What Is the Shape of a Raindrop? An Answer from Radar Measurements

Eugenio Gorgucci Istituto di Scienze dell’Atmosfera e del Clima (CNR), Rome, Italy

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Luca Baldini Istituto di Scienze dell’Atmosfera e del Clima (CNR), Rome, Italy

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V. Chandrasekar Colorado State University, Fort Collins, Colorado

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Abstract

In a previous study, Gorgucci et al. showed the potential advantage of using together polarimetric radar measurements of reflectivity factor, differential reflectivity, and specific differential propagation phase, in order to gather information about the calibration of radar systems. Scarchilli et al. generalized this concept in the self-consistency principle, which stated that, given a drop-shape model to describe the form of raindrops, the corresponding radar measurements are constrained on this three-dimensional surface. In this work the self-consistency principle is collapsed onto a two-dimensional domain defined by the variables: 1) the ratio between specific differential phase and reflectivity factor, and 2) differential reflectivity. In this space the scatter of drop size distribution (DSD) variability is minimized in such a way that drop-shape variability shows up. This methodology is used to observe for the first time the predominant shape of raindrops directly from the radar measurements. The radar polarimetric data were collected in two different climatological regions as central Florida and northern Italy. The significant result shows that the underlying mean axis ratio approaches the model established by Pruppacher and Beard, and the relationship described by Beard and Chuang forms a sort of border for the sphericity of the drop shape.

Corresponding author address: Dr. Eugenio Gorgucci, Istituto di Scienze dell’Atmosfera e del Clima (CNR), Area di Ricerca Roma-Tor Vergata, Via Fosso del Cavaliere, 100–00133 Rome, Italy. Email: gorgucci@radar.ifa.rm.cnr.it

Abstract

In a previous study, Gorgucci et al. showed the potential advantage of using together polarimetric radar measurements of reflectivity factor, differential reflectivity, and specific differential propagation phase, in order to gather information about the calibration of radar systems. Scarchilli et al. generalized this concept in the self-consistency principle, which stated that, given a drop-shape model to describe the form of raindrops, the corresponding radar measurements are constrained on this three-dimensional surface. In this work the self-consistency principle is collapsed onto a two-dimensional domain defined by the variables: 1) the ratio between specific differential phase and reflectivity factor, and 2) differential reflectivity. In this space the scatter of drop size distribution (DSD) variability is minimized in such a way that drop-shape variability shows up. This methodology is used to observe for the first time the predominant shape of raindrops directly from the radar measurements. The radar polarimetric data were collected in two different climatological regions as central Florida and northern Italy. The significant result shows that the underlying mean axis ratio approaches the model established by Pruppacher and Beard, and the relationship described by Beard and Chuang forms a sort of border for the sphericity of the drop shape.

Corresponding author address: Dr. Eugenio Gorgucci, Istituto di Scienze dell’Atmosfera e del Clima (CNR), Area di Ricerca Roma-Tor Vergata, Via Fosso del Cavaliere, 100–00133 Rome, Italy. Email: gorgucci@radar.ifa.rm.cnr.it

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 (Zh), differential reflectivity (Zdr), and differential propagation phase (Φdp). 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.

Wind tunnel measurements by Pruppacher and Beard (1970) found that the raindrop shape can be defined in terms of the axial ratio (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
i1520-0469-63-11-3033-e1
while 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
i1520-0469-63-11-3033-e2

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).

From laboratory measurements, Andsager et al. (1999) showed that droplets with diameters between 2.9 and 4.0 mm exhibited a mean axis ratio larger than that at the theoretical equilibrium and obtained a second-order polynomial fit explaining the generally higher average axis ratio of the raindrops given as
i1520-0469-63-11-3033-e3
Starting from the fact that raindrops in real clouds may be even less oblate for a given size than those obtained by the equilibrium relationship of Beard and Chuang (1987), and arguing on the potential weakness of each dataset from which the axial ratios were obtained, Keenan et al. (2001) used different empirical and observational relationships and deduced another equation based on a third-order least squares polynomial fit
i1520-0469-63-11-3033-e4
Goddard et al. (1982) compared dual-polarization radar measurements of rain with ground-based disdrometer measurements and noted that, for distribution with predominantly small drops, Zdr disdrometer estimates exceeded radar measurements whereas the Zh 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:
i1520-0469-63-11-3033-e5
Brandes et al. (2002), assuming that the discrepancy between experimental and theoretical relations is due to the drop oscillations that determine mean shapes more spherical than the equilibrium values, used experimental data to establish the impact of different drop shapes on the value of disdrometer-derived Zdr. An axis ratio relationship representing more spherical drop shapes was determined by combining different equations as
i1520-0469-63-11-3033-e6
Equation (6) yields axis ratios that are significantly more spherical than were found Pruppacher and Beard (1970), particularly for drops with 1 < 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.

