## 1. Introduction

Polarimetric radar measurements are used to retrieve properties of raindrop distributions. The common procedure assumes that raindrops are represented by a family of drop size distributions and retrieves the parameters from an assumed relationship among the shape and size of raindrops and measurements of radar reflectivity (*Z _{h}*), differential reflectivity (

*Z*

_{dr}), and differential propagation phase (Φ

_{dp}). There has been extensive research conducted on microphysical retrievals such as hydrometeor classification or hail detection (Aydin et al. 1986). However, there has not been much research into polarimetric radar based deduction of raindrop shape even though the shape of raindrops plays a critical role in rainfall estimation from polarimetric radars.

From a theoretical viewpoint, an equilibrium shape of raindrops as the result of a balance of surface tension, aerodynamic forces, and hydrostatic and internal pressures has been assumed in earlier studies, although it was assumed that the average shape departs significantly from that of the equilibrium shape. With the increasing importance of polarimetric measurements, this departure can no longer be neglected. There has been considerable discussion about the deviation of the drop axis ratio from its equilibrium value (Goddard et al. 1982; Beard and Chuang 1987; Chandrasekar et al. 1988; Bringi et al. 1998; Andsager et al. 1999; Gorgucci et al. 2000b; Keenan et al. 2001; Brandes et al. 2002; Thurai and Bringi 2005). All of the above studies have produced a plethora of models emphasizing that a nonlinear relationship should be better than a linear one rather than fixing these new relationships in order to obtain more robust polarimetric estimates. Another point that should be stressed is how different climatological regimes affect the drop shape.

In this work, the retrieval of the drop shape directly from polarimetric radar measurements is presented and evaluated. The use of radar data collected from regions with different climatological regimes (here, Florida and Italy) allows a comparison between the retrieved drop shapes.

## 2. The shape of the raindrop

The equilibrium shape of a raindrop falling at its terminal fall speed is determined by the balance among the forces of surface tension, hydrostatic pressure, and aerodynamic pressure from airflow around the drop.

For approximating the shapes of raindrops falling at terminal velocities, Green (1975) used a simple hydrostatic model representing the droplets as oblate spheroids with axis ratios determined by the balance of surface tension and hydrostatic forces.

*b/a*) between the vertical (

*b*) and the horizontal axis (

*a*) of the raindrop with the equivolumetric drop diameter

*D*> 0.5 mm by the empirical equation

*b/a*= 1 for

*D*< 0.5 mm. Hereafter, diameter units will be expressed in millimeters. A more precise model for the shape of raindrops falling at their terminal velocities was obtained by Beard and Chuang (1987), which introduced to the equilibrium condition the contribution made by aerodynamic pressure. The model is able to explain the drop shape with its characteristic flattened base that increases with drop size and can be expressed in terms of a polynomial

The shapes of raindrops have also been studied experimentally in natural rainfall using aircraft probes by Chandrasekar et al. (1988) and by Bringi et al. (1998). The experimental results were consistent with the model results of Beard and Chuang (1987).

*Z*

_{dr}disdrometer estimates exceeded radar measurements whereas the

*Z*showed good agreement. They attributed this effect to the fact that the radar measured the smaller raindrops as less oblate shapes than those given by the Pruppacher and Beard (1970) relation and explained this discrepancy by hypothesizing that smaller drops were more easily perturbed from the predicted shape by turbulence that could induce canting or vibration on the droplets. Taking account of this, they suggested the following polynomial:

_{h}*Z*

_{dr}. An axis ratio relationship representing more spherical drop shapes was determined by combining different equations as

*D*< 4 mm, and agrees quite well with the relationship of Andsager et al. (1999) for

*D*< 4 mm.

Raindrop oscillations and mechanisms capable of maintaining oscillation against viscous decay (Tokay and Beard 1996) have been the subject of many laboratory and field experiments because they are supposed to play a crucial role in determining drop shape (Testik and Barros 2006).

The use of 2D video disdrometers that can determine the shapes and velocity of droplets gives a stimulus to more studies on drop size distribution. In a recent study using a 2D video disdrometer, Thurai and Bringi (2005) show that droplets for diameters ranging from 1.5 to 9 mm, experimentally obtained by artificial water drops generated from water source on a bridge 80 m above the disdrometer, present a mean axis ratio that decreases with increasing drop diameter in agreement with Beard and Chuang’s (1987) equilibrium shape model. Moreover, they found that the predominant oscillation mode is the oblate–prolate axisymmetric mode.

