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.
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.
Our ability to compute
solutions of the elastic wave equation allows us to both model and image
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.
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
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.
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
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
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.
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