## Introduction

The purpose of this paper is to provide a new synthesis of relevant information to deduce dominant or bulk^{1} hydrometeor types and bulk amounts from polarimetric radar (PR) data. Thus, the paper lays a foundation for developing semiempirical, rule-based algorithms to deduce dominant hydrometeor types and bulk amounts automatically with computers. The information presented includes PR capabilities, basic hydrometeor characteristics, and PR data signatures necessary for hydrometeor discrimination and quantification. A fuzzy classification algorithm that builds upon this foundation will be discussed in a forthcoming paper.

Some past reviews of PR data interpretations include those by Hall et al. (1984), Herzegh and Jameson (1992), and Doviak and Zrnić (1993). Soon, it will be almost a decade since a comprehensive compilation and synthesis of hydrometeor identification and quantification from PR data at a 10-cm wavelength has been presented. Much has been learned to update and enhance the work done between the 1980s and the early 1990s. Recently, Illingworth and Zrnić (1995), Zrnić (1996), Meischner et al. (1997), and Zrnić and Ryzhkov (1999) have highlighted the increased use of PR for research, and soon there will be a prototype Weather Surveillance Radar 1988 Doppler (WSR-88D) PR (Zrnić 1996; Zahrai and Zrnić 1997; Doviak et al. 2000). The prospects for PR modification on at least some operational WSR-88Ds in the coming decade will expose many in the community to this type of radar and its capabilities. This increase in exposure and growing research use of these radars motivates us to provide a solid presentation, balancing engineering and meteorological aspects of PR use and precipitation physics interpretations, in a single paper. In preparing this paper, so that it, we hope, would remain relevant for some time and be complete enough for use by scientists with varying backgrounds, the topics in this synthesis necessarily include descriptions of essentially all of the useful PR variables, physical attributes of the hydrometeor types that permit them to produce signatures detectable by PR, and values of the PR signatures that are associated with the various types of hydrometeors. Many past and recent references are provided to trace the roots of some issues that cannot be covered because of space limitations. Last, because of the deep cross-disciplinary aspects of this topic and our goal to address a multidisciplinary audience, the paper necessarily is extensive. We hope that this extensiveness will enhance its usefulness as a comprehensive synthesis article.

Some of the difficulties in developing procedures to deduce dominant hydrometeor types and bulk amounts from PR data are caused by 1) the lack of a thorough understanding of radar signatures of specific hydrometeor types, 2) the need for information about size distributions and characteristics of hydrometeors, 3) the ambiguities in hydrometeor identifications (several hydrometeor types identified or no type identified), 4) the need for complete sets of quantitative and qualitative observations for rigorous validation, and 5) the occurrence of artifacts in the data and uncertainties in radar calibration. Nevertheless, significant insights already have been obtained concerning the evolution of hydrometeors in convective storms (e.g., Wakimoto and Bringi 1988; Tuttle et al. 1989; Bringi et al. 1991; Fulton and Heymsfield 1991; Herzegh and Jameson 1992; Holler et al. 1994; Bringi et al. 1996; Jameson et al. 1996;Straka 1996; Bringi et al. 1997; Lopez and Aubagnac 1997; Meischner et al. 1997; Carey and Rutledge 1998;Hubbert et al. 1998) and stratiform precipitation events (e.g., Herzegh and Jameson 1992; Zrnić et al. 1993a; Ryzhkov and Zrnić 1998a). In addition, there is promise that PR measurements might help to improve quantitative estimates of liquid and solid forms of precipitation (e.g., Seliga and Bringi 1976; Sachidananda and Zrnić 1986, 1987; Aydin et al. 1990; Balakrishnan and Zrnić 1990a; Chandrasekar et al. 1990; Aydin et al. 1995; Ryzhkov and Zrnić 1995a,b, 1996ab; Zrnić and Ryzhkov 1996; Ryzhkov and Zrnić 1998ac; Ryzhkov et al. 1998). So far, there have been few, though apparently successful, attempts to describe the bulk hydrometeor distributions, primarily in convective cloud systems with PR data using physical and semiempirical rules (e.g., Hall et al. 1984; Zrnić et al. 1993b; Holler et al. 1994; Straka 1996; Lohmeier et al. 1997; Lopez and Aubagnac 1997; Meischner et al. 1997).

