## 1. Introduction

Polarimetric radar measurements of rainfall depend on raindrop size and shape distributions. Use of the dual-polarization radar measurements (Seliga and Bringi 1976) for rain-rate retrievals therefore would be affected by changes in orientation and shape of raindrops. It was shown by Brussaard (1976) and Beard et al. (1983) that wind shear may change mean orientation angle of raindrops. There are several causes for actual raindrop shapes to differ from the equilibrium shape (Pruppacher and Beard 1970; Green 1975). Beard et al. (1983) have shown that in moderate-to-heavy rainfall, raindrop collisions can cause oscillations of raindrops. The other possible reason for raindrop oscillations is the resonant response to eddy shedding as shown in Beard and Kubesh (1991).

Goddard et al. (1982) have found that radar retrieved rainfall rate agreed more to the rainfall rate retrieved from disdrometer measurements, when raindrops that are smaller than 3 mm were assumed as more spherical than given by Pruppacher and Pitter (1971). Chandrasekar et al. (1988) have analyzed two-dimensional (2D) precipitation imaging probe measurements of raindrop shapes in light-to-moderate rainfall and have found that raindrops smaller than 4 mm are more spherical than expected from raindrops in equilibrium. Laboratory observations of raindrop shapes exhibit a similar trend as shown in Andsager et al. (1999). Bringi et al. (2003), Thurai and Bringi (2005), and Ryzhkov and Schuur (2003) have used a 2D video disdrometer to assess raindrop shapes. These studies also have shown that raindrops are generally more spherical than predicted by using equilibrium shapes (Beard and Chuang 1987). As a result a modified raindrop size–shape relation was proposed in Brandes et al. (2002, 2004) that takes into account both theoretical calculations and experimental observations of size–shape relations.

Direct observations of effective raindrop shapes from dual-polarization radar measurements, however, are not straightforward. To interpret dual-polarization radar measurements one would need to know not only raindrop size–shape relation but also drop size distribution (DSD). Goddard et al. (1982) have used disdrometer data to address this problem. Recently, Gorgucci et al. (2000) have introduced an additional shape parameter, *β*, into a polarimetric rain-rate retrieval algorithm that represents the equivalent slope of the size–shape relationship. This approach uses reflectivity (*Z*_{h}), differential reflectivity (*Z*_{dr}), and specific differential phase (*K*_{dp}) measurements to estimate *β*. Use of *K*_{dp} restricts applicability of the proposed technique to moderate-to-heavy rain events. Moreover, use of *K*_{dp} implies reduction of the range resolution, and therefore results in range averaged estimates of *β*.

A number of studies have been presented in literature that describe the retrieval of DSD parameters and ambient air velocity from Doppler power spectra. Hauser and Amayenc (1981) have shown that assuming no spectral broadening, Doppler power spectra measurements of precipitation at vertical incidence can be used to retrieve vertical air motion and two parameters of exponential DSD. Nonetheless, it was acknowledged that spectral broadening would be one of the major sources of errors in this retrieval. Recently, Williams (2002) has shown that the nonlinear least squares optimization procedure can be used for a joint retrieval of three parameters of gamma DSD, vertical air motion, and spectral broadening from vertical incident profiler Doppler measurements. Therefore, by combining Doppler spectra observations taken at a sufficiently high elevation angle with dual-polarization measurements of precipitation one should be able to retrieve both DSD information and raindrop size–shape relation on small time and spatial scales.

To facilitate the investigation of *β* on small time and spatial scales in this study we introduce a new retrieval method of raindrop size–shape relation that is based on dual-polarization spectral measurements of precipitation. In this method the drop size distribution information, spectrum broadening, and ambient air velocity are obtained from Doppler power spectra measurements, and given this information, the *β* parameter is obtained from differential reflectivity measurements. The measurements should be carried out at a sufficiently high elevation angle for correct DSD retrieval and at a rather low elevation angle for accurate dual-polarization measurements. Based on simulations, we have studied the sensitivity of the proposed retrieval method to the radar antenna elevation angle and have determined the window of suitable elevation angles. Moreover, the sensitivity analyses of the proposed technique to spectrum broadening and a number of fast Fourier transform (FFT) samples are also performed.

