J. L. G. Pallero

Particle Swarm Optimization and Uncertainty Assessment in Inverse Problems

Most inverse problems in the industry (and particularly in geophysical exploration) are highly underdetermined because the number of model parameters too high to achieve accurate data predictions and because the sampling of the data space is scarce and incomplete; it is always affected by different kinds of noise. Additionally, the physics of the forward problem is a simplification of the reality. All these facts result in that the inverse problem solution is not unique; that is, there are different inverse solutions (called equivalent), compatible with the prior information that fits the observed data within similar error bounds. In the case of nonlinear inverse problems, these equivalent models are located in disconnected flat curvilinear valleys of the cost-function topography. The uncertainty analysis consists of obtaining a representation of this complex topography via different sampling methodologies. In this paper, we focus on the use of a particle swarm optimization (PSO) algorithm to sample the region of equivalence in nonlinear inverse problems. Although this methodology has a general purpose, we show its application for the uncertainty assessment of the solution of a geophysical problem concerning gravity inversion in sedimentary basins, showing that it is possible to efficiently perform this task in a sampling-while-optimizing mode. Particularly, we explain how to use and analyze the geophysical models sampled by exploratory PSO family members to infer different descriptors of nonlinear uncertainty.

The Effect of the Noise and the Regularization in Inverse Problems: Geophysical Implications

The solution of geophysical inverse problems has an intrinsic uncertainty that is mainly caused by noise in data, incomplete data sampling and simplified physics. This paper analyzes the roles of noise in data and that of the regularization for nonlinear inverse problems. We prove that noise deforms the topography of the cost function non-homogeneously, generally decreasing the regions of low misfits. As a result of this deformation, finding the global optimum by direct search methods becomes a more difficult task. Nevertheless, noise acts similarly to a regularization when local optimization methods are used. Tikhonov’s regularization transforms the linearized hyper-quadric of equivalence from an elliptical cylinder to a very oblong ellipsoid in the directions that originally spanned the kernel of the linearized forward operator in absence of regularization, and also deforms anisotropically the regions of equivalence. Prior models in the regularization term serves to inform the components of the solution that locally belongs to the kernel of the Jacobian. Unfortunately regularization does not cause the disappearance of the nonlinear equivalent models. Thus a full nonlinear uncertainty analysis is still needed.