A statistical version of the solution of the inverse problem for nonabsorbing transparent media

Abstract

Numerous applications of light-scattering methods to the investigation of weakly absorbing transparent objects [I-4] reduce to the solution of the inverse problem with various versions of the regularization conditions. The idea behind the regularization conditions is to achieve by one method or another a unique solution to the inverse problem with the most general algorithm providing for stability of the computational scheme and of the results of reading and accidental errors in an experiment. As a rule, the generality of the solution algorithm is provided for by an a priori choice of the function describing the distribution of scattering centers (SC) with respect to one or another parameter, for example, size. Restrictions associated with the approximate notion of a scattering coefficient suggested by Van de Hulst [6] are imposed in solving the inverse problem from measurements of the transmission coefficients of two-phase objects [2, 5]. These restrictions are so severe that the problem is soluble, as a rule, only for two-phase objects of a polydispersed system of particles having a constant index of refraction (rap) differing insignificantly from the refractive index of the surrounding medium (me) , i.e., for relatively small average values of the dimensionless cross section/3 ("soft particles"). It is known that one can solve the inverse problem when the Van de Hulst approximation is rather little in error if m = mp/m e -< 1.45 in the case of exact solutions of the inverse problem [2] or m = mp/m e -< 1.25 for approximate solutions [5]. The methods of solving the inverse problems for two-phase objeets are applicable, in principle, to single-phase objects with density (or refractive index) fluctuations in the volume if the following conditions are fulfilled: take m e = I; apply the exact solutions for the scattering coefficients K(/3, m) over wide ranges of values of /3 and m; and assume that single-scatterin g occurs. The use of a well-known theorem from mathematical statistics, according to which one can determine for every general set of parameters representing particular solutions of the problem a finite-sized sample adequate to it and consisting of those same solutions, occupies a special place among the conditions for solution of the problem. This theorem permits a significant reduction in the number of solutions necessary for a complete description of the object being investigated and excludes the process of inverting an integral equation [2]. Upon fulfilling the enumerated conditions, one can obtain a solution of the inverse light-scattering problem from the transmission coefficients for a single-phase object with density fluctuations in its volume and negligibly small absorption after having solved the problem for a model representing a system of spheres localized within the object's geometrical boundaries whose diameters and refractive indices are varied within a rather large range of values. Taking the enumerated conditions into account, the transmission equation [7] [ 3 K(~'m)c~[] T = exp 2 D (i)