1. Introduction

The objective of this paper is to derive an algorithm to estimate the mean shape of raindrops from polarimetric radar data. The paper is organized as follows. Section 2 defines the mean shape model for raindrops, whereas section 3 describes the effect of raindrop shape on polarimetric radar measurements. The estimator for mean raindrop shape from radar measurements is developed and its accuracy and sensitivity are evaluated in section 4. The estimator developed in this paper is applied to data collected by the CSU–CHILL radar during the 28 July 1997 Fort Collins, Colorado, flood and the results are presented in section 5. Section 6 summarizes the important results of the paper.

2. Mean raindrop shape model

Polarimetric radar measurements, wind tunnel measurements, as well as in situ observations using airborne 2D probes indicate that the shape of raindrops can be approximated by oblate spheroids described by semimajor axis a and semiminor axis b. The axis ratio of the raindrop (r) is given by

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The equivolumetric spherical diameter is defined by equating the volume of the spheroid to that of a sphere by

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As noted before, the shape versus size relation can be approximated by a straight line given by

rβD.(3)

In Eq. (3) r = 1 when D ⩽ 0.03/β, where β is the magnitude of the slope of the shape–size relationship given by

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The approximation given by (1) corresponds to β = 0.062 mm⁻¹, which is close to the equilibrium shape–size relation, and therefore we denote it by β_e. We note β > β_e indicates that raindrops are more oblate than equilibrium, whereas β < β_e indicates raindrops are less oblate (or closer to spherical) than equilibrium.

3. Polarimetric radar measurements: Sensitivity to shape–size relation

The three commonly used polarimetric radar parameters are reflectivity factor at horizontal polarization (Z_H), differential reflectivity (Z_DR), and specific differential propagation phase (K_DP). Both the cloud model and measurements of raindrop size distribution (RSD) at the surface and aloft show that a gamma distribution model adequately describes many of the natural variations in RSD (Ulbrich 1983):

NDn_cfD⁻³⁻¹(5)

where N(D) is the number of raindrops per unit volume per unit size interval, n_c is the concentration, and f(D) is the gamma probability density function (pdf), given by

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where Λ and μ are parameters of the gamma pdf, and Γ indicates gamma function (Abramovitz and Stegun 1970). The parameter N₀ defined by Ulbrich (1983) is related to n_c as

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The volume-weighted median drop diameter D₀ can be defined as

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The diameter D₀ can be written in terms of the parameters Λ and μ as

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The reflectivity factor Z_H,V at horizontal (H) and vertical (V) polarization can be expressed as

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where σ_H,V denote the radar cross sections at the two linear polarizations; λ is the wavelength; and k = (ε_r − 1)/(ε_r + 2), where ε_r is the dielectric constant of water. Similarly, the differential reflectivity (Z_DR) and specific differential phase (K_DP) can be expressed as

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where f_H, f_V are the forward scatter amplitudes at H and V polarization states. It can be seen from (10a–c) that for a given RSD, Z_H, Z_DR, and K_DP can change with shape–size relationship for raindrops.

According to (1), raindrops become more oblate when the size is large. Therefore, the effect of varying shape–size relationship should be more evident in the presence of larger drops. The volume-weighted median drop diameter D₀ is a good indicator of the mean size of drops in the distribution. The effect of varying shape–size relations of raindrops is illustrated by the following analysis. For a given RSD and at S-band frequency, we compute the radar measurements Z_H, Z_DR, and K_DP for various β in the range of 0.02–0.1 in steps of 0.01. The various shape–size relationships studied here are shown in Fig. 1, where the dash–dotted line represents the equilibrium relation (1). In Fig. 2 the behavior of Z_DR (in linear scale) is shown as a function of D₀ for different values of β. It can be noted that Z_DR increases as D₀ increases for any value of β (Seliga and Bringi 1976);moreover, for a given D₀, Z_DR increases with β. Similar behavior can also be obtained for K_DP. As shown in Fig. 2, the sensitivity of Z_DR to β is most dependent on D₀. Figure 3a shows the normalized variation of Z_DR (in linear scale) with respect to Z_DR obtained from the equilibrium relation (1) as a function of β for different values of D₀. For nearly spherical particles (β ≅ 0), the Z_DR value should be 0 dB or unity in linear scale. The normalized bias for β ≅ 0 in comparison to β_e is determined by the value of Z_DR at β = 0.062 so that it increases as D₀ increases (see Fig. 2). The range of Z_DR difference between nearly spherical drops (β = 0.02) and equilibrium-shape drops varies between 0.84 and 1.89 dB depending on D₀. Similar arguments can also be made when β > 0 so that normalized bias of Z_DR increases with D₀ as we move farther from β ≅ 0. Figure 3b shows similar analysis for K_DP. For nearly spherical particles (β ≅ 0) and D₀ ⩽ 1 mm K_DP is approximately zero and then the ratio between K_DP with respect to K_DP at equilibrium axis ratio is nearly zero and the normalized bias has the maximum negative value equal to −1. By increasing D₀, K_DP increases and then the normalized bias decreases. Similar results can be obtained for β > 0 so that the normalized bias decreases by increasing D₀. The reflectivity factor is fairly insensitive to raindrop shape–size relationships for β < β_e as shown in Fig. 3b, whereas for β > β_e the change in Z_H with β is within 10%.

