This report presents a study on the determination of the required number of azimuth (Fourier terms) and zenith (Gaussian µ) integration points in order to obtain a cloud reﬂectance as calculated by the Doubling Adding KNMI (DAK) radiative transfer model with an absolute accuracy within 0.005. Cloud reﬂectance calculations at 0.632 and 1.605 µm are performed for a plane parallel water cloud having
an optical thickness, τ , of 10, overlying a surface with albedo 0.10, for effective radii of 5-24 µm. Relative azimuth angles are 90, 120, 150, and 175◦ and (cosines of) zenith angles are 0.2-0.8. A two-
parameter Gamma size distribution is applied. Fourier terms and Gaussian µ points are varied from 20-200 for each zenith and relative azimuth angle. Results indicate that accurate cloud reﬂectance
calculations at 0.632 µm can be performed by taking 60-80 Fourier terms and 40-60 Gaussian µ points for most effective radii and viewing geometries investigated. However, for effective radii of 16 and
24 µm, 140-180 Gaussian µ points are required. For the reﬂectance at 1.605 µm, 40-60 Fourier terms and 40-60 Gaussian µ points sufﬁce. At both wavelengths, the given values strongly increase when the backscatter viewing geometry is approached. In addition to the convergence study, the effect of a change in the variance of the size distribution, veff , and the error due to neglecting linear polarization in modeled cloud reﬂectances is examined. An absolute change in v eff of 0.05 results in a maximum relative difference in calculated cloud reﬂectance of ∼4%. Neglecting linear polarization in the 0.632 µm cloud reﬂectance calculations causes an averaged relative difference
of 0.1-0.3%, with a maximum of ∼1%, depending on the effective radius.
ELA Wolters, RA Roebeling, P Stammes. Cloud reflectance calculations using DAK: study on required integration points
KNMI number: TR-292, Year: 2006, Pages: 17