Using the self-consistency of the polarimetric radar measurements, Gorgucci et al. (2000b) proposed a model to describe the shape–size relationship of raindrops in terms of the linear slope (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
i1520-0469-63-11-3033-e7
In (7) a/b = 1 when D < 0.03/β, where β is the magnitude of the slope of the shape–size relationship given by
i1520-0469-63-11-3033-e8
Equation (7) for 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 βe. It should be noted that β > βe indicates that raindrops are more oblate than at equilibrium, whereas β < βe 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 (Zh,v), differential reflectivity (Zdr), which is the ratio of reflectivities at h and v polarization states (Seliga and Bringi 1976), and specific differential phase (Kdp), which is due to the propagation phase difference between h and v polarization states (Seliga and Bringi 1978).

Ulbrich (1983) stated that a gamma drop size distribution (DSD) adequately describes many of the natural variations. The DSD describes the number density of raindrops, which can be written in terms of the probability density of the raindrops as
i1520-0469-63-11-3033-e10
where N(D) is the number of raindrops per unit volume per unit size interval (D to D + ΔD), nc is the concentration, and fD(D) is the probability density function (Chandrasekar and Bringi 1987)
i1520-0469-63-11-3033-e11
where Λ and μ are parameters of the gamma probability density function (PDF) and Γ indicates gamma function. The volume-weighted median diameter D0 can be defined as
i1520-0469-63-11-3033-e12
The gamma density consists of two parameters with the addition of the number concentration nc. The three parameters Λ, μ, and nc describe the DSD; D0 and Λ are related by
i1520-0469-63-11-3033-e13
Testud et al. (2000) fully developed a normalizing concept suggested by Srivastava (1971) and Willis (1984), to scale the variations due to widely varying water content, so that the inherent DSD shape can be observed. The alternative form of normalizing the DSD with respect to water content can be written as
i1520-0469-63-11-3033-e14
where Nw is the intercept parameter of a normalized gamma DSD and it is related to the nonnormalized one by
i1520-0469-63-11-3033-e15a
i1520-0469-63-11-3033-e15b
Once the DSD is established, the reflectivity factor Zh,υ at horizontal and vertical polarization can be expressed as
i1520-0469-63-11-3033-e16
where σh,υ denote the radar cross sections at the two linear polarizations, λ is the wavelength, and k is the dielectric constant of water. Similarly, the differential reflectivity and specific differential phase can be expressed as
i1520-0469-63-11-3033-e17
and
i1520-0469-63-11-3033-e18
where fh,υ are the forward scatter amplitudes at the polarization states. It can be seen from (1)(7) that for a given DSD, Zh, Zdr, and Kdp 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.
For a normalized gamma model and assuming Rayleigh scattering the reflectivity and water content can be expressed as
i1520-0469-63-11-3033-e19a
i1520-0469-63-11-3033-e19b
where ρw is the density of water. The specific differential phase can be expressed in terms of water content W and raindrop shape as
i1520-0469-63-11-3033-e20a
where C ≈ 3.75 is dimensionless and independent of wavelength (Bringi and Chandrasekar 2001) and rm is the mass weighted raindrop axis ratio defined as
i1520-0469-63-11-3033-e20b
where r is the axis ratio. Similarly, the differential reflectivity is expressed as the inverse (7/3)rd moment of the raindrop axis ratio, as
i1520-0469-63-11-3033-e21
where 〈r7/3〉 can be defined as the reflectivity-factor weighted mean of r7/3.
The normalized intercept Nw can be eliminated by constructing the ratio (Kdp/Zh) as
i1520-0469-63-11-3033-e22
where Dm is the mass weighted mean diameter defined as
i1520-0469-63-11-3033-e23
Similarly to the relation between Dm and Kdp, a relation between reflectivity weighted mean diameter (DZ) and Zdr can be derived
i1520-0469-63-11-3033-e24a
i1520-0469-63-11-3033-e24b
For the linear raindrop-shape model (7) introduced by Gorgucci et al. (2000b), DZ can be expressed as
i1520-0469-63-11-3033-e25
Substituting the expression for Dm through DZ in the expression (22) and using (24a) this reduces to
i1520-0469-63-11-3033-e26
where 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 (Kdp/Zh) and Zdr 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 (Kdp/Zh) and Zdr 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:

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

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

  3. 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;

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

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

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

  7. a linear shape–size relation with the slope β = 0.04 mm−1 (hereafter β4);

  8. a linear shape–size relation with the slope β = 0.05 mm−1 (hereafter β5);

  9. a linear shape–size relation with the slope β = 0.07 mm−1 (hereafter β7).