4. Algorithm to estimate raindrop shape–size relation

The result of section 3 indicates that the observations of Z_DR and K_DP are sufficiently sensitive to β, so that it can be turned around into measurement. The estimator of β is developed using the following procedure. First, large number of Gamma RSD is simulated over a wide range of the parameters N₀, D₀, and μ, as suggested by Ulbrich (1983), chosen randomly in the following intervals:

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In addition, for each RSD, Z_H, Z_DR, and K_DP are computed for various values of β ranging between 0.02 and 0.1. The above computations are done at S-band frequency. Subsequently, nonlinear regression analysis is performed to evaluate various functional forms to estimate β. The above analysis yields the estimator for β at the S band given (Gorgucci et al. 1999a)

β̂Z^−0.377_HK^0.396_DP10^0.093Z_DR(12)

A scattergram between β̂ and the true value of β is shown in Fig. 4; it can be noted that (12) estimates β fairly well. The data used in Fig. 4 have a correlation of 0.996 with a normalized standard error (the root-mean-square error normalized with the mean) of 3.6%.

a. Shape–size relation estimate in the presence of measurement errors

The estimate given by (12) uses Z_H, Z_DR, and K_DP. These three measurements have completely different error structure. 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 and is the differential power estimate between Z_H and Z_V. It can be estimated to an accuracy of 0.2 dB. The K_DP is the slope of the range profile of the differential propagation phase Φ_DP, which can be estimated to an accuracy of a few degrees. The subsequent estimate of K_DP depends on the procedure used 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. 1999b)

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where Δr is the range resolution of the Φ_DP estimate and N is the number of range samples within the path. For large N we can see that σ(K_DP) decreases as N^−3/2. For a typical 150-m range spacing, and with 2.5° accuracy of Φ_DP, the K_DP can be estimated, over a path of 3 km, with a standard error of 0.32° km⁻¹. Thus, the three measurements Z_H, Z_DR, and K_DP have completely different error structure. In addition, the measurement errors of Z_H, Z_DR, and K_DP are nearly independent. In the following we use simulations to quantify the error structure of the estimate of β. The simulation is done as follows. Various rainfall values are simulated varying the parameters of the gamma RSD over a wide range of values, as suggested by Ulbrich (1983). For each RSD the corresponding Z_H, Z_DR, and K_DP are evaluated using (10a–c). The random measurement errors are simulated using the procedure described in Chandrasekar et al. (1986). The principal parameters of our simulation are as follows: 1) wavelength λ = 11 cm; 2) sampling time PRT = 1 ms; 3) number of samples pairs M = 64; 4) Doppler velocity spectrum width σ_υ = 2 m s⁻¹; 5) cross correlation between H and V signals ρ_HV = 0.99; 6) range sample spacing over the path where K_DP is estimated is 150 m; and 7) K_DP is estimated over a path of 50 range samples, as a least squares fit on Φ_DP measurements. Figure 5 shows the scatter diagram of β̂ given by (12) versus β in the presence of measurement errors, using K_DP values greater than 0.4° km⁻¹. The scatter diagram of the data in Fig. 5 gives a correlation coefficient of 0.97 and a normalized standard error of 9%. Finally, Fig. 6 shows the normalized standard error of β̂ as a function of β, where normalized standard error is defined as the root-mean-square error normalized with respect to the mean. The results of Fig. 6 show that the slope of the shape–size relation β can be estimated to an accuracy of about 9% in the presence of measurement errors in Z_H, Z_DR, and K_DP. The appendix shows variance computations of β̂ only due to measurement errors.

b. Sensitivity of mean shape estimation to bias in Z_H and Z_DR

Bias errors in Z_H and Z_DR can affect the estimate of β. Bias errors in the measurements of Z_H and Z_DR will remain even if extensive averaging is performed. The term Z_DR is a differential power measurement and its bias can be estimated and removed easily (Gorgucci et al. 1999b). However, Z_H is based on absolute power measurement and it is difficult to get the absolute calibration. Typically this can be known to an accuracy of 1 dB. Thus K_DP is based on phase measurement and is immune to calibration biases in fairly uniform rain medium. The bias in β̂ due to bias errors in Z_H and Z_DR can be defined as

