Click to view images in PDF format.
GCSeismic
Modeling and Imaging
- Making Waves*
By
Phillip Bording1 and Larry Lines2
Search and Discovery Article #40066 (2002)
*Adapted for online presentation from the Geophysical Corner column in AAPG Explorer December, 2000, entitled “Seismic Modeling Makes Waves,” and prepared by the authors. Appreciation is expressed to the author, to R. Randy Ray, Chairman of the AAPG Geophysical Integration Committee, and to Larry Nation, AAPG Communications Director, for their support of this online version.
1Consultant, Hazel Green, Alberta, Canada
2University of Calgary, Alberta ([email protected])
Exploration seismology essentially involves dealing with seismic wave equations. We record seismic waves, process digital seismic signals and attempt to interpret and understand the meaning of these signals in geological terms. Discontinuities in subsurface rock formations give rise to seismic reflections, or “echoes.” These signals provide us with information about the location of geological structures and, consequently, allow us to search for hydrocarbon traps.
The key to
successful seismic exploration lies in deriving meaningful images of
subsurface geology. In order to do this, our computer imaging
codes need
to use accurate mathematical descriptions of waves.
|
Click here for sequence of the snapshots of an expanding seismic wavefield.
Click here for sequence of Figure 3a and 3b.
ModelingOur 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:
In the equation:
·
The symbol
· "u" is the wavefield. (If we are recording with hydrophones, we would consider pressure wavefields.) · "v" represents the wave velocity in the medium.
·
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
Figure 1 shows movie snapshots of wave propagation passing through
an
An even more
useful application of seismic wave computations involves the
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 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
Reverse-time
wave
First of all,
consider recorded seismic traces for positions along the
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
The
In order to
convince the explorationist of the power of reverse-time
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
Summary
For real data,
We should not
give the impression that reverse-time migration is restricted to seismic
In essence, “making waves” to produce useful images is a worthwhile occupation in many scientific pursuits. 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
Whitmore, N. Dan, and Larry R. Lines, 1986, Vertical seismic profiling
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. |