1. Introduction

Ever since the introduction of differential reflectivity (Z_dr) measurement, one of the long standing goals of polarimetric radar has been the estimation of the raindrop size distribution (DSD). Seliga and Bringi (1976) showed that Z_dr, for an exponential DSD, is directly related to the median volume diameter (D₀). Careful intercomparisons between radar measurements of Z_dr and D₀ derived from surface disdrometers and airborne imaging probes have shown that D₀ can be estimated to an accuracy of about 10%–15% (see, for example, Aydin et al. 1987; Bringi et al. 1998). A general gamma distribution model was suggested by Ulbrich (1983) to characterize the natural variation of the DSD. The nonspherical shape of raindrops results in anisotropic propagation of electromagnetic waves, with a difference in the propagation constant at horizontal and vertical polarization states. The specific differential propagation phase (K_dp) is a forward scatter measurement (Seliga and Bringi 1978; Sachidananda and Zrnić 1987) whereas the radar reflectivity (Z_h) and Z_dr are backscatter measurements. The weighting of the DSD by Z_dr and K_dp is controlled by the variation of mean raindrop shape with size. A combination of the three radar measurements (Z_h, Z_dr, and K_dp) can be utilized to estimate the DSD, specifically the parameters of a parametric form of the DSD such as the gamma DSD. This paper presents algorithms for the estimation of parameters of a gamma DSD from polarimetric radar measurements. The paper is organized as follows. Section 2 describes the raindrop size distribution and its parameters, whereas section 3 describes the shape of raindrops and its implication for polarimetric radar measurements. Estimators of the DSD parameters are presented in section 4, and the impact of measurement errors on the estimates are discussed in section 5. Validation of the algorithms using disdrometer data are presented in section 6. Important results of this paper are summarized in section 7.

2. Raindrop size distribution

The raindrop size distribution describes the probability density distribution function of raindrop sizes. In practice, the normalized histogram of raindrop sizes (normalized with respect to the total number of observed raindrops) converges to the probability density function of raindrop sizes. A gamma distribution model can adequately describe many of the natural variations in the shape of the raindrop size distribution (Ulbrich 1983). The gamma raindrop size distribution can be expressed as,

NDn_cf_DD⁻³⁻¹(1)

where N(D) is the number of raindrops per unit volume per unit size interval (D to D + ΔD), n_c is the number concentration, and f_D(D) is the probability density function (pdf). When f_D(D) is of the gamma form it is given by

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where Λ and μ are the parameters of the gamma pdf. Any other gamma form such as the one introduced by Ulbrich (1983),

NDN₀D^μe^−ΛD(3)

can be derived from this fundamental notion of raindrop size distribution. It must be noted that any function used to describe N(D) when integrated over D must yield the total number concentration, to qualify as a DSD function. This property is a direct consequence of the fundamental result that any probability density function must integrate to unity. When μ = 0, the gamma DSD reduces to the exponential form as N(D) = n_cΛe^−ΛD. The relation between D₀, μ, and Λ is given by (Ulbrich 1983)

D₀μ,(4)

where D₀ is the drop median diameter defined as

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Similarly, a mass-weighted mean diameter D_m can be defined as

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where E stands for the expected value. Using (4), f_D(D), the gamma pdf described by (2), can be written in terms of D₀ and μ as

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The above form makes the normalized diameter (D/D₀) as the variable rather than D. Several measurables such as water content (W) and rainfall rate (R) can be expressed in terms of the DSD as

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where R is the still-air rainfall rate and υ(D) is the terminal velocity of raindrops (Gunn and Kinzer 1949). The conventional unit of rainfall rate is millimeters per hour. Converting to this unit, rainfall rate is expressed as

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where n_c is in cubic millimeters, υ(D) in millimeters per second, and D in millimeters.

In order to compare the pdf of D [or, f_D(D)] in the presence of varying water contents, the concept of scaling the DSD has been used by several authors (Sekhon and Srivastava 1971; Willis 1984; Testud et al. 2000). The corresponding form of N(D) can be expressed as

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where N_w is the scaled version of N₀ defined in (3):

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with f(0) = 1 and f(μ) is a unitless function of μ. One interpretation of N_w is that it is the intercept of an equivalent exponential distribution with the same water content and D₀ as the gamma DSD (Bringi and Chandrasekar 2001). Thus N_w, D₀, and μ form the three parameters of the gamma DSD.

