Summary
There is a class of inverse problems in applied geophysics that is well described by a parameter estimation approach. Such problems are often characterized by strong nonlinearity, multimodal objective functions, and parameter bounds that can be well-defined in some cases and unknown in others. We present a robust parametric inversion algorithm designed for such problems based on an adaptive quenched simulated annealing. The method combines successive refinement of the parameter search space and adaptive bound constraints to cover some the difficulties associated with multi-modality in the objective function and to improve the numerical efficiency. We illustrate the method using examples from ensemble fitting in potential-field methods, and joint parametric inversion of gravity and magnetic data with strong remanent magnetization.
Introduction
Much of the recent effort in applied geophysical inversion has focused on the generalized inversion to recover distributions of physical properties or the boundaries between geological units. There remains a significant need for parametric inversions that seek to recover a set of distinct parameters presenting either a set of mixed geometrical and physical properties describing a geological entity, or a set of parameters describing a system model. Examples of the former include the geometry and magnetization of the isolated magnetic source; and the examples of latter include the parameters in an ensemble representation of a radial power spectrum of potential-field data or the parameters in the Cole-Cole model of complex conductivity.
Such parametric inversions are typically nonlinear and the associated objective functions are defined by a data misfit that often have multiple local minima. Gradient-based approaches may have difficulties finding the solutions. Additionally, parameters have well defined bounds, but for many cases we may not know the bounds for the parameters corresponding to a given data set a priori. To overcome these difficulties, many authors have developed different approaches to solve these problems, such as genetic algorithm and simulated annealing We present a robust and efficient approach based on quenched simulated annealing (QSA) that relies only on efficient forward modeling and can adaptively "shrink" the bounds or search interval so the minimization can rapidly reach the convergence radius of the global minimum and complete the solution thereafter.
