Abstract

Stable, explicit depth-extrapolation filters can be used to propagate plane waves corresponding to the qP and qSV (quasi-P and quasi-SV propagation) modes for transversely isotropic (TI) media. Here, I discuss and compare results of two different methods for obtaining the filters for TI media with a vertical axis of symmetry (VTI). The first, a modified Taylor series method, is used to calculate the N-coefficients of a finite-length filter such that the Taylor expansion around vertical propagation matches the spatial Fourier transform of the downward-continuation operator for VTI media. Second, a least-squares method is used to calculate the filter coefficients such that the amplitude and phase departures from the ideal response of the downward-continuation operator for VTI media are minimized over a range of frequencies and propagation angles. In both methods, the amplitude response of the filter is forced to be less than unity in the evanescent region to achieve stability.In general, as exemplified in all the cases studied here, the constrained least-squares method produced filters with accurate wavefield extrapolation for a wider range of propagation angles than that obtained for the modified Taylor series method. In both methods, the maximum angle that can be accurately propagated depends on the ratio of frequency to vertical phase velocity (f/V v ), and on the length of the filter. However, for a fixed filter length and for a given ratio f/V v , the maximum angle propagated with accuracy depends on the elastic constants of the medium. The accuracy of the filters degrades as the degree of anisotropy becomes more extreme.

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