*β*) approximating the implied shape–size function. The estimation of

*β*from polarimetric radar data collected by the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar in Colorado yielded values close to the Pruppacher and Beard (1970) shape relationship. This relationship can be approximated by a straight line given by

*a*/

*b*= 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}is close to the equilibrium shape–size relation of Pruppacher and Beard (1970), and therefore such value is denoted by

*β*

_{e}. It should be noted that

*β*>

*β*

_{e}indicates that raindrops are more oblate than at equilibrium, whereas

*β*<

*β*

_{e}indicates raindrops are less oblate (or closer to spherical) than at equilibrium.

## 3. Polarimetric radar measurement: DSD and drop-shape relation

Raindrop size distribution and drop shapes are of central importance in determining the electromagnetic scattering properties of rain-filled media (Bringi and Chandrasekar, 2001). These effects, in turn, are embodied in the radar parameters of interest such as reflectivity factor (*Z*_{h,v}), differential reflectivity (*Z*_{dr}), which is the ratio of reflectivities at h and v polarization states (Seliga and Bringi 1976), and specific differential phase (*K*_{dp}), which is due to the propagation phase difference between h and v polarization states (Seliga and Bringi 1978).

*N(D)*is the number of raindrops per unit volume per unit size interval (

*D*to

*D + ΔD*),

*n*is the concentration, and

_{c}*f*is the probability density function (Chandrasekar and Bringi 1987)

_{D}(D)*μ*are parameters of the gamma probability density function (PDF) and Γ indicates gamma function. The volume-weighted median diameter

*D*

_{0}can be defined as

*n*. The three parameters Λ,

_{c}*μ*, and

*n*describe the DSD;

_{c}*D*

_{0}and Λ are related by

*N*is the intercept parameter of a normalized gamma DSD and it is related to the nonnormalized one by

_{w}*Z*at horizontal and vertical polarization can be expressed as

_{h,υ}*σ*denote the radar cross sections at the two linear polarizations,

_{h,υ}*λ*is the wavelength, and

*k*is the dielectric constant of water. Similarly, the differential reflectivity and specific differential phase can be expressed as

*f*are the forward scatter amplitudes at the polarization states. It can be seen from (1)–(7) that for a given DSD,

_{h,υ}*Z*,

_{h}*Z*

_{dr}, and

*K*

_{dp}can change with shape–size relationships for raindrops. In general, raindrops become more oblate when the size is large and then the effect of the varying shape–size relationship should be more evident in the presence of larger drops. The commonly used approximation for a raindrop shape is oblate spheroid. If

*b*and

*a*are the semimajor and semiminor axes of an oblate spheroid model of a raindrop, the corresponding axis ratio is

*r*=

*b*/

*a*.

*ρ*is the density of water. The specific differential phase can be expressed in terms of water content

_{w}*W*and raindrop shape as

*C*≈ 3.75 is dimensionless and independent of wavelength (Bringi and Chandrasekar 2001) and

*is the mass weighted raindrop axis ratio defined as*r

_{m}*r*is the axis ratio. Similarly, the differential reflectivity is expressed as the inverse (7/3)rd moment of the raindrop axis ratio, as

*r*

^{7/3}〉 can be defined as the reflectivity-factor weighted mean of

*r*

^{7/3}.

*N*can be eliminated by constructing the ratio (

_{w}*K*

_{dp}/

*Z*) as

_{h}*D*is the mass weighted mean diameter defined as

_{m}*D*and

_{m}*K*

_{dp}, a relation between reflectivity weighted mean diameter (

*D*) and

_{Z}*Z*

_{dr}can be derived

*D*can be expressed as

_{Z}*D*through

_{m}*D*in the expression (22) and using (24a) this reduces to

_{Z}*g(μ)*combines all the constants as well as functions of

*μ*in the process. It should be noted that

*g(μ)*is nearly constant and does not vary too much with

*μ.*From the above expression (26), it can be clearly seen that the relation between (

*K*

_{dp}/

*Z*) and

_{h}*Z*

_{dr}is dependent only on the shape model. Though the above expression was derived assuming a linear model for raindrop shape–size relation, it is a general result, that the relation between (

*K*

_{dp}/

*Z*) and

_{h}*Z*

_{dr}is completely dependent on the raindrop-shape model, and nearly independent of the DSD parameters.