3. Raindrop shape and implication for polarimetric radar measurements

The equilibrium shape of raindrops is determined by a balance of hydrostatic, surface tension, and aerodynamic forces. The commonly used model for raindrops assumes oblate spheroidal shapes, with the axis ratio b/a, where b and a are the semiminor and the semimajor axis lengths, respectively. Pruppacher and Beard (1970) give a simple model for the axis ratio (r) based on a linear fit to wind tunnel data as

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Rotating linear polarization data in heavy rain (Hendry et al. 1987) has indicated that raindrops fall with the mean orientation of their symmetry axis in the vertical direction. The large swing in the crosspolar power in their data implies a high degree of orientation of drops with the standard deviation of canting angles estimated to be around 6° assuming a Gaussian model. It is reasonable to assume that the standard deviation of canting angles is in the range 5°–10° (Bringi and Chandrasekar 2001).

a. Differential reflectivity

The differential reflectivity can be written as (Seliga and Bringi 1976)

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where the symbol E represents expectation and σ_hh and σ_υυ are the cross sections at horizontal and vertical polarizations, respectively.

Seliga and Bringi (1976) showed that for an exponential distribution and axis ratio given by (11), Z_dr can be expressed as a function of the median volume diameter D₀. This microphysical link between a radar measurement and a parameter of the DSD is important. More fundamentally, ξ⁻¹_dr may be related to the reflectivity factor–weighted mean of r^7/3 (Jameson 1985). For a more general gamma form, an approximate power-law fit can be derived assuming −1 ≤ μ ≤ 5, 0.5 < D₀ < 2.5 mm, and N_w chosen to be consistent with thunderstorm rain rates. Using the fit recommended by Andsager et al. (1999) for the Beard and Chuang (1987) equilibrium shapes, power-law fits to D₀ and D_m can be derived as

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where Z_dr is in decibels and the fits are valid at S-band frequency (near 3 GHz; Bringi and Chandrasekar 2001).

b. Specific differential phase

The relation between specific differential phase (K_dp) and the water content and raindrop axis ratio was described by Jameson (1985). Following Bringi and Chandrasekar (2001) a simple approach based on Rayleigh–Gans scattering is described here to derive this relation.

The specific differential phase can be expressed as

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where k₀ is the free space propagation constant, and f_h, f_υ are the forward scatter amplitudes at horizontal and vertical polarization, respectively. For Rayleigh–Gans scattering, (14) reduces to

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where ϵ_r is the dielectric constant and the depolarizing factor λ_z is given by

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The above expectation can be substantially simplified by recognizing that

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where c is approximately constant varying between 3.3 and 4.2 with r from 1 to 0.5. This range of c is valid for ϵ_r of water at microwave frequencies in the range 3–30 GHz. Substituting (16) in (15a) results in

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where W is the rainwater content, ρ_w is the water density, E stands for expectation over the DSD, and λ is the wavelength. The ratio of expectations in (17c) can be defined as the mass-weighted mean axis ratio r_m. In terms of conventional units for W in grams per cubic meters, ρ_w = 1 g cm⁻³, and λ in meters, (17c) can be reduced to

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where c ≅ 3.75 is both dimensionless and independent of wavelength. This result links the specific differential phase with parameters of the DSD (Jameson 1985). If the equilibrium axis ratio model given in (11) is used in (18) then K_dp is given by

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Thus, K_dp is related to the product of D_m and water content. Though the above result was obtained using the Rayleigh–Gans approximation, it is valid up to 13 GHz (Bringi and Chandrasekar 2001).

c. Mean raindrop shape derived from polarimetric radar measurements

Field studies of Tokay and Beard (1996) indicate that raindrops from 1 to 4 mm oscillate. Andsager et al. (1999) show that oscillations result in an upward shift of the mean axis ratio versus diameter curve, specially in the 1- to 4-mm range. Gorgucci et al. (2000) assumed a simple linear model for axis ratio versus size of the form

rβD(20)

and derived radar-based estimators of β. They also showed that β decreases slightly with increasing reflectivity (or, on the average, the axis ratio is smaller than the equilibrium axis ratio) perhaps indicating raindrop oscillations.

