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 A second-order scattering theory is derived for uniformly oriented spheroidal hydrometeors with a general drop size distribution in a single layer. Effects of backscattering enhancement are evaluated for a nadir pointing radar with a finite beam width at millimeter wave frequencies. Spheroids of uniform size are first studied to characterize the deviations from the spherical theories. For water spheroids, the difference between the cross term (backscattering enhancement term) and the ladder term (conventional multiscattering term) becomes larger as the spheroids deviate from the spherical shape. On the contrary, for ice spheroids, this difference is negligible. Marshall-Palmer distributed rains with Pruppacher-Pitter-form drops are studied using the spheroidal approximation. The results confirm the validity of the spherical approximation used in the preceding works within the accuracy limit of measurement. In the Marshall-Palmer distributed rains, the second-order scattering effect becomes serious as the footprint radius increases. Hence the second-order scattering is to be taken into account for accurate measurement of hydrometeor reflectivity for a spaceborne radar because of its large footprint radius.
 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.  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. .
 The backscattering enhancement under the finite beam width was measured in a controlled laboratory experiment by Ihara et al.  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 . 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.
 For remote sensing applications, Marzano et al.  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.  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. , 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.
 In this paper, a theory of finite beam width by Kobayashi et al.  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. . 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.  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.
 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.
 The scattering geometry is illustrated in Figure 1. To reflect the backscattering enhancement in the formalism, a bistatic radar configuration with a scattering angle θs ≪ 1 is assumed. Uniformly oriented spheroidal hydrometeors are randomly distributed with a general drop size distribution in a layer of thickness d. Point O is the center of the footprint at the top surface of the medium layer. Unit vectors i and s denote the directions of the incident wave along the transmitting antenna axis TO and the scattered wave along the receiving antenna axis RO, respectively. ri and rs denotes the incident and scattered distances from points T and R to the top surface, respectively. Because of θs ≪ 1, it is assumed that ri ≈ rs. The incident spherical wave EA transmitted from the transmit antenna at T is first scattered by a scatterer at point rb in the medium, and scattered again at point ra, eventually being received by the receive antenna at R. In general, the propagation wave number in a medium of spheroidal particles oriented in a preferential direction, is anisotropic in space so that its propagation Green function has a dyadic form. To simplify the treatment of the dyadic Green function, two coordinate systems, the “conventional frame” and the “reference frame,” are used throughout this paper. In Figure 2a, the Cartesian coordinates of the conventional frame is denoted by (, , ) axes. An arbitrary direction P is defined by the polar coordinate (θ, ϕ). Furthermore, the symmetry axis of the spheroids is assumed to be in the direction defined by (θB, ϕB). Then, the reference frame is defined so that the B axis of the principal axes (B, B, B) coincides with the symmetry axis of the spheroids. In the reference frame, the polar coordinate corresponding to direction P can be defined by (Θ, Φ)R, and one eigen polarized direction of a wave propagating in direction P coincides with direction , which is tangent to arc BP. The other eigen direction is in direction orthogonal to . In Figure 2b, the incident direction I is represented by (π, 0) in the conventional frame, or (π − θB, 0)R in the reference frame, while the scattered direction S is represented by (θs, 0), or (Θs, Φs)R, respectively. In the same manner as for point P in Figure 2a, the eigen polarized directions for the incident (I) and scattered (S) waves can be defined although not marked in Figure 2b. Using these eigen polarized directions, the propagation Green function from point rb to point ra (Figure 1) in the medium can be represented in the following diagonal form
In equation (1), the Dirac notation of matrix ∣β〉 exp[ikβ∣ra− rb∣] 〈β∣ is used (see section A1). The index β represents one of directions and to specify a diagonal component of matrix with its element exp[ikβ∣ra − rb∣]. The complex wave number kβ in the element is a function of polar coordinate Θ and the eccentricity of spheroid ε. Using the Foldy-Oguchi-Twersky approximation [Oguchi, 1973], kβ can be represented as
where k is the wave number in air, Fββ(, ; D, ε) denotes a ββ component of the forward scattering amplitude matrix in direction = (Θ, Φ)R for a spheroidal hydrometeor equivolumic to a sphere of diameter D, and N(D) is a drop size distribution function of particles equivolumic to spherical diameter D. Note that kβ in equation (2) depends only on angle Θ in the reference frame because of its axial symmetry. In rigor, equation (2) is valid, only when the particle distribution is completely uniform (no density correlation), and the ergodicity is satisfied [Mishchenko, 2002]. Inversely, when the radar illuminates a small sampling volume with continuous wave like in this formalism, we should carefully use equation (2). For the sake of simplicity, in the rest of paper, the complete uniformity of a medium will be assumed even in the small volume.
