Three-dimensional numerical modelling and inversion of magnetometric resistivity data

We develop an algorithm to model magnetometric resistivity (MMR) response over an arbitrary 3-D conductivity structure and a method for inverting surface MMR data to recover a 3-D distribution of conductivity contrast. In the forward modeling algorithm, the second-order partial differential equations for the scalar and vector potentials are discretized on a staggered gird using the finitevolume technique. The resultant matrix equations are consequently solved using the bi-conjugate gradient stablizing (BiCGSTAB), combined with the symmetric successive over relaxation (SSOR) preconditioning. In the inversion method, we discretize the 3-D model into a large number of rectangular cells of constant conductivity, and the final solution is obtained by minimizing a global objective function composed of the model objective function and data misfit. Since 1-D conductivity variations are an annihilator for surface MMR data, the model objective function uses relative conductivity with respect to a reference model. A depth weighting that counteracts the natural decay of the kernels is incorporated. All minimizations are carried out with the GaussNewton algorithm and model perturbations at each iteration are obtained by a conjugate gradient least squares method (CGLS), in which only the sensitivity matrix and its transpose multiplying a vector are required. Through synthetic model studies, we have shown that there are two forms of fundamental ambiguities for recovery of conductivity. For a body buried in a uniform host medium, we can determine only the relative conductivity contrast, not the absolute values. The choice of a constant reference model has no effect on the reconstruction of the relative conductivity. For a 3-D body in a 1-D layered medium, there is additional ambiguity in determining conductivities of the 1-D medium and the target, if receivers are at the surface. A favorable scenario is when the overburden is thin and moderately conductive (less than 10 times the conductivity of underlying basement). Multi-sources and multi-component data will greatly improve the inversion result. We demonstrate our inversion method with tests on both synthetic examples and on a field data set.