Pareto-optimal solutions of the inverse gravimetric problem in the class of three-dimensional contact surfaces

Abstract

In geophysical inverse problems, there are two approaches to data inversion. The first is the search for a number of unknowns by minimizing the residual function. The second is through probabilistic modeling of the posteriori of the probability density function in the framework of the Bayesian interpretation of the inverse problem. In most cases, the data—model relationship is non-linear, and the corresponding minimization or modeling becomes difficult due to the multimodality of the residual function.

This article discusses an approach related to improbability methods for solving inverse problems of geophysics. Its essence consists in direct modeling of a parametric space with a further search for Pareto-optimal solutions based on a priori information. A priori information is formalized through fuzzy sets. The model example demonstrates the use of the improbable direct search and the gradient method of speedy descent in solving the nonlinear gravimetric inverse problem in the class of three-dimensional contact surfaces, and also evaluates the effectiveness of both methods.

An analysis of the performed tests shows that if there is sufficient a priori information, both methods give a completely unambiguous accurate result. The search for Pareto-optimal solutions can have a faster convergence compared to the gradient descent method, although it is determined by many factors — the number of points of the initial population, the threshold value ε, and the required level of data correspondence. Also, the algorithm is resistant to falling into local minima, since it uniformly explores the parametric space.

The algorithm allows us to obtain completely satisfactory solutions already at the stage of searching for the initial Pareto set. This is a consequence of selective modeling under the control of a priori information. A subsequent direct search in the vicinity of the Pareto-optimal points leads to a significant decrease in the residual function and to the deviation of some local minima.

In conditions of a lack of a priori information, a set of Pareto-optimal solutions can serve as a basis for further extraction of useful data on anomalous sources using other geophysical interpretation methods.

We also note that the described approach to solving the inverse problem may be of interest in solving a wide range of other optimization geophysical problems.