 The incident wave ψ(rb) at point rb, and the Green function 01(ra) from point ra to the receiving point R can be rewritten as
where Pt and G0 denote the transmitting power and the center gain of the antennas, respectively. Note that the transmitting and receiving gains have been assumed to be equal. ra and rb denote the transverse lengths of points ra and rb, respectively as shown in Figure 1. σr is the footprint radius defined by
in which large ranges ri ≈ rs ≫ d, and a small 3-dB aperture angle θd ≪ 1 have been assumed (Figure 1). The vector ψ0 in equation (3) represents the unit vector of the initial polarization. The propagation wave numbers in the incident and receiving directions in equations (3) and (4) can be represented on the basis of work by Ishimaru and Tsang  as
in which the subscripts i and s denote the incident and scattered waves, respectively. The incident direction i in equation (6) can be written as
The scattered direction s and complex wave number ksβ in equation (7) have the same expressions as those in equation (8) and (9) after replacing the subscript i with s. The imaginary parts of the wave numbers in equations (6) and (7) are defined with the incident angle θi = π and receiving angle θs as
 In the analytical wave theory, the intensity of the first-order scattering is represented by the first-order ladder term, the scattering diagram and geometry of which are illustrated in Figures 3a and 3b, respectively. The intensity of this term (IL(1)) is equivalent to the returned intensity predicted by the radar equation in the single scattering regime as proven by Kobayashi et al. . The intensity of the second-order scattering is represented as a sum of those of the second-order ladder (IL(2)) and cross (IC(2)) terms. The diagram of IL(2) is represented in Figure 4, and the corresponding geometry is depicted in Figure 1. The conjugate field to the field EA is the field EA itself. Hence the intensity IL(2) is equal to that calculated from the radiative transfer theory. The second-order cross term is seen in Figure 5. The conjugate field to the field EA is the field EB as shown in (b) for a general scattering angle θs ≠ 0. The field EB takes the time reversal path of EA only in the case of the direct backscattering θs = 0 as shown in Figure 3c, resulting in a strong correlation of intensity. As θs increases, Ic(2) reduces to zero because of decorrelation caused by the break of time reversal condition. As a consequence, the total second-order intensity IL(2) + IC(2) has a sharp peak at θs = 0 with a higher intensity than that calculated from the radiative transfer theory (i.e., IL(2)). This is the reason we call it ‘backscattering enhancement’.
 In the system of spherical particles, the relation IL(2) = IC(2) was proven by Kobayashi et al. . However, in the case of spheroidal particles, the property of the scattering amplitude matrix may not guarantee the above relation. Furthermore, the propagation anisotropy may give effects on the intensities. Using the propagation anisotropy–included Green functions (equations (1) and (4)) and the incident wave (equation (3)), the intensities IL(1), IL(2), and IC(2) can be calculated for uniformly oriented spheroids by following the formalism of Kobayashi et al. . Derivation and results are found in sections A2–A4.
 In the following sections, only the direct backscattering (i.e., θs = 0) will be considered because the monostatic radar is our main concern. Analysis associated with a small and finite θs can be found in the work by Kobayashi et al. .
