## 1 Introduction

[2] Precipitation remote sensing instruments in the microwave regime are commonly used to observe falling snow. Recently, the motivation to understand the scattering properties of snowflakes has been driven by the growing interest in using millimeter-wave radars to measure snowfall remotely. Such radars are used, in particular, for global observations by satellites such as the currently orbiting CloudSat, operated by the National Aeronautics and Space Administration (NASA), Global Precipitation Measurement (GPM) under development by NASA and the Japan Aerospace Exploration Agency (JAXA), and EarthCARE, which is being developed by the European Space Agency (ESA) and JAXA.

[3] A number of recent studies using accurate computational scattering models and detailed shapes [*Ishimoto*, 2008; *Botta et al*., 2010; *Petty and Huang*, 2010; *Tyynelä et al*., 2011] have indicated that at sizes that are large compared to the wavelength, the use of homogeneous spherical or spheroidal shape models leads to a misestimation of the backscattering cross section of snowflakes. On the other hand, at the small-particle limit, such models are valid (except for the effect of particle non-sphericity, especially with regard to polarimetric variables) under the Rayleigh scattering law, and their applicability has also been successfully extended further using other theoretical methods and found valid in practice [e.g., *Leinonen et al*., 2011; *Hogan et al*., 2012]. *Leinonen et al*. [2012] found that the spheroid-snowflake model is sometimes, though not always, incompatible with experimental results. In the worst case, the error in the backscattering cross section can be as large as orders of magnitude.

[4] The cross section that is obtained by using the spheroidal shape model depends on the selection of the effective size of the equivalent spheroid. There are a number of different ways to determine the effective size, including the maximum diameter, the equivalent-volume diameter, and the radius of gyration [see, e.g., *Hogan et al*., 2000; *Donovan et al*., 2004; *Westbrook et al*., 2006]. In practice, the maximum diameter has been commonly used because it is readily measurable. However, the results by *Petty and Huang* [2010], *Kneifel et al*. [2011], and *Leinonen et al*. [2012] indicate that an effective radius cannot be chosen consistently with any method if more than two wavelengths are used.

[5] Because of the complexity of the shapes of snowflakes, parametrization of their scattering properties in terms of their physical properties is needed in order to interpret remote sensing measurements. The number of particles contained in the typical volume of interest (a single radar bin) is large, and thus, it is often sufficient to consider the average scattering properties. A very large number of parameters is usually required for a complete description of an aggregate particle, but it may be possible to simplify the particle model greatly as it is enough to consider only the average backscattering.

[6] One hypothesis that has been made in various forms [e.g., *Fabry and Szyrmer*, 1999] is to make spheroidal shape models more realistic by accounting for the decreasing average density of the snowflake as a function of the distance from its center. This feature is not captured by the fully homogeneous sphere and spheroid models. We show in this paper that this is not sufficient at large wavelength-size ratios; instead, the small-scale inhomogeneities of the internal structure of the snowflakes must be considered in order to reproduce the backscattering cross section. The structure can be understood in terms of the density autocorrelation function. Based on the Rayleigh-Gans approximation, we demonstrate a simple parametrization of this function in terms of the particle structure and provide theoretical justification for its form.