The Mirror Reflection Principle and Probabilistic Invariants in the Theory of Multiple Photon Scattering

Abstract

In the framework of the statistical interpretation of multiple photon scattering as a stationary Markov process, which was given in well-known works by V.V. Sobolev and S. Ueno, the author has formulated the principle of mirror spatial-angular symmetry for total probabilities of photons exiting a homogeneous slab of finite optical thickness $\tau_{0} < \infty$. Based on this principle, linear second kind Fredholm integral equations have been obtained, as well as linear singular integral equations, for new objects of the classical radiative transfer theory, namely probabilistic invariants in the case of arbitrary distribution and power of primary energy sources in the medium. Besides, a unified probabilistic function for photons exiting a homogeneous slab was constructed. For finding unique regular solutions of the obtained linear singular integral equations, by analogy with the classical theory, additional integral relations are given, depending on the presence or absence of roots or pseudo-roots in respective characteristic equations. Given the analytic connection between probabilistic and photometric invariants, it has been shown that their calculations by standard computational methods, such as discretization technique or method of successive approximations, lead to substantial savings of computer resources and provide a more convenient form for representing theoretical data and results of numerical modelling of radiation fields of an atmosphere in the visible spectrum range of 0.6–0.8 µm.