Three-dimensional numerical modelling and inversion of magnetometric resistivity data

We develop an algorithm to model the 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 modelling algorithm, the second-order partial differential equations for the scalar and vector potentials are discretized on a staggered-grid using the finite-volume technique. The resulting matrix equations are consequently solved using the bi-conjugate gradient stabilizing (BiCGSTAB), combined with symmetric successive over relaxation (SSOR) pre-conditioning. 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 is formulated in terms of relative conductivity with respect to a reference model. A depth weighting that counteracts the natural decay of the kernels is shown to be essential in typical problems. All minimizations are carried out with the Gauss-Newton 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. For surface MMR data, there are two forms of fundamental ambiguities for recovery of the conductivity. First, magnetic field data can determine electrical conductivity only to within a multiplicative constant. Thus for a body buried in a uniform host medium, we can find 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. The second ambiguity arises from the fact that surface MMR cannot distinguish between a homogeneous half-space and a 1-D conductive medium. For a 3-D body in a 1-D layered medium, it is still difficult to obtain information concerning the general background 1-D medium, if sources and receivers are at the surface. Overall, the surface MMR technique is useful so long as significant current flows through the body. This happens when the overburden is thin and moderately conductive (less than 10 times the conductivity of the underlying basement) and if the current sources are placed so there is good coupling with the body. Our inversion method is applied to synthetic examples and to a field data set. The low-resolution image obtained from using traditional MMR data, involving one source and one magnetic component, illustrates the need for acquiring data from multiple sources if 3-D structure of complex geometries are sought.