Convergence of the Monte Carlo method in calculating the normal spectral albedo of optically thick snow and cloudy ice

Abstract

The Monte Carlo (MC) method is often used to obtain a benchmark numerical solution for radiative transfer problems that can also be solved using approximate methods. The present study is based on accurate analytical solutions for the spectral normal albedo of optically thick snow cover and light-scattering ice sheets. These analytical solutions are used to study the convergence of the MC simulation to the analytical solutions as the number of stochastic rays increases. This is a rare and unusual problem statement. One might expect that the numerical procedure would converge slowly in the visible spectral range, where the absorption of radiation by snow and cloudy ice is very low compared to scattering. This is a typical effect of the “scattering order problem” known in astrophysics. However, this does not apply to the integral characteristics of the radiation field, such as the normal-hemispherical reflectance (normal albedo) of scattering media. The calculations demonstrate good convergence of the Monte Carlo simulation to accurate values of the normal albedo in both the visible and infrared spectral ranges, which are important for solving geophysical problems. At the same time, a more sophisticated physical model based on differential approximations for the diffuse component of radiation intensity is preferable for both snow cover and cloudy ice layers. The results obtained are expected to be useful for selecting a method for radiative transfer calculations in diverse natural or artificial media of relatively strong scattering.