There are a number of scientific and operational reasons for attempting to develop algorithms to deduce hydrometeor types and amounts from PR data. These include 1) calibration of precipitation rates from nonpolarimetric radars such as WSR-88D (Next-Generation Radar), 2) determination of interactions between microphysics and kinematics in severe storms and mesoscale systems, 3) estimation of latent heating for global energy budgets by discriminating between ice and liquid precipitation using spaceborne PR radars, 4) evaluation of advertent and inadvertent weather modification, 5) investigation of lightning production in deep convective clouds, 6) initialization of hydrometeor types and amounts in storm-scale and mesoscale numerical models, 7) determination of detrainment rates in hybrid-cumulus parameterization schemes (e.g., Frank and Cohen 1987; Straka 1994), 8) improvement and verification of microphysical parameterizations in cloud and mesoscale models (Straka 1996), and 9) verification of quantitative precipitation forecasts (Fritsch et al. 1998), among others.

In section 2, we begin by reviewing some of the observed and computed PR variables. Next, we describe hydrometeor characteristics as they are relevant to PR discrimination and justify the use of various PR variables to identify hydrometeor types in section 3. When presented information about hydrometeor types, logical questions a scientist might ask are: “What is the evidence?,” “What are the limitations of the information?,” and “What is the amount?” The former two questions justify the need for sections 2 and 3, whereas the latter question justifies the need for section 4, in which we present methods for precipitation characterization and amount quantification. To provide complete and critical answers to the question, “What are the limitations of the information?” for all of the information presented in this paper would be a very difficult and arduous task at this time because of limited in situ data to compare with theoretical calculations, scattering-model results, and PR data. Nevertheless, the question is of the utmost importance to consider when using PR data for meaningful studies of cloud and precipitation physics. It is suggested that the reader might address these questions and similar ones on a case-by-case basis by reviewing articles cited herein. Estimation errors are not reviewed either; they can be obtained from simulations (Galati and Pavan 1995) or analytic formulas (Ryzhkov and Zrnić 1998c). Last, this paper is closed with a short summary in section 5.

The basis of our hydrometeor classification and quantification algorithm, described in detail along with examples in a forthcoming paper, is fuzzy characterization or fuzzy logic [see Mendel (1995) for a review]. A brief description of the algorithm aimed at a broad audience is in a recent *Bulletin of the American Meteorological Society* article (Vivekanandan et al. 1999), and, in the same volume, there is an example of classification in a severe hail storm (Zrnić and Ryzhkov 1999). The information provided herein also might be useful for construction of other types of rule-based algorithms.

## Polarimetric radar variables

Several of the PR variables have been obtained with radars in the United States such as the Cimarron radar in Oklahoma (Zahrai and Zrnić 1993), the Phillips Laboratory (successor to the Air Force Geophysics Laboratory) radar in Sudbury, Massachusetts (Metcalf et al. 1993), the National Center for Atmospheric Research S-band Doppler dual-polarization radar (S-POL) in Colorado, or the Colorado State University–University of Chicago and Illinois State Water Survey (CSU-CHILL) radar in Colorado (Bringi et al. 1993). Polarimetric radar measurements also are available from European (e.g., Schroth et al. 1988; Meischner et al. 1997; Blackman and Illingworth 1993), Japanese (e.g., Uyeda et al. 1991), Australian (May et al. 1999b; Keenan et al. 1998), and other radars.

We consider dual, linear switchable polarization systems with reception of both copolar and cross-polar components. There are two fundamental kinds of variables available from dual, linear polarization radars. Intrinsic variables provide information about backscatter from hydrometeors in a resolution volume; herein, it is assumed that these variables are not biased by propagation effects (i.e., attenuation and cross coupling). Propagation variables provide information about hydrometeors between the radar and a resolution volume. These variables can be combined to obtain information about dominant, bulk hydrometeor types and amounts both in a resolution volume and between the radar and a resolution volume. Moreover, combining them provides a powerful means for building sets of relations for bulk hydrometeor classification. Basic equations for the most commonly used PR variables are listed in appendix A. An analytical discussion of these variables and others (not used herein) is presented by Zrnić (1991) and Doviak and Zrnić (1993), both of which provide many references. Examples of polarimetric data fields, including signatures of weather events and biological scatterers, are presented by Zrnić and Ryzhkov (1999). A brief, qualitative description of the known useful variables follows as they pertain to hydrometeor identification and quantification. In the rest of this paper, all discussion applies to a 10-cm wavelength radar at quasi-horizontal (elevation angle less than 30°) incidence (except where indicated).

