Joint inversion for correlated models in linear inverse problems
Creators: name: Liu, Yidong
Publisher: name: University of British Columbia
Year: 2009

description: Inverse problems arise in many fields. They are usually ill-posed since they often violate one or more of Hadmard's three conditions for well-posedness: existence, uniqueness and stability. In this thesis, we propose a new method for computing approximate solutions in certain linear inverse problems. We consider linear inverse problems based on integral equations of the first kind. Analysis of Picard's condition reveals that such equations may lead to ill-posed problems which may have no solution satisfying the observed data exactly and stably, but may have infinitely many solutions satisfying the data approximately. To get a unique and stable solution to this kind of inverse problem, we use Tikhonov's Regularization Method. To obtain the best possible approximation to the true model, we should use any and all available information regarding the true model, although we can not expect to get sufficient data. For example, it is standard practice to use the positivity of the model in inverting magnetic and IP data, and to use special weighting functions in solving magnetic problems. The key feature of the present work is a method that exploits the correlation between different model parameters in inverting the geophysical data. To keep different parameters in suitable confidence regions, a new methodology, Combined Inversion, is developed. In combined inversion, different kinds of data are inverted simultaneously. The objective functional imposing the correlation requirement may be neither convex nor quadratic, so corresponding algorithm and code are developed. When the objective functional is not quadratic, we use an iterative method to solve it and approximate the functional with its second order Taylor approximation in each iteration. When the objective functional is not convex, it may have more than one local minimum. To get the minimum which well approximates the true model we should begin with a good initial model. In our case we produce the initial model by solving the combined problem with no correlation requirement. We introduce our method in the context of two practical geophysical inverse problems: the magnetic problem and the Induced Polarization (IP) problem. As we expect, regularization smooths the inverted models, so some model characteristics are lost in the recovered models. Our numerical examples confirm the smoothing effects of the regularization operators. Since magnetic susceptibility and chargeability are negatively correlated, we introduce a nonquadratic, nonconvex "correlation function", whose sub-level sets define confidence regions for the vector of susceptibility and chargeability. Then we require our recovered models to be in the confidence region. The recovered models from combined inversion method are significantly better than those from independent inversion. This method should be useful in practical prospecting when several kinds of data are available and there is some correlation among the parameters. This is the case in mining industry where several kinds of geophysical data are usually measured at the same time and the different parameters producing the data are known to be correlated. If we approximate the correlation with, a reasonable functional, we may reconstruct models satisfying the corresponding correlation.