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.

We present a semi‐analytical, unifying approach for modelling the electromagnetic response of 3‐D bodies excited by low‐frequency electric and magnetic sources. We write the electric and magnetic fields in terms of power series of angular frequency, and show that to obey Maxwell’s equations, the fields must be real when the exponent is even, and imaginary when it is odd. This leads to the result that the scattering equations for direct current fields and for fields proportional to frequency can both be explicitly formulated using a single, real dyadic Green’s function. Although the underground current flow in each case is due to different physical phenomena, the interaction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and electric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic derivation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the solution of a system of linear equations. The matrix of the system is real and does not depend on the type of source or receiver. We present modelling results for different arrays and apply the algorithm to the interpretation of field data. We assume the standard dipole–dipole resistivity array for the direct current case, and vertical and horizontal magnetic dipoles for induction measurements. In the case of magnetometric resistivity we introduce a moving array composed of an electric dipole and a directional magnetometer. The array has multiple separations for depth discrimination and can operate in two modes. The mode where the predominant current flow runs along the profile is called MMR‐TM. This mode is more sensitive to lateral variations in resistivity than its counterpart, MMR‐TE, where the mode of conduction is predominantly perpendicular to the profile.

This paper presents a theoretical yet practical study of electromagnetic (EM) soundings at low induction numbers for vertical and horizontal magnetic dipoles. The physical model is a heterogeneous half‐space with arbitrary vertical conductivity variations. The study comprises a novel approach for solving forward problems, analytical formulas for inversion, and a practical algorithm for recovering conductivity variations from field measurements. The basis of the theoretical approach is a series representation of the EM field in terms of ascending powers of frequency. At low induction numbers only two terms are required. When substituted into Maxwell's equations, one term in the series can be obtained in terms of the other. Furthermore, if the electrical conductivity varies only with depth, the imaginary part of the field can be obtained from its real part through a differential equation. The real part, which corresponds to zero frequency, plays the role of a distributed source for the frequency‐dependent imaginary part. In the case of vertical magnetic dipoles, the approach applies directly to the real and imaginary components of the magnetic field, while for horizontal dipoles one must use the Hertz potential, but the procedure is exactly the same. In each case this leads to a statement of the forward problem as the solution of a real differential equation. The solutions are integral expressions valid for arbitrary conductivity profiles. Assuming that these expressions represent integral equations for conductivity, analytical inverse formulas are derived for both vertical and horizontal dipoles. These formulas ensure a unique recovery of the conductivity profile under ideal conditions. An algorithm based on linear programming offers a variety of practical advantages for the inversion of field data. Numerical experiments and applications to field data illustrate the performance of the algorithm.

We expressed electromagnetic measurements at low induction numbers as spatial averages of the subsurface electrical conductivity distribution and developed an algorithm for the recovery of the latter in terms of the former. The basis of our approach is an integral equation whose averaging kernel is independent of the conductivity distribution. That is, the recovery of conductivity from the measurements leads to a linear inverse problem. Previous work in one and two dimensions demonstrated that using a kernel independent of conductivity leads to reasonably good results in quantitative interpretations. This study extended the approach to 3D models and to data taken along several profiles over a given area. The algorithm handles vertical and horizontal magnetic dipoles with multiple separations for appropriate depth discrimination. The approximation also handles issues like negative conductivity measurements, which commonly appear when crossing near-surface conductors. This happens particularly when using vertical magnetic dipoles; whose averaging kernel has significant negative weights in the space between the dipoles, something that does not happen for the horizontal dipoles. In general, the more complex the kernel, the more complicated the signature of any given anomaly. This makes qualitative interpretations of pseudosections somewhat difficult when dealing with more than one conductive or resistive body. The algorithm was validated using synthetic data for imaging data from horizontal or vertical coils or from a combination of them. Imaging of field data from a mine tailings site recovered a shallow 3D conductive anomaly associated with the tailings.

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