### Reflectivity

Reflectivity factors for horizontally and vertically polarized waves *Z*_{h} and *Z*_{υ} [Eqs. (A1–A2)] are proportional to the hydrometeor’s cross section integrated over a volume. For a particle of given size, ice produces lower *Z*_{h} and *Z*_{υ} than does liquid because of lower dielectric effects; the dielectric constant is about 20% that of liquid for high-density ice and can be less than 5% that of liquid for low-density ice (function of size and density of ice—lower-density ice can be associated with lower dielectric effects). It is important to recognize that *Z*_{h} and *Z*_{υ} are sensitive to calibration, and, even at 10 cm, wavelength can be affected by attenuation in heavy precipitation (Ryzhkov and Zrnić 1995c). The variables *Z*_{h} and *Z*_{υ} together, and combined with other PR variables, are very useful to discriminate hydrometeor types (e.g., Aydin et al. 1986a,b; Leitao and Watson 1984; Golestani et al. 1989; Balakrishnan and Zrnić 1990a,b; Walsh 1993; and Ryzhkov and Zrnić 1998a; as described next).

### Differential reflectivity

Differential reflectivity *Z*_{dr} [Eq. (A3)], which is obtained from the ratio of *Z*_{h} and *Z*_{υ}, can be related to the axis ratio and size of hydrometeors (e.g., Seliga and Bringi 1976; axis ratio is defined as *a*/*b,* where *a* is the horizontal axis radius, and *b* is the vertical axis radius). To be more specific, *Z*_{dr} is a measure of the reflectivity-weighted mean axis ratio of hydrometeors in a volume. For scatterers that are small in comparison with the radar wavelength (Rayleigh conditions) and oriented with their symmetry axis vertical in the plane of polarization, axis ratios less than unity produce positive *Z*_{dr}. Conversely, axis ratios larger than unity produce negative *Z*_{dr}. Canting affects *Z*_{dr} because of changes in effective lengths of the scatterers along the directions of orthogonal polarized transmitted electric fields. Numerous larger-size hydrometeors can strongly influence *Z*_{dr} signals because they produce large reflectivities. The dielectric constant affects *Z*_{dr} much less if the hydrometeors are ice than if they are composed of or are coated by liquid. Differential reflectivity is independent of calibration and total concentration but can depend on how the concentration is distributed among various sizes. Also, differential reflectivity is not immune to propagation effects.

### Reflectivity difference

In addition to *Z*_{dr}, the reflectivity difference *Z*_{dp} [Eq. (A4)] is another convenient combination of *Z*_{h} and *Z*_{υ}. Unlike *Z*_{dr}, because *Z*_{dp} is obtained from the difference of *Z*_{h} and *Z*_{υ}, it depends on hydrometeor concentration and can be used to compute the ice and liquid contributions to *Z*_{h} from a rain and ice mixture (Golestani et al. 1989; Tong et al. 1998). Golestani et al. (1989) showed that *Z*_{dp} indicates the anisotropy (mean apparent shape tends toward oblate or prolate spheroids) of the shapes of hydrometeors; nonspherical, oriented hydrometeors produce different *Z*_{h} and *Z*_{υ}, and, in the difference, the contribution from statistically isotropic (mean apparent shape tends toward a sphere) hydrometeors vanishes.

### Differential phase and specific differential phase

The differential phase *ϕ*_{dp} [Eq. (A5)] is the only propagation variable that is easy to measure and to use. In a volume filled with horizontally oriented hydrometeors such as rain or ice crystals, a horizontally polarized wave has larger phase shifts (per unit length) and propagates more slowly than a vertically polarized wave does; the opposite holds for vertically oriented hydrometeors. The specific differential phase *K*_{dp} [Eq. (A6)] is the difference between propagation constants for horizontally and vertically polarized waves (*k*_{hh} and *k*_{υυ}). In theory, *K*_{dp} allows discrimination between statistically isotropic and anisotropic hydrometeors; isotropic hydrometeors produce similar phase shifts for horizontally and vertically polarized waves. Therefore, differences are due to anisotropic constituents. In general, the magnitude of *K*_{dp} increases as both oblateness (or prolateness) and dielectric constant increase. The advantages of using *K*_{dp} to estimate precipitation rates of anisotropic hydrometeors (Zrnić and Ryzhkov 1996) include that it is 1) independent of receiver/transmitter calibration, 2) independent of attenuation, 3) less sensitive than *Z*_{h} or *Z*_{υ} are to variations of size distributions, 4) immune to partial beam blockage, and 5) not biased by the presence of statistically isotropic hydrometeors such as randomly oriented hail. The effects of reflectivity gradients within the beam affect *K*_{dp} more than they do other polarimetric variables (Ryzhkov and Zrnić 1998b), and a method to identify these gradients needs to be developed. Specific differential phase also is dependent on hydrometeor number concentration.