It was shown in section 3a that ξ⁻¹_dr is related to the reflectivity weighted axis ratio. Similar dependence on K_dp can be derived from (18). Let p(r) be the probability density function of the axis ratio for a given diameter. The expression for K_dp can be generalized as (Bringi and Chandrasekar 2001)

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where E(r) is the mean value of r, and c* is a constant. The functional dependence of E(r) versus D may be modeled as in (20). Using the linear model in (20), Gorgucci et al. (2000) showed the variations of Z_dr and K_dp with respect to β, and in turn derived an estimator for β based on polarimetric radar measurements. This can be used subsequently in algorithms relating Z_dr and K_dp to the parameters of the DSD, which gives rise to a methodology for estimating the gamma DSD parameters based on radar measurements.

4. Estimators of the gamma DSD parameters

Seliga and Bringi (1976) showed that for an exponential distribution, the two parameters of the DSD, namely N_w and D₀, can be estimated using Z_dr and Z_h. They used a two-step procedure where they estimated D₀ using an equilibrium raindrop shape model and subsequently used that in the expression for Z_h to estimate N_w. This procedure can essentially be applied for a gamma DSD, and generalized to account for raindrop oscillations using the linear model in (20). The procedure for estimating the gamma DSD parameters is as follows: first, estimate β using the algorithm described by Gorgucci et al. (2000) and, subsequently, estimate D₀, N_w, and μ, recognizing the right β value.

a. Estimation of D₀

It was noted in section 3a that D₀ can be estimated from Z_dr as a simple power-law expression (13a). This parameterization was based on the Beard and Chuang (1987) equilibrium axis ratios and essentially corresponds to a fixed equivalent β. Gorgucci et al. (1994) obtained approximate parameterizations for Z_h and Z_υ of the form

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where c_h, c_υ, g_h(μ), g_υ(μ), α_h, and α_υ are constants that depend on h and υ polarizations. From the above and with some modest algebra, it can be shown that a parameterization for D₀ can be pursued of the form

D̂0a1Zb1hξdrc1(24)

where ξ_dr = 10^0.1Z_dr is the differential reflectivity in linear scale, and Z_h is the reflectivity factor at horizontal polarization (in mm⁶ m⁻³). Though the above parameterization form was obtained from the approximation in (23), the coefficients in (24) can be derived from the simulation of gamma DSDs directly as follows. Once the gamma DSD is given in the form in (9), it is straightforward to compute radar parameters such as Z_h, Z_dr, and K_dp. The mean axis ratio versus D relation is modeled by (20). Under these conditions and at a temperature of 20°, Z_h, Z_dr, and K_dp are computed for widely varying DSD by randomly varying N_w, D₀, and μ over the following ranges:

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with the constraint R < 300 mm h⁻¹. While D₀ and μ are varied randomly over their respective ranges, log₁₀N_w is randomly varied over its range. This range falls within the range of parameters suggested by Ulbrich (1983). Once Z_h, Z_dr, and K_dp values are simulated, a nonlinear regression analysis is performed to estimate the coefficients a₁, b₁, c₁. Though these coefficients are accurate for a single β, c₁ changes significantly with β. Figure 1 shows the plot of the coefficients a₁, b₁, c₁ as a function of β to demonstrate the sensitivity. This variation of c₁ with β can be further parameterized by fitting power-law expressions. These coefficients are (valid for S band)

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In summary, D₀ can be estimated by first estimating β using the approach of Gorgucci et al. (2000) as

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and then using coefficients (26) in (24). For the equilibrium axis ratios (24) reduces to

D̂₀Z^0.064_hξ^1.245_dr(28a)

whereas, when β = 0.0475 (typical for tropical rain, discussed in section 6),

D̂₀Z^0.064_hξ^1.817_dr(28b)

