately determine velocities for imaging. One of the most advanced
tools for depth imaging is full-waveform inversion. Prestack
seismic full-waveform inversion is very challenging because
of the nonlinearity and nonuniqueness of the solution. Combined with multiple iterations of forward modeling and residual wavefield back propagation, the method is computer intensive, especially for 3D projects. We studied a time-domain, plane-wave implementation of 3D waveform inversion. We found that plane-wave gathers are an attractive input
to waveform inversion with dramatically reduced computer
run times compared to traditional shot-gather approaches.
The study was conducted on two synthetic data sets — Marmousi2 and SMAART Pluto 1.5 — and a field data set. The
results showed that a velocity field can be reconstructed well
using a multiscale time-domain implementation of waveform inversion. Although the time-domain solution does not
take advantage of wavenumber redundancy, the method is
feasible on current computer architectures for 3D surveys.
The inverted velocity volume produces a quality image for
exploration geologists by using numerous iterations of waveform inversion.
INTRODUCTION
Two tools based upon the two-way wave-equation algorithm have
proven useful with imaging seismic data. One of these applications
is forward modeling. This tool can be used to design wide-azimuth
surveys and to help with cost considerations for acquisitions. The
other tool, reverse time migration 共RTM兲, further enhances the final
prestack-depth-migration 共PSDM兲 volume. RTM combines the benefits of imaging multiple arrivals under the salt as well as steep and/
or overturned reflectors. The success of RTM raises the requirements for velocity models with even higher resolution. Another new
tool, full-waveform inversion, has the potential to provide velocity
models with significantly higher resolution and to further advance
depth-imaging capability.
Throughout the years, substantial research has been dedicated to
full-waveform inversion. Lailly 共1983兲, Tarantola 共1984, 1987兲, and
Mora 共1987, 1988兲 introduced the gradient-based full-waveform inversion algorithm to exploration geophysicists. The methods of
Lailly 共1983兲 and Tarantola 共1984, 1987兲 have been improved significantly, increasing efficiency and considerably reducing computational requirements.
There are three key steps to the inversion. Step one is calculating
the differences between the acquired seismic data and the current
model estimate through forward modeling. Most of this residual data
set exhibits the accuracy of the current model. In step two, the backpropagated residual wavefield is crosscorrelated with the corresponding forward-source-propagated wavefield at each time step
and summed over all time steps to produce the gradient volume. Finally, in step three, the amplitude at each spatial point becomes proportional to the velocity change.
To reduce computations for waveform inversion, authors limit
frequencies 共Sirgue and Pratt, 2004; Operto et al., 2007兲, use the logarithmic domain 共Shin and Min, 2006兲, or use multiple grids in the
time domain 共Bunks et al., 1995兲. Because the gradient calculation
for waveform inversion is similar to RTM, we can use an efficient
plane-wave method 共Vigh and Starr, 2006兲. Plane-wave implementations have also reduced computations for other migration methods
共Whitmore and Garing, 1993; Rietveld and Berkhout, 1994; Duquet
et al., 2001; Zhang et al., 2005; Chemingui et al., 2007兲.
Full-waveform inversion can be performed in either the frequency or the time domain. Frequency-domain implementations typically are solved for many shots using LU decomposition of a large,
sparse matrix. This method may be efficient with numerous shots,
Manuscript received by the Editor 4 January 2008; revised manuscript received 6 May 2008; published online 1 October 2008.
1
Staag Imaging, L.P., Houston, Texas, U.S.A. E-mail: [email protected]; [email protected].
© 2008 Society of Exploration Geophysicists. All rights reserved.
VE135
VE136
Vigh and Starr
but it requires large memory requirements, making the method more
suitable to 2D or smaller 3D problems 共Operto et al., 2007兲. Another
benefit of this method is the possibility of the need for only a few select frequencies, but this depends on target depth and maximum offset 共Sirgue and Pratt, 2004兲. Most subsalt targets are deep and many
current acquisitions have a somewhat small finite offset, so dozens
of frequencies may be required, rendering time-domain implementations more attractive.
