The singularity of the potential occurring at the source location is a key point of electrical resistivity forward modelling because it might lead to large numerical errors. To tackle this problem a classical method consists of splitting the total potential into a primary part containing the singularity and a secondary part. The primary potential is defined analytically for flat topography but requires numerical computation in the presence of topography. In that case, an accurate solution happens to be computationally expensive. For any geometry we propose to keep for the primary potential the analytic solution defined for homogeneous models and flat topography, and to modify accordingly the free surface boundary conditions for the secondary potential. The primary potential still contains the singularity and new free surface conditions ensure that the total potential still satisfies the Poisson equation. The modified singularity removal technique thus remains fully efficient even in the presence of topography, without additional numerical computation. The modified secondary potential in a homogeneous model is not null in the case of topography as it would be in the classical approach. We implement the approach with a Finite Difference method. We present potential distributions computed with this technique to illustrate its versatility.


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