## 4. 4. Data sources and methodology

### a. Radar measurements

The polarimetric radar data used in this paper were collected by the National Center for Atmospheric Research (NCAR) S-band dual-polarized (S-POL) radar during two campaigns in different climatological regions: 1) in the subtropical climate of central Florida and 2) in the predominantly Mediterranean climate of Italy.

Texas and Florida Underflights Experiment TEFLUN-B, conducted in central Florida in 1998 from 1 August through 30 September, was a field experiment in support of the ground validation component of the Tropical Rainfall Measuring Mission (TRMM) satellite, which carries as one of its precipitation measuring instruments the first spaceborne precipitation radar (TRMM PR). The S-POL was deployed to evaluate the quality and accuracy of the TRMM rainfall products.

The Mesoscale Alpine Program (MAP) was an international research initiative devoted to the study of atmospheric and hydrological processes over mountainous terrain in order to improve knowledge of weather and climate and forecast capabilities over complex topography. A large-scale experimental field in the Alpine region took place in 1999 from 7 September to 15 November. The S-POL was located at the southern end of the Lake Maggiore (Vergiate, Italy).

The data used for this study were carefully selected to avoid contamination of the rain radar profile from ground clutter, bright-band, and anomalous propagation effects, etc. The TEFLUN-B profiles were selected from the entire dataset while the MAP profiles refer to only one intense observation period (IOP 2, 19–21 September 1999) characterized by the passing of a frontal cloud system with embedded convective elements that yielded heavy rainfall throughout the region.

The chosen rain radar profiles are all paths containing 100 range bins distanced 0.150 m apart with an increasing differential phase along the path greater than 10 degrees. The number of profiles considered in this study is 4982 and 1280 for Florida and Italy, respectively.

### b. Simulation of radar measurements for various drop-shape models

Simulations are used here to establish mean curves in order to study the sensitivity of radar measurements to DSD and drop-shape variation. In this data-based simulation, different axis ratio relationships are assumed between those commonly used in the literature for raindrop-shape models. The relationships are the following:

the experimental linear relation of Pruppacher and Beard (1970) [Eq. (1), hereafter PB];

the equilibrium relation of Beard and Chuang (1987) [Eq. (2), hereafter BC];

the experimental relation of Andsager et al. (1999) [Eq. (3), hereafter ABL]. This relation as recommended by the authors has been used for 1 < D < 4; outside this interval the BC relation was taken;

the cubic-polynomial fit by Keenan et al. (2001) [Eq. (4), hereafter KCZM];

the experimental relation found by tuning radar measurements to the disdrometer data by Goddard et al. (1982) [Eq. (5), hereafter GCB];

the relation obtained by combining different relations by Brandes et al. (2002) [Eq. (6), hereafter BZV];

a linear shape–size relation with the slope

*β*= 0.04 mm^{−1}(hereafter*β*4);a linear shape–size relation with the slope

*β*= 0.05 mm^{−1}(hereafter*β*5);a linear shape–size relation with the slope

*β*= 0.07 mm^{−1}(hereafter*β*7).

Figure 1 shows the axis ratio of oblate drops for the different relations 1–9 as a function of equivalent volume diameter. The plot is characterized essentially by three regions. For *D* < 2 mm, the nonlinear relation values are in the range corresponding to the linear relation values 0.04 < *β* < 0.05; for 2 < *D* < 4 mm, the nonlinear relations approach the PB equilibrium relation with increasing diameter; while, for *D* > 4 mm, they substantially approximate the PB relation.

*Z*,

_{h}*Z*

_{dr}, and

*K*

_{dp}are simulated using the constraints of (10 log

_{10}

*Z*) < 55 dB and

_{h}*R*< 300 mm h

^{−1}at 3 GHz for S-band. In the simulation, it is assumed that the drops are canted with the mean canting angle equal to zero and the width of the canting angle distribution 10°. In conclusion, for a wide range of DSD triplets (100 000), a dataset of simulated radar measurements was built for each of the above drop-shape models.

