Electromagnetic Inversion

There are three famous uniqueness theorems for the electromagnetic inverse problem (Berdichevsky and Zhdanov, 1984) ... 

Zhdanov, M. S. and S. Fang, 1996, 3-D quasi-lincar electromagnetic inversion Radio Science, 3, No. 4, 741-754. 

In this Chapter I will introduce the basic equations governing the electromagnetic field in inhomogeneous conductive media, and review the basic physical laws important in developing electromagnetic inverse theory. 

We shall use these formulae (9.34) and (9.36) in electromagnetic inversion. 9.1.5 Integral equation method in three dimensions... 

Formulae (9.59) and (9.60) play an important role in electromagnetic inversion. 9.1.7 Frechet derivative calculation using the differential method We now present another way of calculating the Frechet derivative using the differential method, proposed by McGillivray et al., 1994. 

The last formulae serve as a basis for the method of quasi-analytical electromagnetic inversion, which I will present in the next Chapter. 

In this chapter we will outline the basic principles of electromagnetic inversion and imaging based on integral representations of the electromagnetic field developed in Chapter 9. 

The most common approach to the solution of nonlinear electromagnetic inverse problems is based on linearization of the forward modeling operator. This approach has found wide practical application because of the ease of its implementation, the accessibility of software for linear inversion, and the speed of numerical calculations. Linearization uses the family of linear and nonlinear approximations based on the Born method, described in Chapter 9. 

The most common approach to the solution of the electromagnetic inverse problem is based on linearization of the integral equations (10.2) and (10.3) using a Born approximation ... 

Quasi analytical approximations (9.90) and (9.91) provide another tool for fast and accurate electromagnetic inversion. This approach leads to a construction of the quasi-analytical (QA) expressions for the Prechet derivative operator of a forward problem, which simplifies dramatically the forward EM modeling and inversion for inhomogeneous geoelectrical structures. ... 

Zhdanov, M, S., Fang, S., and G. Hursan, 2000, Electromagnetic inversion using quasi-linear approximation Geophysics, 65, No. 5, 1501-1513. 

Formulation of the electromagnetic inverse problem as a minimization of the energy flow functional... 

We have learned in the previous chapters of this book, that an electromagnetic inverse problem is ill-posed. Therefore, direct minimization of the energy flow functional (11.13) could lead to an unstable solution. In order to generate a regularized solution, we have to minimize the parametric functional ... 

The general electromagnetic inverse problem can be formulated as follows. We are given the observed total electromagnetic field on the surface of the earth and... 

In this section we introduce first the migrated anomalous electromagnetic field and show how it can be calculated from the anomalous field. In the following sections we will demonstrate the connections between the migrated electromagnetic fields and the solution of the electromagnetic inverse problem. 

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