A Variational Bayesian Approach to Self-Tuning Robust Adjustment for Joint Inversion of Nonlinear Volcano Source Model with t-Distributed Random Errors

Abstract

Variance component estimation (VCE), herein called joint inversion, is a widely used approach to weigh the contributions of different data sets. Traditionally, the random errors of observations in VCE are modeled as Gaussian. However, in many geodetic measurements and sensor technologies, the observation data are non-Gaussian; therefore, the joint inversion with a more general heavy-tailed error model is preferred. Another issue is that the VCE deduced from the existing approaches may be not an interior solution, which means that the estimates may lie outside of the parameter space. Although there are some works on VCE in the Gaussian error model with equality or inequality constraints to mitigate this effect, to the best of our knowledge, there does not exist any work addressing the interior solutions of variance components for the heavy-tailed error model. In this article, we consider these issues for the first time and describe the behavior of multiple data sets using the joint functional model, which allows for the nonlinear modeling through nonlinear (differentiable) observation functions, where the random errors are modeled as Student’s t-distributed. To address the estimation problem, an iteratively reweighted least squares (LS) approach to self-tuning robust estimation of joint functional model parameters, the variance components, and the degree of freedom (df) of the Student’s t-distribution is derived based on a variational generalized expectation-maximization (GEM) algorithm. The proposed algorithm is computationally cheap and easy to implement. The performance of the algorithm is evaluated by means of Monte Carlo simulations of the joint volcano source model. Furthermore, the suitability of the research model and the proposed variational GEM algorithm is investigated within a numerical experiment involving the multisource modeling and adjustment of real data sets of the 2015 Calbuco volcano eruption.