3D inversion of magnetic induced polarization data

The magnetic induced polarization (MIP) is an exploration technique used to obtain information relating to the induced polarization characteristics of the subsurface through measurements of the primary magnetic field associated with steady-state current flow in the earth. According to Seigel, the polarization magnetic field due to polarization current can be expressed as a sum of the products of chargeability and the derivative of primary magnetic field, due to ohmic current, with respect to the logarithmic conductivity (or sensitivity). The magnetic field and the sensitivity matrix can be computed by subsequently solving Poisson's equation and a magnetostatic problem in terms of potentials using a finite-volume algorithm. The MIP response is a function of chargeability difference (eta-eta(0)) and relative conductivity (sigma/sigma(0)), where eta(0) and sigma(0) are constants. When solving the inverse problem we need to impose positivity of the solution but the fact that MIP responses depend only upon the difference in chargeability, means we have options regarding how we set up the inversion. We can: (1) invert for eta without constraints and add a constant to the final result, (2) invert for eta while imposing positivity, or (3) work with log eta. We compare all three methods here. Our inversion problem is formulated as an optimisation problem where the objective function of the model is minimised subject to the constraints that the model adequately reproduces the data. We use a Gauss-Newton method to obtain the model perturbation at each iteration. The system of equations is solved using conjugate gradient least squares method. In order to make the inversion produce depth or distance information, a depth weighting or sensitivity-based weighting is required. Through synthetic model studies, we have shown that the conductivity ratio between a target and its host has a large effect on the MIP response. Ratios greater than two orders of magnitude difference will eventually make the MIP response undetectable. However, if the ratio is in the range of 0.1 to 10, the effect on the recovered chargeability is limited. The inversion algorithm is demonstrated by inverting the data set from Binduli, Australia.