## 1. Introduction

*Z*

_{H}), differential reflectivity (

*Z*

_{DR}), and specific differential propagation phase (

*K*

_{DP}). The equilibrium shape of a raindrop, falling at its terminal fall speed, is determined by the balance between the forces due to surface tension, hydrostatic pressure, and aerodynamic pressure from airflow around the drop. The shapes of raindrops have been studied theoretically by Green (1975) and Beard and Chuang (1987), experimentally in wind tunnels by Pruppacher and Beard (1970), and in natural rainfall using aircraft probes by Chandrasekar et al. (1988) and Bringi et al. (1998). The experimental results of Chandrasekar et al. (1988) and Bringi et al. (1998) were fairly consistent with the model results of Beard and Chuang (1987). All of the above studies as well as polarimetric radar measurements at multiple polarizations show that the shape of raindrops can be approximated by an oblate spheroid, described with an axis ratio (

*b*/

*a*) and equivolumetric spherical diameter

*D,*where

*a*and

*b*are the major and the minor axes of the drop, respectively. A commonly used approximation relating the axis ratio of a raindrop to the diameter is given by (Pruppacher and Beard 1970):

*b*

*a*

*D.*

*D*< 3 mm. However, for

*D*> 4.5 mm the mean axis ratios were smaller than those given by (1). The above results were obtained after careful and tedious analysis of aircraft-mounted 2D imaging probe data. It would be very useful to obtain an estimate of the mean shape–size relation from polarimetric radar measurements in order to study any variability in the mean shape of the raindrops in different storms as well as different regions of storms.

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

*a*and semiminor axis

*b.*The axis ratio of the raindrop (

*r*) is given by

*r*

*βD.*

*r*= 1 when

*D*⩽ 0.03/

*β,*where

*β*is the magnitude of the slope of the shape–size relationship given by

*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

*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):

*N*

*D*

*n*

_{c}

*f*

*D*

^{−3}

^{−1}

*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

*μ*are parameters of the gamma pdf, and Γ indicates gamma function (Abramovitz and Stegun 1970). The parameter

*N*

_{0}defined by Ulbrich (1983) is related to

*n*

_{c}as

*D*

_{0}can be defined as

*D*

_{0}can be written in terms of the parameters Λ and

*μ*as

*Z*

_{H,V}at horizontal (H) and vertical (V) polarization can be expressed as

*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

*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 *D*_{0} 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 *D*_{0} for different values of *β.* It can be noted that *Z*_{DR} increases as *D*_{0} increases for any value of *β* (Seliga and Bringi 1976);moreover, for a given *D*_{0}, *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 D_{0}. 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 *D*_{0}. 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 *D*_{0} 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 *D*_{0}. Similar arguments can also be made when *β* > 0 so that normalized bias of *Z*_{DR} increases with *D*_{0} as we move farther from *β* ≅ 0. Figure 3b shows similar analysis for *K*_{DP}. For nearly spherical particles (*β* ≅ 0) and *D*_{0} ⩽ 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 *D*_{0}, *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 *D*_{0}. 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

*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

*N*

_{0},

*D*

_{0}, and

*μ,*as suggested by Ulbrich (1983), chosen randomly in the following intervals:

*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)

*β̂*

*Z*

^{−0.377}

_{H}

*K*

^{0.396}

_{DP}

^{0.093ZDR}

*β̂*

*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

*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)

*r*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*

*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*

*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*

### b. Sensitivity of mean shape estimation to bias in *Z*_{H} and *Z*_{DR}

*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*

*Z*

_{H}and

*Z*

_{DR}can be defined as

*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*

*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,**Z,* 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**Z* < *Z*_{H} < 50 dB*Z.* The mean *β̂**β̂**β*_{e} perhaps due to drop oscillations (Beard et al. 1983). Similar stratification was continued for reflectivity ranging between 50 and 53 dB*Z* and for *Z*_{H} > 53 dB*Z.* Figure 8c shows the estimate of mean *β̂*

## 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.

## Acknowledgments

This project was supported by the National Science Foundation (ATM-9413453), by the National Group for Defense from Hydrological Hazards (CNR, Italy), by Progetto Strategico Mesoscale Alpine Program (CNR, Italy), by the Italian Space Agency (ASI), and by the NASA TRMM program. The CSU–CHILL is supported by the National Science Foundation (ATM-9500108). The gauge data were collected and archived by the Colorado Climate Center, and the radar data were collected by Bob Bowie of the CSU–CHILL facility. The authors are grateful to A. Mura and P. Iacovelli for assistance rendered during the preparation of the manuscript.

