Method comparison for estimation of distributed parameters in permit

The use of reflectance spectroscopy is a current area of study for the non-invasive evaluation of complex materials such as ceramic matrix composites. In order to model the reflectance, one must specify a model for the complex permittivity. In this work we compare two methods for modeling the complex permittivity of a heterogenous material. In one approach, we impose a probability distribution on a subset of the dielectric parameters. This approach leads to an infinite dimensional optimization problem over the space of probability measures. We approximate this space with a finite dimensional space by using either a Dirac approximation method or a linear spline approximation method. The second approach is to assume a number of oscillators in the permittivity model, and then use a convolution with a normal distribution. We compare both of these approaches on simulated data sets as well as data obtained from inorganic glasses. Each of these methods are able to fit the data well, yet the ease in interpreting the estimation results of imposing a probability distribution on parameters, as well as the tight mathematical results [2, 7] guaranteeing convergence under the Prohorov metric, lead us to favor the first approach.