Simulations can also be utilized to evaluate the performance of the estimator of D₀ in (24). Figure 2a shows a scatterplot of D̂₀ versus true D₀, for widely varying β, and gamma DSD parameters as given by (25). Quantitative analysis of the scatter gives a correlation coefficient of 0.963. It can be seen from Fig. 2a that D₀ is estimated fairly well with negligible bias over a wide range. Figure 2b shows the normalized standard deviation (NSD) of D̂₀ as a function of D₀, where NSD is defined as

View Expanded

where SD indicates standard deviation. Figure 2b shows that D₀ can be estimated to an accuracy of about 10% when D₀ > 1 mm. A similar estimate of D₀ can be derived using K_dp and Z_dr as

D̂0a2Kb2dpξdrc2(30)

where

View Expanded

This estimator is similar to the estimator in (24) except that K_dp estimates are difficult to obtain at low rain rates. On the other hand, this estimator is immune to variations in absolute calibration of the radar system. Error analysis of the estimator given by (30) yields a correlation coefficient of 0.963. The normalized standard deviation of the estimate of D₀ given (30) is also shown in Fig. 2b. It can be seen that in the absence of any measurement errors these two estimates are comparable. For equilibrium axis ratios this estimator for D₀ reduces to

D̂₀K^0.076_dpξ^1.439_dr(32a)

whereas, for tropical rain with β ≅ 0.0475 (discussed in section 6),

D̂₀K^0.076_dpξ^1.864_dr(32b)

b. Estimation of N_w

Once D₀ is estimated, N_w can be easily estimated using one of the moments of the DSD such as Z_h or K_dp. For example, it was shown in (19) that K_dp is proportional to the product of W and D_m (or approximately D₀); Z_h can be written in terms of the gamma DSD parameters as (see also Ulbrich and Atlas 1998)

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where

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For an exponential distribution (μ = 0),

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Thus it can be seen that N_w can be estimated in terms of D₀. However, the estimate of D₀ can be obtained in terms of Z_h and Z_dr (or K_dp and Z_dr). Therefore, a direct estimate of N_w can be pursued of the form

10Nwa3Zb3hξc3dr(34)

The variability of a₃, b₃, c₃ can be parameterized in terms of β as

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In summary, the estimator for N_w is obtained as follows. Using Z_h, Z_dr, and K_dp, first estimate β as given in (27). Subsequently, calculate the coefficients in (35) and use in (34) to estimate N_w. Figure 3a shows a scatterplot of log₁₀N̂_w versus true log₁₀N_w, where log₁₀N̂_w is estimated using (34). It can be seen from Fig. 3a that log₁₀N̂_w is estimated fairly well. Quantitative analysis of the scatter yields a correlation coefficient of 0.831. Figure 3b shows the normalized standard deviation of log₁₀N_w as a function of log₁₀N_w. It can be seen, from Fig. 3b, that log₁₀N_w is estimated to a normalized standard deviation of better than 7% when log₁₀N_w > 3.5. Note that due to the wide variability of N_w, log₁₀N_w is the preferred scale of comparison (similar to dB scale for reflectivity). For equilibrium axis ratios, (34) reduces to

₁₀N̂_wZ^0.058_hξ^−1.094_dr(36a)

whereas for tropical rain with β ≅ 0.0475 (discussed in section 6)

₁₀N_wZ^0.058_hξ^−1.585_dr(36b)

Similarly, another estimate of N_w can be derived using K_dp and Z_dr as

10Nwa4Kb4dpξdrc4(37)

This variability of a₄, b₄, c₄ can be parameterized in terms of β as

View Expanded

For equilibrium axis ratios, (37) reduces to

View Expanded

whereas for tropical rain (β ≅ 0.0475)

₁₀N̂_wK^0.06_dpξ^−1.44_dr(38b)

The normalized standard deviation in the estimate of log₁₀N_w given by (37) is also shown in Fig. 3b. It can be seen from Fig. 3b that the two estimators for log₁₀N_w are comparable in the absence of the measurement errors.

c. Parameterization of μ

The parameter μ describes the overall shape of the distribution. Once D₀ is estimated, μ can be estimated from the following parameterization, which was constructed empirically as

