Approximate depth averages of electrical conductivity from surface magnetotelluric data

This paper presents a simple non-linear method of magnetotelluric inversion that accounts for the computation of depth averages of the electrical conductivity profile of the Earth. The method is not exact but it still preserves the non-linear character of the magnetotelluric inverse problem. The basic formula for the averages is derived from the well-known conductance equation, but instead of following the tradition of solving directly for conductivity, a solution is sought in terras of spatial averages of the conductivity distribution. Formulas for the variance and the resolution are then readily derived. In terms of Backus-Gilbert theory for linear appraisal, it is possible to inspect the classical trade-off curves between variance and resolution, but instead of resorting to linearized iterative methods the curves can be computed analytically. The stability of the averages naturally depends on their variance but this can be controlled at will. In general, the better the resolution the worse the variance. For the case of optimal resolution and worst variance, the formula for the averages reduces to the well-known Niblett-Bostick transformation. This explains why the transformation is unstable for noisy data. In this respect, the computation of averages leads naturally to a stable version of the Niblett-Bostick transformation. The performance of the method is illustrated with numerical experiments and applications to field data. These validate the formula as an approximate but useful tool for making inferences about the deep conductivity profile of the Earth, using no information or assumption other than the surface geophysical measurements.