Inversion and Forward Modelling of EM Induction in Folded Sheet Conductors: Theory and Practise

Abstract

Application of Integral Equation Method to calculate the electromagnetic induction in multiply folded sheet conductors (many folds) is simplified by replacing the conductor with trial source currents (two dimensional polynomials) of unknown amplitude. Using the Galerkin method to solve the integral equation reduces the problem to inverting for the amplitudes of the current basis (trial) functions. This results in calculation of two matrices, one the resistance matrix and only a function of the sheets dimension and conductivity, the inductance matrix related to the self and mutual inductance of the trial currents, and only function of sheet?s geometry, and a vector describing the interaction of the primary magnetic field with each trial function. In comparison to the solution for a flat (not folded) sheet conductor, the folded conductor solution involves changes to the inductance matrix. Using these solutions, computing the EM induction (forward model) requires less than one second of CPU time using current computing units. Including this forward model solution in an inversion scheme to produce parameters of multiply folded and plunging sheet conductors is easy to apply and results is inversion solutions requiring (typically) less that one minute of CPU time using a 2 GHz processor. This is expected to be an orders of magnitude improvement on any inversion scheme using for example smooth model voxel (cells) or finite element inversion algorithms. By using approximate solutions to show that at appropriate sampled times or frequencies, EM response of multiply folded sheet conductors in a layered medium, could be largely controlled by the changes of the primary magnetic field at the conductor, similar quick forward models and inversion algorithms can be applied to sheet conductors in conductive layered earth. A number of forward models and practical inversions of field data are used to demonstrate the effectiveness of the developed forward modelling and inversion algorithms.