## 1. Introduction

[2] Multiple scattering in hydrometeors has been studied through the radiative transfer theory [*Ishimaru et al.*, 1982; *Kuga et al.*, 1989; *Oguchi and Ito*, 1990; *Shin and Kong*, 1981] and analytical wave theory [*Ishimaru and Tsang*, 1988; *Ishimaru*, 1997; *Mandt et al.*, 1990; *Mandt and Tsang*, 1992; *Tsang and Ishimaru*, 1985], and it is expected to play an important role in weather radar studies for operational frequency over 30 GHz. As an experimental demonstration of the second-order scattering effect, *Ito et al.* [1995] measured the linear depolarization ratio (LDR) in return signals from rains with an airborne 35 GHz radar, showing time-dependent behavior of multiple scattering. *Marzano et al.* [2003] performed time-dependent Monte-Carlo simulation based on radiative transfer theory and reported that overestimation of reflectivity due to multiple scattering can reach nearly 20 dBZ in convective precipitation for a 35 GHz spaceborne radar. However their method was practically limited to the plane-wave incident case, i.e. the case of infinite footprint radius. Thereafter, *Kobayashi et al.* [2005] derived an analytical formula of second-order scattering for the radar of a finite footprint radius. They found that the reflectivity of second-order scattering, including the backscattering enhancement, increases as a function of the ratio of footprint radius to mean free path of a random medium, and that this reflectivity asymptotically approaches the values that the plane wave incident theory [*Mandt et al.*, 1990] predicts. However, since this method was derived from the time-independent analytical vector wave theory (Green function method), it cannot be generally applied to the pulsed radar except for the special case described in their appendix. Recently, *Battaglia et al.* [2005] performed another Monte Carlo simulation in which the finite beam effect is explicitly taken into account. Their results reported the multiple scattering effect for a spaceborne radar can reach to 10–20 dBZ at 35 GHz, while almost negligible for an airborne radar due to its small footprint size.

[3] The time-dependent analytical formalism is advantageous to characterize the effect of multiple scattering for various pulsed-weather-radar operations. However, a few studies have been reported. *Oguchi and Ito* [1990] derived a time-dependent solution of radiative transfer theory using a two-frequency mutual coherence function [*Ishimaru*, 1997; *Hong and Ishimaru*, 1976]. Thereafter, *Ito and Oguchi* [1994] and *Ito et al.* [1995] derived analytical formulas of power-returns for circularly and linearly polarized incidences, respectively, based on expansions of the generalized spherical harmonics [*Oguchi*, 1980]. In those studies, however, they assumed a plane wave incidence followed by capture at infinite distance, and no effect of finite beam width was reflected in the formalisms.

[4] In this paper, the second-order scattering approximation based on *Ito et al.* [2007] is extended to a theory for pulsed radars. Our main interest is in W-band radars. For the sake of simplicity, a single layer of random medium is assumed to consist of spherical particles of uniform size. When considering a general particle distribution of spherical particles, the same conclusion will be derived by taking ensemble average over an absorption coefficient and a scattering matrix. Furthermore, the method can be extended to higher-order multiple scatterings, and also to multiple layers of hydrometeors. On the other hand, the formalism in this paper is based on the radiative transfer theory so that the solutions can not include the effects of cross terms, i.e. backscattering enhancement, which is included in the analytical wave theory [*Kobayashi et al.*, 2005]. Since the magnitudes of the cross terms are comparable to those of the ladder terms (intensities in the radiative transfer theory), these effects should be taken into account in the future.