### Backscatter differential phase

In the absence of propagation effects, the backscatter differential phase *δ* is obtained from the argument of the correlation coefficient |*ρ*_{hυ}(0)| (Doviak and Zrnić 1993), which is defined next. In general, nonzero values of *δ* can indicate resonance scattering [scattering beyond the Rayleigh regime (McCormick et al. 1979; Doviak and Zrnić 1993)] by partially aligned hydrometeors. Aydin and Giridhar (1992) show that hydrometeors larger than one-tenth the radar wavelength can produce a sharp *δ* discontinuity. In the resonance regime, *ϕ*_{dp} contains contributions from the backscatter differential phase *δ* that can be estimated by filtering *ϕ*_{dp} data along range (Hubbert and Bringi 1995). For resonance regime scatterers, *δ* can depend on the size of nonspherical hydrometeors. Modeling of a hydrometeor’s backscatter demonstrates that there is a change in the sign of *δ* over narrow ranges of particle sizes (Balakrishnan and Zrnić 1990b). As a result, it is hypothesized that *δ* could be used to interpret the size and type of nonspherical hydrometeors (Zrnić et al. 1993a). At a 10-cm wavelength, smaller particles (diameter *D* < 10 mm) generally should not produce significant *δ.*

### Correlation coefficient

The degree of decorrelation as measured using the correlation coefficient at zero lag |*ρ*_{hυ}(0)| [Eq. (A7)] between horizontally and vertically polarized echoes results, for Rayleigh scatterers, from variability in the horizontal and vertical sizes of hydrometeors. This relation is because the backscatter intensity for the Rayleigh scattering depends monotonically on the dimension of hydrometeors in the direction of the electric field. Use of |*ρ*_{hυ}(0)| is much more complicated for resonance-regime scatterers, because then backscatter differential phase shift is not zero. Decorrelation physically occurs if the horizontal and vertical backscatter fields do not vary similarly. This situation might occur when changes in the horizontal and vertical backscatter fields, caused by each particle in a resolution volume, are not proportional to each other and the particles reorient and/or when there is a change in the number of particles. In support of this reasoning, modeling and observation studies show that |*ρ*_{hυ}(0)| decreases with increasing diversity of hydrometeor orientations and shapes (e.g., Jameson 1989; Balakrishnan and Zrnić 1990b; and (Zrnić et al. 1993a). Decorrelation also can be more significant when particles are wet or when they are large and irregular in shape. Moreover, |*ρ*_{hυ}(0)| is lower when there are mixtures of hydrometeor types rather than when just one type is present (Jameson 1989). The lowest values of |*ρ*_{hυ}(0)| theoretically should occur when there are mixtures of equal amounts of two different types, especially when the size of one varies predominantly in the horizontal and the other varies in the vertical direction. Values of |*ρ*_{hυ}(0)| are independent of radar calibration and hydrometeor concentration. In addition, |*ρ*_{hυ}(0)| is immune to propagation effects.

