Data kit inversion and uncertainty analysis

Abstract

Linear and nonlinear discrete inverse problems are by genesis ill-posed, that is, several solutions exist compatible with the prior information and fit the observed data within the same error bounds (equivalent models). This fact and the presence of noise and modelling errors cause their solution to be uncertain. Uncertainty analysis and solution appraisal is a crucial step in inverse modelling, and it is typically accomplished via random sampling methodologies and Bayesian approaches (provided a good knowledge of the probabilistic distributions of the model and data spaces is at disposal) or via exploratory global optimization algorithms that perform an approximate posterior sampling of the nonlinear equivalence region. This paper demonstrates that a simple method to sample the uncertainty space in linear and nonlinear problems consists in performing different inversions of random data bags. This novel methodology is called data kit inversion and it is inspired in the bootstrapping technique in statistical inference, switching in this case to the existence of other equivalent models, instead of using in estimating the properties of a given estimator. We present its application to different synthetic linear and nonlinear inverse problems and also to the inversion of a real dataset of gravimetric inversion in the Atacama Desert and also to 1D seismic and AVO inversion. We show that using partial information (random data bags) it is possible to sample different models of the equivalence region (uncertainty space) in order to explore the existence of different plausible geophysical scenarios according to the geophysical model that it is adopted. This algorithm has a general character and can be used in different fields of science and technology.