Two datasets were used to illustrate performance of the proposed dual-polarization spectral analysis. In both cases raw time series data were collected. The first data were collected by the S-band Transportable Atmospheric Radar (TARA; Heijnen et al. 2000) of the Delft University of Technology, Netherlands. The second dataset was collected by the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar (Brunkow et al. 2000).

## 2. Modeling Doppler power spectra of precipitation

### a. Microphysics

*ρ*

_{0}and

*ρ*are the air densities at the sea level and the altitude of a considered range gate respectively,

*υ*

_{0}is the ambient air radial velocity,

*D*is the equivolume raindrop diameter given in millimeters, and

*θ*is the antenna elevation angle. Assuming that there is no spectral broadening (e.g., due to turbulence, oscillations, or cross winds), the Doppler power spectrum density,

*S*

_{hh}(

*υ*)

*dυ*, in rain for the “hh” polarization setting, can be written as

*λ*is the radar wavelength,

*K*is the dielectric factor,

_{r}*D*is the equivolume diameter,

*θ*is the radar elevation angle,

*σ*

_{hh}is the backscattering cross section,

*υ*is the radial velocity of a raindrop, and

*N*(

*D*) is the DSD. It should be noted that (2) can be rewritten for different polarization settings just by using appropriate radar cross sections. The DSD,

*N*(

*D*), is assumed to be described by a gamma distribution (Bringi and Chandrasekar 2001):

*N*is the intercept parameter of the distribution,

_{w}*D*

_{0}is the median volume diameter of a raindrop, and

*μ*is the shape parameter.

*σ*is the broadening kernel width. Assuming that separate contributions to spectral broadening are independent we can write the kernel width as

_{b}*σ*

^{2}

_{b}=

*σ*

^{2}

_{turb}+

*σ*

^{2}

_{cross-wind}+

*σ*

^{2}

_{oscill}+

*σ*

^{2}

_{shear}, where

*σ*

^{2}

_{turb}is due to turbulent air motion,

*σ*

^{2}

_{cross-wind}is due to the cross wind,

*σ*

^{2}

_{oscill}is due to raindrop oscillations, and

*σ*

^{2}

_{shear}is due to a wind shear. As one can see

*σ*alone is sufficient to represent all these components in the model.

_{b}### b. Scattering

*β*is the slope parameter. Often (e.g., Matrosov et al. 2002; Ryzhkov and Schuur 2003) another linear relation is used:

Changes in canting angle distribution of raindrops have a similar effect on the copolar backscattering cross section as changes in the magnitude of *β*. Therefore, changes in *β* would correspond to both changes in the canting angle distribution and to the oscillation of raindrops. Since raindrop size–shape relations are generally nonlinear, *β* values also depend on DSD parameters. To study whether observed raindrops have equilibrium shapes we have also used Beard and Chuang’s (1987) relation to compare our retrievals to, as later discussed.

## 3. Retrieval of effective slope of linear size–shape relation

### a. Retrieval of the DSD parameters

The model of a Doppler power spectrum as described in (2), (4), and (5) depends on six parameters: three DSD parameters, spectrum broadening, ambient air velocity, and slope of the drop size–shape relation. In Fig. 1, simulated Doppler spectral densities are shown as functions of *μ*, *D*_{0}, *σ _{b}*, and

*β*. It can be seen that

*β*has a negligible effect on a Doppler spectrum and therefore can be omitted from our considerations at this stage of the retrieval. Moreover, one can observe that changes in

*μ*and

*D*

_{0}would result in changes of the spectrum shape. The spectrum broadening widens and smooths the spectrum. The effect of

*υ*

_{0}and

*N*on Doppler spectra, though not depicted in the Fig. 1, is rather easy to imagine. The ambient air velocity

_{w}*υ*

_{0}shifts the velocity axis according to (1). And changes in

*N*would result in scaling along the reflectivity axis.

