Multiple scattering returns from spherical water particles of uniform diameter are studied for pulsed radar operation. The Gaussian transverse beam-profile and the rectangular pulse-duration are taken into account. A second-order analytical solution is derived for a single layer structure, based on the time-dependent radiative transfer theory. The method itself can be extended to higher-order scatterings in a multiple layered structure with general drop size distributions. The calculation results show that the effect of second-order scattering becomes larger, as the footprint radius, range, and range resolution, which are normalized by the mean free path, increase.
 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.  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.  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.  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.  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.
 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  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  and Ito et al.  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.
 In this paper, the second-order scattering approximation based on Ito et al.  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.
 In this section, time-dependent solutions of the first and second-order scatterings are derived based on Ito et al. .
2.1. Time-Dependent Radiative Transfer Equation
 Suppose that a single layer of hydrometeors is isotropic with respect to propagation, which is always satisfied for spherical-shaped hydrometeors. A time-dependent vector radiative transfer equation is written [Ito and Oguchi, 1994; Ito et al., 1995] in the form
in which J represents the 4-dimensional specific intensity propagating in a direction that is observed at a point x and a time t. z and ρ represent the longitudinal and transverse coordinates of point x, respectively; c is the speed of light. e denotes the extinction rate given by the Foldy-Oguchi-Twersky formula [Oguchi, 1973]. Ψ(, ) denotes the 4 × 4 scattering matrix defined in a wave-propagation medium. In rigor, the Foldy-Oguchi-Twersky formula is valid, only when the particle distribution is completely uniform (no density correlation), and the ergodicity is satisfied [Mishchenko, 2002]. For the sake of simplicity, throughout the paper, the complete uniformity of a medium will be assumed even in the small volume. When adopting the perturbation method, the solution of the N-th order multiple scattering can be represented with that of the (N-1)-th order multiple scattering as described in Ito et al.  and also in Appendix A1:
in which the directional cosine μ ≡ Ωz has been defined to denote the longitudinal component of , and ΩT designates the transverse component vector of . Inversely transforming a general solution of equation (A4) in Appendix A1, one obtains the zeroth order intensity in a general form
in which C0(ρ, , t) denotes an arbitrary function of ρ, and t. Iarb denotes an arbitrary 4-d constant vector.
2.2. Gaussian Radar Beam
Figure 1 is a schematic of a layer of hydrometeors of thickness d. A radar antenna at point An with a 3-dB beam width θd ≪ 1 is located at a distance R from the top of the medium. The origin of position coordinate O is set at the beam center on the surface. Suppose that the antenna An transmits a pulse of duration time T with linear polarization. The origin of time t is set at the moment when the transmitted pulse reaches the top surface of the medium. Furthermore the transmitting and receiving antenna gains are assumed to be equal, denoted by G. In the following derivation, the modified Stokes parameters defined in chapter 3 in Tsang et al.  are adopted to describe the specific intensity and the scattering matrix.
 The incident specific intensity, shown below, satisfies the general form of equation (3) as proved in Appendix A2:
in which the constant Ii and a, and the unit vector Iinc are defined as
The incident direction i0 is represented by the polar coordinates i0 and ρ as functions of z and ρ
Using equation (2), the first-order 4-d specific intensity in a general direction = (, ) with the constraint of μ ≡ cos < 0 can be written
in which the subscript 1b of J means that the first-order scattering ′1′ propagates in the backward direction ′b′ i.e. μ < 0. In equation (13), the incident direction for the first-order scattering can be represented as
through the transverse coordinate ρ′ defined as
along with the transverse component of
The explicit forms of θi1 and ϕρ′ can be given by replacing ρ with ρ′, and z with z′ in equations (9) and (10). To calculate the received power by the antenna at An, a polar coordinate system = (Θ, Φ)ω is introduced at point An. When the extinction rate from the top surface of medium to point An is represented by κar, the 4-d intensity received by the antenna at time t + R/c can be written:
in which J1b has been defined in equation (13). ρs(Θ, Φ) is defined only in the plane z = 0 as a function of Θ and Φ:
Note that ρs is defined as a function of Θ and Φ, while ρ denotes the transverse components of an arbitrary point x. Hence, as shown in Appendix A3, two parameters representing the infinitesimal solid angle d will be transformed from (Θ, Φ) to (ρs, Φ). The direction r(Θ, Φ) in equation (17) is defined
(N.B. the superscript ∀ on left shoulder means an arbitrary value). When cT/2 ≪ R, κe = 0, ct/2 ≤ d, and c(T − t)/2 ≥ 0 are satisfied, equation (21) reduces to the conventional radar equation.
