A Bayesian approach to solving the non-invasive time domain reflectometry inverse problem

Abstract

Non Invasive Time Domain Reflectometry (TDR) may be used to estimate the volumetric moisture content, θv, with depth for a variety of sample materials. The forward physical model is couched in terms of a moment method where integration is performed over a descretized sample space to estimate the measured propagation time tp down a pair of parallel transmission lines. We show that the inverse solution to this, which recovers relative permittivity and thus θv, is greatly facilitated by a simplification of the system geometry via, 1) realistically modeling the prior density of the sample, 2) using this prior with the inherent system symmetry to reduce the number of required discretization cells, and 3) determining a physically meaningful reduction operator to allow a coarse discretization mesh to be used. The observational equation is expressed in the Bayesian paradigm with the most accurate and robust solution obtained using the Conditional Mean of the posterior distribution constructed via a Monte Carlo method. Results of simulation show that the method is capable of providing accurate estimates of the moisture density profile down to a depth of 100 mm with an error of less than 4 %. Further, the reduction in the number of descretized cells required to accurately estimate these profiles means that the inversion procedure is quick enough to enable the real time application of the equipment, a fundamental requirement in the development.