Our first investigation (Grenfell and Warren, 1999) compared the equivalent-sphere formulation to ensembles of randomly oriented infinitely long circular cylinders of ice, because an exact solution is available for all cylinder radii. The equal-V/A spheres were highly successful at mimicking single-scattering and multiple-scattering properties of the cloud of cylinders, for wavelengths 0.2 to 50 micrometers, cylinder radii 1 to 500 micrometers, and ice water paths (IWP) 0.4 to 200,000 grams per square meter.

In a second test, accurate computations of scattering and absorption for hexagonal prisms are made using the geometric optics method for large size parameters and the finite-difference-time-domain (FDTD) method for small size parameters. Such perfectly shaped prismatic crystals do exist in nature in the Antarctic winter, but they are rare elsewhere.

The extinction efficiency and single-scattering albedo are matched well at all wavelengths for all length-to-width ratios (c/2a) of the prisms. The asymmetry factor g is matched well at absorptive wavelengths. At visible wavelengths g is matched adequately for columns, but for plates it is larger than that of the equal-V/A spheres and for equidimensional crystals it is smaller than that of the spheres. The largest multiple-scattering errors are for equidimensional crystals at visible wavelengths, as large as 0.13 in reflectance for certain combinations of size and IWP. Errors in transmittance, reflectance and absorptance at other wavelengths and for other aspect ratios are generally much smaller.

The errors in g are due to the perfect shapes of the model prisms, with large parallel faces on the plates and a preponderance of 90-degree edges on the equidimensional crystals. This suggests that the representation by equal-V/A spheres will be more successful for the irregular crystals that are most common in cirrus clouds. This will be a subject for future research.

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