 Iterate use of equation (2) followed by the same procedure as the derivation of equation (17) gives the 4-d power return of the second-order scattering received by the antenna at An at time t + R/c (details in section A4):
in which I0 denotes the first kind modified Bessel function of order 0, and the footprint radius is represented as a function of z
The upper integral limit ′d′ in the integral ∫ dz′ in equation (25) can be replaced with min[ct/2, d].
3. Results and Discussions
 Both the radiative transfer theory [Ishimaru, 1997; Tsang et al. 1985] and the analytical wave theory [Ishimaru and Tsang, 1988; Mandt et al., 1990] show that the multiple scattering can be well characterized as a function of the optical thickness of a random medium. As mentioned in section 1, Kobayashi et al.  showed that the multiple scattering also depends on a footprint radius normalized by the mean free path lfree of medium. For these reasons, a layer-thickness d, a range resolution 2−1cT, and a footprint radius σr will be represented as quantities normalized by lfree throughout the paper, namely defined as τd = d/lfree, τp = 2−1cT/lfree, and Ξr = σr/lfree. In Kobayashi et al. , the thickness of random medium d was assumed to be much smaller than the top surface range R, i.e. d ≪ R, which is well satisfied for spaceborne weather radars (R ⪆ 400 km, d < 20 km, θd ∼ 10−3 − 10−2 radian). With this assumption, equations (21) and (25) can be simplified further by changing R + z′ and R + z″ to R, and also changing σr(z) to σr(0). As proved in Tsang and Kong , the validity of the second-order theory becomes degraded due to increase in higher-order scatterings, when the optical thickness is much beyond 2. Hence in this section the optical thickness of 2 is used for calculation.
Figure 2 illustrates the time-dependent return profile of a pulsed radar calculated for the normalized footprint radius Ξr = ∞ (i.e. plane wave incident approximation). A layer of spherical water particles of uniform diameter D = 1 mm is assumed to have the normalized thickness τd = 2. The carrier frequency in the pulse is set at 95 GHz, and the normalized range resolution is τp = 0.1. The return signals of the first and second-order scattering at time t + R/c are given by equations (21) and (25), respectively. Without loss of generality, we can eliminate the effect of distance R from the receiving time t + R/c, redefining the receiving time as t. Furthermore, we can represent the receiving time t by using the normalized range τr = 2−1ct/lfree. The normalized first-order copolarized power-return l1co and cross-polarized power-return l1cx are defined by dividing equation (21) by equation (22). The normalized second-order copolarized power-return l2co, and cross-polarized power-return l2cx are defined in a similar manner. In the following results, the particle number density N does not appear, however information on N is implicitly included in the mean free path lfree. In Figure 2a, l1co, l2co and l2cx are plotted as functions of τr. Notice l1cx = 0 for the spherical particles. As the pulse enters the medium, l1co increases monotonously from τr = 0 to τr = τp = 0.1 to reach the maximum, followed by the reduction due to the absorption κe with −8.6 dB per unit-thickness, i.e. 10log(e−2). Then, l1co rapidly decreases at the rear edge of medium from τr = τd = 2 to τr = τd+τp = 2.1, as the pulse leaves out the medium. On the other hand, the second-order terms l2co and l2cx show more gentle changes. The maximums of l2co and l2cx appear around τr = 0.6 much delayed from τr = 0.1. It is noted that l2co overcomes l1co about τr = 1.6, when l1co still exists. Even after τr = 2.1, l2co and l2cx remain for long time. In Figure 2b, the normalized total copolarized power-return l1co+l2co and total cross-polarized power-return l2cx are plotted as functions of τr, which corresponds to Figure 2 in Ito et al. . In Figure 2c, the second-order ratio (SR) in copolarized return, defined as l2co/(l1co + l2co), is plotted as a function of τr. It also shows the linear depolarization ratio (LDR) defined as l2cx/(l1co + l2co). Since l1co vanishes at τr = 2.1, SR turns to be 0 dB, and LDR abruptly increases. To characterize the asymptotic behavior of the second-order scattering, we shall define the maximum effective range τr−max(2) in the second-order scattering. The maximum travel-path in the second-order scattering can be defined as shown in Figure 3 for a very long range, in which the wave incident along one of the edge of footprint is scattered twice at the bottom of layer with the scattering angle of 90°, eventually returning to the antenna along the opposite edge of footprint radius. Thus, τr−max(2) can be calculated as:
