A Source-detector Methodology for the Construction and Solution of the One-dimensional Inverse Transport Equation

A source-detector methodology is presented for the construction of an inverse transport equation that once solved provides estimates for radiative properties and/or internally distributed sources in participating media. From the proper combination of source and detector pairs, a system of non-linear equations is assembled, taking also in consideration experimental data on the exit radiation from the medium. Test case results are also presented. INTRODUCTION The inverse analysis of radiative transfer in participating media has several relevant applications in engineering, medicine, geophysics, astrophysics and other research areas. Ustinov (1978) estimated the extinction coefficient and aerosol particles concentration in atmospheres. Bakirov et al. (1986) proposed a methodology for the determination of mass concentration of soot particles in flames. McCormick (1979), McCormick and Sanches (1981), Ho and Özisik (1988), Sanchez et al. (1990), Subramaniam and Mengüç (1991), Nicolau et al. (1994), and Silva Neto and Özisik (1993, 1995), just to name a few, solved inverse problems for single scattering albedo, optical thickness and/or anisotropic scattering phase function estimation. Yi et al. (1992) estimated the location and strength of a bioluminescent radiation source. Fukshansky et al. (1991) estimated the absorption and scattering coefficients and the asymmetry factor of scattering in living plant leaves. Different types of radiation such as neutrons, gamma-rays and photons have been used for object identification in industry (non-destructive testing), and in medicine (diagnosis and therapy). In many of the techniques used, scattering is neglected, yielding relatively simple reconstruction problems. This is the case in Computerized Tomography and Single Photon Emission Computerized Tomography (SPECT). When scattering has to be taken into account (McCormick, 1993, Mengüç and Dutta, 1994, Roberty and Oliveira, 1995), such as in Near Infrared Optical Tomography (NIROT), the reconstruction model becomes much more complex, non-linear, even requiring the computation of the radiation field. This particular tomographic problem is placed in the same context as radiative heat transfer in participating media and neutron transport in nuclear reactors, being the related physical phenomena (absorption, emission and scattering) modeled by the linearized Boltzmann equation. Silva Neto and Roberty (1998, 1998a) have been working on a source-detector methodology for the estimation of radiative properties and internally distributed sources in participating media. In this work the methodology is presented, as well as test case results for extinction and scattering coefficients estimation in one-dimensional homogeneous media. MATHEMATICAL FORMULATION OF THE DIRECT PROBLEM A plane-parallel, gray, anisotropically scattering slab of thickness L, with transparent boundaries is subjected to an external collimated radiation source that may be positioned in different locations around the medium, as shown in Fig.1. The mathematical formulation of this steady-state onedimensional radiative transfer problem is given by: ) , x ( S ' d ) ' , x ( ) , ' , x ( ) , x ( ) x ( x ) , x ( a a a k , a 1 1 s 2 1 k , a t k , a μ = μ μ φ μ μ σ − μ φ σ + ∂ μ φ ∂ μ − 1 1 , L x 0 in ≤ μ ≤ − < < (1a) ) ( f A ) , b ( a a k , b ab k , a μ δ = μ φ (1b)