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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.
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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
In the equation:
·
The symbol
· "u" is the wavefield. (If we are recording with hydrophones, we would consider pressure wavefields.)
·
"v" represents the wave
·
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. 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
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
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 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”
Reverse-time
wave imaging or 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
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
In order to
convince the explorationist of the power of reverse-time depth
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
Summary
For real data,
depth
We should not
give the impression that reverse-time 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
Fink, Mathias, 1999, Time-reverse acoustics: Scientific American, November 1999.
McMechan, G., A, 1983,
Whitmore, N. Dan, 1983, Iterative depth
Whitmore, N. Dan, and Larry R. Lines, 1986, Vertical seismic profiling
depth
Wu, Yafai, and G.A. McMechan, 1998, Wave extrapolation in the spatial
wavelet domain with application to poststack reverse-time |