## 3 Computational Results
### Zenith Angle Discretizations  
In this section, different zenith angle discretization schemes were tested. As previously mentioned, the zenith angle of photon propagation trajectory (from 0\({}^{\circ}\) to 180\({}^{\circ}\)) was divided into 180 intervals. Three different discretization schemes were utilized: (1) Uniform discretization. In this approach, each interval covered a uniform 1\({}^{\circ}\); (2) Forward emphasizing approach. In this scheme, a finer resolution was applied for a small zenith angle range, for example, from 0\({}^{\circ}\) to 30.0\({}^{\circ}\), and a coarse resolution was used for the rest of the angular range; and (3) Forward and backward emphasizing approach. In this scheme, a finer resolution was applied for both the forward and backward zenith angles, for example, from 0\({}^{\circ}\) to 15.0\({}^{\circ}\) and from 165\({}^{\circ}\) to 180.0\({}^{\circ}\), and a coarse resolution was used for the rest of the angular range. The OD was assumed to be uniform and the value of the OD was also set to be either 2 or 5. Figure 8 depicts the absolute relative error of the Markov chain approximations compared with the Monte Carlo simulations with zenith angle discretization scheme 2. To show the performance of the zenith angle discretization scheme, the phase function 3 was selected for its complex angular structure. For zenith angle discretization scheme 2, the angular resolution was 0.25\({}^{\circ}\) for \(\theta<30^{\circ}\) and 2.5\({}^{\circ}\) for \(\theta>30^{\circ}\). As such, 180 meshes were used to cover the angular range from 0\({}^{\circ}\) to 180\({}^{\circ}\), which kept the computational cost the same as the first discretization scheme. Uniform OD discretization was utilized in Figure 8. It can be seen in Figure 8 that the absolute relative error for both OD = 2 and OD = 5 cases was reduced from 1% to 0.2% in the angular range of 0\({}^{\circ}\) to 30\({}^{\circ}\). Therefore, a proper zenith angle discretization scheme can indeed improve the predictive performance in certain regimes. However, the trade-off is the sacrifice of prediction accuracy in the angular regime from 30\({}^{\circ}\) to 180\({}^{\circ}\). As can be seen, the relative error near 90\({}^{\circ}\) and 180\({}^{\circ}\) was more significant. Furthermore, adopting a different OD discretization scheme could not remarkably reduce the peak error for the new zenith angle discretization scheme. \(a\) and \(b\) values recommended in Section 3.2 were still valid; however, the values only reduced the peak error from 6% to 4.5% (OD = 2) and 3% to 2% (OD = 5). This observation suggested that with a coarse zenith angle mesh, angular discretization was the dominating source of the computing error, especially for the regimes with coarse mesh. However, it should be noted that experimental measurement can only be achieved within a limited angular regime, i.e., from 0\({}^{\circ}\) to 30\({}^{\circ}\). Improving the accuracy in this regime was critical for practical applications whilst preserving the accuracy in the range from 30\({}^{\circ}\) to 180\({}^{\circ}\) was not necessary. Therefore, a proper zenith angle discretization was meaningful for angular distribution-based algorithms and potential spectroscopy methods. Figure 7: Averaged absolute relative error of Markov chain angular distribution predictions compared to Monte Carlo in different cases. Angular range was set to 0–30\({}^{\circ}\) (transmitted photons) and 150–180\({}^{\circ}\) (reflected photons). Figure 9 closely examines the performance of zenith angle discretization scheme 2 in the range from 0\({}^{\circ}\) to 30\({}^{\circ}\). In this comparison, phase function 3 was selected and OD = 5 for Figure 9. Panel (**a**) of Figure 9 shows the transmitted angular distribution predictions for both uniform zenith angle distribution (scheme 1) and non-uniform zenith angle distribution (scheme 2).