_{w}*β*has a negligible effect on a Doppler spectrum, the resulting problem is a five-parameter nonlinear least square problem:

*υ*and

_{a}*υ*are lower and upper bounds of the spectrum region used for the fit. There are two possible approaches for defining the fitting problem. One approach would be to use logarithms of power spectra (Williams 2002) and the other is to fit in linear space. May et al. (1989) have shown that fitting in linear space improves detectability of the signal, but as a drawback spectral coefficients have large uncertainty. In this work we have chosen to use log values of the spectral coefficients. Use of logarithm scale equalizes fit residuals from different spectral lines and therefore allows for unbiased retrieval. This approach, however, might result in poor performance for signals with a small signal-to-noise ratio (SNR; May et al. 1989). A more detailed description of the optimization problem is given in the appendix, where it is shown that this problem can be simplified to a three-parameter (

_{b}*D*

_{0},

*μ*, and

*σ*), nonlinear least squares problem.

_{b}### b. Retrieval of β

Given dual-polarization observations of precipitation and knowing corresponding DSD parameters, one can infer information about drop size–shape relations. For this purpose, measurements of differential reflectivity *Z*_{dr}, that is ratio of hh and *υυ* reflectivities, can be used. The *Z*_{dr} values depend on DSD parameters, *μ* and *D*_{0}, as well as on *β*. In Fig. 2 the dependence of *Z*_{dr} on *β* for different values of *μ* and *D*_{0} is shown. These calculations were done for 45° radar elevation angles.

*β*results in an increase in

*Z*

_{dr}. Moreover, an increase in

*D*

_{0}or a decrease in

*μ*increases

*Z*

_{dr}as well. Therefore, given

*D*

_{0}and

*μ*one can retrieve

*β*value by solving the following minimization problem:

*Z*

^{mod}

_{dr}and

*Z*

^{meas}

_{dr}are, respectively, modeled and measured differential reflectivities. It should be noted that a change of the elevation angle would result in a change of the slopes of the curves in Fig. 2. The lower the elevation angle, the steeper the curves. However, even at elevation angles as high as 70° there would be an observable

*Z*

_{dr}signal for larger values of

*β*,

*D*

_{0}, and/or smaller values of

*μ*.

## 4. Sensitivity analysis of the retrieval technique

### a. Measurement simulation

Furthermore, the simulated measurements were constrained to have reflectivity values larger than 10 dB*Z* and smaller than 55 dB*Z*. Prior to the retrieval, 30 realizations of the hh and vv Doppler power spectra for a given set of input parameters were calculated and averaged to obtain an estimate of the true spectrum. The velocity resolution of 0.078 m s^{−1} and 128 spectral lines were used for this simulation. This simulation coincides with TARA measurements specifications (see Table 1). For these specifications it takes 19-s time series observations to obtain one Doppler spectrum. Then, the estimated spectrum is reduced to the spectral lines where *S*_{hh}(*υ*)*dυ* is larger than −20 dB*Z* and are no more than 30 dB below the peak spectral density. This step simulates clipping of observed spectra to remove spectral lines affected by noise and spectral leakage. Then the optimization procedure was applied to the averaged hh spectrum to retrieve DSD parameters, spectral broadening, and air velocity. In Fig. 3 scatterplots of the retrieved parameters versus the input ones are shown. It should be noted that because accuracy of *N _{w}* can be directly determined from the errors in

*D*

_{0}, results for

*N*retrievals are not shown. For these simulations the antenna elevation angle of 45° was used.

_{w}After *D*_{0}, *μ*, and *σ _{b}* were retrieved, the slope of linear raindrop size–shape relation is estimated. To simulate dual-polarization measurements |

*ρ*

_{co}(0)|, the value of 0.95 was assumed. To estimate

*Z*

_{dr}values, the Doppler power spectra were integrated to obtain

*Z*

_{h}and

*Z*values. And by solving (8),

_{υ}*β*is retrieved. In Fig. 4 scatter graphs of the retrieved

*β*versus the input

*β*is shown. One can observe that

*β*can be estimated with approximately 10% accuracy.