### Linear depolarization ratio

The linear depolarization ratio LDR_{υh} [Eq. (A8)] is the logarithm of the ratio of the cross-polar power received to the copolar power received. For a horizontally polarized transmitted wave, a spherically shaped hydrometeor would reflect a likewise (horizontally) polarized wave, resulting in LDR_{υh} equal to −∞ dB. The same applies to axially symmetric particles for which the axis of symmetry is vertical or horizontal in the polarization plane; otherwise, there would be cross-polar power returned. The hydrometeor characteristics associated with depolarization of transmitted energy include hydrometeor shape, shape irregularity, thermodynamic phase, dielectric constant, and canting in the plane of polarization (Herzegh and Jameson 1992). In addition, randomly oriented symmetric particles produce a minimum in |*ρ*_{hυ}(0)|, which is related to LDR_{υh} by |*ρ*_{hυ}(0)|_{min} = 1–2 × ^{(0.1LDRhυ)}_{υh} tends to be more susceptible to noise than other PR variables such as *Z*_{dr}, because the cross-polar signal is typically two to three orders of magnitude smaller than the copolar signal. From a practical point, though, the lowest observable value for LDR_{υh} with, for example, the S-POL radar is about −30 dB (determined by the authors from examinations of the S-POL data collected in 1998 in Florida). With the CSU-CHILL radar, the lowest possible LDR_{υh} is about −34 dB (Hubbert et al. 1998). LDR_{υh} is independent of number concentration and radar calibration. LDR_{υh} unfortunately is not immune to propagation effects. There is no quantitative information about precipitation amounts available from LDR_{υh}. Because most previous use of LDR_{υh} has been with 3-cm-or-shorter-wavelength radar, there are few comparisons between 10-cm-wavelength LDR_{υh} values and in situ observations. Exceptions include observations and modeling studies by Frost et al. (1989, 1991) for various basic types of hydrometeors [see Fig. 8.29 in Doviak and Zrnić (1993)]. Also, Vivekanandan et al. (1994) have modeled LDR_{υh} for various hydrometeors at 10-cm wavelength. More recently, Carey and Rutledge (1998) and Hubbert et al. (1998) have presented comparisons between LDR_{υh} signatures obtained with the CSU-CHILL radar and hydrometeor observations made at the ground.

## Classification

Identification of hydrometeor types using PR data is accomplished by associating different bulk hydrometeor characteristics with the unions of subsets of values of the various PR variables. In principle, several methods are available to achieve this goal. The classic approach is the statistical decision theory whereby regions in the PR variable space are sought such that the probability of correct classification is maximized for a given probability of a wrong classification. This approach requires statistical information that is not yet available. Another method, based on neural networks, also could be devised. Although powerful, this method needs a verified training set of considerable size that currently does not exist. Furthermore, it is unlikely that such a set will become available in the foreseeable future. On the other hand, rule-based methods are not prone to these shortcomings because they are tied to physical principles. Thus, it is possible to evolve the rule-based methods in step with the progress in understanding the physical principles. Consequently, we lay foundations for these methods by partitioning the PR variable spaces into subsets corresponding to specific, bulk hydrometeor types. Partitions of individual variables are presented in tables. Thus, even if only one of the variables is available, a crude classification still can be made. Where there are known relations between pairs of variables, partitions are presented in graphs. Admittedly, these graphs sometimes are difficult to construct such that they all agree exactly with each other at boundaries because of the uncertainties in hydrometeor identification with PR data. Also, inherent statistical errors increase the uncertainty of the boundaries. Therefore, a conservative philosophy in constructing graphs is generally adopted. The titles of the tables refer to the general species or habits of hydrometeors, and the PR variable boundaries of this general class (e.g., rain) are listed in the first row. In subsequent rows, we present the boundaries of subdivisions of the general species of hydrometeors (e.g., small drops, medium drops, and large drops for rain).

Ambiguities in hydrometeor identification with PR data might be reduced, by a yet-to-be-determined amount, by invoking arguments based on physical considerations. Another source of information for identifying hydrometeor types and, possibly, for reducing identification ambiguities comes from self-consistency among the PR variables; if two or more PR variables suggest a certain hydrometeor type, the identification procedure should be more reliable. Much support for understanding hydrometeor scattering properties has come from modeling studies. A brief review of the philosophy behind the use of scattering models, in the framework of observing and simulating PR-variable signatures of hydrometeors, is presented by Aydin and Zrnić (1992). As shown in Fig. 1, precipitation models are used to specify characteristics of hydrometeors such as size distributions, concentrations, shapes, orientations, dielectric constants, and others, which all are contained in the vector **X**. Scattering models are used to compute the forward-scattering and backscattering amplitudes of individual hydrometeors. The PR variables (e.g., *Z*_{h}, *Z*_{dr}, LDR_{υh}, *K*_{dp}, *ϕ*_{dp}, |*ρ*_{hυ}(0)|) are computed using the precipitation model (stored in the vector **X**) and the scattering amplitudes. Results from this stage then are compared with observations for validation. Explanations of the differences between observed and computed PR variables might be used to update the precipitation and scattering models.

