Three‐dimensional modelling and inversion of dc resistivity data incorporating topography – II. Inversion

SUMMARY 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.