### b. Effect of spectral broadening, σ_{b}

Part of the optimization problem (7) can be considered as an inversion problem of finding *S*_{hh}(*υ*) from S_{broad}(*υ*) * *S*_{hh}(*υ*) or solving the integral in Eq. (4). That is a Fredholm integral equation of the first kind. Since the spectral broadening kernel *S*_{broad}(*υ*), (4), is smooth and monotonic from *υ* = 0 to *υ* = ∞, one can expect an increase in the instability of the solution with the relative increase of the width of the kernel with respect to the width of *S*_{hh}(*υ*) (Twomey 1977). In our case this instability would translate into higher errors in the retrieved DSD parameters and ambient air velocity values. Since retrieval of *β* requires knowledge of *D*_{0} and *μ*, it also would be affected by the instability of the inversion. In Figs. 5 and 6 the resulting root-mean-square errors (RMSEs) of retrieved *D*_{0}, *μ*, *υ*_{0}, *σ _{b}*, and

*β*are shown for different values of

*σ*; the simulations were carried for 45° elevation angles. As expected the RMSE values increase with an increase in

_{b}*σ*. One can observe that for the

_{b}*σ*values larger than 1 m s

_{b}^{−1}the retrieved values of

*β*become unreliable. It should be noted that the accuracy of the retrieved

*σ*values is not affected.

_{b}Each point on the graphs shown in Figs. 5 and 6 was estimated from 100 simulated Doppler spectra. In these simulations input parameters were varied in the range given in the previous section, except that *σ _{b}* was varied from 0 to 3 m s

^{−1}and was fixed for each set of simulations.

### c. Elevation angle dependence

Dependence of the DSD parameters retrieval on the elevation angle is caused by two opposing effects. First, the lower the elevation angle, the smaller the difference between fall velocities radial projections of differently sized raindrops; therefore, spectrum broadening would have a larger effect on the retrieval at smaller elevation angles. Second, for the same reason, a number of spectral lines available for the retrieval would be smaller at lower elevation angles if the same number of samples is used to calculate Doppler spectra. Therefore, one would expect that accuracy of the DSD parameters retrieval would increase with the increase of the elevation angle as confirmed by our calculations and shown in the Fig. 7.

On the other hand, the influence of raindrop shapes on the *Z*_{dr} measurements is larger for smaller elevation angles. Therefore, accuracy of the *β* estimate for different elevation angles would be a trade-off between accuracy of the retrieval of DSD parameters and influence of raindrop shapes on the *Z*_{dr} measurements. This effect is shown in the Fig. 8. From this figure we can observe that useful elevation angles for this retrieval belong to the interval between 30° and 70°.

Similar to the sensitivity studies discussed in the previous sections, each point on the graphs in Figs. 7 and 8 was obtained from 100 simulations, where the input parameters to these simulations were randomly taken in the intervals given in (9).

### d. Dependence on FFT length

Changes in the number of FFT samples will result in changes in velocity resolution and therefore in accuracy of *υ*_{0} arid *σ _{b}*. Accuracy of the retrieval of these two parameters is related to errors in the other four parameters. To study these dependencies on FFT length, we have simulated the power spectra using 64, 128, 256, and 512 FFT samples and model parameters in (9). For each of these simulations 250 spectra were simulated. Each spectrum was calculated by averaging 15 spectra. The setup of this simulation coincides with CSU–CHILL measurement specifications, except for variable FFT length and elevation angle. In Fig. 9 results of these study are shown. It should be noted that pulse repetition time of 1 ms and the alternating polarization scheme was assumed for these simulations. Moreover, the simulations were carried out for two elevation angles: 30° and 45°.

In Fig. 9 one can observe that there is a strong dependence of the retrieval errors on FFT length. This dependence is more pronounced for observations taken at the 45° elevation angle. It should be noted that differences in resulting accuracy for this simulation and simulations shown in previous sections can be explained by difference in number of averages; that is, 15 and 30 spectra were, respectively, used for averaging.