3. Spheroidal Particles of Water and Ice With a Drop Size Distribution of Uniform Size
Kobayashi et al.  mainly used spherical water particles of uniform diameter D = 1 mm with a particle number density N0 = 5 × 103 m−3 for calculations. To illustrate differences between the spherical and spheroidal formalisms, water spheroids which are equivolumetric to the sphere of diameter D = 1 mm with N0 = 5 × 103 m−3 are first considered at 95 GHz. The eccentricity = b/a, in which a and b are respectively the horizontal and vertical diameters of spheroid, is assumed to have values from 0.4 to 1 because this range sufficiently covers eccentricities that are typical for raindrops [Oguchi, 1983]. The horizontal diameter a of spheroid is determined by the equation
In the following analysis, it is assumed that the direction of the incident wave represented by point I in Figure 2b has the initial polarization in direction, and the symmetry axis of the spheroid is oriented in the direction B (θB = 30°, ϕB = 30°). The footprint radius σr is assumed to be σr = ∞ so that the plane wave incidence theory is applied. When the layer thickness is set at d = 100 m, the corresponding optical thickness τd varies from τd = 1.3 at ɛ = 1 to τd = 1.8 at ɛ = 0.4. For this configuration, normalized power returns are plotted in dB unit as functions of ɛ in Figure 6. l1co/cx and l2co/cx represent the first and second-order ladder terms in copolarized/cross-polarized returns, calculated from equations (A6) and (A8), respectively, and are normalized by the radar constant B given in equation (A7). These four terms correspond to the first and second-order scattering terms calculated through radiative transfer theory. c2co and c2cx calculated from equation (A11) along with normalization by equation (A7) are the second-order cross terms in copolarized and cross-polarized returns, respectively, which cause the backscattering enhancement. For reference, the normalized total second-order returns are plotted with black dotted lines for copolarization (l2co + c2co) and cross-polarization (l2cx + c2cx) returns. The scattering amplitude matrices F in equations (A6), (A9), and (A12) are calculated by a T matrix code [Mishchenko, 2000; Mishchenko et al., 2002] and a least squares point-matching code [Oguchi, 1973]. The power returns at ɛ = 1 correspond to the values of a sphere of diameter D = 1 mm, and hence are independent of the orientation of spheroidal symmetry axis. These values have been used to cross check the present formalism with the more convenient theory for sphere [Kobayashi et al., 2005]. As seen from Figure 6, the relation l2co = c2co is always satisfied as proven by Mishchenko [1991, 1992]. On the other hand, l2cx and c2cx are equal only for ɛ = 1 (i.e., sphere), and the discrepancy of these two terms get larger as ɛ deviates from 1, reaching l2cx/c2cx = 3 dB at ɛ = 0.4. The proof of l2cx = c2cx at ε = 1 is described by Kobayashi et al. . As also shown in Figure 6, l1co reaches the maximum at ɛ ≈ 0.7 although it appears rather insensitive to change in ɛ. This extremum can be explained as follows. Equation (12) implies that when seeing the noncanting spheroid (θB = 0°) from above, its projected area represented by π · (a/2)2 = 4−1πD2ɛ−2/3 decreases with ɛ, resulting in a decreasing function l1co with ɛ. On the other hand, at θB = 90°, the projected area becomes πab/4 = 4−1πD2ɛ1/3, and l1co turns out to be an increasing function with ɛ. The above fact suggests that for a given canting angle, l1co may exhibit an extremum at some value of ɛ. It is further noted that the effect of azimuthal rotation ϕB on l1co is not so large as that of θB.