View Expanded

The variability of a₅, b₅, c₅, and d₅ can be parameterized in terms of β as

View Expanded

D₀ calculated from either (24) or (30) can be utilized in (39) to estimate μ. Figure 4a shows the scatterplot of μ̂ given by (39) versus μ under the assumption that D₀ is known. The results of Fig. 4a indicate that μ can be parameterized of the form given by (39) (though it appears complicated). Figure 4b shows the corresponding standard deviation in the estimate of μ̂, which is about 0.3. However, in practice D₀ has to be estimated using (24) or (30), using Z_h, Z_dr, and K_dp. Estimating μ under such conditions will result in higher error than that projected by Fig. 4b. Estimating μ accurately under practical conditions, especially in the presence of measurement errors is very difficult using the procedures discussed here.

5. Impact of measurement error on the estimates of D₀ and N_w

Estimators of D₀ given by (24) and (30) as well as N_w given by (34) and (37) use measurements of Z_h, Z_dr, and K_dp. Any error in the measurement of these three parameters will directly translate into errors in the estimates of D₀ and N_w. The three measurements Z_h, Z_dr, and K_dp have completely different error structures.

The Z_h is based on absolute power measurement and has a typical accuracy of 1 dB. The Z_dr is a relative power measurement that can be estimated to an accuracy of about 0.2 dB. The slope of the range profile of the differential propagation phase Φ_dp is K_dp, which can be estimated to an accuracy of a few degrees. The subsequent estimate of K_dp depends on the procedure used to compute the range derivative of Φ_dp such as a simple finite-difference scheme or a least squares fit. Using a least squares estimate of the Φ_dp profile, the standard deviation of K_dp can be expressed as (Gorgucci et al. 1999)

View Expanded

where Δr is the range resolution of the Φ_dp estimate and N is the number of range samples along the path. For a typical 150-m range spacing, and with 2.5° accuracy of Φ_dp, K_dp can be estimated over a path of 3 km, with a standard error of 0.32° km⁻¹.

The measurement errors of Z_h, Z_dr, and K_dp are nearly independent. In the following, simulations are used to quantify the error structure of the estimates of D₀ and N_w. The simulation is done as follows. For each gamma DSD the corresponding Z_h, Z_dr, and K_dp are computed. The random measurement errors are simulated using the procedure described in Chandrasekar et al. (1986). The principal parameters of the simulation are as follows: 1) wavelength λ = 11 cm; 2) pulse repetition time (PRT) 1 = ms; 3) number of sample pairs used in the estimates is 64; 4) Doppler velocity spectrum width σ_υ = 2 m s⁻¹; 5) copolar correlation between horizontally and vertically polarized signals ρ_hυ = 0.99; 6) range sample spacing is 150 m; and 7) K_dp is estimated as a least squares fit over a path consisting of 50 range samples. Subsequently, estimates of D₀ and N_w are obtained using the simulated measurements of Z_h, Z_dr, and K_dp. There are some practical issues associated with the estimation of D₀ and N_w from radar measurements (or simulated radar measurements). At low rain rates, K_dp is small, and in the presence of measurement errors, K_dp estimates could be very small (fluctuating around zero). Under this condition (say, when K_dp < 0.2 deg km⁻¹), (27) cannot be used to estimate β. Therefore, when K_dp is small the following procedure is adopted: whenever K̂_dp < 0.2° km⁻¹ the equilibrium model for axis ratios are assumed and (28a) and (36a) are used for estimating D₀ and N_w, respectively. It can be noted that (13a) also could be used for estimating D₀, followed by either (33c) or (36b) for N_w. In light rain all these algorithms provide similar results for estimates of N_w and D₀.

