We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process.
We use a Gauss–Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented.
We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today's standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data.
As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.