1. Introduction

An S-band (10-cm wavelength) bistatic radar system consisting of a monostatic (transmitter and receiver collocated) radar and a remotely located nonscanning bistatic receiver has been successfully demonstrated for measuring dual-Doppler vector wind fields (Wurman et al. 1993; Wurman et al. 1994; Wurman 1994). In this system, the transmitter and both receivers operate at a single polarization. It is of interest to determine the information that may be obtained about the hydrometeors in the scattering volume if the bistatic receiver is dual polarized. Dual-linear-polarization measurements with a monostatic radar have been effective in differentiating liquid and ice phase hydrometeors, detecting hail, and estimating rainfall rate (Seliga and Bringi 1976, 1978; Bringi et al. 1984; Bringi et al. 1986a; Bringi et al. 1986b; Goddard et al. 1982; Radio Science, special issue, 1984, Vol. 19, Issue 1; Jameson 1985; Aydin et al. 1986; Aydin et al. 1990; Aydin and Zhao 1990; Illingworth et al. 1986; Vivekanandan et al. 1990;Balakrishnan and Zrnić 1990a,b; Zrnić et al. 1993a; Zrnić et al. 1993b; Doviak and Zrnić 1993).

Bistatic radar systems have previously been investigated for use in observing clouds, precipitation, and clear air echoes (Atlas et al. 1968; Eccles and Rogers 1968; Doviak and Weil 1972; Crane 1974; Shupyatsky 1974; Olsen and Lammers 1978; Awaka and Oguchi 1982a,b; Dibbern 1987). Of particular interest are the work on the use of polarization in bistatic radar systems. Shupyatsky (1974) proposed depolarization measurements at various bistatic angles for differentiating large particles from smaller ones (relative to the wavelength). He reported successfully tracing the formation of large particles in cumulonimbus development with such measurements. Awaka and Oguchi (1982a,b) calculated bistatic radar reflectivities of raindrops at 5.33, 14.3, and 34.8 GHz for studying the interference caused by rain between two communication links operating at the same frequency. Based on the work of Shupyatsky (1974), they explored the idea of measuring the drop size distribution aloft using co- and cross-polarized reflectivities. They decided that bistatic azimuth angles near 90° and vertically polarized transmission would provide the best observations for this purpose. Dibbern (1987) also focused on bistatic scattering from rain. He investigated the relationships between median drop diameter and radar parameters such as differential reflectivity, and linear and circular depolarization ratios at various bistatic angles. Based on calculations and measurements at 33 GHz with a continuous wave (CW) bistatic radar, he concluded that scattering angles at 90°–100° were best suited for investigating rainfall, including the estimation of the drop size distribution.

This paper presents results from a computational study of bistatic scattering from rain and hail at 10.9-cm (S-band) and 5.4-cm (C-band) wavelengths. These are the most common frequency bands used around the world for observing weather phenomena. The potential of bistatic polarimetric radar parameters for differentiating rain and hail and estimating hailstone size distribution will be investigated.

2. Rain and hail models

Raindrops and hailstones are both modeled as oblate spheroids. The size-dependent equilibrium axial ratios of raindrops are calculated using Green’s formula (Green 1975). Since there is no established relationship for hailstones, their axial ratios are assumed to be independent of size with a value of 0.8 (Knight 1986). The effects of changing the axial ratio to 0.7 and 0.9 are also considered.

The orientation of an oblate spheroid is determined by the polar and azimuthal angles of its symmetry axis, assuming the vertical direction to be the z axis and the origin to be at the spheroid’s center. For both raindrops and hailstones the azimuthal angle is assumed to be uniformly distributed between 0° and 360°. Their polar angles are assumed to have a Gaussian distribution. For raindrops, which are known to be highly oriented, a mean value of 0° and a standard deviation of 5° is used. For hailstones, which may wobble and gyrate as they fall (Knight and Knight 1970; Roos and Carte 1973; List et al. 1973; Kry and List 1974; Matson and Huggins 1980), a mean value of 0° (90° is considered later in section 5) and a standard deviation of 30° is used. A 45° standard deviation did not lead to any significant changes in the radar parameters compared to 30°.

