Numerical analysis of the 45‐degree finite‐difference equation for migration

Abstract

Migration is now most commonly performed by means of a finite‐difference solution of the wave equation in the space‐time domain (although alternative approaches such as f-k, Kirchhoff, finite‐difference in the space‐frequency domain have strong adherents). Claerbout’s derivation of the 15‐degree paraxial ray equation and its iteration to the 45‐degree equation are well documented. On the other hand, the transcription of the differential equation to a finite‐difference scheme has accreted with practical computing experience and is only mentioned piecemeal (when at all) in the literature. The full expression is reviewed here, as used in a typical production code. A numerical stability analysis of the von Neumann type is applied to the complete finite‐difference equation. It proves that the computer algorithm is stable, at least for the values of the computational parameters in normal use (the sensitivity to the values of these parameters is illustrated.) Thus, any perceived noisiness of migrated sections cannot be blamed on computational precision. The shortcomings are entirely caused by deficiencies in the analytic framework and the modeling.