 We next consider ice hydrometeors that have large single-particle albedos with a variety of values of ɛ. In Figure 7a, the normalized power returns are calculated for ice with the same N0, D, θB and ϕB as those in Figure 6, and by assuming infinite footprint radius. The range of ɛ is limited from 0.3 to 3, because both the T matrix code and the point-matching method code yield unreliable results for out of this range. The ice layer thickness d = 500 m is chosen so as to give a similar optical thickness τd to the water hydrometeor case of Figure 6. With this layer thickness, τd for the ice hydrometeors varies between 1.4 at ɛ = 0.3 and 0.6 at ɛ = 3. Notice in Figure 7a that, in copolarization, the normalized total second-order power return l2co + c2co is larger than the first-order power return l1co over the entire range of ɛ. In cross-polarized returns, l2cx and c2cx, although not equal except for ɛ = 1, have very close magnitudes to each other for all ε. Figure 7b is calculated for a finite footprint radius σr = 100 m, while the other parameters are kept the same as those in Figure 7a. Note that the first-order terms l1coand l1cx are invariant for the change in σr, while the second-order terms l2co, l2cx, c2co and c2cx decrease from those of Figure 7a by 2–3 dB. As a consequence, l2co + c2co becomes larger than l1co only in the range of ɛ ≥ 1.8. Unlike l1co for the water hydrometeor case in Figure 6 which exhibits a maximum at ɛ> ≈ 0.7, l1co for ice hydrometeors in Figure 7 monotonously decreases with ε for the same canting geometry. In general, the range of canting angle θB in which l1co exhibits an extremum depends on the azimuthal orientation angle ϕB, particle size D and the physical state of hydrometeors such as dielectric constant. In fact, it was confirmed that l1co exhibits a maximum at a larger canting angle (e.g., θB = 60°).
 In Figures 8a and 8b, the normalized power returns are plotted as functions of footprint radius σr for ɛ = 0.5 and 2.5, respectively, with the other parameters being kept the same as those in Figure 7. σr is normalized by the average of the two orthogonal mean free paths in the incident direction (lfree), and lfree are calculated as 408 m and 758 m for ɛ = 0.5 (Figure 8a) and 2.5 (Figure 8b), respectively. In Figures 8a and 8b, all the second-order terms decrease rapidly for σr/lfree ≲ 1. As σr/lfree increases, these terms asymptotically approach the values predicted by the plane wave incident model that is given by substitution of σr/lfree = ∞. It is noted that a similar behavior was also reported for spherical hydrometeors in the work by Kobayashi et al. .
4. Marshall-Palmer Distributed Rains
 In an experimental study of multiple scattering in rains performed by Ito et al.  using a 35 GHz radar, large values of linear depolarization ratio (LDR) were observed and, attributed to second-order scattering rather than to first-order scattering from nonspherical raindrops. In their formalism, however, spherical particles were assumed instead of deformed raindrops. In this section, the spheroidal formulation is adopted to validate their observed results at millimeter wave frequencies. For this purpose, equations (A6), (A8), and (A11) are applied to Marshall-Palmer distributed rains with raindrops deformed according to the model of Pruppacher and Pitter . On the basis of the works of Oguchi , the Pruppacher-Pitter deformed drops can be approximated by spheroids with eccentricity:
where D is the diameter of an equivolumic sphere in mm. Since the stable raindrop was reported to have D ≲ 6 mm [Beard et al., 1989], the corresponding range of ɛ can be calculated as 0.7 ≲ ɛ ≤ 1. The Marshall-Palmer drop size distribution N(D) is defined as
where R is the rain rate in mm/hr. A radiation frequency of 95 GHz is used for analysis for the following reasons. First, the effect of multiple scattering from deformed raindrops will be more pronounced at 95 GHz than at 35 GHz. Second, there is a demand of interpretation of light rain measurements by the W band spaceborne radar for the upcoming CloudSat mission [Im et al., 2005; Stephens et al., 2002].
 In Figure 9a, the normalized reflectivities of the second-order scatterings from the Marshall-Palmer distributed rains are plotted as functions of R. L2co/cx and C2co/cx denote the second-order reflectivities of the ladder and cross terms in copolarized/cross-polarized returns, which are defined through normalization by the first-order copolarized power return, respectively:
The rain layer thickness is assumed to be 500 m, and the symmetry axes of spheroids are assumed to be noncanted, i.e., θB = 0°. Using equations (A6), (A8), and (A11) along with equations (13)–(16), the spheroidal approximation of L2co(= C2co) is shown by the dash-dotted line (overlapping the solid line), and L2cx(∼C2cx) is shown by the dotted line. The corresponding results of the sphere approximation obtained by substituting ɛ = 1 into equations (A6), (A8), and (A11), are shown for L2co(= C2co) by the solid line, and for L2cx(∼C2cx) by the dashed line. It is noted that the two approximations in copolarized returns (L2co) yield almost the same values. For cross-polarized returns, the spherical approximation is slightly higher than the spheroidal approximation. Before detailed comparisons between these approximations are considered, some remarks are noted. In the noncanting case (θB = 0°), since the symmetry axis of the spheroids is parallel to the incident and scattered directions, the first-order cross-polarized reflectivity L1cx is zero. Furthermore, the sum L2cx + C2cx roughly represents LDR, because L2coand C2co are much smaller than L1co = 1.