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.

Assuming the DSD parameters varying in a wide range as
i1520-0469-63-11-3033-e27
for each drop-shape relation, radar measurements Zh, Zdr, and Kdp are simulated using the constraints of (10 log10 Zh) < 55 dB and 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 Zh and Zdr 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 Zh, Zdr, Kdp nearly lie on a three-dimensional surface when the drop-shape model is fixed. Therefore, once Zh and Zdr are specified, the choice of possible Kdp values falls in a narrow range. For each pair of Zh and Zdr 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 Zh, Zdr, and Kdp 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 Zdr as a function of the median volume diameter (D0) for linear and nonlinear drop-shape models; Zdr is a relative measurement and it is easy to observe the effect of drop-shape variability. Equation (18) shows that Kdp is influenced not only by the drop shape but also by number concentration. The effect of drop shape on Kdp can be seen by normalizing with Nw as is shown in Fig. 2b.

In Fig. 3 the mean value of Zdr is shown as a function of Zh. 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, Kdp is water content multiplied by mass weighted axis ratio whereas Zdr 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 Kdp/Zh versus Zdr 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 Kdp/Zh as a function of Zdr 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 Zh and Zdr must be the same as the estimation using Kdp, 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 Zh, Zdr, and Kdp 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: Zh is based on absolute power measurement and has a typical accuracy of 1 dB; Zdr 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 Φdp, which can be estimated to an accuracy of a few degrees, is Kdp. This estimate depends on the procedure used to compute the range derivative of Φdp such as a simple finite-difference scheme or a least squares fit. In the presence of a constant Zh path, using a least squares estimate of the Φdp profile, the standard deviation of Kdp is better than that obtained by the finite-difference procedure (Gorgucci et al. 2000a). On the contrary, a presence of Zh gradients along the path will affect both the value of Kdp and its standard deviation.

In general, the main difference between the Kdp measurement and the two other radar measurements is that Kdp refers to the path over which it is estimated whereas Zh and Zdr are point measurements that refer to the radar resolution volume. To make Zh, Zdr, and Kdp comparable, the three measurements are related to a fixed path. In this way, Zh represents the mean power along the path in logarithm scale, Zdr the ratio between the average power at horizontal and vertical polarization expressed in dB, and Kdp 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 Kdp/Zh and Zdr.

Using the reconstructed rain profile dataset for the PB model, all Zh, Zdr, and Kdp pointwise measurements of each range bin are used to compute the two quantities Kdp/Zh and Zdr. For each 0.1 dB interval in which Zdr has been divided, the average value of Kdp/Zh has been computed in the range 0.5 < Zdr<2 dB. Figure 5 shows the resulting curve (marked A) of Kdp/Zh as a function of Zdr. A similar curve has been generated doing exactly the same computation for 15-km pathwise measurements and the corresponding Kdp/Zh versus Zdr 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 Kdp/Zh values as a function of Zdr 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 Zh, Zdr, and Kdp measurements. In fact, also in the occurrence of the DSD variability along the path revealed by the presence of a Zh gradient, the difference between point- and pathwise values of Kdp/Zh and Zdr 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 Kdp/Zh versus Zdr relation is the bias on Zh and Zdr estimates. Here, Zdr 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 Zh cannot be removed easily. Any error on Zh is directly converted into a shift of the relationship between Kdp/Zh and Zdr. Figure 5 also shows the shift in the Kdp/Zh versus Zdr relation obtained assuming a bias on Zh of +1 and −1 dB, respectively.

