Estimation of the inverse acoustic transmission operator of a heterogeneous region directly from its reflection operator

Abstract

A method is derived for estimating the inverse acoustic transmission operator of a heterogeneous subsurface region directly from its reflection operator (without determining the subsurface model itself). As such, it is an alternative way of formulating the seismic near-surface inverse problem and is an extension of widely used processing techniques for removing near-surface effects from seismic data (like gapped-deconvolution methods or methods for static corrections). The method is deterministic and based on two general concepts: the conservation of acoustic energy and an auxiliary relation derived from causality arguments. The different steps are illustrated by considering the case of a depth-dependent near-subsurface model for which the reflection response due to an impulsive line source is known. It is shown that a meaningful estimate is obtained for the inverse transmission operator, the most important step in the estimation procedure turns out to be the solution of a linear set of equations with a Levinson recursion scheme. In this way, near-surface effects can be removed without first having to estimate the near-surface model itself (this estimation would be a nonlinear, and in many cases, ill-posed problem). The example presented serves as an illustration and a 'proof of principle'. In order to fully assess the method, its sensitivity to imperfect knowledge of the source wavelet, to noise and to imperfect knowledge of the reflection operator has to be investigated.