### c. Reconstructed S-band rain profiles

Following Chandrasekar et al. (2004), the selected profiles of *Z _{h}* and

*Z*

_{dr}obtained from TEFLUN-B and MAP were used to generate realistic profiles of DSD parameters. It was shown by Scarchilli et al. (1996) that triplets of measurements

*Z*,

_{h}*Z*

_{dr},

*K*

_{dp}nearly lie on a three-dimensional surface when the drop-shape model is fixed. Therefore, once

*Z*and

_{h}*Z*

_{dr}are specified, the choice of possible

*K*

_{dp}values falls in a narrow range. For each pair of

*Z*and

_{h}*Z*

_{dr}in the S-POL profiles, a search in each dataset of the simulated radar measurements corresponding to the drop-shape models described above provides a possible choice of DSDs that satisfy the observations. One of those DSDs is randomly chosen to establish the reconstructed S-band rain observation profile corresponding to a fixed drop-shape model.

## 5. Drop-shape retrieval methodology

The datasets of the simulated radar measurements have been used to study the sensitivity of the radar measurements *Z _{h}*,

*Z*

_{dr}, and

*K*

_{dp}to the variation of the DSD and drop shape. The reflectivity factor is almost insensitive to the drop shape whereas it is very sensitive to the DSD. Figure 2a shows the mean value of

*Z*

_{dr}as a function of the median volume diameter (

*D*

_{0}) for linear and nonlinear drop-shape models;

*Z*

_{dr}is a relative measurement and it is easy to observe the effect of drop-shape variability. Equation (18) shows that

*K*

_{dp}is influenced not only by the drop shape but also by number concentration. The effect of drop shape on

*K*

_{dp}can be seen by normalizing with

*N*as is shown in Fig. 2b.

_{w}In Fig. 3 the mean value of *Z*_{dr} is shown as a function of *Z _{h}*. It is important to focus on the fact that Figs. 2 and 3, obtained by the simulated radar measurements, represent the result of DSD variation weighted by the different axis ratios as shown by (21) and (22). Here,

*K*

_{dp}is water content multiplied by mass weighted axis ratio whereas

*Z*

_{dr}is (3/7) moment of reflectivity weighted axis ratio. The standard deviation of the parameters presented in Fig. 3 (not shown) has variability of the same order as the mean value. As a result, the contributions from variations of DSD and drop shape are combined such that it is not possible to separate one from the other.

The discussion of section 3 has demonstrated that *K*_{dp}/*Z _{h}* versus

*Z*

_{dr}curves depend on the drop shape. This method of using radar measurements and enforcing consistency to make inferences is sustained by (26).

Figure 4 shows the scatterplot of *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for widely varying DSDs and for different drop-shape relations. The DSD variability results in points along the tight curve and any change in drop-shape model moves the curve up and down.

Starting from the assumption that for the same radar measurement volume the estimation of rain rate obtained using *Z _{h}* and

*Z*

_{dr}must be the same as the estimation using

*K*

_{dp}, for the first time Gorgucci et al. (1992) proposed the synergistic use of the three measurements. Scarchilli et al. (1996) generalized this concept in the self-consistency principle. It is based on the fact that polarization diversity measurements of rainfall, namely the reflectivity factor, differential reflectivity, and specific differential propagation phase, vary in a constrained three-dimensional space for a fixed microphysical model. Keeping all this in mind, Fig. 4 is essentially a two-dimensional manifestation of the self-consistency principle, cast in a way that the variability in drop-shape model is enhanced while suppressing DSD variability.

## 6. Practical considerations

The radar measurements *Z _{h}*,

*Z*

_{dr}, and

*K*

_{dp}are affected by measurement errors that will directly translate into an error of the parameter that must be estimated. These three radar measurements have completely different error structures; in addition, these errors are nearly independent:

*Z*is based on absolute power measurement and has a typical accuracy of 1 dB;

_{h}*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}, which can be estimated to an accuracy of a few degrees, is

*K*

_{dp}. This estimate 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. In the presence of a constant

*Z*path, using a least squares estimate of the Φ

_{h}_{dp}profile, the standard deviation of

*K*

_{dp}is better than that obtained by the finite-difference procedure (Gorgucci et al. 2000a). On the contrary, a presence of

*Z*gradients along the path will affect both the value of

_{h}*K*

_{dp}and its standard deviation.