 We next define the following ratio in copolarized second-order reflectivity to compare the spheroidal approximation with the spherical approximation:
Here, L2co(θB, ϕB; ε < 1) given by equation (15) represents the spheroidal approximation with the orientation angles θB, ϕB, and L2co(ε = 1) represents the value in the spherical approximation. In the same manner, the ratio in cross polarization can be defined as
Our analysis primarily focuses on canting angles θB between 0° and 45°, because this range covers over 90% of rainfall canting angle at low altitude [Oguchi, 1983]. In Figure 9b, the results of Qco/cx(0°, ϕB) (noncanting case) and Qco/cx(45°, 45°) (45° canting case) are plotted as solid lines for copolarization, and as dashed lines for cross polarization, where ϕB represents an arbitrary angle between 0° and 360°. For reference, Qcx(90°, 45°) in cross polarization is also plotted as the dotted line. However, its counterpart in copolarization, i.e., Qco(90°, 45°), which almost overlaps Qco(45°, 45°), is not shown in Figure 9b for clarity purpose. Note that for ϕB between 0° and 360°, Qco(45°, ϕB) changes by less than ±0.01 dB, and Qcx(45°, ϕB) changes by at most 0.04 dB. On the contrary, for θB between 0° and 45°, Qco(θB, 45°) and Qcx(θB, 45°) vary by as much as 0.04 dB and 0.29 dB, respectively. Thus the effect of ϕB is much smaller than that of θB in both polarizations.
 In Figure 9b, the following three facts are worth mentioning regarding the differences between the spheroidal and spherical approximations: (1) Qco(θB, ϕB) is less sensitive to change in θB than Qcx(θB, ϕB) is; (2) Qco(θB, ϕB) is closer to 1 (0 dB) than Qcx(θB, ϕB) is, (e.g., Qco(0°, ϕB) = −0.04 dB, Qcx(0°, ϕB) = −0.32 dB at R = 10 mm); (3) Qcx(0°, ϕB) shows the largest deviation from the spherical approximation. The proof are described in section A5. In Summary, the maximum deviation between the spherical and spheroidal approximations will be limited to 0.32 dB at R = 10 mm/hr. Since 0.32 dB difference in hydrometeor reflectivity measurements is generally considered to be well within the accuracy limitation of the measuring equipments, the above results confirms the validity of adopting the spherical approximation for the study of multiple scattering in light rains at low altitude with millimeter wave radars, such as the 95 GHz Cloud Profiling Radar for the CloudSat Mission [Im et al., 2005; Stephens et al., 2002], and the 35 GHz radar used in the multiple scattering experiment [Ito et al., 1995].
 Since the canting angle θB = 45° is sufficiently large, one would expect L1cx to be comparable to L2cx(∼C2cx). However, the calculated L1cx turns out to be much lower than L2cx(∼C2cx) by 20 dB at 95 GHz. As a reference, L1cx at 35 GHz is smaller than L2cx(∼C2cx) by 10 dB. The above suggests that, at millimeter wave frequencies, cross-polarized returns from raindrops in nadir operation are dominated by multiple scattering rather than by the first-order cross-polarized return caused by non spherical effects.
 The normalized total copolarized and cross-polarized reflectivities from Marshall-Palmer distributed rains are represented by 1 + L2co + C2co and L2cx + C2cx, respectively. In Figure 10, these reflectivities at 35 and 95 GHz are calculated as functions of rain rate R by using the spherical approximation. Notice that if multiple scattering is not taken into account, the rain reflectivity at 10 mm/hr would be overestimated by as much as 2 dB at 95 GHz, and 0.8 dB at 35 GHz, respectively. Furthermore the LDR, which reaches −12 dB at 95 GHz and −15 dB at 35 GHz would be attributed to the single scattering from extraordinarily deformed raindrops.
