3D electromagnetic inversion based on quasi-analytical approximation

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 (Zhdanov M S, Dmitriev V I, Fang S and Hursan G 1999 Geophysics at press). It is based on ideas similar to those developed by Habashy et al (Habashy T M, Groom R W and Spies B R 1993 J. Geophys. Res. 98 1759-75) for a localized nonlinear approximation, and by Zhdanov and Fang (Zhdanov M S and Fang S 1996a Geophysics 61 646-65) 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 Geophysics 64 874-87). 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.