Figure Captions
Figure 1. A
series of model snapshots of an expanding seismic wavefield at 200 ms
time intervals for a surface seismic source above a salt dome model.
The
figure is from a paper by Whitmore and Lines (1986) and is used with the
authors’ permission.
Click here for
sequence of the snapshots of an expanding seismic wavefield.
Figure 2. This
cartoon shows the seismic experiment, and the concept of the “exploding
reflector” used in the reverse-time migration experiment. The upper left
part of the figure shows the seismic reflection experiment for a
coincident seismic source (flag) and receiver (triangle). Seismic energy
travels from the source, down to the reflector and returning back to the
receiver. In the upper right we see the "exploding reflector" model that
is almost always equivalent to this seismic reflection experiment. In
this model, we consider seismic waves to travel a one-way path from the
reflector to the surface receiver (or half the reflected distance) at a
velocity that is half the velocity of the medium. Reverse-time migration
images the seismic data by reversing the path of the exploding reflector
model. Recorded seismic energy is propagated backward in time to its
point of origin.
Figure 3. An
example of reverse-time migration for a salt pillow model from Bording
and Lines 1997 SEG publication “Seismic Modeling and Imaging With the
Complete Wave Equation.” Figure 3a shows an unmigrated stacked section -
few reflection events can be discerned on the seismogram. Figure 3b
shows the depth image of the salt pillow obtained by reverse-time
migration. The dipping seismic reflectors and the salt pillow between
offsets of 4-8 kilometers and depths of 3-5 kilometers can be readily
interpreted.
Click here for
sequence of Figure 3a and 3b.
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Modeling![Next Hit](/images/arrow_right.gif)
Our ability to
compute solutions of the elastic wave equation allows us to both model
and image seismic waves. In an elastic medium, the wave equation is
based on two fundamental laws of physics:
·
One is
Newton's
Second Law of Motion, which states that the acceleration of a body
equals the force acting on the body divided by the mass of the body.
·
The other law is Hooke's Law of elasticity, which states that the
restoring force on a body is proportional to its displacement from
equilibrium.
By combining
these two laws, we obtain the elastic wave equation. In the simplest
case of a homogeneous rock body, the wave equation is given by:
![](index_files/image001.jpg)
In the
equation:
·
The symbol
is the Laplacian, which represents the sum of second derivatives of the
wavefield with respect to spatial variation.
·
"u" is the wavefield. (If we are recording with hydrophones, we would
consider pressure wavefields.)
·
"v" represents the wave velocity in the medium.
·
represents
the second derivative of the wavefield with respect to time. The
velocity term is required to scale the equation properly.
To compute
solutions numerically to the wave equation, we need to evaluate second
derivatives in space and time. This evaluation basically amounts to the
use of finite differences of the wavefield in space and time. If we set
up a stencil of points in the space and consider digital values of the
seismic wave in time, we can compute the wavefield by finding numerical
solutions to the wave equation. In other words, we can model or simulate
seismic wave propagation - we can examine wave propagation as a movie of
waves traveling through the earth .
Figure 1 shows movie snapshots of wave propagation passing through
an earth model consisting of layers onlapping on a salt dome. This
allows us to model or simulate numerically the seismic wave response in
an earth model. The model response is termed a synthetic seismogram.
These models are useful for seismic survey design and for examining how
we might illuminate subsurface features by seismic experiments. Forward
modeling allows us to predict how our experiments might aid in
exploration.
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Imaging![Next Hit](/images/arrow_right.gif)
An even more
useful application of seismic wave computations involves the imaging of
actual data that we have recorded. We can do this by essentially running
the seismic wave propagation movie backward in time. Let's examine
applications of this type of imaging .
In order to
understand the ability of seismic wavefield computations to image
subsurface geology, consider the simple example in
Figure 2, where we consider the case of a coincident source and
receiver. The seismic experiment, as shown in the figure, displays a
wave emanating from a surface source, traveling through the earth at the
seismic velocity of the earth and hitting a geologic discontinuity. Upon
hitting the discontinuity, reflected seismic energy travels back to the
surface at the seismic velocity, where it is recorded by a receiver.
For this
experiment, we could equivalently also consider the wave to be generated
by a pulse that was initiated at the geologic reflector and traveled at
half of the velocity of the medium to the receiver. This is the
so-called “exploding reflector model.” which was an ingenious idea of
the late Dan Loewenthal who pioneered its use in various migration
algorithms.