 By adopting spherical approximation, the normalized reflectivities L2co(= C2co) and L2cx(= C2cx) of Marshall-Palmer distributed rains at R = 10 mm/hr are plotted in Figure 11 as functions of the footprint radius σr normalized by its mean free path lfree = 526 m. The other parameters are kept the same as those in Figure 9a. As similar to the results in Figure 8, the reflectivities are practically constant for σr/lfree > 3, and asymptotically approaching the value predicted by the plane wave incidence theory of Mandt and Tsang . For σr/lfree ≲ 1, the reflectivities rapidly decrease with σr/lfree.
 In this paper, the second-order scattering for ice particles with a realistic drop size distribution was not calculated, although Figure 7 and 8 imply that these effects will be large for ice particles. For ice hydrometeors, the orientations of the symmetry axes are rather complex. For instance, the symmetry axes of needle-shaped ice particles have been reported to be completely randomly oriented in the horizontal plane in the case of free falling [Bringi and Chandrasekar, 2001]. However, if these symmetry axes were assumed to be uniformly oriented in the horizontal plane, the real parts of the two eigen values of propagation wave numbers in the nadir direction could be very different, because of their asymmetric shapes based on the mean field theory (i.e., equation (2)). In this situation, the first-order cross-polarized returns would be comparable to the first-order copolarized returns due to phase rotation, as the layer thickness increases. On the contrary, when the randomness is taken into account, the real parts of the two eigenwave numbers become equal to each other, and the cross-polarized returns will be much smaller than the copolarized return as expected in measurement.
 The derivation on uniformly oriented spheroids in this paper can be easily extended to randomly oriented particles of arbitrary shapes, if the two eigen polarized directions of propagation are roughly aligned in the directions and about a certain axis. In this case, the direction B in Figure 2 does not always coincide with a direction of the symmetry axis of a particle. For instance, when considering needle-shaped ice particles free falling in air, calculation can be proceeded by first taking ensemble average of the scattering matrix over azimuthal rotation about the vertical axis. If the two eigen polarized directions of propagation are roughly aligned in the directions and about a certain axis, this axis can be considered as the principal axis direction B in Figure 2, and the formulation in section 2 can be used directly.
5. Summary and Future Works
 The second-order scattering theory associated with backscattering enhancement [Kobayashi et al., 2005] has been extended to cover a single layer of spheroidal particles with a general drop size distribution. The spheroids are assumed to be uniformly oriented in a direction so that anisotropic propagation of wave should be included in the formulation. Effects of backscattering enhancement have been quantitatively evaluated for a 95 GHz nadir pointing radar with a finite beam width.
 Spheroids of uniform size are first studied to characterize the deviations from the spherical theories [Kobayashi et al., 2005; Mandt et al., 1990; Mandt and Tsang, 1992]. For the water particles with eccentricity ɛ < 1, the second-order ladder l2cx and cross c2cx terms are generally unequal, which is in agreement with a general theory of Mishchenko [1991, 1992]. The inequality l2cx > c2cx reduces as ɛ approaches 1, and becomes equal for ɛ = 1 (i.e., sphere). On the contrary, for ice particles, the differences between l2cx and c2cx are negligible. In Figure 7a, an infinitely large footprint radius is assumed, and the normalized total second-order copolarized power return l2co + c2co is larger than the normalized first-order copolarized power return l1co in the entire range of ɛ. These large values of second-order scattering can be attributed to large albedo (≈1) of an ice particle of diameter of 1 mm. When the footprint radius is reduced to 100 m as shown in Figure 7b, l2co + c2co exceeds l1co only in the range ɛ ≥ 1.8 because of reduction in reflectivity of the second-order scattering. The dependences of the normalized power returns on footprint radius are explicitly shown in Figure 8. Notice that the second-order power return rapidly decreases for σr/lfree < 1, while for a large σr/lfree, it asymptotically approaches the values predicted by the plane wave incident case (σr/lfree = ∞).
