Series approximation of the Kirchhoff method for wavefield computation on a rough subsurface

Abstract

ABSTRACT Computation of the wavefield due to reflection from an irregular surface is carried out for subsurfaces with large radii of curvature. The Kirchhoff approximation is proved to be sufficiently accurate provided that the acoustic wavelength is sufficiently small with respect to the asperities of the rough surface. For cases where the irregular surface does not fulfil this condition, a series solution is proposed. The first term of this series appears to be the result obtained by conventional Kirchhoff approximation. The series, initially developed in the space–wavenumber domain by Meecham, is transformed into the space–time domain, and the general expression for the series is obtained by calculation of the normal derivative of the field function. The series solution, restricted to the first two terms, is illustrated by application to three synthetic examples. Applications show that the series approximation obtained by the Kirchhoff method contributes significantly to the modelling of narrow, steep and deep structures and consequently it appears that the second term in the series cannot be ignored in the computation of the wavefields arising from a rough surface.