UBC Theses and Dissertations
Exploring magnetotelluric nonuniqueness using inverse scattering methods Whittall, Kenneth Patrick
I present two algorithms which solve the one-dimensional magnetotelluric (MT) problem of finding the electrical conductivity σ(z) as a function of depth in the earth. Together, these algorithms restrict and explore the nonuniqueness of the nonlinear MT inverse problem. They accept constraints which limit the space of acceptable conductivity models and they construct diverse classes of σ(z) in order to explore this space. To avoid pitfalls during interpretation, it is essential to investigate the extent of the nonuniqueness permitted by the MT data. Algorithm 1 is a two-stage process based on the inverse scattering theory of Weidelt. The first stage uses the MT frequency-domain data to construct an impulse response analogous to a deconvolved seismogram. Since this is a linear problem (a Laplace transform), numerous impulse responses may be generated by linear inverse techniques which handle data errors robustly. I minimize four norms of the impulse response in order to construct varied classes of limited structure models. Two least-squares norms minimize the energy in the impulse response or the energy in its derivative with respect to depth. Two least absolute value norms minimize the magnitudes of the response or its derivative. It is possible to use other norms. The different classes sample the range of acceptable models and the minimum structure criterion is unlikely to allow models with spurious features. The second stage of Algorithm 1 constructs the conductivity model from the impulse response using any of four Fredholm integral equations of the second kind. I evaluate the performance of each of the four mappings and recommend the Burridge and Gopinath-Sondhi formulations. I also evaluate three approximations to the second-stage equations. One of these is equivalent to the Born approximation which assumes the impulse response has negligible multiple reflections. The approximation that includes first-order multiple reflections is the most accurate and gives conductivity models similar to those given by the integral equations. Algorithm 2 solves an integral form of a nonlinear Riccati equation relating the measured frequency-domain data to a function of the conductivity. The iterative solution scheme sacrifices the efficiency of a direct inversion process such as Algorithm 1 for the advantages of incorporating localized conductivity constraints. The linear programming formulation readily accepts a wide variety of equality and inequality constraints on σ(z). I use these constraints in two ways to combat the nonuniqueness of this nonlinear inverse problem. First, I impose physical constraints derived from external sources to restrict the nonuniqueness and construct σ(z) models that are closer to reality. Second, I impose constraints specifically designed to estimate the extent ofthe nonuniqueness and explore the range of acceptable σ(z) profiles. The first technique enhances the reliability of an interpretation and the second assesses the plausibility of particular conductivity features. The convergence of Algorithm 2 is good because Algorithm 1 provides varied initial σ(z) which already fit the data well.
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