What are the optimum discrete angles to use in thermal‐infrared radiative transfer calculations?

Abstract As computer power increases, there is a need to investigate the potential gains of using more than two streams in the radiative transfer calculations of weather and climate models. In this article, seven quadrature schemes for selecting the zenith angles and weights of these streams are evaluated rigorously in terms of the accuracy of thermal‐infrared radiative transfer calculations. In addition, a new method is presented for generating “optimized” angles and weights that minimize the thermal‐infrared irradiance and heating‐rate errors for a set of clear‐sky training profiles. It is found that the standard approach of applying Gauss–Legendre quadrature in each hemisphere is the least accurate of all those tested for two and four streams. For clear‐sky irradiance calculations, “optimized” quadrature is between one and two orders of magnitude more accurate than Gauss–Legendre for any number of streams. For all‐sky calculations in which scattering becomes important, a form of Gauss–Jacobi quadrature is found to be most accurate for between four and eight streams, but with Gauss–Legendre being the most accurate for 10 or more streams. No single quadrature scheme performs best in all situations, because computing irradiances involves two different integrals over angle and the relative importance of each integral depends on the amount of scattering taking place. Additional optimized quadratures for clear‐sky and all‐sky calculations with four to eight streams are presented that constrain the relationships between angles in a way that reduces the number of exponentials that need to be computed in a radiative transfer solver.