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.