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Figure 7 shows contours of bias in the estimate of β as a function of bias in Z_H and Z_DR. The contours line marked 1 indicates no bias, and lines marked different from 1 indicate overestimation (>1) and underestimation (<1). Typically in a well-maintained system, bias error in Z_DR is less than 0.2 dB and bias in Z_H is less than 1 dB. Therefore, from Fig. 7 it can be seen that β̂ can be estimated within 10% accuracy under those biases of Z_H and Z_DR.

5. Data analysis

On the evening of 28 July 1997, the city of Fort Collins was hit by a flash flood that caused fatalities and extensively property damage. Mesoscale analysis of this flood is described in Petersen et al. (1999). CSU–CHILL radar recorded continuous data over the event, collecting multiparametric measurements over 5 h. The radar recorded measurements of Z_H, Z_DR, and K_DP. The characteristics of the CSU–CHILL radar that are relevant for this paper are listed in Table 1. The application of algorithm (12) is fairly straightforward, but numerous details are important. A linear least squares fit was done on the Φ_DP observations to obtain one K_DP estimate for a 3-km path, whereas Z_H and Z_DR are computed as the mean value of Z_H and Z_DR measurements on the same path. These values of Z_H, Z_DR, and K_DP were used in (12) to estimate β. Only data from regions with K_DP > 0.4° km⁻¹ were used to ensure good accuracy in the estimate of β. A histogram of the various observed values of β̂, for reflectivity in the range of 40 to 45 dBZ, is shown in Fig. 8a, where the mean and standard deviation of data are 0.061 and 0.01 respectively. The standard deviation of data in Fig. 8a is fairly close to measurement standard deviation, as shown in the appendix. Therefore, most of the spread in β̂ is due to measurement error. In addition, it can be seen that the mean slope of shape–size relation shown in Fig. 8a is close to the theoretical predictions as well as to experimental observation reported in the literature so far (Beard and Chuang 1987; Chandrasekar et al. 1988; Bringi et al. 1998). Figure 8b shows similar results for data corresponding to 45 dBZ < Z_H < 50 dBZ. The mean β̂ and standard deviation of the data shown in Fig. 8b are 0.057 and 0.008, respectively. Once again it can be seen that most of the standard deviation is due to measurement error, and the mean β̂ of 0.057 indicates that the drops are less oblate than β_e perhaps due to drop oscillations (Beard et al. 1983). Similar stratification was continued for reflectivity ranging between 50 and 53 dBZ and for Z_H > 53 dBZ. Figure 8c shows the estimate of mean β̂ and its standard deviation as a function of reflectivity. The standard deviation was computed for each case and was found to range between 18% and 13% around the mean. Figure 8d shows the observed mean shape–size relations, stratified with reflectivity. It can be seen from Figs. 8c and 8d that the axis ratios become progressively slightly less oblate in comparison to equilibrium axis ratios probably due to raindrop oscillations.

6. Summary and conclusions

The mean shape–size relation of raindrops plays an important role in the interpretation of polarimetric radar measurements. The polarimetric radar algorithms available in the literature have been developed for equilibrium axis ratios. A simple model was developed to describe the shape–size relation of raindrops in terms of the slope (β) of the linear approximation to the shape–size function. Subsequently, theoretical analysis was utilized to quantify the variability in Z_H, Z_DR, and K_DP due to changes in β. The sensitivity of Z_H, Z_DR, and K_DP to deviation from equilibrium shape–size relation β_e was studied. It was found that both Z_DR and K_DP were fairly sensitive to changes in β, whereas Z_H was insensitive as expected. There was enough sensitivity to β in Z_DR and K_DP that it could be turned around to a measurement. An algorithm to estimate the slope of the shape–size relation was derived. The algorithm can be used to estimate β from measurements of Z_H, Z_DR, and K_DP. Error analysis of the algorithm demonstrated that the algorithm estimates β on the average to an accuracy of 9%, when K_DP is estimated over a path of 50 range bins with a range spacing of 150 m. Polarimetric radar data collected by the CSU–CHILL radar was used to evaluate the algorithm developed in this paper. The estimation of β from radar data yielded values very close to the equilibrium shape–size relation of raindrops. When the data were stratified with reflectivity, the results indicated that the drops became less oblate as reflectivity increases, an indication of possible raindrop oscillation.