A method for solving an inverse problem with unknown parameters from two sets of relative measurements

Abstract

This work deals with an ill-posed inverse problem in which a distribution function, f(x), is estimated from two independent sets of non-negative relative measurements. Each measurement set is modeled through a Fredhohn equation of the first kind, with unknown parameters in its kernel. While the first measurement model only includes a scalar unknown parameter, p0, the second model contains a vector of unknown parameters, p. The proposed method consists of the following steps: (i) to obtain a first estimate of f(x) and p0 from the first measurement; (ii) to estimate the vector p from the second measurement and the previous estimate of f(x); and (iii) to estimate an improved f(x) by simultaneously using both measurements and the estimated parameters in a unique combined problem. The proposed algorithm is evaluated through a numerical example for simultaneously estimating the particle size distribution and the refractive index of a polymer latex, from combined measurements of elastic light scattering and turbidity.