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Seismic Modeling Makes Waves

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.

Modeling

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:

In the equation:

  • The symbol -- 2 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 numerically compute solutions 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.

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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.

Modeling

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:

In the equation:

  • The symbol -- 2 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 numerically compute solutions 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 travelling 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 numerically simulate 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.

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, Geophysical Prospecting), Whitmore (1983 SEG abstracts), and Baysal et al. (1983 Geophysics).

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.

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 - since the unmigrated data does not have the dipping reflectors in their correct subsurface position. 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.

Summary

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.

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