## REFERENCES

Abramovitz, M., and A. Stegun, 1970:

*Handbook of Mathematical Functions.*Dover, 1043 pp.Andsager, K., K. V. Beard, and N. F. Laird, 1999: Laboratory measurements of axis ratios for large raindrops.

*J. Atmos. Sci.,***56,**2673–2683.Beard, K. V., and C. Chuang, 1987: A new model for the equilibrium shape of raindrops.

*J. Atmos. Sci.,***44,**1509–1524.——, D. B. Johnson, and A. R. Jameson, 1983: Collisional forcing of raindrop oscillations.

*J. Atmos. Sci.,***40,**455–462.Bringi, V. N., V. Chandrasekar, and R. Xiao, 1998: Raindrop axis ratio and size distributions in Florida rainshafts: An assessment of multiparameter radar algorithms.

*IEEE Trans. Geosci. Remote Sens.,***36,**703–715.Chandrasekar, V., V. N. Bringi, and P. J. Brockwell, 1986: Statistical properties of dual polarized radar signals. Preprints,

*23rd Conf. on Radar Meteorology,*Snowmass, CO, Amer. Meteor. Soc., 154–157.——, W. A. Cooper, and V. N. Bringi, 1988: Axis ratios and oscillation of raindrops.

*J. Atmos. Sci.,***45,**1325–1333.Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1999a: Estimation of mean raindrop shape from polarimetric radar measurements. Preprints,

*29th Int. Conf. on Radar Meteorology,*Montreal, PQ, Canada, Amer. Meteor. Soc., 168–171.——, ——, and ——, 1999b: A procedure to calibrate multiparameter weather radar using properties of the rain medium.

*IEEE Trans. Geosci. Remote Sens.,***37,**269–276.Green, A. W., 1975: An approximation for the shapes of large raindrops.

*J. Appl. Meteor.,***14,**1578–1583.Petersen, A. P., and Coauthors, 1999: Mesoscale and radar observations of the Fort Collins flash flood of 28 July 1997.

*Bull. Amer. Meteor. Soc.,***80,**191–216.Pruppacher, H. R., and K. V. Beard, 1970: A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air.

*Quart. J. Roy. Meteor. Soc.,***96,**247–256.Seliga, T. A., and V. N. Bringi, 1976: Potential use of the radar reflectivity at orthogonal polarizations for measuring precipitation.

*J. Appl. Meteor.,***15,**69–76.Ulbrich, C. W., 1983: Natural variations in the analytical form of raindrop size distributions.

*J. Climate Appl. Meteor.,***22,**1764–1775.

## APPENDIX

### Variance in the Estimate of Mean Shape–Size Relation (*β*)

*β*is given by

*β̂*

*cZ*

^{a1}

_{H}

*K*

^{a2}

_{DP}

^{−a3ZDR}

*a*

_{1},

*a*

_{2},

*a*

_{3}, and

*c*are the coefficients given by (12). The variance of

*β*normalized to the mean value can be expressed from perturbation analysis as

*Z*

_{H}can be measured to an accuracy of better than 1 dB,

*Z*

_{DR}can be measured to an accuracy of 0.2 dB, and standard deviation in the estimate of

*K*

_{DP}is given by (12). Assuming 20 range bins with range spacing of 0.15 km and for a mean value of

*K*

_{DP}of 0.86° km

^{−1}, the normalized standard error (standard deviation normalized with respect to the mean) of

*β̂*

Averaged value of differential reflectivity (in linear scale), as a function of median drop diameter (*D*_{0}) for different values of *β,* for various RSD

Citation: Journal of the Atmospheric Sciences 57, 20; 10.1175/1520-0469(2000)057<3406:MOMRSF>2.0.CO;2

Averaged value of differential reflectivity (in linear scale), as a function of median drop diameter (*D*_{0}) for different values of *β,* for various RSD

Citation: Journal of the Atmospheric Sciences 57, 20; 10.1175/1520-0469(2000)057<3406:MOMRSF>2.0.CO;2