Comprehensive in situ measurements of hydrometeor types and amounts to validate PR data are very difficult to obtain. The possible exception to this scarcity is from, perhaps, measurements at the ground (hail size from observing networks and raindrop size distributions from disdrometer measurements). In situ observations of rain, small hail, ice crystals, and graupel are also available from aircraft (Bringi et al. 1986a, 1991; Aydin et al. 1993; Brandes et al. 1995; Bringi et al. 1997; and Ryzhkov et al. 1998; just to name a few). Most aircraft observations, however, are point measurements or alongline measurements with very limited temporal and spatial resolutions. Very promising are comparisons between vertically looking radars and polarimetric radars (May et al. 1999a). In these measurements, the resolution volume sizes are compatible, and Doppler spectra from the vertically looking radars can reveal the distribution of drops, hail, and, perhaps, some ice crystals. This kind of research has just begun and hopes are high that it will provide some clues; still, it is not likely that the comprehensive observations needed fully to validate PR analyses of hydrometeor types and amounts will be available in the near future.

In the remainder of this section, we first present specifics about hydrometeor characteristics (required for modeling and interpreting PR data) and then present relations between the values of PR variables and bulk hydrometeor types. The general hydrometeor types considered are hail, graupel/small hail, rain, and crystals/snow aggregates (snow hereinafter generally refers to aggregates). Also included for classification purposes is temperature *T* because it can be estimated from a proximity sounding (from which updraft and downdraft temperatures can be crudely determined) or possibly from thermodynamic retrievals (Gal-Chen 1978) when three-dimensional winds can be approximated from multi–Doppler radar data or single-Doppler data (e.g., Shapiro et al. 1995; Sun and Crook 1996). The use of temperature is most important in minimizing some unreasonable ambiguities. For example, ice crystals would not be expected at 15°C, and rain would not be expected at −30°C. There are various ways the boundaries of PR variables and temperature thresholds in the tables and graphs can be applied for hydrometeor type classification. A simple way is to assume the boundaries are rigid and classify according to majority rule; that is, the class that satisfies the thresholds for the most PR variables is declared to be present. Ambiguous signatures also could be specified. Better classifiers can be designed if the boundaries are “porous”; that is, a hydrometeor type is allowed to exist on either side of a boundary (Straka 1996; Vivekanandan et al. 1999). We adopt this definition: the boundaries presented are “fuzzy,” and the confidence of hydrometeor classification at the threshold boundaries is 0.5 on a scale of 0–1. Various functions, such as sigmoid, bell, Gaussian, trapezoid, or triangle can be used to prescribe the confidences in the vicinity of the boundaries, with values ranging between 0.0 and 1.0 (e.g., Straka 1996). It is important to realize that, depending on what is on the other side of the boundary, the relative significance of different segments for a specific hydrometeor type can be very different. For example, a segment of a boundary between rain and hail is much more significant than a segment that delineates rain from a forbidden region (i.e., a region where it is not possible for hydrometeors to produce PR signals). It is important to specify well the former, whereas, for the latter, it suffices to make sure that it encompasses all the PR data corresponding to rain.

