Abstract During the last century, electrical geophysics has been transformed from a simple resistivity method to a modern technology that uses complex data-acquisition systems and high-performance computers for enhanced data modeling and interpretation. Not only the methods and equipment have changed but also our ideas about the geoelectrical models used for interpretation have been modified tremendously. This paper describes the evolution of the conceptual and technical foundations of EM methods. It outlines a framework for further development, which should focus on multitransmitter and multireceiver surveys, analogous to seismic data-acquisition systems. Important potential topics of future research efforts are in the areas of multidimensional modeling and inversion, including a new approach to the formulation and understanding of EM fields based on flux and voltage representation, which corresponds well to geophysical experiments involving the measurement of voltage and flux of electric and magnetic fields.

Summary Parameter estimation in geoelectromagnetics aims to obtain the most important parameters of a well-defined conductivity model of the Earth. These parameters are features of typical geological structures, such as depth and size of conductive or resistive targets, angle of dike inclination and its length, and conductivity of anomalous bodies. We develop this approach through regularized nonlinear optimization. We use finite differences of forward computations and Broyden’s updating formula to compute sensitivities (Frechet or partial derivatives) for each parameter. To estimate the optimal step length, we apply line search, with a simple and fast parabolic correction. Our inversion also includes Tikhonov’s regularization procedure. We use our method to study measurements of the magnetic fields from a conductive body excited by a loop source at the surface. Keeping the depth of the body constant, we estimate the horizontal coordinates of the body from three components of the magnetic field measured in a borehole. These measurements accurately determine the direction to the conductive target.

Summary 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.