When tens of iterations are required to derive subsurface velocity
structures, a rapid propagation and modeling algorithm should be
used as the core of the inversion. Time-domain implementations are
straightforward to achieve and relatively fast, and they require less
memory. The disadvantage of the method is that it may require significant computation time if the forward-modeling time increment is
small or if many shots are needed, such as for 3D surveys. The timedomain method may also have cycle-skipping sensitivity because
both low and high frequencies are inverted simultaneously. Low-frequency estimation can also be a problem because of the time derivatives in the gradient calculation 共Sirgue and Pratt, 2004兲. To avoid
the need for a large number of shots, we use a plane-wave scheme.
We minimize the cycle-skipping and low-frequency issues by performing the inversion in a multiscale manner.
By performing waveform inversion in a multiscale manner, the
lower frequencies can provide us with several scalable options, such
as changing the number of planes in conjunction with incrementing
the bin sizes in all directions. These features are essential to run the
inversions relatively quickly in three dimensions. In addition, the inversion can be carried out in a layer-stripping style to accommodate
the traditional model-building flows, controlling and speeding up
the time needed to derive a model.
We first present the methodology for the time-domain, full-waveform inversion using plane-wave gathers as input. We then apply
full-waveform inversion to two 2D synthetic data sets and one 3D
marine data set. The 3D marine data set is compared to a PSDM result derived with the workflow used in practice. Finally, run-time
comparisons are shown to support the acceleration by using planewave gathers instead of shot gathers as input to the inversion.
condition between the source and receiver plane-wave wavefields.
To prevent aliasing, the shot interval and p range should be
⌬s ⱕ
1
2f max p
,
共2兲
where ⌬s is the shot interval and f max is the maximum frequency for
the current waveform inversion iteration. In addition, the extreme p
values should be chosen to image steep dips properly within the section.
Because plane-wave imaging involves the superposition of many
shots, cross-term artifacts occur as a result of crosscorrelation of the
source wavefields with the receivers of different shots during the imaging condition. This noise can be minimized by selecting a sufficiently large number of p sections. Other workers 共Stork and Kapoor, 2004; Etgen, 2005; Zhang et al., 2005; Grion and Jakubowicz,
2006; Chemingui et al., 2007兲 suggest that the number of p traces
should be approximately
Np ⱖ
Wf max共sin ␪ 2 ⳮ sin ␪ 1兲
,
vs
共3兲
where W is a distance measure, ␪ 1 ⱕ ␪ s ⱕ ␪ 2 are the surface angles,
and Vs is the surface velocity. The value of W can be calculated many
ways, such as by using the maximum source-receiver offset or the
migration aperture, but in reality we accept some small amount of
noise by reducing this value 共Etgen, 2005兲.
Waveform inversion
Seismic waveform inversion is based on minimizing a cost function that measures the difference between the calculated and acquired data. To minimize the cost function, several forward-modeling and residual back propagations are required to update the velocity field gradually. Our optimized time-domain approach in the
plane-wave domain makes waveform inversion feasible for large 3D
surveys.
Forward modeling uses a higher-order, staggered-grid, finite-difference approximation of the acoustic-wave equation:
冋 冉 冊 冉 冊 冉 冊册
METHOD
1 1 ⳵ 2P
⳵ 1 ⳵P
⳵ 1 ⳵P
⳵ 1 ⳵P
Ⳮ
Ⳮ
2 ⳱
2
V ␳ ⳵t
⳵x ␳ ⳵x
⳵y ␳ ⳵y
⳵z ␳ ⳵z
Plane-wave gathers
A 3D survey can be composed of thousands of shot gathers, making time-domain, full-waveform inversion challenging. To reduce
the number of calculations, we compose the shot records into a series
of plane-wave gathers. The plane-wave gathers are formed by a linear slant stack operating on common-receiver gathers followed by a
time delay,
where P共x,y,z,t兲 is the pressure field, ␳ 共x,y,z兲 is the density,
V共x,y,z兲 is the interval velocity, and S共x,y,z,t兲 is the source. In the
current method, we only invert for velocity while relating the density
to the velocity using the equation 共Gardner et al., 1974兲
⌬t jk共p兲 ⳱ px共xsj ⳮ x0兲 Ⳮ py共y sk ⳮ y 0兲,
␳ ⳱ ␳ 0V k0 ,
Ⳮ S,
共4兲
共1兲
applied at plane-wave source locations 共xsj,y sk兲, where j ⳱ 1,2,
. . . ,Nx and k ⳱ 1,2, . . . ,Ny, using the ray parameter vector p
⳱ 共px,py兲 and the plane-wave origin 共x0,y 0兲 at the surface. To create
the proper py components, we would need a y-offset 共hy兲 component.