The normalized standard deviation in the estimates of D₀ and N_w including the effect of measurement error are evaluated and shown in Figs. 5 and 6, respectively. Figure 5 shows the NSD in the estimates of D₀ given by (24) and (30). Comparing Fig. 5 to Fig. 2b it can be seen that in general, there is about a 10% increase in the NSD of D₀ estimate, computed from (24), due to measurement error. The NSD of D₀ estimate from (30) gets worse for smaller D₀ primarily due to the error in K_dp. The NSD of the N_w estimates, given by (34) and (37) in the presence of measurement errors, are shown in Fig. 6. Again comparing these to the NSD computations without measurement error (Fig. 3b), a 4% to 16% increase is noted depending on the value of N_w. For an N_w value of 8000 mm⁻¹ m⁻³ the NSD of log₁₀N_w is about 12% in the presence of measurement errors. Once again the estimate of log₁₀N_w from (37) has higher standard deviation when N_w < 20 000 mm⁻¹ m⁻³. Thus, D₀ and N_w can be estimated fairly well from radar measurements at least for convective rainfall with R ≥ 5–10 mm h⁻¹. These errors can be further reduced using other techniques such as spatial averaging whenever possible, which may be especially useful for stratiform rain. The following section presents evaluation of the algorithms developed here using disdrometer observations.

6. Evaluation of the algorithms using disdrometer data

The algorithms developed in this paper to estimate D₀ and N_w are applied to data collected with a J–W impact disdrometer (Joss and Waldvogel 1967) during a rainfall season (covering about 3 months) from Darwin, Australia. This dataset was collected by the Bureau of Meteorology Research Center (BMRC) and includes a variety of rainfall types from a tropical regime with rain rates between 1 and 150 mm h⁻¹. The disdrometer data consists of measurements of N(D) in discrete intervals of ΔD at 30-s intervals, which are subsequently averaged over 2 min. While several methods are available to fit the measured N(D) to a gamma form (e.g., Willis 1984), the method used here is based on Bringi and Chandrasekar (2001). First, D_m is estimated using the definition in (5b), that is, as a ratio of the fourth to third moments of the measured N(D). Next the water content W is estimated from the definition in (7). The parameter N_w in (9) is then calculated as the intercept of the equivalent exponential DSD that has the same W and D_m as the measured N(D), as

View Expanded

(where W is in g cm⁻³, D_m is in mm, and the water density ρ_w is in g cm⁻³). Finally, the parameter μ is estimated by minimizing the absolute deviation between observed log₁₀N(D) and that given by (9). Here, D₀ is estimated from D_m as (Ulbrich 1983)

View Expanded

Once the set of (N_w, D₀, μ) parameters are obtained, the radar observables Z_h, Z_dr, and K_dp are simulated based on the following assumptions:

1. axis ratio versus D relation based on the fit proposed by Andsager et al. (1999) for D up to 4 mm; beyond 4 mm, the equilibrium axis ratios of Beard and Chuang (1987) are used;

2. Gaussian canting angle distribution with mean of 0° and standard deviation 10°; and

3. truncation of the gamma DSD at D_max = 3.5 D_m [see Ulbrich and Atlas (1998) for a discussion of the drop truncation of the DSD].

The simulated set of radar observables (Z_h, Z_dr, and K_dp) when used in (27) gives an “effective” β of 0.0475 (for comparison, the equilibrium β is 0.062).

Note that the algorithms for D₀ and N_w are constructed to be insensitive to the actual value of β, so that the details of the assumptions used in simulating the set of radar observables are not of particular relevance, and this fact is indeed the power of the proposed D₀ and N_w algorithms in (24), (30), (34), and (37). In order to evaluate these algorithms using disdrometer measurements, the simulated values of Z_h, Z_dr, and K_dp are used in (24), (34), and (39) to calculate D̂₀, N̂_w, and μ̂, which are then compared against D₀, N_w, and μ estimated by gamma fits to the set of measured N(D). Once again, to be consistent when K_dp < 0.2° km⁻¹, the estimates are computed using the practical approximation discussed in section 5. Figure 7a shows the D₀ comparisons while Fig. 7b shows the NSD. Note that the D̂₀ algorithm can retrieve the “true” D₀ quite accurately (NSD < 7%) especially for D₀ > 1 mm. As expected the D₀ estimates get very accurate for higher values. The log₁₀(N_w) comparison are shown in Fig. 8a, while Fig. 8b shows the NSD. The scatter in Fig. 8a shows that the accuracy in the retrieval of log₁₀N_w is quite high (<5%) for N_w > 1000 mm⁻¹ m⁻³ (for reference, the Marshall–Palmer value for N_w is 8000 mm⁻¹ m⁻³). Figure 9a shows the μ comparison, while Fig. 9b shows the corresponding standard deviation. The results of Fig. 9 show that it is difficult to retrieve μ with any reasonable accuracy with the current algorithms, though it may be possible to distinguish between certain ranges of μ, for example, μ = 0 as opposed to μ > 5, which may be sufficient in practice.