When considering τr−max(2), l1co is not included in the total returned signal. Furthermore, since the extinction rate κe is common for both l2co and l2cx in the isotropic medium, the LDR corresponding to τr-max(2), referred to as LDRaym, is determined only by scattering amplitude matrix, regardless of values of τr-max(2). LDRaym can be explicitly calculated through equation (28) in Kobayashi et al.  with θ = 90°. Hence, at 95 GHz, spherical water particles of diameter 1 mm yield LDRaym = −6.4 dB for an arbitrary τr-max(2). In Figure 2, τr-max(2) is infinite so that the LDR in Figure 2c slowly approaches to its asymptotic value of LDRaym = −6.4 dB, theoretically at τr = ∞, and l2co and l2cx in Figure 2a would remain forever if no absorption κe = 0 was assumed.
 As a reference, in the Rayleigh scattering regime, LDRaym is always calculated as 1/3 = −4.8 dB, which is a good indicator to check a given second-order scattering code.
 To see the effect of a finite footprint radius, a normalized footprint radius Ξr = 0.2 is adopted in Figure 4 with the other parameters kept same as those in Figure 2. In Figure 4a, the first-order term l1co is equal to that of Figure 2a, being invariant of change in Ξr. On the other hand, l2co and l2cx in Figure 4a become lower than those in Figure 2a, and especially l2co becomes always lower than l1co, because the second-order scatterings with scattering angles near θ′ = 90° (lateral scattering) can not be effectively captured with a footprint radius much smaller than the mean free path. In other words, most of lateral scattering goes out of sight in the radar. Inversely, the second-order scattering in the case of Ξr = 0.2 and τr = 2 is strongly dominated by the forward-backward-type scattering (beam-line direction). In this situation, the second-order path tends to go close to the first-order path to be subject to having equal absorption, which explains that SR and LDR in Figure 4c show approximately constant values around τr = 1 − 2. Alternately, this inefficient capture of lateral scattering can be characterized by its small effective maximum range τr-max(2) = 2.2. In Figure 4c, LDR quickly approaches to the asymptotic value LDRaym = −6.4 dB at τr > 2.2 (NB: if the rectangular transverse profile is used instead of the Gaussian one, the convergence is obtained exactly at τr-max(2) = 2.2). Furthermore, the rapid decreases in l2co, l2cx (Figure 4a) and l1co + l2co, l2cx (Figure 4b) after τr = 2.1 can be also explained by the small τr−max(2) = 2.2. In measurement, the values at τr > 2.1 in Figures 2 and 4 can be in noise area so that we can hardly observe the calculated behaviors.
 In Figure 5a, the second-order power-returns l2co and l2cx from Figure 2a and Figure 4a are compared as functions of the normalized range τr from 0 to the rear edge τd = 2. The black solid and dashed lines represent the values of l2co and l2cx for Ξr = ∞, respectively. The gray solid and dashed lines represent the values of l2co and l2cx for Ξr = 0.2, respectively. In both polarizations, differences between Ξr = ∞ and Ξr = 0.2 are negligible at τr < 0.2. However these differences increase, as τr increases. For further consideration, we shall define the second-order reflectivities as
where the superscript ⊕ represents one of co and cx. L2co and L2cx are plotted in Figure 5b. For Ξr = ∞, as τr increases, the reflectivities L2co (black solid line) and L2cx (black dashed line) monotonously increase. However, for Ξr = 0.2, the curves of L2co (gray solid line) and L2cx (gray dashed line) are almost flatten in the region of τr ⪆ 1 and τr ⪆ 0.5, respectively. This fact is related to the ineffective capture of lateral scattering for a small footprint radius as mentioned for Figure 4a; l2co and l2cx are controlled mainly by the longitudinal absorption like the first-order return l1co, giving very small changes in L2co and L2cx as the range increases. Furthermore, since the contributing ratio of lateral scattering to the forward-backward-type scattering is larger in cross-polarization than in copolarization [Kobayashi et al., 2005], the effect of footprint radius becomes larger in L2cx than in L2co. This is the reason that the curve of L2cx starts flattening at τr ⪆ 0.5, much before that of L2co does at τr ⪆ 1. As a consequence, we can summarize that, as τr increases, the effect of the normalized footprint radius Ξr on the second-order scattering (i.e. the differences between Ξr = ∞ and Ξr = 0.2) becomes more evident, especially in cross polarization.
