Asymptotic error analysis of constant‐velocity viscoacoustic migration

Abstract

The deterioration of prestack migration images caused by an erroneous velocity or dissipation model can be represented by a space‐variant filter. We have derived an approximate expression of this spatial filter for the case of a plane horizontal reflector between two homogeneous half‐spaces, using analytical expressions for the Green’s functions and a stationary phase approximation of the Kirchhoff integral. We have used complex, frequency‐dependent velocities to represent a general homogeneous viscoacoustic medium. Hence, the filter can describe the effects of errors in phase velocity and absorption on the image of the reflector for different situations. A pure phase velocity error deforms the reflector, affects the value of the recovered reflectivity, and creates a spurious amplitude versus offset (AVO) effect. Neglecting dissipation destroys the phase of the image, decreases the resolution, and causes an underestimation of the reflectivity. To have a true‐amplitude, good‐resolution migration with Claerbout’s crosscorrelation imaging principle, the forward extrapolation of the downgoing wavefield should be done with the complex conjugate of the velocity in the viscoacoustic medium, and the backward extrapolation of the upgoing wavefield should be done with the complex velocity. In both cases, this amplifies the wavefields to compensate for the effects of attenuation on wave propagation.