7. Summary and conclusions

One of the long-standing goals of polarimetric radar has been the estimation of the parameters of the raindrop size distribution. Estimators for the parameters of a three-parameter gamma model, namely D₀, N_w, and μ, are developed in this paper based on the radar observations Z_h, Z_dr, and K_dp. The behavior of the three radar observations Z_h, Z_dr, and K_dp are influenced by the underlying DSD, and the mean shape of raindrops. Reflectivity Z_dr is proportional to the reflectivity-weighted axis ratio, whereas K_dp is proportional to the volume-weighted deviation of the axis ratio from unity. In addition, reflectivity is proportional to the sixth moment of the DSD, with corresponding variability due to polarization. Thus, the different polarimetric radar observations weight the DSD differently. It should be noted that the DSD estimates computed here correspond to radar measurements from the radar resolution volume. Among the three measurements (Z_h, Z_dr, and K_dp), Z_dr is the most closely related to a parameter of the DSD, namely D₀. Gorgucci et al. (2000) described a procedure to estimate the mean shape–size relation of raindrops based on a simple linear model. Therefore, after the prevailing shape–size relation is established, Z_dr can be used to estimate D₀ directly. This concept is implemented in this paper as an algorithm to estimate D₀ from Z_h, Z_dr, and K_dp. Statistical analysis of the estimator of D₀ indicates that it can be estimated to an accuracy of 10% when D₀ is 2 mm (and similar accuracies at the other D₀ values). Once D₀ is estimated, other measurements such as Z_h or K_dp can be used to estimate N_w, to a normalized standard deviation of about 6.5% when N_w = 8000 mm⁻¹ m⁻³, and similar order at the other values. The estimation of μ is not easy because of the least influence of this parameter on the three measurements Z_h, Z_dr, and K_dp. Therefore, the parametric estimates of μ derived are not as accurate. Measurement errors in Z_h, Z_dr, and K_dp play a key role in the final accuracy of DSD estimates. Reflectivities Z_h and Z_dr are based on backscatter power measurement whereas K_dp is a forward scatter phase measurement. In addition, Z_dr is a differential power measurement between two correlated signals, and can be measured accurately. This high degree of accuracy in Z_dr translates to high accuracy in D₀. However, to estimate the prevailing mean shape–size relation, K_dp is needed that is relatively noisy at low rainrates. A hybrid approach is implemented in this paper such that when K_dp ≤ 0.2 deg km⁻¹ the equilibrium shape model is used to estimate D₀. This procedure yields estimates of D₀ to an accuracy of the order of 15%. Similarly, log₁₀N_w can be estimated in the presence of measurement error to an accuracy of 15% when N_w = 8000 mm⁻¹ m⁻³. This accuracy deteriorates to about 20% when N_w is of the order 1000 mm⁻¹ m⁻³ but improves to 10% if N_w is of the order 40 000 mm⁻¹ m⁻³. At low rainrates the best estimate of D₀ or N_w is still the original estimates by Seliga and Bringi (1976). At low rainrates accurate estimates of Z_dr can be obtained by doing sufficient areal averaging, which can then be used in (13a) to estimate D₀ and a subsequent exponential distribution algorithm given by (33c) to estimate N_w. In the presence of measurement error, μ is difficult to estimate using the procedure described here in a meaningful manner. However, it may be possible to distinguish between μ ≈ 0 versus μ > 5, which may be sufficient in practice. The algorithms developed here were applied to one rainy season of disdrometer data collected in Darwin, Australia. The disdrometer analysis indicates that the algorithms work fairly well for the estimation of D₀ and N_w. In summary, the algorithms presented in this paper can be used to estimate the parameters of the raindrop size distribution, from polarimetric radar data at a frequency near 3 GHz (S band).