Because we have limited hy for narrow-azimuth data, we can use 3D
methods such as delayed shot or cylindrical wave on 2D plane-wave
sections 共Liu et al., 2006兲. For wide-azimuth data where we have
enough crossline offsets, we can compute a full py section.
When composing plane-wave sections, care must be taken to
avoid artifacts caused by a large shot interval within common-receiver gathers and cross-term artifacts resulting from the imaging
共5兲
where ␳ 0 is 0.31 g/cm3 and k0 is 0.25.
Denote P共xr,y r,zr,t兲 as the pressure data recorded at locations xr.
The velocity is determined by minimizing the misfit function:
E⳱
1
兺兺
2 p r
冕
dt关Pobs共xr,t兲 ⳮ Pcal共xr,t兲兴2 ,
共6兲
where Pobs are the observed plane-wave data and Pcal are the data calculated using equation 4 with the current velocity and density model.
The value E is minimized iteratively by calculating the gradient or
steepest descent direction 共Tarantola, 1984兲 at iteration n by
3D plane-wave, full-waveform inversion
␥n ⳱
1
兺
V3n p
冕
dt共⳵ t P f 兲共⳵ t Pb兲,
共7兲
where Vn is the velocity field at iteration n, P f is the forward-propagated plane-wave source wavefield, and Pb is the wavefield using reverse-time propagation of the residuals 共Pobs ⳮ Pcal兲 acting as if they
were plane-wave sources at the receiver locations. The gradient of
the misfit function can be computed by zero-lag correlation of the
forward-propagated wavefields with the back-propagated residual
wavefield. Both wavefields are propagated using the current velocity
and density fields.
To speed up convergence, a preconditioned conjugate-gradient
method 共Zhou et al., 1995兲 can be used as follows:
␭n ⳱ ⳮFn␥n ,
共8兲
where the preconditioning term Fn is the diagonal inverse to the approximate Hessian of the misfit function E. The value Fn is then calculated as
H␣ ⳱ JTJ,
共9兲
Fn ⳱ 共diag Ha Ⳮ ␬ I兲ⳮ1 ,
共10兲
VE137
proach, the inversion results from the lower frequencies are passed
on to the next higher range of frequencies; the lowest frequency used
in each scale step is held constant from scale step to scale step. In this
case, our time-domain method does not take advantage of wavenumber redundancy that frequency-domain methods use to limit frequencies. However, by using redundant coverage, it may make our
multiscale method more robust. For instance, redundant coverage
may be needed to improve the signal-to-noise ratio 共S/N兲.
Nonlinearity of waveform inversion problem
All waveform inversions using a gradient method should provide
a long-wavelength velocity model that is within half a period of the
dominant wavelength to keep convergence 共Figure 2兲 to a global
minimum. This issue of nonlinearity in the offset-time 共x-t兲 domain
between the observed and the model data might be seen as cycle
skipping.
When convergence to local minima occurs, one must use lower
frequencies or start with a better higher-frequency velocity model.
To prevent local minima, we can determine how accurate the initial
velocity model and the starting frequency range must be by analyzing the misfit function versus the smoothness, using a known true ve-
where Ha is the approximate Hessian 共Shin et al., 2001兲, J is the
Jacobian matrix of partial derivatives, ␬ is a damping factor to avoid
singular values, and I is the identity matrix. The diagonal terms of Ha
are proportional to the source and receiver illumination. It corresponds to the squared product of the source and receiver Green functions associated with equation 4 using the current velocity and density model.
The preconditioned conjugate gradient is then calculated as
␦n ⳱ ␭n Ⳮ ␤ n␦nⳮ1 ,
共11兲
where ␤ n is the conjugate scaling factor calculated using the PolakRibière method 共Gilbert and Nocedal, 1992兲, given by
␤n ⳱
具␭n,␭n ⳮ ␭nⳮ1典
储␭nⳮ1储2
.
共12兲
Figure 1. Step-length line search using three points.