In general, the main difference between the *K*_{dp} measurement and the two other radar measurements is that *K*_{dp} refers to the path over which it is estimated whereas *Z _{h}* and

*Z*

_{dr}are point measurements that refer to the radar resolution volume. To make

*Z*,

_{h}*Z*

_{dr}, and

*K*

_{dp}comparable, the three measurements are related to a fixed path. In this way,

*Z*represents the mean power along the path in logarithm scale,

_{h}*Z*

_{dr}the ratio between the average power at horizontal and vertical polarization expressed in dB, and

*K*

_{dp}the mean value obtained from the finite difference between the end and the beginning of the differential propagation phase profile. Simulation analysis was done to evaluate the impact of the path integration on the parameters

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}.

Using the reconstructed rain profile dataset for the PB model, all *Z _{h}*,

*Z*

_{dr}, and

*K*

_{dp}pointwise measurements of each range bin are used to compute the two quantities

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}. For each 0.1 dB interval in which

*Z*

_{dr}has been divided, the average value of

*K*

_{dp}/

*Z*has been computed in the range 0.5 <

_{h}*Z*

_{dr}<2 dB. Figure 5 shows the resulting curve (marked A) of

*K*

_{dp}/

*Z*as a function of

_{h}*Z*

_{dr}. A similar curve has been generated doing exactly the same computation for 15-km pathwise measurements and the corresponding

*K*

_{dp}/

*Z*versus

_{h}*Z*

_{dr}is shown in Fig. 5 (marked B). The comparison between the curves A and B shows very little difference even with the presence of a reflectivity gradient along the path. For comparison, the performance of the different averaged

*K*

_{dp}/

*Z*values as a function of

_{h}*Z*

_{dr}for each drop-shape model considered in this study are also shown in Fig. 5. The key result obtained from Fig. 5 is that, in the hypothesis of a rain filled medium following a fixed drop-shape model, the model can be retrieved using pathwise

*Z*,

_{h}*Z*

_{dr}, and

*K*

_{dp}measurements. In fact, also in the occurrence of the DSD variability along the path revealed by the presence of a

*Z*gradient, the difference between point- and pathwise values of

_{h}*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}variables is negligible. Figure 6 shows the frequency of the reflectivity excursion, defined as the difference between the maximum and the minimum value of reflectivity in each path. A cumulative frequency analysis reveals that reflectivity excursions greater than 10, 15, and 20 dB have a frequency of 40%, 15%, and 5%, respectively to be encountered along the path.

Another aspect that should be addressed regarding the sensitivity of *K*_{dp}/*Z _{h}* versus

*Z*

_{dr}relation is the bias on

*Z*and

_{h}*Z*

_{dr}estimates. Here,

*Z*

_{dr}is a relative measurement and the bias can be easily removed by several techniques available in the literature (Gorgucci et al. 1999). However, absolute bias on

*Z*cannot be removed easily. Any error on

_{h}*Z*is directly converted into a shift of the relationship between

_{h}*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}. Figure 5 also shows the shift in the

*K*

_{dp}/

*Z*versus

_{h}*Z*

_{dr}relation obtained assuming a bias on

*Z*of +1 and −1 dB, respectively.

_{h}## 7. Application of the methodology for retrieving drop shape from polarimetric radar measurements

The methodology developed in this paper was applied to retrieve drop shape from polarimetric radar measurements from two datasets. Both these datasets were collected by the NCAR S-POL radar in two different climatic regions: Florida and Lombardia (Italy). The test data were obtained over 15-km-long profiles where there was precipitation echo; *Z _{h}* was computed as the mean power along the path,

*Z*

_{dr}was obtained as the ratio between the average power at horizontal and vertical polarization, and the

*K*

_{dp}mean value was obtained from the finite difference between the end and the beginning of the differential propagation phase profile Φ