 Assuming Marshall-Palmer distributed rains of Pruppacher-Pitter-type spheroid particles [Oguchi, 1983], the second-order power returns have been calculated for the spherical and spheroidal approximations. When the reflectivities defined in equations (15) and (16) are adopted at 10 mm/hr, the difference between the spherical and spheroidal approximations is negligible in copolarization, while it is limited within 0.4 dB in cross polarization. This result confirms the validity of the spherical approximation as used by Ito et al. . Furthermore it is found that if the multiple scattering is not taken into consideration, the rain reflectivity would be overestimated, and the observed large LDR could not be reasonably interpreted. The dependences of the normalized reflectivities on footprint radius are also calculated for rains at 10 mm/hr in Figure 11, which shows similar results to those in Figure 8. For a spaceborne radar mission in millimeter wavelength frequencies, the footprint radius is comparable or superior to the mean free path of hydrometeors. Hence the multiple scattering should be taken into account in light rain measurements.
 The theory in this paper is time-independent, and it cannot be applied directly to pulsed radars. However, when the range resolution of a pulsed radar lres is larger than the mean free path lfree in a medium (i.e., lres > lfree), the works of Kobayashi et al.  showed that the time-independent theory can estimate the power return from a pulsed radar near the top surface of the medium. Very recently, Ito et al.  and Kobayashi et al.  derived time-dependent formulas of the second-order scattering from spherical particles illuminated by a pulsed radar with a finite beam width. In the future, these time-dependent formula should be extended to apply to spheroidal particles by taking into account the propagation anisotropy. When it is done, the time-independent theory in this paper can be used to cross check the value of the time-dependent theory near the top surface of medium with the condition lres > lfree.
 Throughout this paper, the optical thickness has been assumed to be at most 2, within which the second-order approximation is valid [Tsang and Kong, 2001]. When the optical thickness is very large, the diffusion approximation may work as a good approximation. For the scalar wave (e.g., sound wave), a diffusion approximation including both ladder and cross terms was constructed from the analytical wave theory for anisotropic scatterers [Tsang and Kong, 2001]. However, it is not so direct to extend their theory to vector waves such as electromagnetic waves. Although the millimeter wavelength radar is rarely used in the regime of the diffusion approximation, this topic will give a theoretical interest.
Appendix A:: Mathematical Details
A1. Dirac Notation
 As long as the orthonormal base system is concerned, ‘bra and ket method’ or ‘Dirac notation’ developed by P.A.M. Dirac is advantageous to perform a series of matrix multiplication, especially when a transformation of frame system such as rotation is accompanied. Simple and clear explanation of Dirac notation should be referred to Sakurai . Here, only essentials are described. In Dirac notation, the vector α is represented by ∣α. In the orthonormal base system, the dual correspondence to vector ∣α is simply complex conjugate, which is defined as α∣. Hence the inner product of vector ∣α with vector ∣β can be represented as β∣α. To represent a matrix operator T, we shall introduce a set of base vector ∣a〉 or ∣a′〉, which represents one of the base vectors. For instance, in the Cartesian system, ∣a〉 represents one of , , . The representation of matrix T can be written in the form of T = ∣a〉 〈a∣T∣a′〉 〈a′∣. When a and a′ are fixed, 〈a∣T∣a′〉 means the (a, a′) component of T. For instance, 〈∣T∣〉 represents (xz) component of T in the Cartesian system. The multiplication of matrix T with vector ∣α〉 can be written as T∣α〉 = ∣a〉 〈a∣T∣a′〉 〈a′∣α〉, which reduces to the linear combination of base vectors ∣a〉 multiplied by complex numbers 〈a∣T∣a′〉 〈a′∣α〉, as expected.
