UBC Theses and Dissertations
Approximate sensitivities for the multi-dimensional electromagnetic inverse problem Farquharson, Colin Glennie
Inversion of measurements from a geophysical electromagnetic survey to produce a two- or three-dimensional conductivity model of the Earth is computationally demanding. The classical approach is to linearise the inverse problem and iterate towards the solution. A typical, modern version of this approach is described in the initial part of this thesis, and applied to the one-dimensional inversion of time-domain electromagnetic data. One of the most time-consuming parts of a linearised, iterative inversion procedure is the generation of the Jacobian matrix of sensitivities. In this thesis, I have developed an approximate method for generating these sensitivities. The approximation is based on the adjoint-equation method in which the sensitivities are obtained by integrating, over an individual cell, the scalar product of an adjoint field (the adjoint Green’s function) with the electric field produced by the sources for the geophysical survey. Instead of calculating the adjoint field in the multi-dimensional conductivity model, an approximate adjoint field is computed in a homogeneous or layered halfspace, or using the Born approximation. This approximate adjoint field is significantly quicker to compute than the true adjoint field and leads to considerable reductions in the time required to generate the Jacobian matrix of sensitivities. The time-differences were found to be of one or two orders of magnitude for the small examples considered in this thesis, and this saving will further increase with the size of the problem. The approximate sensitivities were compared to the accurate sensitivities for two and three-dimensional conductivity models, and for both artificial sources and the plane wave source of magnetotellurics. In all the examples considered, the approximate sensi tivities appeared to be sufficiently accurate to allow an iterative inversion algorithm to converge to an acceptable model. This was emphasised by the successful inversion, using approximate sensitivities, of two sets of magnetotelluric data: a synthetic data-set and a sub-set of the COPROD2 data.
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