The model is then updated using
VnⳭ1 ⳱ Vn Ⳮ ␣ ␦n ,
共13兲
where ␣ is the optimal step length derived from a line search along
the gradient direction using a parabolic fit 共Nash, 1979兲 共Figure 1兲.
The line search involves evaluating several misfit functions
whereby we need two additional values of ␣ such that ␣ 1 is less than
␣ 2 and E共1兲 is less than both E共0兲 and E共2兲. The value E共0兲 is already
evaluated from the gradient calculation. Each additional misfit function evaluation needs a forward-modeling run to calculate the residuals required for the function. Once we have three valid values, we
use the minimum of the parabolic function describing these points.
Inversion proceeds in a multiscale approach 共Bunks et al., 1995兲
from lower to higher frequencies to improve the chances that the global minimum is reached and not a local one. The multiscale temporal
decomposition uses a windowed finite-impulse-response 共FIR兲 lowpass sinc filter 共Oppenheim and Schafer, 1975兲 applied to the planewave source and the observed receiver data. In our multiscale ap-
Figure 2. Observed and calculated data within half a period of the
dominant wavelength.
VE138
Vigh and Starr
locity model as shown in Figure 3. In this figure, we progressively
increase the smoothing on a known velocity model and measure the
misfit for different frequency ranges. When the slope of a curve proceeds downward to the right, we will converge to local minima.
Data preconditioning for the inversion
The success of the inversion relies upon matching the amplitudes
between the observed and the modeled data. One must pay attention
to the data preconditioning prior to the inversion. Several factors are
important.
Attenuation
Forward modeling is based on the acoustic-wave equation, so we
must deal with nonacoustic factors prior to the waveform inversion.
Of these factors, the correct attenuation is important. Based on Liao
and McMechan 共1997兲, the linear attenuation function T共f r兲 can be
described as
再 冎
T共f r兲 ⳱ exp ⳮf r
␲t
,
Q
共14兲
where the attenuation factor Q is constant, t is the traveltime, and f r is
the frequency content of the recorded data. The value T共f r兲 shapes
the S共f r兲 signal spectra into the output spectra R共f r兲 as follows:
R共f r兲 ⳱ S共f r兲T共f r兲.
Multiple attenuation
Free-surface or internal multiples, in principle, can be used for imaging and waveform inversion. We have achieved superior results by
attempting to remove free-surface and internal multiples. One can
claim the multiples should be included for the model derivation;
however, we traditionally remove multiples. Waveform inversion is
a migration process for the first iteration. Therefore, if the migrated
result does not show signs of noise, the inversion can be run without
the imprint of possible artifacts.
Wavelet factors
Phase- and amplitude-controlled preconditioning factors that cannot be explained by the source wavelet in the inversion should also
be carried out. For example, bubbles must be removed if marine data
are used for the inversion process. The source wavelet for the inversion should match the real wavelet in the seismic data as close as possible. Therefore, either the source needs to be estimated or the data
need to be modified to resemble the source wavelet.
Plane-wave transformation
Prior to transforming the data to the plane-wave domain, the shots
should be interpolated to avoid spatial aliasing as described by equation 1. To minimize the mismatch between the modeled and observed plane-wave gathers, as well as to reduce offset edge effects
during the transformation, split-spread receiver gathers are generated using the reciprocity principle 共Liu et al., 2004兲.
共15兲
FIELD DATA EXAMPLES
Random and coherent noise
Noise, either random or coherent, is an enemy to the waveform inversion process. Forward modeling is free of random and coherent
noise, so it is best to remove these phenomena from the observed
data set before undertaking the inversion. If enough noise is present,
the waveform inversion process will attempt to incorrectly change
the inverted attribute so the modeled data match the observed data in
a least-squares sense. Random noise seems to have less impact on inversion than coherent noise. However, enhancing the signal prior to
the inversion will improve the results. Examples of coherent noise
that need to be removed are swell noise and seismic interference.
Next, we show some synthetic and field data examples where the
solutions are focused on the suprasalt, salt geometry, and subsalt regions. Two of the examples are salt related to address Gulf of Mexico
imaging issues where forward modeling and reverse-time migration
are commonly used to verify and/or enhance migrated images.
