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# 40_04 - T h e Craft of Wavefield Extrapolation T his...

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The Craft of Wavefield Extrapolation This chapter attends to those details that enable us to do a high-quality job of downward continuing wavefields. There will be few new seismic imag- ing concepts here. There will, however, be interesting examples of pitfalls. And in order to improve the quality of seismic images of the earth, several new and interesting mathematical concepts will be introduced. Toward the end of the chapter a program is prepared to simulate and compare various migration methods. The Magic of Color The first thing we will consider in this chapter is signal strength. Echoes get weaker with time. This affects the images, and requires compensation. Next, seismic data is colored by filtering. This filtering can be done in space as well as time. Time-series analysis involves the concept of enhancing the signal-to-noise ratio by filtering to suppress some spectral regions and enhance others. Spectral weighting can also be used on wavefields in the space of o and k . In the absence of noise, wave-equation theory tells us what filters to use. Loosely, the wave equation is a filter with a flat amplitude response in (o, k )-space and a phase response that corresponds to the time delays of propagation. The different regions of (o, k)-space have different amounts of noise. But the different regions need not all be displayed at the strength proposed by the wave equation, any more than data must be displayed with Ax = Az. An example of the mixture of filter theory and migration theory is pro- vided by the behavior of the spatial Nyquist frequency. Because seismic data is often spatially aliased, this example is not without practical significance. Think of an impulse function with its Nyquist frequency removed. The remo- val has little relative effect on the impulse, but a massive relative effect on the zeroes surrounding the impulse. When migrating an impulse by frequency domain methods, spatial frequencies just below the spatial Nyquist are treated

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CRAFT 4.0 Craft of Wavefield Extrapolation much differently from frequencies just above it. One is treated as left dip, the other right dip. This discontinuity in the spatial frequency domain causes a spurious, spread-out response in the space domain shown in figure 1. The spurious Nyquist noise is readily suppressed, not by excluding the Nyquist frequency from the display, but by a narrow band filter such as used in the display, namely (1 + cos kz Ax )/(I + .85 cos k, Ax ) which goes smoothly to zero at the spatial Nyquist frequency. This filter has a simple tri- diagonal representation in the x -domain. FIG. 4.G1. Hyperbola amplified to exhibit surrounding Nyquist noise (top) removed by filtering (bottom). Survey of Migration Technique Enhancements In our quest for quality, we will also recall various approximations as we go. Now is the time to see how the use of approximations degrades results, and to discover how to improve those results. Five specific problems will be considered: 1 The frequency dispersion that results from the approximation of differential operators by difference operators
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40_04 - T h e Craft of Wavefield Extrapolation T his...

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