Semi-analytic reference solutions for 3D controlled-source electromagnetics: a conductive sphere beneath Gaussian topography in a layered earth

Klaus Lippert 

Three-dimensional forward codes for controlled-source and transient electromagnetics are verified mostly by comparison with one another when the model contains topography and a compact body in a layered host. We present terrasphere_csem, an open-source semi-analytic reference for this coupled problem. A Gaussian hill or depression forms the surface of a horizontally layered earth, and a conductive, optionally magnetically permeable sphere lies in any layer below. Grounded wires, grounded electric dipoles and current loops on the flat ground serve as sources; all components of the electric and magnetic fields are computed on the terrain, in the frequency domain and as transients.

The formulation combines plane-wave representations of the layered earth with vector spherical-wave expansions of arbitrary degree and order for the sphere, connected through identities from scattering theory. Topography enters through an expansion in the terrain height formulated entirely in the magnetic field: at each order the problem reduces to a flat-surface layered-earth solution with a low-rank correction for the sphere.

Particular emphasis is placed on independent verification and reproducibility. Each part is tested against a solution it was not derived from: closed-form limits, an independent direct-current solution, a one-dimensional code and electromagnetic reciprocity. Coupled terrain–sphere responses are compared with a three-dimensional finite-element code in the frequency domain; the transform to transients is validated separately. Numerical examples quantify the non-additive interaction of topography and the buried sphere.

For terrain of moderate slope, the framework provides an independently verified, reproducible reference for assessing the accuracy and convergence of three-dimensional CSEM forward solvers in models that combine topography, layering and a compact conductive target, and of transient solvers for such targets in a layered earth without topography. The software is available under the MIT licence.