Quantitative accuracy assessment of empirical phase functions for the bidirectional reflectance distribution function in particulate media

The scattering phase function is a fundamental determinant of radiative transfer in particulate media, yet the predictive accuracy of widely used empirical phase function models remains poorly quantified, directly limiting the reliability of optical simulations and quantitative remote sensing inversion. Here, we systematically compare four models (HG, CS, RM, and TTRM) across non‑absorbing, weakly absorbing, and strongly absorbing particle systems, employing rigorous Lorenz–Mie theory as the benchmark and Monte Carlo simulations to provide, for the first time, a comprehensive quantification of their effects on both directional‑hemispherical reflectance and angularly resolved BRDF. The influences of size parameter, particle size distribution, and optical thickness are evaluated. Using exact Mie calculations as the reference, errors in directional-hemispherical reflectance and angularly resolved BRDF are systematically quantified. The Monte Carlo results show that the TTRM model reproduces the angularly resolved BRDF with the highest fidelity in the vast majority of cases. For the polydisperse systems considered in this work, the relative errors of its BRDF·cos θ_r are typically below 5%, while the absolute errors range from several times to two orders of magnitude smaller than those of the HG, CS, and RM models. For the hemispherical reflectance, the TTRM maintains relative errors below 3% under polydisperse conditions, while the HG, CS, and RM models produce maximum errors of 168.37%, 70.59%, and 129.93%, respectively. These findings confirm that, within the range of conditions investigated, the TTRM model exhibits superior performance and high reliability for radiative transfer simulations and BRDF calculations compared to the other empirical models tested. This work provides a quantitative basis for the selection of scattering phase function models and for improving the accuracy of remote sensing inversions.