The raindrop and hailstone temperatures are assumed to be 5° and 0°C, respectively. Both dry (solid ice) and melting (ice–water mixture) hailstones are considered. The melting hailstones are modeled as mixtures of 90% ice–10% water and 70% ice–30% water (percentages based on volume), and for convenience they will be referred to as “slightly wet” and “wet,” respectively. The dielectric constants for ice and water are determined at 10.9- and 5.4-cm wavelengths (Ray 1972) and the Maxwell–Garnet theory is used to calculate the average dielectric constants for the spongy hailstones (Bohren and Battan 1982). A summary of the dielectric constants is given in Table 1.

The exponential size distribution N(D) = N₀ exp(−ΛD) is used for both rain and hail, where D is the equivolume spherical diameter of a particle. The median volume diameter D₀ ≅ 3.67/Λ. The maximum sizes of raindrops and hailstones are chosen as 8 and 50 mm, respectively. The specific raindrop size distribution model used here is for thunderstorms (Joss et al. 1968) with N₀ = 1400 mm⁻¹ m⁻³ and Λ = 3R^−0.21 mm⁻¹, where R is the rainfall rate in millimeters per hour. Here, Λ is varied from 1.14 to 3 mm⁻¹, which corresponds to R over the range 1–100 mm h⁻¹ and D₀ from about 1.2 to 3.2 mm. For hail the model suggested by Cheng and English (1982) is used with N₀ = 115Λ^3.63 mm⁻¹ m⁻³. Here, Λ is varied between 0.15 to 0.8 mm⁻¹, which corresponds to D₀ ranging from about 4.5 to 24 mm.

3. Bistatic radar parameters

The bistatic scattering computations for a single particle are performed with the T-matrix method, originally known as the extended boundary condition method (Waterman 1969). Following a notation similar to that of Doviak and Zrnic (1993), the scattering amplitude matrix of a single particle relates the incident (i) field to the scattered (s) field in a given bistatic direction as

View Expanded

where the harmonic time dependence exp(jωt) is assumed and the first and second subscripts of the matrix elements denote the polarization of the scattered and the incident fields, respectively. Also, k₀ = 2π/λ₀, where λ₀ is the free space wavelength in millimeters. The bistatic radar cross sections of these particles are related to the scattering amplitude matrix elements in (1) as

σ_pqπS_pq²²pq(2)

The bistatic radar parameters of interest are the effective reflectivity factors at horizontal (h) and vertical (v) polarizations

Z_vCσ_vv⁶⁻³(3a)

and

Z_hCσ_hh⁶⁻³(3b)

the bistatic-to-backscattering reflectivity ratios (Aydin and Park 1996; Park 1996)

View Expanded

and linear depolarization ratios

View Expanded

the cross-polarized correlation coefficients

View Expanded

the differential reflectivity

View Expanded

and the copolarized correlation coefficient

View Expanded

where C = λ⁴₀/(π⁵|K_w|²) and |K_w|² ≅ 0.93 for water at S- and C-band frequencies (Battan 1973). Here σ_vv(180°) and σ_hh(180°) denote the backscattering cross sections. The brackets 〈 〉 denote averaging over the size and orientation distributions, and the symbol * indicates complex conjugation. Hailstones from 1 to 50 mm in 1-mm intervals and raindrops from 0.1 to 8 mm in 0.1-mm intervals were used for integration over the size distribution. Integration over the polar and azimuthal canting angles were performed with 4.5° and 3° step intervals, respectively.

With a bistatic radar system that transmits a single polarization and receives two orthogonal polarizations, only four of the parameters in (3) would potentially be measurable. For v-polarized transmission these are Z_v, BBR_v, LDR_v, and ρ_v, and for h-polarized transmission they are Z_h, BBR_h, LDR_h, and ρ_h. If the transmission is dual polarized, then in addition to these eight parameters, Z_DR and ρ_hv would also be measurable.

