Three - dimensional quasi - linear electromagnetic inversion

One of the most challenging problems of electromagnetic (EM) geophysical methods is developing three-dimensional (3-D) EM inversion techniques. This problem is of utmost importance in practical applications because of the 3-D nature of the geological structures. The main difficulties in 3-D inversion are related to (1) limitations of 3-D forward modeling codes available and (2) ill-posedness of the inversion procedures in general. The multidimensional EM inversion techniques existing today can handle only simple models and typically are very time consuming. We developed a new approach to a rapid 3-D EM inversion. The forward scattering problem is solved using a new quasi linear (QL) approximation of the existing integral equation algorithms, developed for various sources of excitation. The QL approximation for forward modeling is based on the assumption that the anomalous field is linearly related to the normal field in the inhomogeneous domain by an electrical reflectivity tensor. We introduce also a modified material property tensor which is linearly proportional to the reflectivity tensor and the complex anomalous conductivity. The QL approximation generates a linear equation with respect to the modified material property tensor. The solution of this equation is called "a quasi-Born inversion". We apply the Tikhonov regularization for the stable solution of this problem. The next step of the inversion includes correction of the results of the quasi Born inversion: after determining a modified material property tensor, we use the electrical reflectivity tensor to evaluate the anomalous conductivity. Thus the developed inversion scheme reduces the original nonlinear inverse problem to a set of linear inverse problems, which is why we call this approach" a QL inversion". Synthetic examples (with and without random noise) of inversion demonstrate that the algorithm for inverting 3-D EM data is fast and stable. Introduction is due to the fact that inverse algorithms in general require more effort both in their theoretical develop­ During the past decade, considerable advances have ment as well as in computational resources. The lit­ been made in the forward modeling of electromag­ erature on EM inversion deals primarily with oneor netic (EM) fields [Tripp, 1990; Wannamaker, 1991; two-dimensional inversions. Three-dimensional (3­ Xiong, 1992; Xiong and Tripp, 1993; Druskin and D) EM inversion is only a recent adventure which Knizhnerman, 1994]. has attracted relatively few workers [Eaton, 1989; In contrast to the rapid developments in forward Madden and Mackie, 1989; Smith and Booker, 1991; modeling, progress in solving the EM inverse problem Xiong and Kirsh, 1992; Tripp and Hohmann, 1993; in multiple dimensions has been much slower. This TorresVerdin and Habashy, 1994J. However, the de­ mand for the development of three-dimensional in­ verse algorithms continues to be strong enough, as one could see, for example, at the Progress in Elec­ Copyright 1996 by the American Geophysical Union. tromagnetic Research Symposium (PIERS) in July Paper number 96RS00719. 1995 [TorresVerdin and Habashy, 1995b; Xie and 0048-6604/96/96RS-00719$11.00 Lee, 1995; Newman and Alumbaugh, 1995J.