The synthetic examples include two data sets: Marmousi2 and
SMAART Pluto 1.5. The Marmousi2 data set represents a complex
suprasalt velocity structure with faults, thin layers, and low-velocity
anomalies fairly close to the seafloor. The SMAART Pluto data set
has a complex subsalt velocity regime segmented by faults below the
salt bodies. Even though the synthetic examples are two-dimensional, they best demonstrate the typical issues in the Gulf of Mexico environment. The 3D field data example of approximately 36 km2 is
from the Green Canyon area in the Gulf of Mexico. It measures
28,230 ⫻ 1250 m with a maximum cable length of 8000 m. The iterations were performed on 200 nodes of two AMD 64-bit Opteron dual-core processors with 8 GB of memory. Run-time comparisons between shot-record and plane-wave inversion on the example data
sets are described in the examples below and summarized in Table 1.
Marmousi2
Figure 3. Smoothing factor versus misfit function over different frequency ranges using a known velocity from the Pluto 1.5 data set.
The Marmousi2 model 共Martin et al., 2006兲 is an elastic overthrust example that extends the original Marmousi model by introducing a deeper water layer, small velocity anomalies in the shallow
sedimentary section, and a salt body in the deep part of the original
velocity profile. This model may address near-seafloor gas pockets
combined with thin-layered faults above the salt structures, showing
how difficult sedimentary regimes can be in a marine environment.
3D plane-wave, full-waveform inversion
VE139
first, low-frequency step ranged from 0 to 13 Hz. The second, midStreamer shot records were generated using the acoustic-wave
dle-frequency step ranged from 0 to 18 Hz. The final frequency
equation without a free surface and using the known P-wave velocity
VP and density models. A total of 851 shots were
generated using a shot-receiver interval of 20 m.
Table 1. Run-time comparison summary between shot record and plane-wave
Forty ray-parameter planes, sampled at equal ininversion for the example data sets. Run times are cumulative for one iteration.
The shot inversion time for the Gulf of Mexico example is estimated from one
tervals between ⳮ333 and 333 ␮s/m, were gensail line.
erated using a tau-p transform from the synthetic
shot gathers. This model has a known source
wavelet 共2–40-Hz Ricker兲 and no noise, so preprocessing was ignored.
Example
Number
data set
of shots
The starting velocity model 共Figure 4a兲 was a
smoothed version of the true velocity model. This
Marmousi2
851
model has small, subtle anomalies, so we need to
Pluto 1.5
1387
capture the resolution of the anomalies in the
shallow velocity model. The inversion was split
Gulf of
5600
Mexico
into five scale steps with the output of the inverted
velocity from one step used as input for the next
step. The frequency bands were 0–8, 0–13, 0–19,
0–24, and 0–29 Hz. Each frequency step had 12 iterations for a total
of 60 iterations. The final inverted velocity model is shown in Figure
4c as a comparison to the true velocity model 共Figure 4b兲. The misfit-function reduction over all frequency bands is shown in Figure 5.
The jump of the residual as a result of the frequency-band change is
easy to observe. Figure 6 shows the data-space residuals corresponding to the starting and final velocities. Interestingly, although most of
the residual energy has been absorbed in the final velocity model,
some unrecovered residual energy remains. The unrecoverable residual energy could not be explained, as well as the other features of
the model. A more elaborate optimization scheme could have recovered this energy better.Avisual comparison of the true and recovered
velocities in Figure 4b and c with the data residual in Figure 6 indicates the extent of velocity resolution.
For this example, we ran a shot-record inversion as well. The inversion showed comparable quality to the plane-wave inversion;
however, the run time was 10 times slower.
Number
of planes per
sailline
Plane-wave
inversion time
共hours兲
Shot
inversion time
共hours兲
Speedup
40
40
40
0.66
16.00
1644.00
7
224
5096*
10
14
⬃3
a)
b)
SMAART Pluto 1.5
The original Pluto 1.5 data set from SMAART was produced using an elastic finite-difference algorithm with a P-wave velocity
model, an S-wave velocity model, and a density model typical of the
Gulf of Mexico. Our research has determined that we can improve
results if an absorbing boundary is used on all sides of the finite-difference grid. Thus, the free-surface multiples should be removed using, for example, surface-related multiple elimination 共SRME兲 before transforming the data to the plane-wave domain.