Each radar parameter in (3) depends on the incidence angles (θ_i, ϕ_i) of the transmitted radiation and the bistatic scattering angles (θ_s, ϕ_s). These angles are measured in a coordinate system with the z axis along the vertical direction, the origin centered in the radar resolution volume, and the incidence azimuth angle ϕ_i = 180° (Fig. 1). This indicates that the backscattering direction is θ_s = θ_i, ϕ_s = 180°. It is clear that there are innumerable possibilities for sets of incidence and scattering angles. Several selected angles will be used to illustrate the characteristics of the radar parameters considered in this study.

4. Scattering from single particles

Before discussing the radar parameters that involve averaging over orientation and size distributions, it is worthwhile to observe some single scattering parameters of the raindrop and hailstone models with their symmetry axes along the vertical direction. These results should be useful in gaining insight into the behavior of radar parameters for various combinations of angles other than those used in the next section. Figure 2 shows S-band (λ₀ = 10.9 cm) results for a raindrop with equivolume spherical diameter D = 4 mm and axial ratio 0.773 and for three hailstone models (dry, slightly wet, and wet) of the same size and shape, with D = 16 mm and axial ratio 0.8. Consider the normalized size parameter X = |m|k₀D/2, where m is the index of refraction of the homogeneous particle. Here, X = 0.82, 1.05, 1.37, and 2.13 for dry hail, raindrop, slightly wet hail, and wet hail, respectively. None of these cases strictly fall into the Rayleigh scattering regime, which requires that X ≪ 1; however, the raindrop and dry hail models exhibit features resembling Rayleigh scattering.

For the 4-mm raindrop, σ_vv and BBR_v change very little as the azimuth angle varies from the backward to the forward direction (first column in Fig. 2), while a large dip in both σ_hh and BBR_h occurs at ϕ_s = 90°, and both are nearly symmetric about this angle. On the other hand, σ_vv and BBR_v for the hailstones steadily increase from the backward to the forward direction (Bohren and Huffman 1983); this is more pronounced for the wet hailstones, which have larger dielectric constants compared to the dry ones. Also, the dip in both σ_hh and BBR_h is not as deep as that of the raindrop (except for the dry hailstone model) and is shifted in the backward direction. Since the oblate spheroidal particles are not canted, LDR_v and LDR_h are equal to 0 (−∞ dB) on the ϕ_s = 0° and 180° planes that include the forward and backward scattering directions (Chylek 1977). Both LDRs reach a peak near ϕ_s = 90°; this peak shifts to the backward direction with increasing size parameter X. The peak in LDR_h is mainly due to the dip in σ_hh, which can create difficulty in its measurement at this angle if small particles (including raindrops and dry hail) fill the scattering volume. Here, Z_DR follows the combined trends of σ_hh and σ_vv, with a dip near ϕ_s = 90°. Also note that all of these parameters are significantly affected by the degree of wetness of the hailstones.

The observed dips in σ_vv and BBR_v, and peaks in LDR_v and Z_DR, as a function of the scattering polar angle (second column in Fig. 2) and incident polar angle (third column in Fig. 2), are a result of the relative orientation of the incident wave and the observed scattered wave polarizations. For θ_s = 105° and ϕ_s = 60° LDR_v is very sensitive to changes in θ_i when θ_i < 120° (i.e., α_i < 30°), whereas LDR_h is not. On the other hand, for θ_i = 95° and ϕ_s = 60° LDR_h is much more sensitive to θ_s when θ_s < 120° compared to LDR_v.

Most of the trends observed at S band are also present at C band (not shown here). However, since the model hydrometeors are larger compared to the wavelength at C band, resonance scattering characteristics are dominant. The dips and peaks are smoother and shift to the backward direction with increasing size parameter X, which corresponds to increasing wetness of the hailstone; however, the shift reverses to the forward direction for hail with the largest X. Also, note that at S-band BBR_v and BBR_h increase as the hailstone becomes more wet. However, at C-band BBR_v and BBR_h increase going from dry to slightly wet hailstones and decrease as the hailstone becomes more wet.