_{dp}. Moreover, in order to reduce the influence of the Φ

_{dp}fluctuation error on the corresponding

*K*

_{dp}path mean value, only profiles that had at least 10° of increase in Φ

_{dp}were chosen. A total of 4982 and 1280 profiles for Florida and Italy, respectively, was obtained. Figures 7a,b show scatterplots between the ratio

*K*

_{dp}/

*Z*as a function of

_{h}*Z*

_{dr}for data collected in Florida. As a reference, the behavior of the different averaged

*K*

_{dp}/

*Z*path values as a function of

_{h}*Z*

_{dr}are also shown for the different drop-shape models considered in this study. Figure 7a shows the scatterplot obtained using the data collected on 8 August. The points are essentially located around the PB model. Figure 7b shows the scatterplot for data collected on 20 September where the points are limited in the space bordered by

*β*= 0.05 and the PB model, showing a drop shape less oblate than the equilibrium. It should be emphasized here that, in the plane defined by the quantities

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}, these values for the entire datasets are mostly located within the region bordered by the BC and

*β*= 0.07 models. More precisely, for the Florida dataset, the distribution shows that 95.9% of the measurements are placed in that surface whereas 3.9% and 0.2% present axis ratios greater than

*β*= 0.07 and less than of BC model, respectively. For the Italian dataset the values are 93.2%, 6.7%, and 0.1%, respectively. It should be noted that both datasets present a very small percentage of measurements with axis ratios less oblate than the BC ones.

Assuming a linear relationship for the underlying drop shape, it is possible to find, interpolating each pair of *K*_{dp}/*Z _{h}* and

*Z*

_{dr}measurements with

*β*= 0.05–0.07 models, a value of

*β*in the (

*K*

_{dp}/

*Z*,

_{h}*Z*

_{dr}) domain. It is important to recognize that, even if the original drop axis ratio is a nonlinear function, it is possible to define an equivalent linear model that results in the same

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}pair. Moreover, it should be emphasized that this slope takes account of the effects due to the average canting angle and oscillation of raindrops.

Figures 8a,b show the histograms of *β* obtained from the Florida dataset and from the Italy dataset, respectively. It must be pointed out that the two histograms are not comparable because the Florida dataset is composed of radar measurements spread over three months whereas the Italy dataset is built up over only one event. However, Fig. 8b reveals a nonsymmetrical distribution of the drop shape showing higher frequencies for the more oblate drop shapes.

An analysis of the mean *β* as a function of time has been performed for the Florida dataset. The result is shown in Fig. 9, which shows that the mean *β* ranges between 0.058 and 0.066 mm^{−1} as a function of the day. It is also evident that *β* oscillates around two averaged values, 0.064 mm^{−1} up to 14 August, and then 0.059 mm^{−1}. Range variability of the *β* standard deviation confirms two different behaviors, showing a tight distribution around the mean value before 14 August and a broader distribution thereafter.

A larger amount of information about the underlying drop shape can be obtained by comparing the distribution of the *K*_{dp}/*Z _{h}* and Z

_{dr}values with the corresponding values of the other models considered in this study. Figures 10a,b show the distribution of the

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}values for the two datasets of Florida (Fig. 10a) and Italy (Fig. 10b). Figure 10a shows that the different contours are centered around a parallel line to the PB curve, denoting a linear relation as in the PB model, except that the droplets are more spherical. The same argument can be made for the distribution shown in Fig. 10b except that the droplets appear more oblate that the PB equilibrium model.

## 8. Summary and conclusions

The paper presents a technique by which it is possible to reveal information about the mean actual axis ratio corresponding to the polarimetric radar measurements. This technique follows the methodology introduced by Gorgucci et al. (1992): to use in synergy the polarimetric radar measurements—namely, the reflectivity factor, differential reflectivity, and specific differential phase—and taking into account the generalization offered by Scarchilli et al. (1996) with the self-consistency principle of the polarimetric radar measurements. The methodology presented in this paper is a two-dimensional manifestation of the 3D self-consistency principle developed to examine raindrop shape. This is achievable by considering a two-dimensional domain defined by the two variables *K*_{dp}/*Z _{h}* and

*Z*

_{dr}. Using simulation, it can be seen that for fixed drop-shape models the different

*K*

_{dp}/

*Z*and

_{h}*Z*

_{dr}pairs, obtained widely varying DSD are constrained over a well-defined curve. As a result, any changes from the curve depend on variability of the drop shapes. Plotting on this domain the curves relative to the different drop-shape models considered in this study, it is possible to examine the prevailing drop shape related to

*Z*,

_{h}*Z*

_{dr}, and

*K*

_{dp}measurements.

