Migration by finite differences

 

In the last chapter we learned how to extrapolate wavefields down into the earth. The process proceeded simply, since it is just a multiplication in the frequency domain by $\exp [ ik_z ( \omega , k_x ) z ]$.Finite-difference techniques will be seen to be complicated. They will involve new approximations and new pitfalls. Why should we trouble ourselves to learn them? To begin with, many people find finite-difference methods more comprehensible. In (t,x,z)-space, there are no complex numbers, no complex exponentials, and no ``magic'' box called FFT.

The situation is analogous to the one encountered in ordinary frequency filtering. Frequency filtering can be done as a product in the frequency domain or a convolution in the time domain. With wave extrapolation there are products in both the temporal frequency $\omega$-domain and the spatial frequency k_x-domain. The new ingredient is the two-dimensional $( \omega , k_x )$-space, which replaces the old one-dimensional $\omega$-space. Our question, why bother with finite differences?, is a two-dimensional form of an old question: After the discovery of the fast Fourier transform, why should anyone bother with time-domain filtering operations?

Our question will be asked many times and under many circumstances. Later we will have the axis of offset between the shot and geophone and the axis of midpoints between them. There again we will need to choose whether to work on these axes with finite differences or to use Fourier transformation. It is not an all-or-nothing proposition: for each axis separately either Fourier transform or convolution (finite difference) must be chosen.

The answer to our question is many-sided, just as geophysical objectives are many-sided. Most of the criteria for answering the question are already familiar from ordinary filter theory where a filter can be made time-variable. Time-variable filters are useful in reflection seismology because the frequency content of echoes changes with time. An annoying aspect of time-variable filters is that they cannot be described by a simple product in the frequency domain. So when an application of time-variable filters comes along, the frequency domain is abandoned, or all kinds of contortions are made (stretching the time axis, for example) to try to make things appear time-invariant.

All the same considerations apply to the horizontal space axis x. On space axes, a new concern is the seismic velocity v. If it is space-variable, say v(x), then the operation of extrapolating wavefields upward and downward can no longer be expressed as a product in the k_x-domain. Wave-extrapolation procedures must abandon the spatial frequency domain and go to finite differences. The alternative again is all kinds of contortions (such as stretching the x-axis) to try to make things appear to be space-invariant.

Fourier methods are global. That is, the entire dataset must be in hand before processing can begin. Remote errors and truncations can have serious local effects. On the other hand, finite-difference methods are local. Data points are directly related only to their neighbors. Remote errors propagate slowly.

Some problems of the Fourier domain have just been summarized. The problems of the space domain will be shown in this chapter and chapter . Seismic data processing is a multidimensional task, and the different dimensions are often handled in different ways. But if you are sure you are content with the Fourier domain then you can skip much of this chapter and jump directly to chapter where you can learn about shot-to-geophone offset, stacking, and migration before stack.