To simplify the process, we regenerated the synthetic data set
without the free-surface multiples and used a known Ricker wavelet
with a frequency range of about 2–40 Hz. The acquisition geometry
of the source and receivers is the same as in the original data set. In
all, 1387 shot records with a source-receiver interval of 23 m were
generated with the acoustic-wave equation and using the known VP
and density models. We created 40 planes with ray parameters sampled at equal intervals between ⳮ333 and 333 ␮s/m. This model has
a known source wavelet and no noise, so preprocessing was ignored.
Inversion proceeded in three scale steps with the inverted velocity
that was output from one step used as input for the next step. The
c)
Figure 4. Marmousi2 velocity model: 共a兲 starting model, 共b兲 true velocity model, 共c兲 after waveform inversion.
VE140
Vigh and Starr
range was from 0 to 24 Hz. The starting velocity model was a linear
velocity gradient with a smoothed salt body. The water bottom was
assumed to be known; thus, the water layer contained a constant for
the velocity and density. Each frequency step had 20 iterations for a
total of 60 iterations.
The starting velocity is shown in Figure 7a; the true velocity is
shown in Figure 7b, which should be compared with the inversion
results shown in Figure 7c. The misfit-function reduction over all
frequency bands is shown in Figure 8. Jumps in the misfit function
occur at the start of a new frequency-range step because of introducing additional higher-frequency data. The data-space residuals cor-
a)
b)
Figure 5. Marmousi2 misfit function versus iteration.
a)
c)
b)
Figure 6. Marmousi2 data residuals at a near ray parameter for 共a兲 the
starting velocity and 共b兲 the final inverted velocity. The same display
gain is applied to both sections.
Figure 7. Pluto velocity: 共a兲 starting velocity, 共b兲 true velocity, 共c兲 after waveform inversion.
3D plane-wave, full-waveform inversion
responding to the starting and final velocities are shown in Figure 9.
Comparisons of the true and inverted velocities at selected commondepth-point locations are seen in Figure 10. The data-space residual
decreased substantially over the iterations. The remaining energy
can be explained by the small discrepancy between the true and final
inverted velocity models, especially at the salt boundary. The inverted model is not as crisp as the true model, which reduces the calculated reflection coefficients. This difference is noticeable on Figure 10,
where the derived vertical interval velocity is overlain on the true velocity function at selected locations along the line.
For this example, we ran a shot-record inversion as well. The shotrecord inversion showed comparable quality to the plane-wave inversion; however, the run time was 14 times slower.
VE141
The misfit function reduction over all frequency bands 共Figure
12兲 was not as good as the results from the synthetic examples. Some
reasons for this can include an inaccurate source wavelet and con-
a)
Gulf of Mexico data set
The 3D field marine data set is typical for the Gulf of Mexico.
Free-surface multiples were removed using an SRME algorithm.
The source wavelet was unknown. In our case, we derived the source
wavelet using the near offsets at the water bottom and computed a
shaping filter to transform the data from about 6 Hz to a maximumfrequency 60-Hz Ricker wavelet. A total of 5600 shot records were
interpolated to an inline and crossline source-receiver interval of
25 m. We composed 40 planes per source line with ray parameters
sampled at equal intervals between ⳮ333 and 333 ␮s/m. Frequency
steps used included 10 iterations from 0 to 18 Hz and 27 iterations
from 0 to 24 Hz for a total of 37 iterations. In this data example, the
velocity model was originally derived through suprasalt tomography combined with salt-geometry picking followed by subsalt velocity scans 共Figure 11a兲. A heavily smoothed version of this model
was used as a starting velocity model 共Figure 11b兲 for the waveform
inversion to initiate the iterative process. The final model after 37 iterations is shown in Figure 11c.
b)
Figure 9. Pluto data residuals at a near ray parameter for 共a兲 the starting velocity and 共b兲 the final inverted velocity. The same display gain
is applied to both sections.
a)
Figure 8. Pluto misfit function versus iteration.
b)
c)
Figure 10. Pluto vertical velocity comparisons between the true velocity and the derived velocity at distances of 共a兲 6530 m, 共b兲
12,820 m, and 共c兲 20,360 m.