To overcome practical issues related to measurement fluctuations and to make comparable pointwise measurements of *Z _{h}* and

*Z*

_{dr}with path measurement

*K*

_{dp}, all the radar measurements are computed for the same path. A dataset was set up using S-band profiles collected by NCAR S-POL radar during measurement campaigns conducted in two different climatic regions: Florida and Lombardia (Italy). From this experimental dataset, a second one was built up by simulation. For each drop-shape model, keeping in mind the self-consistency principle, by simulation from

*Z*and

_{h}*Z*

_{dr}S-POL measurements were found a corresponding possible value of

*K*

_{dp}. Using the simulated dataset, and with

*Z*, Z

_{h}_{dr}, and

*K*

_{dp}as pointwise or path measurements, it is possible to study the influence of the gradient on the technique. From a 15-km path composed by 100 range bins, the influence of the gradient is very minor when compared with the drop-shape variability shown by the different model.

In this two-dimensional domain, assuming a linear drop-shape model underlying each of *K*_{dp}/*Z _{h}* and

*Z*

_{dr}experimental pairs by interpolation with the

*β*curves corresponding to 0.07, 0.062, and 0.05 mm

^{−1}, it is possible to estimate a

*β*value. The resulting

*β*histograms show higher frequency in the range 0.057 <

*β*< 0.061 and 0.061 <

*β*< 0.064, respectively, for Florida and Italy. However, it should be pointed out that the two different behaviors are not strictly comparable because the Florida dataset refers to measurements distributed over three months whereas the Italy dataset is composed by only one event with prevailing convective condition.

From the radar datasets it has been found that the underlying drop-shape model is slightly more spherical than the PB model with the BC model as the boundary. This means the dataset used in this study did not produce any mean raindrop axis ratios less than the axis ratios of BC shape model. In the domain defined by *K*_{dp}/*Z _{h}* versus

*Z*

_{dr}variables the drop-shape models of GCB, BZV, ABL, and KCZM are located outside the BC model boundary.

## Acknowledgments

This research was conducted as part of the NASA Precipitation Measurement Mission (TRMM/GPM) program. The research was partially supported by the National Group by Defense from Hydrological Hazard (CNR, Italy), the European Commission through the Interreg IIIB CADSES “RISK-AWARE” (3B064) project, and the NSF (ATM-0313881).

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Mean values of (a) *Z*_{dr} and (b) *K*_{dp}/*N _{w}* for widely varying DSD as a function of the median volume diameter

*D*

_{0}for the nonlinear relations of BC, ABL, and KCZM (solid lines), and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(dashed lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Mean values of (a) *Z*_{dr} and (b) *K*_{dp}/*N _{w}* for widely varying DSD as a function of the median volume diameter

*D*

_{0}for the nonlinear relations of BC, ABL, and KCZM (solid lines), and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(dashed lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Mean values of (a) *Z*_{dr} and (b) *K*_{dp}/*N _{w}* for widely varying DSD as a function of the median volume diameter

*D*

_{0}for the nonlinear relations of BC, ABL, and KCZM (solid lines), and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(dashed lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Mean values of differential reflectivity for widely varying DSD as a function of reflectivity for the nonlinear relations of BC, ABL, and KCZM (dashed lines), and for the linear relations of PB, *β* = 0.04 mm^{−1}, *β* = 0.05 mm^{−1}, and *β* = 0.07 mm^{−1} (solid lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Mean values of differential reflectivity for widely varying DSD as a function of reflectivity for the nonlinear relations of BC, ABL, and KCZM (dashed lines), and for the linear relations of PB, *β* = 0.04 mm^{−1}, *β* = 0.05 mm^{−1}, and *β* = 0.07 mm^{−1} (solid lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Mean values of differential reflectivity for widely varying DSD as a function of reflectivity for the nonlinear relations of BC, ABL, and KCZM (dashed lines), and for the linear relations of PB, *β* = 0.04 mm^{−1}, *β* = 0.05 mm^{−1}, and *β* = 0.07 mm^{−1} (solid lines).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot of *K*_{dp}/*Z _{h}* ratio for widely varying DSD as a function of

