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# 40_02 - Why S pace and Time I n the previous chapter we...

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Why Space and Time? In the previous 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,(w, k,)~]. 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, 2)-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 w-domain and the spatial fre- quency &-domain. The new ingredient is the two-dimensional (w, kx)-space, which replaces the old one-dimensional uspace. 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 ure 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. Those electrical engineers and old-time deconvolution experts who have pushed themselves into wave processing have turned out to be delighted by it. They hadn't realized their knowledge had so many applications!

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FINITE DIFFERENCING 2.0 Why space and time? Figure 1 illustrates the differences between Fourier domain calculations and time domain calculations. The figure was calculated on a 256x64 mesh to exacerbate for display the difficulties in either domain. Generally, you notice wraparound noise in the Fourier calculation, and frequency dispersion (Section 4.3) in the time domain calculation. (The "time domain" hyperbola in figure 1 is actually a frequency domain simulation - to wrap the entire hyperbola into view). In this Chapter we will see how to do the time domain calculations. A more detailed comparison of the domains is in Chapter 4. FIG. 2.0-1. Frequency domain hyperbola (top) and time domain hyperbola (bottom). Even if you always migrate in the frequency domain, it is worth studying time domain methods to help you choose parameters to get a good time domain response. For example both parts of figure 1 were done in the fre- quency domain, but one simulated the time domain calculation to get a more causal response. Lateral Variation In ordinary linear filter theory, a filter can be made time-variable. This
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40_02 - Why S pace and Time I n the previous chapter we...

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