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Solutions, algorithms and inter-relations for local minimization search geophysical inversion

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This paper presents in a general form the most popular local minimization search solutions for geophysical inverse problems—the Tikhonov regularization solutions, the smoothest model solutions and the subspace solutions, from which the inter-relationships between these solutions are revealed. For the Tikhonov regularization solution, a variety of forms exist—the general iterative formula, the iterative linearized scheme, the Levenberg–Marquardt version, the conjugate gradient solver (CGS) and local-search quadratic approximation CGS. It is shown here that the first three solutions are equivalent and are just a specified form of the gradient solution. The local-search quadratic approximation CGS is shown to be a more general form from which these three solutions emerge. The smoothest model solution (Occam's inversion) is a subset of the Tikhonov regularization solutions, for which a practical algorithm generally comprises two parts...

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