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; Zh was computed as the mean power along the path, Zdr was obtained as the ratio between the average power at horizontal and vertical polarization, and the Kdp mean value was obtained from the finite difference between the end and the beginning of the differential propagation phase profile Φdp. Moreover, in order to reduce the influence of the Φdp fluctuation error on the corresponding Kdp path mean value, only profiles that had at least 10° of increase in Φdp were chosen. A total of 4982 and 1280 profiles for Florida and Italy, respectively, was obtained. Figures 7a,b show scatterplots between the ratio Kdp/Zh as a function of Zdr for data collected in Florida. As a reference, the behavior of the different averaged Kdp/Zh path values as a function of Zdr 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 Kdp/Zh and Zdr, 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 Kdp/Zh and Zdr measurements with β = 0.05–0.07 models, a value of β in the (Kdp/Zh, Zdr) 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 Kdp/Zh and Zdr 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 Kdp/Zh and Zdr values with the corresponding values of the other models considered in this study. Figures 10a,b show the distribution of the Kdp/Zh and Zdr 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 Kdp/Zh and Zdr. Using simulation, it can be seen that for fixed drop-shape models the different Kdp/Zh and Zdr 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 Zh, Zdr, and Kdp measurements.

To overcome practical issues related to measurement fluctuations and to make comparable pointwise measurements of Zh and Zdr with path measurement Kdp, 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 Zh and Zdr S-POL measurements were found a corresponding possible value of Kdp. Using the simulated dataset, and with Zh, Zdr, and Kdp 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 Kdp/Zh and Zdr 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 Kdp/Zh versus Zdr 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).

REFERENCES

  • Andsager, K., K. V. Beard, and N. F. Laird, 1999: Laboratory measurements of axis ratios for large raindrops. J. Atmos. Sci., 56 , 26732683.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., T. A. Seliga, and V. Balaji, 1986: Remote sensing of hail with a dual linear polarization radar. J. Appl. Meteor., 25 , 14751484.

    • Search Google Scholar
    • Export Citation
  • Beard, K. V., and C. Chuang, 1987: A new model for the equilibrium shape of raindrops. J. Atmos. Sci., 44 , 15091524.

  • Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41 , 674685.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 648 pp.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., V. Chandrasekar, and R. Xiao, 1998: Raindrop axis ratio and size distribution in Florida rainshaft: An assessment of multiparameter radar algorithms. IEEE Trans. Geosci. Remote Sens., 36 , 703715.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., and V. N. Bringi, 1987: Simulation of radar reflectivity and surface measurements of rainfall. J. Atmos. Oceanic Technol., 4 , 464477.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., W. A. Cooper, and V. N. Bringi, 1988: Axis ratios and oscillations of raindrops. J. Atmos. Sci., 45 , 13231333.

  • Chandrasekar, V., E. Gorgucci, S. Lim, and L. Baldini, 2004: Simulation of X-band radar observation of precipitation from S-band measurements. Proc. IEEE Int. Geoscience and Remote Sensing Symp., Anchorage, AK, IEEE, 2752–2755.

  • Goddard, J. W. F., S. M. Cherry, and V. N. Bringi, 1982: Comparison of dual-polarization radar measurements of rain with ground-based disdrometer measurements. J. Appl. Meteor., 21 , 252256.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1992: Calibration of radars using polarimetric techniques. IEEE Trans. Geosci. Remote Sens., 30 , 853858.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1999: A procedure to calibrate multiparameter weather radar using properties of the rain medium. IEEE Trans. Geosci. Remote Sens., 37 , 269276.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 2000a: Practical aspects of radar rainfall estimation using specific differential propagation phase. J. Appl. Meteor., 39 , 945955.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, V. Chandrasekar, and V. N. Bringi, 2000b: Measurement of mean raindrop shape from polarimetric radar observations. J. Atmos. Sci., 57 , 34063413.

    • Search Google Scholar
    • Export Citation
  • Green, A. W., 1975: An approximation for the shape of large raindrops. J. Appl. Meteor., 14 , 15781583.

  • Keenan, T. D., L. D. Carey, D. S. Zrnic, and P. T. May, 2001: Sensitivity of 5-cm wavelength polarimetric radar variables to raindrop axial ratio and drop size distribution. J. Appl. Meteor., 40 , 526545.

    • Search Google Scholar
    • Export Citation
  • 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 , 247256.

    • Search Google Scholar
    • Export Citation
  • Scarchilli, G., E. Gorgucci, V. Chandrasekar, and A. Dobaie, 1996: Self-consistency of polarization diversity measurement of rainfall. IEEE Trans. Geosci. Remote Sens., 34 , 2226.

    • Search Google Scholar
    • Export Citation
  • Seliga, T. A., and V. N. Bringi, 1976: Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation. J. Appl. Meteor., 15 , 6976.

