Velocity continuation by spectral methods

Introduction

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 ($+1$ km/sec); the right plot, smaller velocity, ($-1$ km/sec).

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-$\tau$ 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 $t$ to the squared time $\sigma = t^2$, which removes the dependence on $t$ from the coefficients of the velocity continuation equation. Additionally, the Fourier transform in the space (midpoint) variable $x$ 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