An explicit formulation for the inverse transport problem using only external detectors: Part I: computational modelling

Abstract

Abstract. In the present work the inverse problem of identification of radiative properties, the total extinction and scattering coefficients, is analyzed and explicitly formulated based in an elementary semi-group theory. The Chandrasekhar discrete ordinates finite dimensional approximation of the angular variables is used for the direct problem representation with the stationary Linear Transport (Boltzmann) Equation in absorbing and scattering media in matrix form. For the inverse problem we suppose known the albedo operator from measured intensities at the boundaries of the medium. Here we analize the inverse problem for an N -dimensional medium and from the solution of the transmission problem we present an explicit form for the matrix that contains the total extinction and scattering coefficients.Mathematical subject classification: 80A23, 15A29.

An explicit formulation for the inverse transport problem using only external detectors: Part II: Application to one-dimensional homogeneous and heterogeneous participating media

“…In the present work and in the accompanying paper [2] we derive and show the feasibility of a new explicit formulation for the inverse radiative transfer problem. From the knowledge of the imposed incident and measured emergent radiation intensities in a one dimensional homogeneous slab we were able to estimate the anisotropic scattering coefficients, using only external detectors.…”

“…We developed an explicit formulation for the inverse radiative transfer problem for estimating the total extinction and scattering coefficients in an absorbing and anisotropic scattering medium only using external detectors, which is presented in an accompanying article [2]. This formulation uses concepts of the discrete ordinates method and matrix operators for modelling the direct problem.…”

“…Note that τ 1 = τ (1) means the thickness in the direction μ = 1 and that μ = 0 is a singular value for equation (2).…”

“…The steady state inverse radiative transfer problem in one-dimensional participating media is studied as an example of application of the new methodology presented in a accompanying paper by the authors [2]. Spectral methods are used for the appropriate analysis of the direct transport problem.…”

Some Features of Modeling Ultrasound Propagation in Non-Destructive Control of Metal Structures Based on the Magnetostrictive Effect

“…The mathematical basis for the study of such processes is the second-order wave equation of mathematical physics. Next, we will consider the equation in the area G ⊂ R n [44][45][46]:…”