5. Scattering from rain and hail

Figures 3–5 show the S- and C-band radar parameters as a function of the median volume diameter D₀ for several scattering angles with the incidence angles set to θ_i = 95° and ϕ_i = 180°. The first three columns in each figure correspond to θ_s = 95° and three scattering azimuth angles ϕ_s: one in the backward quadrant at 120°, and two in the forward quadrant at 60° and 30°. The last three columns have θ_s = 105° with the same ϕ_s values. All of these parameters are significantly different in the backward and forward quadrants, especially BBR_v and BBR_h. Note that as D₀ increases, the effects of larger particles in the size distribution become significant. For example, S-band BBR_v and BBR_h increase with D₀ for dry and slightly wet hail, but for wet hail they reach a peak value near D₀ = 7.5 mm and decrease afterward; at C band this peak occurs below 5 mm and is not seen in these graphs. The copolarized parameters (BBR_v, BBR_h, Z_v, Z_h, Z_DR, and ρ_hv) are not affected as much as the cross-polarized parameters (LDR_v, LDR_h, ρ_v, and ρ_h) by the change in θ_s from 95° to 105°, which corresponds to a 10° increase in the bistatic elevation angle α_s (Fig. 1). Most of these parameters are significantly affected by the degree of wetness of the hailstones; as a result, the transition from dry hailstones to wet hailstones may be observable in the vertical profiles of some of these parameters.

Figure 6 shows plots of cross-polarized parameters LDR_v–ρ_v and LDR_h–ρ_h, and copolarized parameters Z_v–BBR_v, Z_h–BBR_h. Note that rain and hail generally lie in different parts of these planes, which may be useful for differentiating them. These clustering regions will be different for each set of incidence and scattering angles. In some cases there are regions where rain and hail are very close or overlapping; as a result, discrimination may not always be possible. Furthermore, Z_DR and ρ_hv (not shown here) may be useful in differentiating only the largest hail from rain. It should be noted that other combinations of these parameters may also be used for discrimination purposes.

Figure 7 illustrates the effects of hailstone axial ratios on the bistatic radar parameters. Three axial ratios are considered: 0.9, 0.8, and 0.7 (note that the results thus far were for an axial ratio of 0.8). It is clear that BBR_v, BBR_h, Z_v, and Z_h are the least affected compared to the other parameters. Parameters Z_DR, LDR_v, and LDR_h increase, while ρ_hv, ρ_v, and ρ_h decrease with decreasing axial ratio. In these plots, as the axial ratio decreases from 0.9 to 0.8, or 0.8 to 0.7, the increases in LDR_v and LDR_h range between 0.5 and 3 dB while the decrease in ρ_v and ρ_h varies between 5% and 30%. The increase in Z_DR is about 0.5 dB and changes very little with D₀. Note that BBR_v and BBR_h are not significantly affected (less than 0.5 dB) by changes in the axial ratio.

Figure 8 shows the effects of rotating the mean polar canting angle of the hailstones by 90°; in other words, the symmetry axis, which corresponds to the smaller dimension of an oblate spheroid, becomes parallel to the horizontal plane. In all the previous figures, the symmetry axis was aligned (on average) with the vertical direction. With the symmetry axis rotated 90° the vertical polarization becomes parallel to the larger dimension of the hailstones, whereas the horizontal polarization becomes parallel to both the large and the small dimensions because of the symmetry axis being randomly orientated on the horizontal plane (see section 2). Note that the effect of this orientation change for the hailstones is negligibly small on all the parameters except LDR_v, ρ_v and Z_DR. This is mainly due to the larger change in the scattering of the vertically polarized wave compared to the horizontally polarized wave.

