Quasi-linear approximation and its applications in the solution of three-dimensional electromagnetic forward and inverse problems

A novel approach--quasi-linear (QL) approximation--in three-dimensional (3-D) electromagnetic (EM) forward modeling, and a fast 3-D EM inversion technique--QL inversion--are introduced in this dissertation.; The QL approximation for the EM field is based on the evaluation of the anomalous field E{dollar}sp{lcub}a{rcub}{dollar} by a linear transformation of the normal field: E{dollar}sp{lcub}a{rcub} = lambda{dollar}E{dollar}sp{lcub}n{rcub},{dollar} where {dollar}lambda{dollar} is called the electrical reflectivity tensor. The reflectivity tensor inside inhomogeneities can be approximated by a slowly varying function which can be determined numerically by a simple optimization technique. The new approximation gives an accurate estimate of the EM response for conductivity contrast of even more than one thousand to one and for a wide frequency range.; The accuracy of the EM response can he further improved by considering the higher order QL approximation. This approach can be considered as the natural generalization of the Born series. We use a modified Green's operator with the norm less than 1 to ensure the convergence of the QL series.; By introducing a modified material property tensor which is proportional to the reflectivity tensor of the QL approximation and the complex anomalous conductivity, we generate a linear equation with respect to the modified material property tensor. The solution of this equation is called a "quasi-Born inversion." The Tikhonov regularization and a physical constraint have been introduced to obtain a stable and reasonable solution to this problem. The next step of the QL inversion includes a 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. That is why we call this approach a "QL inversion." The inversion of synthetic 3-D data (with and without random noise) and real data indicates that this algorithm is fast and stable.