    • Search Google Scholar
    • Export Citation
  • Seliga, T. A., and V. N. Bringi, 1978: Differential reflectivity and differential phase shift: Application in radar meteorology. Radio Sci., 13 , 271275.

    • Search Google Scholar
    • Export Citation
  • Srivastava, R. C., 1971: Size distribution of raindrops generated by their breakup and coalescence. J. Atmos. Sci., 28 , 410415.

  • Testik, F. Y., and A. P. Barros, 2006: Towards elucidating the microstructure of rainfall. A survey. Rev. Geophys., in press.

  • Testud, J., E. Le Bouar, E. Obligis, and M. Ali-Mehenni, 2000: The rain profiling algorithm applied to polarimetric weather radar. J. Atmos. Oceanic Technol., 17 , 332356.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., and V. N. Bringi, 2005: Drop axis ratios from 2D video disdrometer. J. Atmos. Oceanic Technol., 22 , 966978.

  • Tokay, A., and K. V. Beard, 1996: A field study of raindrop oscillations. Part I: Observation of size spectra and evaluation of oscillation causes. J. Appl. Meteor., 35 , 16711687.

    • Search Google Scholar
    • Export Citation
  • Ulbrich, C., 1983: Natural variations in the analytical form of the raindrop-size distribution. J. Climate Appl. Meteor., 22 , 17641775.

    • Search Google Scholar
    • Export Citation
  • Willis, P. T., 1984: Functional fits to some observed drop size distributions and parameterization of rain. J. Atmos. Sci., 41 , 16481661.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Axis ratio of oblate drops (b/a) as a function of equivalent volume diameter (D) for the nonlinear relations of BC (dotted line), ABL (dotted line), KCZM (dashed line), GCB (dash–dot line), and BZV (solid line), 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

Fig. 2.
Fig. 2.

Mean values of (a) Zdr and (b) Kdp/Nw for widely varying DSD as a function of the median volume diameter D0 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

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

Scatterplot of Kdp/Zh ratio for widely varying DSD as a function of Zdr 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

Fig. 5.
Fig. 5.

Averaged pointwise A (black dashed line) and pathwise B (gray dashed line) of Kdp/Zh ratio computed from the reconstructed S-band rain profiles as a function of Zdr for the PB model. Gray dashed lines labeled ±1 dB indicate shifts due to the presence of ±1 dB bias on Zh. For comparison, corresponding averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and 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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

Scatterplot between the ratio Kdp/Zh as a function of Zdr 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

Fig. 8.
Fig. 8.

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

Fig. 9.
Fig. 9.

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

Fig. 10.
Fig. 10.

Contours of occurrence frequency of the ratio Kdp/Zh as a function of Zdr 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

Save
  • Andsager, K., K. V. Beard, and N. F. Laird, 1999: Laboratory measurements of axis ratios for large raindrops. J. Atmos. Sci., 56 , 26732683.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., T. A. Seliga, and V. Balaji, 1986: Remote sensing of hail with a dual linear polarization radar. J. Appl. Meteor., 25 , 14751484.

    • Search Google Scholar
    • Export Citation
  • Beard, K. V., and C. Chuang, 1987: A new model for the equilibrium shape of raindrops. J. Atmos. Sci., 44 , 15091524.

  • Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41 , 674685.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 648 pp.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., V. Chandrasekar, and R. Xiao, 1998: Raindrop axis ratio and size distribution in Florida rainshaft: An assessment of multiparameter radar algorithms. IEEE Trans. Geosci. Remote Sens., 36 , 703715.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., and V. N. Bringi, 1987: Simulation of radar reflectivity and surface measurements of rainfall. J. Atmos. Oceanic Technol., 4 , 464477.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., W. A. Cooper, and V. N. Bringi, 1988: Axis ratios and oscillations of raindrops. J. Atmos. Sci., 45 , 13231333.

  • Chandrasekar, V., E. Gorgucci, S. Lim, and L. Baldini, 2004: Simulation of X-band radar observation of precipitation from S-band measurements. Proc. IEEE Int. Geoscience and Remote Sensing Symp., Anchorage, AK, IEEE, 2752–2755.

