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Inversion and appraisal for the one-dimensional magnetotellurics problem

University of British Columbia

The method of magnetotellurics (MT) uses surface measurements of naturally-occurring electromagnetic fields to investigate the conductivity distribution within the Earth. In many interpretations it is adequate to represent the conductivity structure by a one-dimensional (1-D) model. Inferring information about this model from surface field measurements is a non-linear inverse problem. In this thesis, linearized construction and appraisal algorithms are developed for the 1-D MT inverse problem. To formulate a linearized approach, the forward operator is expanded in a generalized Taylor series and second-order terms are neglected. The resulting linear problem may be solved using techniques of linear inverse theory. Since higher-order terms are neglected, the linear problem is only approximate, and this process is repeated iteratively until an acceptable model is achieved. Linearized methods have the advantage that, with an appropriate transformation, a solution may be found which minimizes a particular functional of the model known as a model norm. By explicitly minimizing the model norm at each iteration, it is hypothesized that the final constructed model represents the global minimum of this functional; however, in practice, it is difficult to verify that a global (rather than local) minimum has been found. The linearization of the MT problem is considered in detail in this thesis by deriving complete expansions in terms of Fréchet differential series for several choices of response functional, and verifying that the responses are indeed Fréchet differentiable. The relative linearity of these responses is quantified by examining the ratio of non-linear to linear terms in order to determine the best choice for a linearized approach. In addition, the similitude equation for MT is considered as an alternative formulation to linearization and found to be inadequate in that it implicitly neglects first-order terms. Appropriate choices of the model norm allow linearized inversion algorithms to be formulated which minimize a measure of the model structure or of the deviation from a (known) base model. These inversions construct the minimum-structure and smallest-deviatoric model, respectively. In addition, minimizing I₂ model norms lead to smooth solutions which represent structure in terms of continuous gradients, whereas minimizing I₁ norms yield layered conductivity models with structural variations occurring discontinuously. These two formulations offer complementary representations of the Earth, and in practice, a complete interpretation should consider both. The algorithms developed here consider the model to be either conductivity or log conductivity, include an arbitrary weighting function in the model norm, and fit the data to a specified level of misfit: this provides considerable flexibility in constructing 1-D models from MT responses. Linearized inversions may also be formulated to construct extremal models which minimize or maximize localized conductivity averages of the model. These extremal models provide bounds for the average conductivity over the region of interest, and thus may be used to appraise model features. An efficient, robust appraisal algorithm has been developed using linear programming to extremize the conductivity averages. For optimal results, the extremal models must be geophysically reasonable, and bounding the total variation in order to limit unrealistic structure is an important constraint. Since the extremal models are constructed via linearized inversion, the possibility always exists that the computed bounds represent local rather than global extrema. In order to corroborate the results, extremal models are also computed using simulated annealing optimization. Simulated annealing makes no approximations and is well known for its inherent ability to avoid unfavourable local minima. Although the method is considerably slower than linearized analysis, it represents a general and interesting new appraisal technique. The construction and appraisal methods developed here are illustrated using synthetic test cases and MT field data collected as part of the LITHOPROBE project. In addition, the model construction techniques are used to analyze MT responses measured at a number of sites on Vancouver Island, Canada, to investigate the monitoring of local changes in conductivity as a precursor for earthquakes. MT responses measured at the same site over a period of four years are analyzed and indicate no significant changes in the conductivity (no earthquakes of magnitude greater than 3.0 occurred in this period). Conductivity profiles at a number of sites are also considered in an attempt to infer the regional structure. Finally, a method of correcting linearized inversions is developed. The corrections consist of successively approximating an analytic expression for the linearization error. The method would seem to represent a novel and practical approach that can significantly reduce the number of linearized iterations. In addition, a correspondence between the correction steps and iterations of the modified Newton's method for operators is established.

Text

2011-01-19

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http://hdl.handle.net/2429/30692

Geophysics

University of British Columbia

Graduate