Velocity continuation by spectral methods |
In a recent work (Fomel, 1996,1994), I introduced the process of velocity continuation to describe a continuous transformation of seismic time-migrated images with a change of the migration velocity. Velocity continuation generalizes the ideas of residual migration (Rothman et al., 1985; Etgen, 1990) and cascaded migrations (Larner and Beasley, 1987). In the zero-offset (post-stack) case, the velocity continuation process is governed by a partial differential equation in midpoint, time, and velocity coordinates, first discovered by Claerbout (1986b). Hubral et al. (1996) and Schleicher et al. (1997) describe this process in a broader context of ``image waves''. Generalizations are possible for the non-zero offset (prestack) case (Fomel, 1996,1997).
A numerical implementation of velocity continuation process provides an efficient method of scanning the velocity dimension in the search of an optimally focused image. The first implementations (Fomel, 1996; Li, 1986) used an analogy with Claerbout's 15-degree depth extrapolation equation to construct a finite-difference scheme with an implicit unconditionally stable advancement in velocity. Fomel and Claerbout (1997) presented an efficient three-dimensional generalization, applying the helix transform (Claerbout, 1997).
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Figure 1. Impulse responses (Green's functions) of velocity continuation, computed by a second-order finite-difference method. The left plots corresponds to continuation to a larger velocity ( km/sec); the right plot, smaller velocity, ( km/sec). |
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A low-order finite-difference method is probably the most efficient numerical approach to this method, requiring the least work per velocity step. However, its accuracy is not optimal because of the well-known numerical dispersion effect. Figure 1 shows impulse responses of post-stack velocity continuation for three impulses, computed by the second-order finite-difference method (Fomel, 1996). As expected from the residual migration theory (Rothman et al., 1985), continuation to a higher velocity (left plot) corresponds to migration with a residual velocity, and its impulse responses have an elliptical shape. Continuation to a smaller velocity (right plot in Figure 1) corresponds to demigration (modeling), and its impulse responses have a hyperbolic shape. The dispersion artifacts are clearly visible in the figure.
In this paper, I explore the possibility of implementing a numerical velocity continuation by spectral methods. I adopted two different methods, comparable in efficiency with finite differences. The first method is a direct application of the Fast Fourier Transform (FFT) technique. The second method transforms the time grid to Chebyshev collocation points, which leads to an application of the Chebyshev- method (Lanczos, 1956; Boyd, 1989; Gottlieb and Orszag, 1977), combined with an unconditionally stable implicit advancement in velocity. Both methods employ a transformation of the grid from time to the squared time , which removes the dependence on from the coefficients of the velocity continuation equation. Additionally, the Fourier transform in the space (midpoint) variable takes care of the spatial dependencies. This transform is a major source of efficiency, because different wavenumber slices can be processed independently on a parallel computer before transforming them back to the physical space.
Velocity continuation by spectral methods |