The radar parameters in the form of ratios, (3c)–(3j), are functions of D₀ and are independent of N₀, which may make them useful for estimating D₀. For monostatic radars, Seliga and Bringi (1976) used Z_DR and its sensitivity to the size-dependent axial ratios of raindrops to estimate D₀, which together with Z_h produced an estimate of N₀. Awaka and Oguchi (1982a) suggested the use of bistatic LDR_v and Z_v measurements in rain for the same purpose. The two parameters (D₀ and N₀) of an exponential size distribution for hail may be estimated if its dielectric constant can be assessed. The location within a storm may provide insight into whether or not the hail is dry or has begun melting. As noted earlier, most of these radar parameters change significantly when hail starts melting and acquires liquid water; this may be observable in their vertical profiles. For a given size distribution, BBR_v and BBR_h are the only parameters that are negligibly affected by both the axial ratio and the orientation of (wet or dry) hailstones and may be preferable for estimating D₀ (see Figs. 4–7). Note that in some cases (e.g., at S band for wet hailstones) the same value of BBR_v (or BBR_h) may lead to two different D₀ values. Assuming that D₀ can be obtained, one can determine the ratio Z_v/N₀ or Z_h/N₀ (see Figs. 4–7) and, knowing Z_v or Z_h, this will lead to an estimate of N₀.

Although not shown here, the general behavior and sensitivities noted for all of these bistatic radar parameters are similar at other incidence and scattering angles such as θ_i = 91°, θ_s = 91°, 100°, 110°, and θ_i = 110°, θ_s = 100°, 110°, and 120°. As indicated in the previous section, the single scattering results should provide insight into the variability of these parameters for other combinations of angles as well.

6. Conclusions

S- and C-band bistatic dual-polarization radar parameters were simulated in rain and hail to evaluate their potential use in differentiating the two precipitation types and estimating the parameters of an exponential size distribution for hail. Oblate spheroidal shape models together with canting distributions were used for raindrops and hailstones. Melting hailstones were simulated with several different mixtures of water and ice. For single-polarized transmission there are four parameters that may be useful for the purposes stated above;these are Z_v and the ratio parameters BBR_v, LDR_v, and ρ_v for v-polarized transmission. A similar set exists for h-polarized transmission. However, there may be limitations to using h-polarization due to low signal levels near the azimuth angle of 90° at low elevation angles for raindrops and small hailstones. In the case of dual-polarized transmission, both the v- and h-polarization sets plus the ratio parameters Z_DR and ρ_hv would be available. All of these parameters show significant variability with the incidence and bistatic angles. They are also significantly affected by the amount of liquid water in melting hailstones, which may be useful for estimating the height where melting begins. The least affected parameters from changes in the axial ratios for hailstones are BBR_v, BBR_h, Z_v, and Z_h. All of the parameters except LDR_v, ρ_v, and Z_DR are negligibly affected by a change in the orientation model for hailstones (i.e., rotating the symmetry axis by 90°).

It was shown that rain and hail may be differentiated by observing them on radar parameter planes where they tend to cluster in different regions. Examples of these planes include LDR_v–ρ_v and BBR_v–Z_v for v-polarized transmission, and a similar set for h-polarized transmission, plus Z_DR–ρ_hv for dual-polarized transmission. Other combinations of these parameters also lead to similar results. Using more than one pair of parameters and/or several different bistatic receiver locations may enhance the level of confidence in the discrimination process. It should be noted that on some of these planes there are regions where rain and hail overlap; as a result, their discrimination may not always be possible.

For an exponential size distribution all the ratio parameters depend on the median volume diameter D₀ and not on N₀. Assuming that one knows the amount of water and ice in the hailstones (which may be difficult to determine except for hail that has not started melting), any one of the ratio parameters may be used for estimating D₀ over a limited size range. However, BBR_v and BBR_h appear to be the most promising ones due to their insensitivity to axial ratio and orientation. In some cases though (e.g., at S band for wet hailstones), BBR_v and BBR_h may lead to two different values of D₀, which may not always be resolvable even with the aid of other parameters. If it is possible to estimate D₀, then N₀ can be obtained using Z_v or Z_h.