The first author acknowledges financial support from the German Weather Service (DWD). We also thank the reviewers for many helpful and clarifying suggestions. We also thank F. Jakub for providing numbers of the computational timing.
Bowker, D. E., R. E. Davis, D. L. Myrick, K. Stacy, and W. T. Jones, 1985: Spectral reflectances of natural targets for use in remote sensing studies. NASA Reference Publ. 1139, 181 pp.
Buras, R., and B. Mayer, 2011: Efficient unbiased variance reduction techniques for Monte Carlo simulations of radiative transfer in cloudy atmospheres: The solution. J. Quant. Spectrosc. Radiat. Transfer, 112, 434–447.
Faure, F., L. Bugliaro, T. Zinner, R. Buras, and B. Mayer, 2009: Radiative transfer simulations for the validation of cloud products from MSG. Proc. 2009 EUMETSAT Meteorological Satellite Conf., Bath, United Kingdom, EUMETSAT, Poster Paper 25, 6 pp. [Available online at http://www.eumetsat.int/website/home/News/ConferencesandEvents/PreviousEvents/DAT_2042712.html.]
Frame, J. W., and P. M. Markowski, 2010: Numerical simulations of radiative cooling beneath the anvils of supercell thunderstorms. Mon. Wea. Rev., 138, 3024–3047.
Frame, J. W., J. L. Petters, P. M. Markowski, and J. Y. Harrington, 2009: An application of the tilted independent pixel approximation to cumulonimbus environments. Atmos. Res., 91, 127–136.
Gabriel, P. M., and K. F. Evans, 1996: Simple radiative transfer methods for calculating domain-averaged solar fluxes in inhomogeneous clouds. J. Atmos. Sci., 53, 858–877.
Hogan, R. J., and J. K. P. Shonk, 2013: Incorporating the effects of 3D radiative transfer in the presence of clouds into two-stream multilayer radiation schemes. J. Atmos. Sci., 70,708–724.
Kato, S., T. P. Ackerman, H. M. James, and E. Clothiaux, 1999: The k-distribution method and correlated-k approximation for a shortwave radiative transfer model. Atmos. Chem. Phys., 5, 1855–1877.
Markowski, P. M., and J. Y. Harrington, 2005: A simulation of a supercell thunderstorm with emulated radiative cooling beneath the anvil. J. Atmos. Sci., 62, 2607–2617.
Marshak, A. M., and A. B. Davis, Eds., 2005: 3D Radiative Transfer in Cloudy Atmospheres. Springer-Verlag, 686 pp.
Marshak, A. M., A. B. Davis, W. Wiscombe, and R. Cahalan, 1995: Radiative smoothing in fractal clouds. J. Geophys. Res., 100, 26 247–26 261.
Mayer, B., and A. Kylling, 2005: Technical note: The libRadtran software package for radiative transfer calculations—Description and examples of use. J. Quant. Spectrosc. Radiat. Transfer, 62, 109–121.
Tompkins, A. M., and F. Di Giuseppe, 2007: Generalizing cloud overlap treatment to include solar zenith angle effects on cloud geometry. J. Atmos. Sci., 64, 2116–2125.
Várnai, T., and R. Davies, 1999: Effects of cloud heterogeneities on shortwave radiation: Comparison of cloud-top variability and internal heterogeneity. J. Atmos. Sci., 56, 4206–4224.
Wapler, K., and B. Mayer, 2008: A fast three-dimensional approximation for the calculation of surface irradiance in large-eddy simulation models. J. Appl. Meteor. Climatol., 47, 3061–3071.
Zinner, T., A. Marshak, S. Lang, J. V. Martins, and B. Mayer, 2008: Remote sensing of cloud sides of deep convection: Towards a three-dimensional retrieval of cloud particle size profiles. Atmos. Chem. Phys., 8, 4741–4757.
Zuidema, P., and K. F. Evans, 1998: On the validity of the independent pixel approximation for boundary layer clouds observed during ASTEX. J. Geophys. Res., 103, 6059–6074.
The normalized bias is defined as the difference of the mean irradiance obtained with ICA or any (N)TICA method and the mean 3D Monte Carlo irradiance