2.5-D Electrical Resistivity Forward Modelling with Undulating Topography using a Modified Half-Space Analytical Solution

TL;DR

This paper introduces an improved 2.5-D electrical resistivity forward modelling method that accurately accounts for undulating topography, including sharp corners, by deriving a new analytical primary potential for V-shaped wedges.

Contribution

The study develops a novel analytical primary potential for V-shaped wedges, enhancing accuracy in resistivity modelling over complex terrains without fine mesh refinement.

Findings

Achieves errors below 0.1% on various topographies

Remains accurate with coarse finite element meshes

Valid for sharply varying surfaces and high curvature regions

Abstract

Field measurements for direct current (DC) resistivity imaging, used for subsurface profiling, are frequently conducted over undulating terrain. Accurately incorporating such topographic variations in its forward modelling is essential for reliable inversion and interpretation. Singularity removal techniques provide a computationally efficient framework by analytically representing the singular component of the electric potential. Existing secondary potential formulations use the analytical solution for a flat homogeneous half space, but this assumption is realistic only when the source lies on a locally smooth, flat planar surface. In practice, natural topography often contains sharp corners or regions of high curvature, and additional slope discontinuities arise from linear finite element discretization. These conditions invalidate the flat-surface analytical primary field and lead to substantial modelling errors. These errors originate from a fundamental geometric mismatch between the flat half-space analytical primary field and the true solid angle subtended by the topography at the source. This study presents an improved singularity removal strategy for 2.5-D forward modelling by deriving a new analytical primary potential for a V-shaped wedge. The formulation remains valid for sharply varying surfaces and accurately captures the singular behaviour without requiring geometric smoothing or excessive mesh refinement. By embedding the correct geometric singularity into the primary field, the proposed formulation remains consistent with both the discretized surface geometry and the physical boundary conditions. Numerical experiments on flat, V-shaped trench, and sinusoidal hill-valley models reveal that the proposed method consistently achieves errors below 0.1 per cent, even when using coarse linear finite element meshes.