VE142
Vigh and Starr
verted waves associated with the salt body. The first inverted frequency band’s poor S/N kept the residual energy relatively high. By
opening up the high end of the frequency band for consecutive iterations, the S/N improved, significantly decreasing the misfit function,
as seen in Figure 12 at iteration 11. The data-space residuals corresponding to the starting and final velocities are shown in Figure 13.
Figure 14a shows the result of migrating the data with the nons-
a)
Figure 12. Gulf of Mexico misfit function versus iteration.
a)
b)
c)
Figure 11. Gulf of Mexico data. In all views, the top display is the
vertical section at y ⳱ 625 m and the bottom display is the horizontal slice at z ⳱ 4725 m. 共a兲 Traditionally derived migration velocity.
共b兲 Starting velocity. 共c兲 After waveform inversion.
b)
Figure 13. Gulf of Mexico data residuals at a near ray parameter for
共a兲 the starting velocity and 共b兲 the final inverted velocity. The same
display gain is applied to both sections.
3D plane-wave, full-waveform inversion
VE143
moothed final migration velocity model using RTM. Figure 14b
shows the result of migrating the data with the final inverted velocity
model.
When the final migrated images are compared, the velocity field
determined by the waveform inversion produced visible improvements compared to the one obtained using traditional methods. Theimage enhancement can also be observed by comparing the corresponding common-image gather displays 共Figures 15 and 16兲.
a)
b)
Figure 16. Gulf of Mexico migrated gathers using the velocity derived from waveform inversion.
Figure 14. Gulf of Mexico RTM using 共a兲 the traditionally derived
migration velocity and 共b兲 the velocity derived from waveform inversion.
We did not run a full shot-record inversion for the entire data set.
However, we ran timing on a single sail line and extrapolated it to estimate that the run time for a full-shot-record inversion on this data
set would be about three times slower than a plane-wave inversion.
CONCLUSION
Figure 15. Gulf of Mexico migrated gathers using the original migration velocity.
Plane-wave gathers can be used to perform full-waveform inversion. Using plane-wave gathers can dramatically reduce the number
of computations. Plane-wave gathers can provide us with the ability
to use full-waveform inversion on 3D data. Using plane waves for
RTM has significantly reduced turnaround time, and using the same
domain for full-waveform inversion can make 3D cases feasible,
given new hardware configurations. The waveform inversion process measures the errors or residuals in time, ensuring the most accurate velocity field by a given seismic data set.
In mature exploration areas such as the Gulf of Mexico, fairly accurate velocity fields are derived in typical PSDM work, which can
help us create a relatively good starting-velocity volume to bridge
the possible lack of low frequencies in the acquired data. Promising
new data-collection methods can extend the low-frequency part of
the spectrum by a few hertz, which could help the inversion process.
If the number of iterations can be kept within a reasonable range, 3D
inversions are feasible on the latest hardware configurations. Even if
the human component 共interpretation and/or image-gather analysis兲
of the PSDM processing sequence could be eased and the subjective
human decision could be augmented by full-waveform inversion,
PSDM has its place in the modern imaging workflow.
VE144
Vigh and Starr
ACKNOWLEDGMENTS
We thank Anadarko for its continuing encouragement to pursue
waveform inversion technology, Ente Nazionale Idrocarburi 共ENI兲
for the seismic data, and the Utah Tomography and Modeling/Migration 共UTAM兲 consortium for its research into waveform inversion methods.
REFERENCES
Bunks, C., F. M. Saleck, S. Zaleski, and G. Chavent, 1995, Multiscale seismic waveform inversion: Geophysics, 60, 1457–1473.
Chemingui, N., R. van Borselen, and M. Orlovich, 2007, 3D plane-wave migration of wide-azimuth data: 77th Annual International Meeting, SEG,
Expanded Abstracts, 2195–2199.
Duquet, B., P. Lailly, and A. Ehinger, 2001, 3D plane-wave migration of
streamer data: 71st Annual International Meeting, SEG, Expanded Abstracts, 1033–1036.
Etgen, J. T., 2005, How many angles do we really need for delayed-shot migration?: 75th Annual International Meeting, SEG, Expanded Abstracts,
1985–1988.
Gardner, G. H. F., L. W. Gardner, and A. R. Gregory, 1974, Formation velocity and density — The diagnostic basics for stratigraphic traps: Geophysics, 39, 770–780.
