3D electromagnetic inversion based on quasi-analytical approximation

Abstract

In this paper we address one of the most challenging problems of electromagnetic (EM) geophysical methods: three-dimensional (3D) inversion of EM data over inhomogeneous geological formations. The difficulties in the solution of this problem are two-fold. On the one hand, 3D EM forward modelling is an extremely complicated and time-consuming mathematical problem itself. On the other hand, the inversion is an unstable and ambiguous problem. To overcome these difficulties we suggest using, for forward modelling, the new quasi-analytical (QA) approximation developed recently by Zhdanov et al (ZhdanovMS, DmitrievVI, FangS and HursanG 1999 Geophysics at press). It is based on ideas similar to those developed by Habashy et al (HabashyTM, GroomRW and SpiesBR 1993 J. Geophys. Res.98 1759-75) for a localized nonlinear approximation, and by Zhdanov and Fang (ZhdanovMS and FangS 1996a Geophysics61 646-65) for a quasi-linear approximation. We assume that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through a scalar electrical reflectivity coefficient, which is a function of the background geoelectrical cross-section and the background EM field only. This approach leads to construction of the QA expressions for an anomalous EM field and for the Frechet derivative operator of a forward problem, which simplifies dramatically the forward modelling and inversion. To obtain a stable solution of a 3D inverse problem we apply the regularization method based on using a focusing stabilizing functional introduced by Portniaguine and Zhdanov (Portniaguine O and Zhdanov M S 1999 Geophysics64 874-87). This stabilizer helps generate a sharp and focused image of anomalous conductivity distribution. The inversion is based on the re-weighted regularized conjugate gradient method.