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

IF 0.8 Q3 STATISTICS & PROBABILITY Monte Carlo Methods and Applications Pub Date : 2023-01-27 DOI:10.1515/mcma-2023-2001
P. J. Valades-Pelayo, M. Ramirez-Cabrera, A. Balbuena-Ortega
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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.
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将蒙特卡罗辐射传递算法与辐射传递方程联系起来
摘要本文直接从辐射传递方程(RTE)出发,提出了一条简单的途径来证明广泛使用的蒙特卡罗辐射传递(MCRT)算法。在这方面,本文开始推导一个概率测度,该测度是从无限域中单向点源下RTE的积分公式中获得的。这种推导只需要对扰动展开的前两项进行分析积分。尽管推导是为了澄清MCRT和RTE之间的关系而设计的,但它们往往相当冗长和详细。考虑到从光子物理传播过程的松散概率解释来证明MCRT是多么简单,以前将MCRT与RTE相关的方法的流行程度下降是完全可以理解的。不幸的是,所有这些都给人一种错误的印象,即MCRT和RTE没有那么紧密的联系,以至于最近的作品明确表示它们之间不存在直接联系。这项工作提供了一个更简单的途径,展示了MCRT算法是如何通过马尔可夫链显式地对RTE进行统计采样的,进一步阐明了该方法的基础。尽管紧凑,但这项工作中提出的推导并没有跳过任何基本步骤,在给出特定表达式和函数的同时保持了数学的严谨性。因此,这种推导可以帮助设计有效的方法来对不同场景的RTE进行统计采样,或者当将MCRT方法与传统上基于RTE的其他方法相结合时,例如球面谐波和离散坐标方法。
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来源期刊
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
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