 The normalized power-returns l1co, l2co and l2cx are plotted as functions of the normalized footprint radius Ξr for fixed normalized ranges τr = 0.6 and 1.8 in Figures 6a and 6b, respectively. As mentioned for Figure 4, the first-order term l1co is invariant for change in Ξr. The second-order terms l2co and l2cx decrease strongly for Ξr ⪅ 1. On the other hand, for Ξr ⪆ 2, these values asymptotically approach to the values predicted by the plane wave incidence case that is given by Ξr = ∞. This behavior is consistent with the time-independent theory [Kobayashi et al., 2005]. As also stated for Figure 5, the effect of footprint radius on l2co and l2cx becomes more evident with τr; for τr = 0.6 (Figure 6a), the increments in l2co and l2cx from Ξr = 0.2 to Ξr = 4 are 2.5 dB and 3.6 dB respectively, while for τr = 1.8 (Figure 6b), these increments are enhanced to 6.0 and 8.4 dB, respectively.
 The time-independent theories [Ishimaru and Tsang, 1988; Tsang and Ishimaru, 1985; Mandt et al., 1990; Mandt and Tsang, 1992; Kobayashi et al., 2005] give analytical formulas of multiple scatterings including both the ladder (intensity derived from radiative transfer theory) and cross terms (backscattering enhancement) in a single layer of finite thickness. These theories can be applied only to the steady state, but not for the non-steady state such as for pulsed-radar operation. Kobayashi et al.  derived a condition under which the time-independent formalism can estimate power-return for a pulsed radar, being summarized as; when a pulse resolution is larger than the mean free path of a given medium (i.e. τp > 1), the time-independent theory give a rough estimation of return signals on pulsed-radar operation near the top surface of the medium. Since the occurrence probability of the ladder term, in which intensity is taken into account, is equal to that of the cross term in the second-order theory for spherical particles, we can assess the aforementioned condition by studying only the ladder terms. We shall consider the normalized power-returns l1co, l2co and l2cx from the first range for which the normalized range τr is set equal to the normalized range resolution τp (i.e. τr = τp). l1co, l2co and l2cx are plotted as functions of τp for Ξr = ∞ and Ξr = 0.2 in Figures 7a and 7b, respectively. In the figures, the gray solid line l2ico and the black solid line l2dco represent the values of l2co, calculated by the time-independent theory [Kobayashi et al., 2005] and by the time-dependent theory equation (25), respectively. The corresponding values of l2cx are plotted by the gray dashed line l2icx and by the black dashed line l2dcx. There is no difference in l1co between these two theories as expected. Figures 7a and 7b indicate that, as the normalized footprint radius Ξr increases, it is necessary to adopt a larger τp to approximate the time-dependent value l2dco/cx by the time-independent value l2ico/cx. These calculations quantify the applicability of the time-independent results to the finite-pulse-duration case under the condition derived by Kobayashi et al. .
 A practical usability of the time-independent theory is to cross-check the values calculated by a time-dependent theory/simulation for a single layer of random medium. For this purpose, first, a range resolution should be chosen as τp ⪆ 2 for the time-dependent theory/simulation, and then a return from the first range τr = τp should be compared to that of the time-independent theory.
