Application of probabilistic approach to the solution of inverse problems in nondestructive testing and engineering geophysics

Interpretation of geophysical data often requires the solution of an inverse problem or simply, inversion. Inverse problem is a problem where the observed data is used to infer the characteristics of the system under investigation. There are two general approaches to the solution of inverse problems, deterministic and probabilistic approaches. Traditionally, in engineering geophysics, inversion is carried out using a deterministic approach, where a single set of results is identified as the interpretation outcome. In complex inverse problems, the deterministic solution process is often guided by an interpreter, who uses his information, experience, or judgment to guide the process. The deterministic approach to the solution of inverse problems implicitly assumes that the uncertainties in data and quantitative models are negligible. However, this assumption is not valid in many applications and, consequently, obtaining a single set of results does not provide a complete picture in terms of quantifying the effects of data and/or theoretical uncertainties on the obtained solution. In this paper, a general probabilistic approach to the solution of inverse problems is introduced, which offers the framework required to obtain uncertainty measures and to include some a priori information in the solution process. A technique for the evaluation of the probabilistic solution using Monte Carlo Markov Chains (MCMC) with Neighborhood Algorithm (NA) approximation is introduced and explained. Finally, the application of the presented approach in the health monitoring of transportation infrastructure using non-destructive testing (NDT) is illustrated.