## 1. Introduction

[2] The multiple scattering effect on radar reflectivity for hydrometeors has been extensively studied on the basis of the radiative transfer theory [*Shin and Kong*, 1981; *Ishimaru et al.*, 1982; *Kuga et al.*, 1989; *Oguchi and Ito*, 1990; *Ito et al.*, 1995; *Oguchi et al.*, 1998] and the analytic wave theory (Green function method) [*Ishimaru and Tsang*, 1988; *Mandt et al.*, 1990; *Mandt and Tsang*, 1992; *Barabanenkov et al.*, 1991; *Kravtsov and Saichev*, 1982]. These two methods yield similar results when the scattering angle largely deviates from the direct backscattering. However, when the scattering angle is equal to the direct backscattering angle, the latter method gives nearly twice the values of multiple scattering as the former, which is due to backscattering enhancement caused by the so-called “cross terms” in the Dyson's diagram [*Kuga and Ishimaru*, 1984; *Tsang et al.*, 1985]. These previous works generally assumed that a plane wave is incident onto a layer of randomly distributed particles, and that the reflected wave is received by an antenna at infinite range. Recently, *Kobayashi et al.* [2005] developed a theory for weather radar applications by adopting the spherical incident wave with a finite transverse beam profile, usually approximated as a Gaussian antenna pattern, and by assuming that the reflected wave is captured at a finite range. For a single layer of spherical water particles illuminated by a Gaussian radar beam, the results of Kobayashi et al. showed that the radar reflectivity increases because of second-order scattering, as the ratio of footprint radius to mean free path of a random medium increases. In the limit of a large footprint radius compared to the mean free path, however, the formulas for a Gaussian radar beam give the identical results to that predicted by the conventional plane wave theory of *Mandt et al.* [1990].

[3] The backscattering enhancement under the finite beam width was measured in a controlled laboratory experiment by *Ihara et al.* [2004] by using a 30-GHz radar with the 3-dB beam width of 10°. This experiment was also confirmed by the associated computer simulation by *Oguchi and Ihara* [2006]. At present, their experimental and simulation results have only been obtained for one ratio of footprint radius to mean free path of a random medium. However, this work is currently being extended for several other ratios.

[4] For remote sensing applications, *Marzano et al.* [2003] studied multiple scattering effect for a three-layer system of spherical hydrometeors (ice-graupel, melting, and rain layers) through radiative transfer theory with a Monte Carlo simulation, concluding that overestimation of reflectivity due to multiple scattering can reach 20 dBZ or more in extremely intense convective precipitations for a spaceborne 35 GHz radar. *Battaglia et al.* [2005] revisited the same problem using a different Monte Carlo method to account for the finite beam width effects. Their results, in qualitative agreement with the theoretical prediction of *Kobayashi et al.* [2005], reported that the multiple scattering effect in extremely intense convective precipitations can reach 10–20 dBZ for a 35 GHz spaceborne radar, while it is negligible for an airborne radar because of its smaller footprint size.

[5] The previously mentioned theoretical works [*Ishimaru et al.*, 1982; *Ito et al.*, 1995; *Kobayashi et al.*, 2005; *Kuga et al.*, 1989; *Mandt et al.*, 1990; *Mandt and Tsang*, 1992] and computer simulations [*Battaglia et al.*, 2005; *Marzano et al.*, 2003; *Oguchi and Ihara*, 2006] assumed spherical particles. The shapes of raindrops, however, are known to be better represented by spheroids. In general, change of raindrop shape from sphere to spheroid brings about spatial anisotropy not only in the scattering matrix but also in the propagation wave constants, making a propagation Green function dyadic [*Oguchi*, 1977]. This change must be carefully treated when the second-order scattering is calculated for cross-polarized returned signal, because there is no guarantee that the second-order ladder term is equal to the second-order cross term [*Mishchenko*, 1991, 1992].

[6] Another improvement that is important for weather radar remote sensing incorporates with a general drop size distribution for hydrometeors. Previous analytical wave theories [*Akkermans et al.*, 1986, 1988; *Barabanenkov et al.*, 1991; *Ishimaru and Tsang*, 1988; *Mandt et al.*, 1990; *van Albada and Lagendijk*, 1987] were derived only for particles of a uniform size distribution, until *Mandt and Tsang* [1992] included a general drop size distribution into their theory, which, however, is limited to the plane wave incidence to a layer of spherical particles.

[7] In this paper, a theory of finite beam width by *Kobayashi et al.* [2005] is extended to include a general drop size distribution with spheroid-shaped hydrometeors to account for propagation anisotropy. Throughout the paper, it will be assumed that the symmetry axes of spheroids are uniformly oriented in a direction rather than randomly oriented. There are two reasons for this assumption: the first reason is that, once the formalism is developed for the uniformly oriented particles, it is relatively easy to extend the formalism to some cases of randomly oriented nonspherical hydrometeors as discussed in section 4; the second reason is to tie the result of this paper to an experimental demonstration of multiple scattering in millimeter wavelength weather radars studied by *Ito et al.* [1995]. In their paper, raindrops were assumed as spheres, and large values of measured linear depolarization ratios (LDR) were attributed to the effect of second-order scattering rather than that of first-order scattering from nonspherical raindrops. Their assumption of spherical raindrops looks reasonable. However, when spheroids are adopted to represent raindrops as a better approximation, and if the principal axes of the spheroids are largely canted, then a large change in value may occur for the second-order cross-polarized power returns as reported in the propagation problem [*Oguchi*, 1973, 1983]. Hence it is relevant to confirm the validity of the assumption of *Ito et al.* [1995] by choosing the extreme case that the principal axes of spheroidal raindrops are uniformly oriented in a preferential direction, which may be caused by strong wind.

[8] This paper is organized as follows: in section 2, the formalism for spherical particles of uniform size [*Kobayashi et al.*, 2005] is extended to that for spheroidal particles with a general drop size distribution. In section 3, the derived formalism is applied to water and ice particles of uniform size to illustrate the difference between the spherical and spheroidal approximations. In section 4, Marshall-Palmer distributed rains of uniformly oriented spheroids are calculated and compared with the results based on the spherical approximation [*Ito et al.*, 1995]. Section 5 provides the conclusion and future works.