 Some remarks from our very recent studies are summarized. The second-order theory was compared to a Monte Carlo simulation including higher order scattering [Battaglia et al., 2007]. When the same parameters as those in Figure 2 of this paper were used for the 95 GHz radar, the second-order theory underestimated the total copolarized return-power by 0.1 dB at the normalized range τr = 1 in the plane-wave injection case, and this underestimation was deteriorated to 2 dB at τr = 2. This comparison confirmed the statement in Tsang and Kong  that the second order theory is a good approximation for the optical thickness less than or around 2. When the conventional first-order theory was used instead of the second-order theory, it gave a smaller copolarized-return by 4 dB at τr = 2 (see Figure 2b). The corresponding underestimation at 35 GHz was calculated as 1.6 dB. Beyond the optical thickness of 2, it is necessary to develop a more precise analytical theory for a systematic study of multiple scattering, and also in viewpoint of computational time.
 Another interesting issue is the effect of non-spherical particles. For raindrops, the eccentricity ɛ can be practically limited in 0.4 ≤ ɛ ≤ 1 [Oguchi, 1983]. Using this result, Kobayashi et al.  showed that the dependencies of second-order reflection on ɛ in rains were not so severe in either polarization, even when considering a thick rain layer. On the other hand, since ɛ of ice particles ranges so broadly, they concluded that the non-spherical effect of ice particles on the second order theory should be considered case by case.
 In this paper, the entire formulation has been based on the radiative transfer theory that cannot intrinsically include backscattering enhancement caused by cross-terms in the analytic Green function method. In copolarization, the backscattering enhancement generally doubles the values of second-order scattering since the ladder term (intensity derived from radiative transfer theory) is equal to the corresponding cross term. In cross polarization, these two terms are not always equal [Mishchenko, 1991]. However, Kobayashi et al.  demonstrated that for raindrops and ice particles, the cross term is nearly equal to the ladder term in cross polarization as long as the second-order theory was concerned. For the third and higher order scatterings, Oguchi and Ihara  simulated the ladder and cross terms up to the 7th order in both polarizations by using large water spheres of diameter 12.5 mm at 30 GHz. Their results showed that differences between the ladder and cross terms in cross-polarization increase as the scattering order increases. All the analytic Green function methods in the past studies were performed under steady process. The effect of cross terms in time-dependent process will become a fruitful topic in the future.
 An analytical method has been studied to represent the multiple scattering returns on pulsed radar operation, based on a time-dependent radiative transfer theory [Ito et al. 2007]. The Gaussian transverse beam-profile and the rectangular pulse-duration are taken into account. The second-order returns from a single layer of spherical water particles of uniform diameter have been derived, while the method itself can be extended to higher-order scattering in a multiple layered structure with general drop size distributions. The main focus of this paper is on the effect of finite footprint radius. General conclusions can be stated as follows. (1) The second-order power return/reflectivity increases in both polarizations, as the ratio of footprint radius to mean free path increases. (2) The effect of footprint radius becomes more significant, as the ratio of range to mean free path increases. This effect is more conspicuous in cross-polarization than in copolarization. As a special case, we should note that, when the pulse passes over the rear edge of a medium, the second-order power-return decays more slowly as the footprint radius increases. Including this special case, conclusion (2) can be well represented in linear depolarization ratio (LDR). (3) The second-order power return/reflectivity increases with the range resolution.
Equations (20) and (A16) indicate that the first-order scattering satisfies backscattering as expected. Furthermore, as long as spherical particles are concerned, the directions r and i1 defined in equation (A11) can be replaced with an arbitrary set of directions satisfying the backscattering condition such as equations (23) and (24). Thus we deduct for the first-order scattering:
Now we shall use further approximations:
To perform the integral over d(Θ, Φ) in equation (17), the area element subtended in the direction = (Θ, Φ) at z = 0 is introduced
 In this section, derivation of equation (25) is described. The second-order specific intensity can be obtained by substituting equation (13) defined for μz < 0 and its counterpart for μz > 0 into equation (2). The results can be written at z = 0 for a general direction with the constraint μ = cosθ < 0:
where the subscripts 2bf in equation (A23) and 2bb in equation (A24) follow the notation of Ito et al. ; ′2′ denotes the second-order scattering, and ′bf′ denotes that the first-time scattering is forward ′f′ (μ > 0), followed by the backward second-time scattering ′b′ (μ < 0). The other subscript 2bb can be defined in a similar manner. In equations (A23) and (A24), the incident direction Ωi2 for the second-order scattering can be defined:
in which ϕρ″ is defined in the same manner as equation (10). Using equations (A23) and (A24) under the narrow beam condition, we obtain the second-order returned intensity received by the antenna at time t + R/c: