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

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: (1) determine the subset of the Tikhonov regularization solutions which satisfy the desired tolerance of data fit, (2) choose the solution which best fits the data in the subset. The solution procedure is actually an adaptive Tikhonov regularization solution. The subspace solution is mathematically no more than a dimension-decreased transform of model parameterization in the Tikhonov regularization solution or the smoothest model solution. The crucial step is to choose the basis vectors which span the whole model space in the parameterization transform. A simple form of the transform is developed which yields greater flexibility for geophysical inverse applications.