### e. Effect of radar calibration errors

The proposed method to retrieve the slope of the raindrop size–shape relation depends on the accuracy of the retrieved values of *μ* and *D*_{0} and on the accuracy of the *Z*_{dr} measurements. Since *μ* and *D*_{0} are retrieved from the shape of Doppler spectral density function, the accuracy of these retrieved values do not depend on a radar calibration. Therefore only the *Z*_{dr} calibration affects the accuracy of *β*. It is expected that a well-calibrated dual-polarization radar system is able to measure *Z*_{dr} with 0.2-dB accuracy. This error in the estimation of *Z*_{dr} leads to approximately 0.005 mm^{−1} uncertainty in the *β* estimate for the 45° elevation angle measurements.

## 5. Data analysis

### a. TARA data

The TARA is a frequency modulated-continuous wave (FM-CW) S-band radar capable of dual-polarization Doppler measurements (Heijnen et al. 2000). The rain measurements used in this study were collected by the TARA during a stratiform rain event that took place on 19 September 2001 in Cabauw, Netherlands. It was a light-to-moderate rain event with reflectivities ranging from 20 to 35 dB*Z*. The radar antenna elevation angle was 45°. In total 15 min of time series data were collected. The range resolution of 15 m was used for this measurement. TARA has a single-channel receiver; therefore, the measurements were carried out in alternating polarization mode where hh, *υυ*, and h*υ* observations were collected. Moreover, two offset beam measurements were carried out during the fourth and fifth sweeps; therefore, data was collected in blocks of five sweeps with 1-ms sweep duration. A summary of measurement specifications is given in Table 2. To calculate one Doppler power spectrum, 30 spectra were averaged, where each spectrum was estimated applying an FFT on 128 samples with the Hamming window. It should be noted that prior to averaging all spectra were shifted to have the same mean Doppler velocity to reduce unwanted broadening of the spectra estimates due to air motion variability^{1} (Giangrande et al. 2001). Finally, the estimated Doppler power spectral density was thresholded to remove noisy parts of the spectrum and the resulting spectral density was passed through the optimizer (7). In Fig. 10 an example of the Doppler spectral density measurement and resulting fit are shown. It should be noted that not all the spectra were correctly fitted. In some cases the resulting sum square residuals exceeded the threshold that was set such that the coefficient of determination (Rust 2001) should not be less than 0.9. In these cases the resulting observations were not used for our study. In the remaining cases accuracy of the retrievals should be similar to the previously mentioned simulations.

Then, given the DSD parameters, the *β* values were retrieved by matching modeled and observed *Z*_{dr} values. Here two sets of *β* values were calculated: one using Gorgucci et al.’s (2000) definition of *β* and the other using Matrosov et al.’s (2002) definition. The histograms of the resulting *β* values are shown in Fig. 11. One can see that these two definitions of the effective slope of size–shape relation produce somewhat different results. The mean *β* value retrieved from observations using (5) is equal to 0.04 and the mean value calculated using definition (6) is equal to 0.049. This result is expected since (6) allows the partially mitigate nonlinearity effect of an actual raindrop size–shape relation. Nonetheless, retrieved *β* values, using definitions (5) or (6), do not give a direct information about whether raindrops are affected by oscillations or not. To answer this question we have compared retrieved *β* values to ones calculated using known size–shape relations. For this calculation-retrieved DSDs and size–shape relations were used to calculate *Z*_{dr} values (Beard and Chuang 1987; Brandes et al. 2002). Then by applying *β* retrieval procedure (8) on these *Z*_{dr} values, instead of measured ones, we estimated the effective slope of size–shape relations for a given DSD (Beard and Chuang 1987; Brandes et al. 2002). The results of these calculations are also shown in Fig. 11. We can observe that retrieved *β* values are smaller than equilibrium ones, as one would expect for oscillating raindrops. Also one can observe that National Center for Atmospheric Research (NCAR) size–shape relation (Brandes et al. 2002) gives a good approximation to the observations. It should also be noted that both definitions of *β* confirm this conclusion. This precipitation event is characterized by mean log *N _{w}* value of 3.63, a mean

*D*

_{0}value of 1.3 mm, and a mean

*μ*value of 0.4.

