Three-Dimensional Quasi-linear Electromagnetic Modeling and Inversion
The quasi-linear (QL) approximation replaces the (unknown) total field in the integral equation of electromagnetic (EM) scattering with a linear transformation of the primary field. This transformation involves the product of the primary field with a reflectivity tensor, which is assumed to vary slowly inside inhomogeneous regions and therefore can be determined numerically on a coarse grid by a simple optimization. The QL approximation predicts EM responses accurately over a wide range of frequencies for conductivity contrasts of more than 100 to 1 between the scatterer and the background medium. It also provides a fast-forward model for 3-D EM inversion. The inversion equation is linear with respect to a modified material property tensor, which is the product of the reflectivity tensor and the anomalous conductivity. We call the (regularized) solution of this equation a quasi-Born inversion. The material property tensor (obtained by inversion of the data) then is used to estimate the reflectivity tensor inside the inhomogeneous region and, in turn, the anomalous conductivity. Solution of the nonlinear inverse problem thus proceeds through a set of linear equations. In practice, we accomplish this inversion through gradient minimization of a cost function that measures the error in the equations and includes a regularization term. We use synthetic experiments with plane-wave and controlled sources to demonstrate the accuracy and speed of the method.