Linking the Monte Carlo radiative transfer algorithm to the radiative transfer equation

Abstract

Abstract This manuscript presents a short route to justify the widely used Monte Carlo Radiative Transfer (MCRT) algorithm straight from the Radiative Transfer Equation (RTE). In this regard, this paper starts deriving a probability measure obtained from the integral formulation of the RTE under a unidirectional point source in an infinite domain. This derivation only requires the analytical integration of the first two terms of a perturbation expansion. Although derivations have been devised to clarify the relationship between the MCRT and the RTE, they tend to be rather long and elaborate. Considering how simple it is to justify the MCRT from a loose probabilistic interpretation of the photon’s physical propagation process, the decay in popularity of former approaches relating MCRT to the RTE is entirely understandable. Unfortunately, all of this has given the false impression that MCRT and the RTE are not that closely related, to the point that recent works have explicitly stated that no direct link exists between them. This work presents a simpler route demonstrating how the MCRT algorithm emerges to statistically sample the RTE explicitly through Markov chains, further clarifying the method’s foundations. Although compact, the derivation proposed in this work does not skip any fundamental step, preserving mathematical rigor while giving specific expressions and functions. Thus, this derivation can help devise efficient ways to statistically sample the RTE for different scenarios or when coupling the MCRT method with other methods traditionally grounded in the RTE, such as the Spherical Harmonics and Discrete Ordinates methods.