### b. CSU–CHILL data

The CSU–CHILL data were collected during a stratiform precipitation event that took place during thunderstorm on 23 July 2004. The observed reflectivities were around 35 dB*Z*. During this event, 2 min of time series data were collected. The measurements were carried out in alternating mode (Bringi and Chandrasekar 2001). The measurement specifications are given in Table 2. The Doppler power spectra were estimated by averaging 15 spectra, where each spectrum was estimated from 64 samples using the periodogram approach with the Hamming window. The resulting velocity resolution was 0.39 m s^{−1}. In Fig. 12 an example of the measured spectral density function and corresponding model fit is shown. In Fig. 11 the resulting histogram of the observed *β* values is shown. The same calculations were performed on this data as on the TARA measurements. We can see that as expected two definitions of *β* show different distributions. The mean retrieved equivalent slope of the size–shape relation, defined as (5), is equal to 0.036 and the mean value calculated using definition (6) is equal to 0.046. We should also note that equilibrium *β* values are larger than the observed one and that for this measurement the NCAR size–shape relation (Brandes et al. 2002) gives a good approximation to the observations. For this measurement the mean retrieved log *N _{w}* value is 3.54, the mean

*D*

_{0}value is 1.2 mm, and the mean

*μ*value is −0.4.

## 6. Discussion

In this study a new approach to retrieve raindrop shape information was developed. It was shown that the slope of raindrop shape–size relation can be retrieved from dual-polarization spectral analysis of time series radar data. As a result a finescale analysis of the precipitation microphysical properties can be carried out. As a part of the proposed method an efficient algorithm for DSD parameters retrieval was developed. It is shown that the DSD parameter retrieval method, which usually requires the solution of the five-parameter nonlinear optimization problem, can be simplified to a three-parameter nonlinear least squares problem.

It was shown that the most critical part of this method is the DSD parameter retrieval. The accuracy of this retrieval depends on spectral broadening, FFT length, elevation angle, and the number of spectral averages. To study the error budget of the method a sensitivity analysis of the proposed technique was performed. It was shown that spectral broadening has a very strong influence on the retrieval accuracy. Nonetheless, for spectral broadening widths smaller than 1 m s^{−1} an accurate retrieval of DSD parameters, wind velocity, and slope of the raindrop size–shape relation can be achieved. This is a general limitation applicable to most DSD parameters retrieval methods based on the analysis of Doppler power spectra. Large spectral widths are generally associated with strong wind shears, which are more common for larger radar volumes. In case of high-elevation angle measurements, radar measurements of rainfall are generally collected in the first 2–10 km. This limits the size of a radar volume and therefore reduces the influence of wind shear on measurements.

The FFT length also has a strong effect on the accuracy of the method. For the 45° elevation measurements, changes in FFT length from 64 to 512 samples would reduce the error in *β* estimate twice, from 0.028 to 0.014 mm^{−1}. It was also shown that for the CSU–CHILL measurement setup, *β* can be estimated with an accuracy of 0.024 mm^{−1}. By comparing findings of section 4d with sections 4b,c we can conclude that the number of spectral averages has also a nonnegligible influence on the retrieval errors. It was shown that for 15 averages and TARA measurement setup, *β* can be estimated with an accuracy of 0.014 mm^{−1}. On the other hand, if 30 averages are used, this error would be around 0.006 mm^{−1}, as shown in sections 4b, c. Same observation holds for errors in the retrieved DSD parameters.

The effect of different radar elevation angles on the errors in the retrieved parameters was also studied. It was shown that elevation angles lying in the range between 30° and 70° are optimal for the proposed study.

Since DSD retrieval is the most critical part of this method, it would advantageous to validate the retrieved parameters. The measurement setup during the observations used in this study, however, does not allow us to compare retrieved parameters to measurements taken using some other instrument. An indirect validation of the retrieved DSD parameters, nonetheless, is possible. By using retrieved DSD parameters and equilibrium raindrop shapes (Beard and Chuang 1987) we have calculated *Z*_{dr} values. Goddard et al. (1982) have observed that in this case calculated *Z*_{dr} values would be 0.1–0.3 dB larger than observed ones. That was confirmed by our observations and calculations, which indicate that our retrieval has resulted in reasonable values of DSD parameters.