*Z*

_{dr}for the nonlinear relations of BC, ABL, and KCZM, and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot of *K*_{dp}/*Z _{h}* ratio for widely varying DSD as a function of

*Z*

_{dr}for the nonlinear relations of BC, ABL, and KCZM, and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot of *K*_{dp}/*Z _{h}* ratio for widely varying DSD as a function of

*Z*

_{dr}for the nonlinear relations of BC, ABL, and KCZM, and for the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged pointwise A (black dashed line) and pathwise B (gray dashed line) of *K*_{dp}/*Z _{h}* ratio computed from the reconstructed S-band rain profiles as a function of

*Z*

_{dr}for the PB model. Gray dashed lines labeled ±1 dB indicate shifts due to the presence of ±1 dB bias on

*Z*. For comparison, corresponding averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and linear relations of PB,

_{h}*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged pointwise A (black dashed line) and pathwise B (gray dashed line) of *K*_{dp}/*Z _{h}* ratio computed from the reconstructed S-band rain profiles as a function of

*Z*

_{dr}for the PB model. Gray dashed lines labeled ±1 dB indicate shifts due to the presence of ±1 dB bias on

*Z*. For comparison, corresponding averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and linear relations of PB,

_{h}*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged pointwise A (black dashed line) and pathwise B (gray dashed line) of *K*_{dp}/*Z _{h}* ratio computed from the reconstructed S-band rain profiles as a function of

*Z*

_{dr}for the PB model. Gray dashed lines labeled ±1 dB indicate shifts due to the presence of ±1 dB bias on

*Z*. For comparison, corresponding averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and linear relations of PB,

_{h}*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Occurrence frequencies of reflectivity excursion, defined as the differences between the maximum and the minimum value of reflectivity in the path, computed from the reconstructed S-band rain profiles for the PB model.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Occurrence frequencies of reflectivity excursion, defined as the differences between the maximum and the minimum value of reflectivity in the path, computed from the reconstructed S-band rain profiles for the PB model.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Occurrence frequencies of reflectivity excursion, defined as the differences between the maximum and the minimum value of reflectivity in the path, computed from the reconstructed S-band rain profiles for the PB model.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot between the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected (a) on 8 Aug 1998 and (b) on 20 Sep 1998. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot between the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected (a) on 8 Aug 1998 and (b) on 20 Sep 1998. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Scatterplot between the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected (a) on 8 Aug 1998 and (b) on 20 Sep 1998. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB,

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Histograms of *β* for the entire dataset (a) of Florida and (b) of Lombardia (Italy), assuming a linear relation for the underlying drop shape.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Histograms of *β* for the entire dataset (a) of Florida and (b) of Lombardia (Italy), assuming a linear relation for the underlying drop shape.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Histograms of *β* for the entire dataset (a) of Florida and (b) of Lombardia (Italy), assuming a linear relation for the underlying drop shape.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged (solid line) and standard deviation (dash lines) of *β* as function of the day for S-POL dataset of Florida.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged (solid line) and standard deviation (dash lines) of *β* as function of the day for S-POL dataset of Florida.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Averaged (solid line) and standard deviation (dash lines) of *β* as function of the day for S-POL dataset of Florida.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Contours of occurrence frequency of the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected during (a) Teflun-B (Florida) in 1998 and (b) IOP 2, 19–21 Sep 1999 of MAP. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB, for

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Contours of occurrence frequency of the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected during (a) Teflun-B (Florida) in 1998 and (b) IOP 2, 19–21 Sep 1999 of MAP. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB, for

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1

Contours of occurrence frequency of the ratio *K*_{dp}/*Z _{h}* as a function of

*Z*

_{dr}for S-POL data collected during (a) Teflun-B (Florida) in 1998 and (b) IOP 2, 19–21 Sep 1999 of MAP. For comparison, averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and the linear relations of PB, for

*β*= 0.04 mm

^{−1},

*β*= 0.05 mm

^{−1}, and

*β*= 0.07 mm

^{−1}(solid lines), are also shown.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3781.1