Our ability to
image the subsurface geology would be made possible by “running the wave
propagation movie backward in time” for the exploding reflector
experiment. This would be achieved by moving the recorded seismic
reflections backward in time to the subsurface points from which they
emanated as shown in the reverse-time migration part of the diagram of
Figure 2.
Fortunately, we
are able to "reverse-time propagate" wavefields by using the same wave
equation computations as we used in forward modeling . Wavefields for the
movie progressing backward in time satisfy the wave equation, just as
waves progressing forward in time.
For a brief
historical note, it should be mentioned that this idea had an enormous
practical use in Amoco's exploration of the Wyoming Overthrust Belt in
the 1980s. Dan Whitmore of Amoco Research was probably the first to make
widespread use of “reverse-time” migration in exploration geophysics -
as evidenced by his examples of overthrust imaging and salt dome
imaging , which were shown at the 1982 and 1983 SEG annual meetings.
Reverse-time
wave imaging or migration can be done via the following technique, as
originally explained by three papers in 1983 produced by McMechan
(1983), Whitmore (1983), and Baysal et al. (1983).
First of all,
consider recorded seismic traces for positions along the earth's surface
and reverse the signals in time. These become the time-varying seismic
boundary values at the earth's surface.
Next, propagate
these seismic recordings back into the depths - back to the reflecting
points from which they originated - by using the same wave equation
algorithm that we used in forward modeling . We use half the wave
velocity since the propagation is one-way. In other words, we "depropagate"
the seismic waves back to the reflecting surfaces in depth .
The imaging
method is as general as the form of the wave equation that is used.
Almost all of the complexities of reverse-time wave equation migration
in its various combinations of acoustic, elastic, 2-D, 3-D, anisotropic,
multi-component forms have been described in several papers by George
McMechan and his students at the
University
of Texas
at Dallas (e.g., Wu and McMechan, 1998).
In order to
convince the explorationist of the power of reverse-time depth
migration, we examine the salt pillow model example shown in
Figure 3. The seismogram at the top of this figure is not
interpretable - except possibly for a few flat reflectors - because the
unmigrated data do not have the dipping reflectors in their correct
subsurface positions. Recorded seismic traces are plotted in time
directly below the source-receiver points, which is the correct position
only for the case of flat reflectors.
In order to
unravel the seismic reflector positions and place them in their true
subsurface locations, we migrate the reflection energy back to the point
in the subsurface where it originated. In
Figure 3, the depth image obtained by reverse-time migration
provides a nearly perfect image of the desired geologic model.
For real data,
depth migration is rarely this good due to the fact that we generally
have only estimates of the seismic velocity with which to depropagate
the wavefields. Although reverse-time migration is the most general of
depth migration methods, it is usually the most expensive due to the
fact that a complete solution to the wave equation is computed without
approximations. Nevertheless, it is a beautiful computational technique
that can be coded without great difficulty for general use in both
forward modeling and seismic imaging .
We should not
give the impression that reverse-time migration is restricted to seismic
imaging . In fact, the November 1999 issue of Scientific American
contains a paper titled “Time-Reverse Acoustics” by Mathias Fink, which
describes several applications of acoustic time-reversal mirrors that
have applications in medicine, material testing, and marine acoustic
communication.
In essence,
“making waves” to produce useful images is a worthwhile occupation in
many scientific pursuits.
References
Baysal, E., D.D. Kosloff, and J.W.C. Sherwood, 1983, Reverse time
migration: Geophysics, v. 48, p. 1524-1524..
Fink, Mathias, 1999, Time-reverse acoustics: Scientific American,
November 1999.
McMechan, G., A, 1983, Migration by extrapolation of time-dependent
boundary values: Geophysical Prospecting, v. 31, p. 413-420.
Whitmore, N. Dan, 1983, Iterative depth migration by backward time
propagation: SEG Abstracts, SEG International Meeting and Exposition, v.
1, p. 382-385.
Whitmore, N. Dan, and Larry R. Lines, 1986, Vertical seismic profiling
depth migration of a salt dome flank: Geophysics, v. 51, p. 1087-1109.
Wu, Yafai, and G.A. McMechan, 1998, Wave extrapolation in the spatial
wavelet domain with application to poststack reverse-time migration:
Geophysics, v. 63, p. 589-600.
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