To illustrate the performance of the technique two precipitation cases were studied. In both cases we have observed a stratiform precipitation event with light-to-moderate rain intensities. In these cases we have observed that raindrops appear more spherical than raindrops in equilibrium. We have also observed that Brandes et al.’s (2002) size–shape relation gives a reasonable fit to the observations.

## Acknowledgments

The research was supported by the National Science Foundation (Grant ATM-0313881). The authors wish to thank David Brunkow and Pat Kennedy for collecting CSU–CHILL radar data and Silvester Heijnen for collecting TARA data.

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,*J. Atmos. Oceanic Technol.***22****,**966–978.Twomey, S., 1977:

*Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements*. Elsevier Scientific Publishing, 243 pp.Williams, C. R., 2002: Simultaneous ambient air motion and raindrop size distributions retrieved from UHF vertical incident profiler observations.

,*Radio Sci.***37****,**8-1–8-3.

## APPENDIX

### Optimization Procedure

*β*. Similar to Williams (2002) we have used a nonlinear least squares method (Rust 2003) to fit logarithm values of the modeled spectra to the observed ones. Therefore, the optimization procedure can be formulated as minimization problem of sum squared residuals (SSR

_{ds}) of a fit of the modeled power spectrum to the measured one. Here SSR

_{ds}is defined as

*S*

_{hh,mod}(

*N*,

_{w}*D*

_{0},

*μ*,

*υ*

_{0},

*σ*) is given by (4). Analogously the SSR

_{b}_{zdr}, for the fit of the modeled differential reflectivity to the measured one can be defined as

*β*as follows:

_{ds}, requires solution of a nonlinear least squares problem with five unknowns. If we consider the structure of SSR

_{ds}, however, we can notice that

*N*can be separated from the rest of unknowns using the variable projection method (Golub and Pereyra 2003). The variable projection method is used to solve the separable nonlinear least square problems, where the model function is a linear combination of nonlinear functions. In our case we can rewrite the least squares problem as follows:

_{w}*ɛ*are the random errors. Therefore,

*N*can be directly estimated as follows (Rust 2003):

_{w}**Σ**

^{2}is the error covariance matrix.

*υ*

_{0}from the nonlinear least squares fit. Let us assume that

*N*,

_{w}*D*

_{0},

*μ*, and

*σ*are given, then

_{b}*υ*

_{0}can be estimated by solving the following minimization problem:

*υ*

_{0}value. It should be noted that in most cases this estimation of

*υ*

_{0}is identical to finding the lag at which the cross correlation of modeled and measured spectra is maximum. Therefore, the inverse problem (A2) can be split into four steps and (A2) can be rewritten as follows:

The practical implementation of this procedure is complicated by the discrete nature of measured Doppler power spectra, system noise, and spectral leakage. As a result the objective function SSR_{ds}(*N _{w}*,

*D*

_{0},

*μ*,

*υ*

_{0},

*σ*) is not necessary smooth. Therefore, one should pay special attention to the convergence of the optimization procedure to a global minimum. We have addressed this issue by randomly selecting a number of seed values,

_{b}*D*

_{0},

*μ*, and

*σ*, which were used to initiate the optimization procedure.

