## 1 Introduction
In previous work [21; 22], the authors have introduced a Markov chain approach for modeling transmitted photon angular distribution (i.e., the probability density function of propagation angles of transmitted photons) via multiple scattering based on the Markov chain approximation. There has been some research using the Markov chain method to model the multiple scattering problems. For instance, [PERSON] et al. [23] have used the Markov chain radiative transfer to invert OII 83.4 nm dayglow profiles with multiple scattering considerations. [PERSON] et al. [24] utilized the Markov chain approach to model obliquely incident rays on the surface of participating media for atmospheric research. Although Markov chain methods in dealing with light propagations in multiple scattering medium can be directly derived by discretization of the generalized radiative transfer equations (RTE) [25], the modeling process can be independent from solving the ordinary differential equations (ODE) in typical RTE solution procedures [26]. Such a feature would enable customized Markov chain approximations based on the needs of the investigation without solving the whole RTE system, and the authors' previous work using the Markov Chain has successfully built a precision connection between actual measureable data (angular distribution) and the desired optical properties of the medium, with less computational costs. The discrete Markov chain method discretizes the domain of interests into multiple \"states\", and the transition probability from one state to another can be found for specific problems. A transition matrix \(T\) is then built with each entry \(T_{\bar{y}}\) representing the probability of transition from state \(i\) to state \(j\), so that the Markov chain approach converts the stochastic, random process problem into a mathematical problem involved with matrix manipulations. The Markov chain approximation approach for photon multiple scattering has been verified for both isotropic scattering and anisotropic scattering through the turbid slab and more details regarding Markov chain approximations for multiple scattering can be found in the authors' previous publications. Despite the many advantages of Markov chain approximations over Monte Carlo simulations, the method suffers from a variety of computational complexities. For example, the \"curse of dimensionality\" [27]. Here, the exponential increase in computational cost with higher dimensions occurs due to the size increase of the transition matrix (and the number of states). Techniques have been developed to improve performance at higher dimensions. Illustrations of these techniques include state reduction methods, such as the lump process established by [PERSON] and [PERSON] [28] and the GTH elimination algorithm (proposed by [PERSON], [PERSON], and [PERSON]) [29]. Methods for improving the performance of Markov chain approximations also include investigations into proper discretization of continuous Markov chain problems [30; 31]. For multiple scattering problems, it is necessary to properly discretize the domain of interest to improve the efficiency of Markov chain methods, and thus reduce the computational cost. It is also of interest to find an effective way to discretize the zenith angle (photon propagation angle), so that the transmitted and reflected photon angular distributions can be simulated with improved fidelity. Such investigation is expected to improve the performance of the Markov chain method, for that many scattering applications are in Mie scattering regimes with high anisotropies (\(g\)), indicating that highly forward scatterings are desirable. Finer discretization in the forward direction could potentially better resolve the resultant angular distribution. Furthermore, there is usually a detection limit of the transmitted/reflected photon angles experimentally, for example, up to 580 m/rad (33.2\({}^{\circ}\)) [32; 33]. This issue also suggests an angular discretization that emphasizes finer resolution within the detection limit.