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Introduction

One-way wavefield continuation methods correctly account for traveltimes, but the amplitude and phase of the images they produce can still be improved. Zhang et al. present theoretical formulations of amplitude-improving corrections for shot-profile migration. Vlad et al. (2003) show concrete ways of implementing Zhang's theory for both finite-difference and mixed-domain extrapolators, with applications to constant-velocity, laterally smooth and heterogeneous cases.

The above-described corrections consist of two parts: one that is applied at the z=0 boundary, and one that is applied in the propagation operator. The boundary condition correction depends only on the velocity at the surface, while the propagation correction takes the entire interval velocity model into account and is directly proportional to its vertical gradient, vz, vanishing where this becomes zero. As a result, the effect of the boundary condition correction can be tested in isolation, using constant velocity models. Figure 1 in Vlad et al. (2003) shows that the Zhang boundary condition correction improves the phase and brings the amplitudes very close to the ones computed analytically, especially in the case of the mixed-domain implementation. Valenciano et al. (2004) show the effect of applying it to propagation through the Marmousi model. They also present a more intuitive interpretation of the boundary condition correction, and an alternative method for computing it.

Vlad et al. (2003) show in their Figure 2 that the propagation operator correction is equivalent to the WKBJ one in a v(z) medium. They also apply it to the v(x,z) model presented in Figure 3 of their paper. However, the results (their Figure 4) did not show a visible effect of the correction. In the first section that follows, we explain why that was the case for that very particular velocity model, and we quantitatively show that the propagation operator correction does have visible effects. In the last section of this paper, we discuss the application of the Zhang corrections to the linearized downward continuation operator used in wave equation migration velocity analysis.


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Stanford Exploration Project
5/23/2004