  • Goddard, J. W. F., S. M. Cherry, and V. N. Bringi, 1982: Comparison of dual-polarization radar measurements of rain with ground-based disdrometer measurements. J. Appl. Meteor., 21 , 252256.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1992: Calibration of radars using polarimetric techniques. IEEE Trans. Geosci. Remote Sens., 30 , 853858.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1999: A procedure to calibrate multiparameter weather radar using properties of the rain medium. IEEE Trans. Geosci. Remote Sens., 37 , 269276.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 2000a: Practical aspects of radar rainfall estimation using specific differential propagation phase. J. Appl. Meteor., 39 , 945955.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., G. Scarchilli, V. Chandrasekar, and V. N. Bringi, 2000b: Measurement of mean raindrop shape from polarimetric radar observations. J. Atmos. Sci., 57 , 34063413.

    • Search Google Scholar
    • Export Citation
  • Green, A. W., 1975: An approximation for the shape of large raindrops. J. Appl. Meteor., 14 , 15781583.

  • Keenan, T. D., L. D. Carey, D. S. Zrnic, and P. T. May, 2001: Sensitivity of 5-cm wavelength polarimetric radar variables to raindrop axial ratio and drop size distribution. J. Appl. Meteor., 40 , 526545.

    • Search Google Scholar
    • Export Citation
  • 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 , 247256.

    • Search Google Scholar
    • Export Citation
  • Scarchilli, G., E. Gorgucci, V. Chandrasekar, and A. Dobaie, 1996: Self-consistency of polarization diversity measurement of rainfall. IEEE Trans. Geosci. Remote Sens., 34 , 2226.

    • Search Google Scholar
    • Export Citation
  • Seliga, T. A., and V. N. Bringi, 1976: Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation. J. Appl. Meteor., 15 , 6976.

    • Search Google Scholar
    • Export Citation
  • Seliga, T. A., and V. N. Bringi, 1978: Differential reflectivity and differential phase shift: Application in radar meteorology. Radio Sci., 13 , 271275.

    • Search Google Scholar
    • Export Citation
  • Srivastava, R. C., 1971: Size distribution of raindrops generated by their breakup and coalescence. J. Atmos. Sci., 28 , 410415.

  • Testik, F. Y., and A. P. Barros, 2006: Towards elucidating the microstructure of rainfall. A survey. Rev. Geophys., in press.

  • Testud, J., E. Le Bouar, E. Obligis, and M. Ali-Mehenni, 2000: The rain profiling algorithm applied to polarimetric weather radar. J. Atmos. Oceanic Technol., 17 , 332356.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., and V. N. Bringi, 2005: Drop axis ratios from 2D video disdrometer. J. Atmos. Oceanic Technol., 22 , 966978.

  • Tokay, A., and K. V. Beard, 1996: A field study of raindrop oscillations. Part I: Observation of size spectra and evaluation of oscillation causes. J. Appl. Meteor., 35 , 16711687.

    • Search Google Scholar
    • Export Citation
  • Ulbrich, C., 1983: Natural variations in the analytical form of the raindrop-size distribution. J. Climate Appl. Meteor., 22 , 17641775.

    • Search Google Scholar
    • Export Citation
  • Willis, P. T., 1984: Functional fits to some observed drop size distributions and parameterization of rain. J. Atmos. Sci., 41 , 16481661.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Axis ratio of oblate drops (b/a) as a function of equivalent volume diameter (D) for the nonlinear relations of BC (dotted line), ABL (dotted line), KCZM (dashed line), GCB (dash–dot line), and BZV (solid line), 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).

  • Fig. 2.

    Mean values of (a) Zdr and (b) Kdp/Nw for widely varying DSD as a function of the median volume diameter D0 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).

  • Fig. 3.

    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).

  • Fig. 4.

    Scatterplot of Kdp/Zh ratio for widely varying DSD as a function of Zdr 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.

  • Fig. 5.

    Averaged pointwise A (black dashed line) and pathwise B (gray dashed line) of Kdp/Zh ratio computed from the reconstructed S-band rain profiles as a function of Zdr for the PB model. Gray dashed lines labeled ±1 dB indicate shifts due to the presence of ±1 dB bias on Zh. For comparison, corresponding averaged values for widely varying DSD obtained from the nonlinear relations of BC, ABL, and KCZM, and 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.

  • Fig. 6.

    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.

  • Fig. 7.

    Scatterplot between the ratio Kdp/Zh as a function of Zdr 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.

  • Fig. 8.

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

  • Fig. 9.

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

  • Fig. 10.

    Contours of occurrence frequency of the ratio Kdp/Zh as a function of Zdr 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.

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