Gilbert, J. C., and J. Nocedal, 1992, Global convergence properties of conjugate gradient methods for optimization: SIAM Journal on Optimization, 2,
21–42.
Grion, S., and H. Jakubowicz, 2006, On the influence of offset in plane-wave
migration: 68th Conference and Exhibition, EAGE, Extended Abstracts,
G011.
Lailly, P., 1983, The seismic inverse problem as a sequence of before stack
migrations, in J. Bednar, E. Robinson, and A. Weglein, eds., Inverse scattering theory and application: Society for Industrial and Applied Mathematics 共SIAM兲, 206–220.
Liao, Q., and G. A. McMechan, 1997, Tomographic imaging of velocity and
Q, with application to crosswell seismic data from the Gypsy pilot site,
Oklahoma: Geophysics, 62, 1804–1811.
Liu, F., D. W. Hanson, N. D. Whitmore, R. S. Day, and R. H. Stolt, 2006, Toward a unified analysis for source plane-wave migration: Geophysics, 71,
no. 4, S129–S139.
Liu, F., N. D. Whitmore, D. W. Hanson, R. S. Day, and C. C. Mosher, 2004,
The impact of reciprocity on prestack source plane wave migration: 74th
Annual International Meeting, SEG, Expanded Abstracts, 1045–1048.
Martin, G., R. Wiley, and K. Marfurt, 2006, Marmousi2: An elastic upgrade
for Marmousi: The Leading Edge, 25, 156–166.
Mora, P., 1987, Nonlinear two-dimensional elastic inversion of multioffset
seismic data: Geophysics, 52, 1211–1228.
——–, 1988, Elastic wave-field inversion of reflection and transmission data:
Geophysics, 53, 750–759.
Nash, J. C., 1979, Compact numerical methods for computers: Linear algebra and functions minimization: Pitman Press.
Operto, S., J. Virieux, P. Amestoy, J. l’Excellent, L. Giraud, and H. B. H. Ali,
2007, 3D finite-difference frequency-domain modeling of viscoacoustic
wave propagation using a massively parallel direct solver: A feasibility
study: Geophysics, 72, no. 5, SM195–SM211.
Oppenheim, A. V., and R. W. Schafer, 1975, Digital signal processing: Prentice-Hall, Inc.
Rietveld, W. E. A., and A. J. Berkhout, 1994, Prestack depth migration by
means of controlled illumination: Geophysics, 59, 801–809.
Shin, C., and D. Min, 2006, Waveform inversion using a logarithmic wavefield: Geophysics, 71, no. 3, R31–R42.
Shin, C., K. Yoon, K. Marfurt, K. Park, D. Yang, H. Lim, S. Chung, and S.
Shin, 2001, Efficient calculation of a partial-derivative wavefield using
reciprocity for seismic imaging and inversion: Geophysics, 66, 1856–
1863.
Sirgue, L., and R. G. Pratt, 2004, Efficient waveform inversion and imaging:
A strategy for selecting temporal frequencies: Geophysics, 69, 231–248.
Stork, C., and J. Kapoor, 2004, How many P values do you want to migrate
for delayed-shot wave equation migration?: 74th Annual International
Meeting, SEG, Expanded Abstracts, 1041–1044.
Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266.
——–, 1987, Inverse problem theory: Methods for data fitting and model parameter estimation: Elsevier Science Publishing Co., Inc.
Vigh, D., and E. W. Starr, 2006, Shot profile versus plane wave reverse time
migration: 76th Annual International Meeting, SEG, Expanded Abstracts,
2358–2361.
Whitmore, N. D., and J. D. Garing, 1993, Interval velocity estimation using
iterative prestack depth migration in the constant angle domain: The Leading Edge, 12, 757–762.
Zhang, Y., J. Sun, C. Notfors, S. H. Gray, L. Chernis, and J. Young, 2005, Delayed-shot 3D depth migration: Geophysics, 70, no. 6, E21–E28.
Zhou, C., W. Cai, Y. Luo, G. T. Schuster, and S. Hassanzadeh, 1995, Acoustic
wave-equation traveltime and waveform inversion of crosshole seismic
data: Geophysics, 60, 765–773.