_{b}Dependence of *Z*_{dr} on *β* for different values of (left) *D*_{0} and (right) *μ*. The elevation angle is 45° in both cases.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Dependence of *Z*_{dr} on *β* for different values of (left) *D*_{0} and (right) *μ*. The elevation angle is 45° in both cases.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Dependence of *Z*_{dr} on *β* for different values of (left) *D*_{0} and (right) *μ*. The elevation angle is 45° in both cases.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplots of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values by solving (6) vs input values.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplots of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values by solving (6) vs input values.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplots of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values by solving (6) vs input values.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplot of retrieved *β* values from (7) vs input *β*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplot of retrieved *β* values from (7) vs input *β*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Scatterplot of retrieved *β* values from (7) vs input *β*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values as a function of the input value *σ _{b}*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values as a function of the input value *σ _{b}*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *D*_{0}, *μ*, *υ*_{0}, and *β* values as a function of the input value *σ _{b}*.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *β* as a function of the spectrum broadening kernel width.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *β* as a function of the spectrum broadening kernel width.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved *β* as a function of the spectrum broadening kernel width.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *D*_{0}, *μ*, *υ*_{0} and *σ _{b}* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *D*_{0}, *μ*, *υ*_{0} and *σ _{b}* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *D*_{0}, *μ*, *υ*_{0} and *σ _{b}* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *β* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *β* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved values of *β* as a function of the elevation angle.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved parameters as functions of velocity resolution. Lines with circles correspond to measurements taken at 30° of elevation. Lines with crosses correspond to measurements at 45°.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved parameters as functions of velocity resolution. Lines with circles correspond to measurements taken at 30° of elevation. Lines with crosses correspond to measurements at 45°.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

RMSEs of the retrieved parameters as functions of velocity resolution. Lines with circles correspond to measurements taken at 30° of elevation. Lines with crosses correspond to measurements at 45°.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by TARA. A total of a 128-sample FFT was used to calculate this spectrum and 30 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by TARA. A total of a 128-sample FFT was used to calculate this spectrum and 30 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by TARA. A total of a 128-sample FFT was used to calculate this spectrum and 30 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Histogram of the retrieved *β* values. The gray bars represent *β* values calculated using retrieved DSD parameters from the Beard and Chuang relation. The dot–dashed lines represent *β* values calculated from Brandes et al. (2002) relation. The solid lines depict histograms of *β* values retrieved from measurements. Histograms are derived from CSU–CHILL measurements. (a) Obtained using *β* as defined in Eq. (5). (b) Obtained using the definition of (6). (c), (d) Same as (a), (b), but were derived from TARA measurements.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Histogram of the retrieved *β* values. The gray bars represent *β* values calculated using retrieved DSD parameters from the Beard and Chuang relation. The dot–dashed lines represent *β* values calculated from Brandes et al. (2002) relation. The solid lines depict histograms of *β* values retrieved from measurements. Histograms are derived from CSU–CHILL measurements. (a) Obtained using *β* as defined in Eq. (5). (b) Obtained using the definition of (6). (c), (d) Same as (a), (b), but were derived from TARA measurements.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Histogram of the retrieved *β* values. The gray bars represent *β* values calculated using retrieved DSD parameters from the Beard and Chuang relation. The dot–dashed lines represent *β* values calculated from Brandes et al. (2002) relation. The solid lines depict histograms of *β* values retrieved from measurements. Histograms are derived from CSU–CHILL measurements. (a) Obtained using *β* as defined in Eq. (5). (b) Obtained using the definition of (6). (c), (d) Same as (a), (b), but were derived from TARA measurements.

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by the CSU–CHILL. The spectrum was calculated from 64 samples. A total of 15 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by the CSU–CHILL. The spectrum was calculated from 64 samples. A total of 15 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

An example of Doppler power spectral density measured by the CSU–CHILL. The spectrum was calculated from 64 samples. A total of 15 spectra were averaged to obtain this plot. The gray line represents the measurement and the black line shows the fit to the data obtained by solving (6).

Citation: Journal of Atmospheric and Oceanic Technology 23, 12; 10.1175/JTECH1945.1

Simulation specifications. Here *N*_{av} stands for the number of spectra used for averaging and *N*_{FFT} stands for FFT length. The simulation specifications coincide with TARA measurement specifications (see Table 2).

Measurements specifications.

^{1}

The validity of this step can be shown as follows. The standard deviation of the estimated mean velocity for each spectrum is roughly equal to the size of a velocity resolution cell (Bringi and Chandrasekar 2001). The air motion variability, on the other hand, would often result in the variation of the mean velocity from one spectra to another that exceeds the width of a velocity resolution cell. Therefore, if this processing step is not applied, additional broadening of the spectra and an increase in the retrieval errors can result, as was shown in section 4b.