Effect of multiple scattering on radiation transmission in absorbing-scattering media

Abstract

A successive approximation procedure is developed to determine the scattering correction to the Beer-Lambert law in the evaluation of the geometric mean transmittance in a general multidimensional absorbing and scatter- ing medium. At each step of the approximation, the evaluation of an upper and lower bound of the scattering correction requires only a single integral over the volume of the scattering medium. This represents a great reduc- tion in mathematical complexity as compared to the direct numerical approach. Results for a two-dimensional rectangular absorbing and scattering medium are presented. The procedure is shown to converge rapidly in the optically thin limit. The lower-order results are useful for engineering application to media with arbitrary optical thickness. Some interesting conclusions concerning the qualitative physical behavior of the scattering correction are also generated. EASUREMENT of radiative transmission is a common experimental technique for many engineering applica- tions. In remote sensing, transmission measurements in dif- ferent atmospheric absorption windows are used to determine surface temperatures, surface emissivity, and other important geographical data.1 In the study of thermal insulation, transmission measurements are used to determine effective radiative properties of many porous insulating materials.2 In combustion, radiative transmission measurements are often used to determine flame properties. Most of the existing data reduction works assume that the transmissivity and the medium's optical thickness are related by the Beer-Lambert (B-L) law. The B-L law, however, fails for media that scatter as well as absorb radiation because the scattering process can- not be lumped together with the absorption process without a separate description. The accuracy of the conventional data reduction procedure for scattering media is thus uncertain. In some recent works,3'5 the deficiency of the B-L law assumption for scattering media is recognized. But almost without exception, all of the existing works consider only in- finite or semi-infinite scattering media with a parallel slab geometry. For many practical situations, in which the scatter- ing medium is finite, the applicability of the numerical results and the solution techniques developed in these works appears doubtful. The objective of this work is to present a successive approximation procedure, based on which the scattering cor- rection to the B-L law for media with general geometry can be estimated to within an arbitrary degree of accuracy. Only isotropic scattering is considered. Generalization of the pres- ent work to media with anisotropic scattering is quite straightforward and will be presented in future works. Specifically, successively improved estimates of the upper and lower bound of the scattering correction can ge generated by the present technique with little mathematical complexity. Unlike a straightforwar d numerical computational approach involving multiple-variable integration, which are time con-