Imaging low-frequency and dc electromagnetic fields using a simple linear approximation

We consider that all types of electromagnetic measurements represent weighted averages of the subsurface electrical conductivity distribution, and that to each type of measurement there corresponds a different weighting function. We use this concept for the quantitative interpretation of dc resistivity, magnetometric resistivity, and low-frequency electric and magnetic measurements at low induction numbers. In all three cases the corresponding inverse problems are nonlinear because the weighting functions depend on the unknown conductivity distribution. We use linear approximations that adapt to the data and do not require reference resistivity values. The problem is formulated numerically as a solution of a system of linear equations. The unknown conductivity values are obtained by minimizing an objective function that includes the quadratic norm of the residuals as well as the spatial derivatives of the unknowns. We also apply constraints through the use of quadratic programming. The final product is the flattest model that is compatible with the data under the assumption of the given weighting functions. This approximate inversion or imaging technique produces reasonably good results for low and moderate conductivity contrasts. We present the results of inverting jointly and individually different data sets using synthetic and field data.