Introduction

Analyses of finite-difference migration errors of the approximations to the semi-circular dispersion relation by various one-way wave equations have been given by Claerbout (1976, 1985). However, the effects caused by sampling in time, space, and depth and the interaction between them have been largely ignored. Lynn (1976) performed an analysis of group velocity errors in 15- and 45-degree migration that included the effect of trace spacing but not that of depth step. Beaumont et al. (1987) presented a quantitative analysis of the phase errors of finite-difference migration. A systematic numerical analysis, however, is absent in the literature. This paper presents a numerical analysis of the constant velocity 15-degree space-frequency ($\omega-x$) time migration algorithm as a useful step toward a systematic numerical analysis of finite-difference migration algorithms. The numerical analysis procedures, I presented, can be extended to other finite-difference migration algorithms that are implemented in space-frequency ($\omega-x$) and space-time (t-x) domain. I agree with Nichols (1992) that by optimizing the numerical parameters used in $\omega-x$ downward continuation, we can use the 15-degree scheme to migrate higher angles accurately. The following derivations largely follow Claerbout (1985) in geophysics, and Moin (1992) in numerical analysis.

This paper first explains how the parabolic approximation of the exact one-way wave equation causes undermigration. It then shows how the numerical method based on the eigenvalue structure of the diffraction partial differential equation and the Crank-Nicolson method causes overmigration. Finally, it derives the modified wave equation that the numerical solution actually satisfies for the diffraction equation.