Averaged value of differential reflectivity (in linear scale), as a function of median drop diameter (*D*_{0}) for different values of *β,* for various RSD

Citation: Journal of the Atmospheric Sciences 57, 20; 10.1175/1520-0469(2000)057<3406:MOMRSF>2.0.CO;2

Normalized bias (a) on the differential reflectivity (*Z*_{DR}), in linear scale, with respect to *Z*_{DR}, and (b) on the reflectivity factor (*Z*_{H}) and specific differential phase (*K*_{DP}) with respect to *Z*_{H} and *K*_{DP}, obtained from Pruppacher and Beard relation as a function of the slope *β,* for the values of the median drop diameter *D*_{0} corresponding to 1 mm (solid line), 1.5 mm (dashed line), and 2 mm (dotted line)

Normalized bias (a) on the differential reflectivity (*Z*_{DR}), in linear scale, with respect to *Z*_{DR}, and (b) on the reflectivity factor (*Z*_{H}) and specific differential phase (*K*_{DP}) with respect to *Z*_{H} and *K*_{DP}, obtained from Pruppacher and Beard relation as a function of the slope *β,* for the values of the median drop diameter *D*_{0} corresponding to 1 mm (solid line), 1.5 mm (dashed line), and 2 mm (dotted line)

Normalized bias (a) on the differential reflectivity (*Z*_{DR}), in linear scale, with respect to *Z*_{DR}, and (b) on the reflectivity factor (*Z*_{H}) and specific differential phase (*K*_{DP}) with respect to *Z*_{H} and *K*_{DP}, obtained from Pruppacher and Beard relation as a function of the slope *β,* for the values of the median drop diameter *D*_{0} corresponding to 1 mm (solid line), 1.5 mm (dashed line), and 2 mm (dotted line)

Scatter diagram between the slope *β* and the estimate *β̂*

Scatter diagram between the slope *β* and the estimate *β̂*

Scatter diagram between the slope *β* and the estimate *β̂*

Scatter diagram between the slope *β* and the estimate *β̂*

Scatter diagram between the slope *β* and the estimate *β̂*

Scatter diagram between the slope *β* and the estimate *β̂*

Normalized standard error of the estimate *β̂**β.*

Normalized standard error of the estimate *β̂**β.*

Normalized standard error of the estimate *β̂**β.*

Contours of bias in the estimate of the slope *β* as a function of biases in the reflectivity (*Z*_{H}) and differential reflectivity (*Z*_{DR}). The contour line marked 1 indicate no bias, whereas lines marked >1 indicate overestimation and <1 underestimation, respectively

Contours of bias in the estimate of the slope *β* as a function of biases in the reflectivity (*Z*_{H}) and differential reflectivity (*Z*_{DR}). The contour line marked 1 indicate no bias, whereas lines marked >1 indicate overestimation and <1 underestimation, respectively

Contours of bias in the estimate of the slope *β* as a function of biases in the reflectivity (*Z*_{H}) and differential reflectivity (*Z*_{DR}). The contour line marked 1 indicate no bias, whereas lines marked >1 indicate overestimation and <1 underestimation, respectively

(a) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (b) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (c) The mean value of the estimate *β̂,**Z* and for reflectivity greater than 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (d) Observed shape–size relation for the values of mean *β̂**Z*_{H} < 45 dB*Z,* 45 < *Z*_{H} < 50, and for *Z*_{H} > 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar

(a) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (b) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (c) The mean value of the estimate *β̂,**Z* and for reflectivity greater than 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (d) Observed shape–size relation for the values of mean *β̂**Z*_{H} < 45 dB*Z,* 45 < *Z*_{H} < 50, and for *Z*_{H} > 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar

(a) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (b) Histogram of observed values of the estimate *β̂,**Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (c) The mean value of the estimate *β̂,**Z* and for reflectivity greater than 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar. (d) Observed shape–size relation for the values of mean *β̂**Z*_{H} < 45 dB*Z,* 45 < *Z*_{H} < 50, and for *Z*_{H} > 53 dB*Z.* The data referring to a flash flood that occurred over Fort Collins were collected by the Doppler and polarimetric CSU–CHILL radar

System characteristics of the CSU–CHILL radar