 In the case of complete random distribution (no density correlation), the diagram of the first-order ladder term IL(1) is translated to the integral form:
in which the superscript † indicates the complex conjugate of a dyad or a vector. The coordinate z is taken parallel to the layer thickness of d. F(s, i; D) denotes the scattering matrix of a spheroidal particle equivolumic to a sphere of diameter D, defined by the incident i and scattered s directions. N(D) is a drop size distribution function. Substitution of equations (3) and (4) into equation (A5) gives an explicit form IL(1) along with the notations of equations (A1) and (A2). For the nearly direct backscattering condition θs ≪ 1, IL(1) is simplified:
with the radar constant
A3. Second-Order Ladder Term
 For uniformly oriented spheroids, we can calculate IL(2) in the same manner as in the case of spherical particles [Kobayashi et al., 2005], but just including the propagation anisotropy into the incident field Ψ (equation (3)), and the Green functions 11(equation (1)) and 01 (equation (4)). Taking the ensemble average over the particle diameters D1 and D2, we can write IL(2):
where the new variables η = tanθ and ς = za − zb have been introduced, and the following matrix has been defined as
Furthermore, kβ defined in equation (2) has been decomposed as
The directional vector in equation (A9) is represented by the polar coordinate (θ, ϕ). Note that when performing the integrations over θ and ϕ in equation (A8), the scattering amplitude matrix F in equation (A9) must be calculated after transformation from (θ, ϕ) to (Θ, Φ)B.
A4. Second-Order Cross Term
 The integral form of IC(2) can be calculated in a similar manner to the ladder term:
As the scattering angle θs deviates from the right backscattering angle θs = 0, the value of qβαβα increases, resulting in strong decorrelation of the cross term (A11), which eventually reduces the backscattering enhancement. Equations (A8) and (A11) include the effect of spatial anisotropy caused by spheroidal particles. However, when considering spheres instead of spheroids in equations (A6), (A8), and (A11), it is easily seen that the spatial anisotropy disappears, and all propagation complex wave numbers become independent of propagating direction. As a consequence, in the limit of a large σr, equations (A6), (A8), and (A11) reduce to the forms of Mandt and Tsang .
 To explain facts (1) and (2), we note that the copolarized power returns l1co and l2co are strongly controlled by backward/forward scattering cross sections of raindrops [Kobayashi et al., 2005], so that these two terms show similar dependences on the eccentricity ε (0.7 ≲ ɛ ≤ 1) and the canting angle θB. On the other hand, the cross-polarized power returns l2cx and c2cx are generally much less sensitive to changes in these parameters than l1co, and they have the different dependences on ε (0.7 ≲ ɛ ≤ 1) and θB from those of l1co. As a result, the dependence of l2co and l1co on ε and θB are offset in L2co(θB, ϕB; ε) = l2co/l1co, whereas the ε and θB dependence of l2cx and l1co are not offset in L2cx(θB, ϕB; ε) = l2cx/l1co, resulting in L2cx that is strongly controlled by the denominator l1co. As a consequence, Qco(θB, ϕB) is less sensitive to change in θB, and it is much closer to unity (i.e., less shape dependence). To explain fact (3), we recall that strong dependence of L2cx on l1co means that L2cx is controlled by the backscattering cross section that is proportional to the geometrical cross section viewed at nadir. When θB = 0°, the geometrical cross section for ɛ = 1 is proportional to πD2/4, while that for ɛ < 1 is given by πa2/4 (>πD2/4) (see equation (12)). This relation along with equation (15) yields L2cx(0, ϕB; ε < 1) < L2cx(ε = 1) to give Qcx(0, ϕB) < 1 in equation (18) for an arbitrary ϕB. As θB increases, the cross section approaches πab/4 (<πD2/4) at θB = 90°. This implies that L2cx(θB, ϕB; ε < 1) in equation (18) increases as θB increases, while L2cx(ε = 1) in equation (18) is invariant. Thus Qcx(θB, ϕB) increases with θB as shown in Figure 9b. By considering the relation Qcx(0°, ϕB) = −0.32 dB < Qcx(45°, ϕB) ≲ 0 dB at R = 10 mm/hr together with fact (2), the fact (3) is proven.
 The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The authors thank Toshio Ihara at Kanto Gakuin University for elaborate advice and suggestion.