### Hail

Detecting the presence of hail and its size has been a long-standing goal of radar meteorologists. Several physical characteristics of hail help to make it distinguishable from other hydrometeors in PR radar data. Yet, the orientation of hail in fall is not fully understood or documented, so gauging its size would be difficult, as the following discussion amply demonstrates. For example, List (1986) describes at least a weak association between hail size and shape; that is, hail 5–10 mm in diameter is spherical or conical, hail 10 < *D* < 20 mm is ellipsoidal or conical, hail 10 < *D* < 50 mm is ellipsoidal with lobes and other protuberances along the short axis, and hail 40 < *D* < 100 mm is spherical with small and large lobes and other protuberances. List (1986), however, found no simple relation between protuberance size and number and the size of hail, although larger hail tends to be more irregular. Another observation is that most hailstones are oblate to at least some degree (Barge and Isaac 1973); for example, 83% have axis ratios between 0.6 and 1.0, 15% have axis ratios between 0.4 and 0.6, and less than 2% have axis ratios less than 0.4. In addition, the majority of hailstones observed at ground have axis ratios (minor to major) of 0.8 (Knight 1986; Matson and Huggins 1980). Wet hail typically has an axis ratio of about 0.8, and spongy hail has an axis ratio of 0.6 to 0.8 (Knight 1986). The orientation of falling hail is somewhat questionable. There is evidence that hailstones fall with their maximum dimensions in both the horizontal (Knight and Knight 1970; List et al. 1973; Matson and Huggins 1980) and the vertical (Knight and Knight 1970; Kry and List 1974). List (1986) suggested that ellipsoidal hailstones 10–50 mm in diameter typically fall most stably when oriented in the vertical, which is in agreement with frequent observations by one of the authors of this paper (J. M. Straka). Hailstones also can exhibit gyrating motions (List et al. 1973; Kry and List 1974) and tumbling motions (List et al. 1973; Knight and Knight 1970; Matson and Huggins 1980). Tumbling hail may have an apparent axis ratio of unity; that is, it may appear statistically to be spherical or isotropic. The structure of hail can vary from solid to spongy to porous, and the outer shell can be dry or wet. Hail density typically varies from about 400 to 900 kg m^{−3} for particles smaller than 10 mm and from 700 to 900 kg m^{−3} for particles that are larger. Hail distributions can be represented with some form of exponential or gamma function (Ulbrich and Atlas 1982) and total hail number concentrations range from 10^{−2} to 1 m^{−3} for diameters of 5–25 mm and from 10^{−6} to 10^{−2} m^{−3} for diameters larger than 25 mm (Pruppacher and Klett 1981). A summary of the characteristics of hail is presented in Table 1. When classifying hail types, dry or wet surfaces can be considered, as can size such as small (<20 mm), large (20–40 mm), and giant (>40 mm). The large and giant hail size categories in Table 2 are close to the categorizations used by the National Weather Service (severe when greater than 19 mm and significantly severe when greater than 51 mm). [Our categories differ from those (8.5, 19, and 32 mm) proposed by Lipshutz et al. (1986).]

#### Reflectivity factor

Mason (1971) found that *Z*_{h} in excess of 55 dB*Z* is appropriate to indicate hail. Vertical profiles of *Z*_{h}, the maximum *Z*_{h}, and the height of the *Z*_{h} = 45 dB*Z* level above the melting level (e.g., 1.5 km) also can be used together to indicate hail (Waldvogel et al. 1979). Our lower thresholds for hail are 45 dB*Z* (Table 2), although smaller *Z*_{h} have been found in hail (Walsh 1993), and values larger than 60 dB*Z* indicate the largest sizes of wet hail. Values of *Z*_{h} greater than 80 dB*Z* generally are considered extreme. The low threshold for dry hail is 5 dB lower than that for wet hail. Wetting will enhance most PR signatures of ice because of dielectric effects either on the surface of hailstones or because particle density is increased when liquid penetrates the hail’s ice lattice. Also, values of *Z*_{h} greater than 80 dB*Z* should be considered to be exceptionally rare, even at 10-cm wavelength. The values for small and large wet hail are in accord with recent comparisons between observations and radar measurements (Carey and Rutledge 1998; Hubbert et al. 1998). Note that three-body scattering could produce anomalous *Z*_{h} signatures associated with hail (Zrnić 1987; Doviak and Zrnić 1993; Hubbert and Bringi 1997; Lemon 1998). If multidimensional spatial considerations of *Z*_{h} are attempted, then this anomalous signature could be used for discriminating large hail.

#### Differential reflectivity

The use of *Z*_{dr} to identify hail size is complicated because there are no definitive relations between axis ratio and size. Small hail tends to be more spherical; therefore, *Z*_{dr} ≈ 0 dB. Larger hail might be spherical, or it could be prolate to produce *Z*_{dr} < 0 dB. Tumbling motions can make nonspherical hail shapes of any size appear to be isotropic or spherical in the mean, so *Z*_{dr} ≈ 0 dB.

Ground-based radar and in situ observations, as well as scattering-model studies, of hail are numerous. Ulbrich and Atlas (1984), Aydin et al. (1986b), Bringi et al. (1986), Aydin et al. (1990), Balakrishnan and Zrnić (1990b), and Zrnić et al. (1993b) provide examples from both observations and models of −0.5 > *Z*_{dr} > −2 dB for hailstones of 20 < *D* < 40 mm, assuming minor axes randomly oriented in the horizontal plane (vertically oriented oblates) and larger values of *Z*_{dr} for smaller hailstones. Illingworth and Caylor (1986), Illingworth et al. (1987), Bringi et al. (1986b), and Husson and Pointin (1989) also found negative *Z*_{dr} associated with maxima of *Z*_{h} in regions of hail. Hail pad measurements by Husson and Pointin showed that hail with *D* ≈ 23 mm was associated with negative *Z*_{dr}. Theory and observations suggest that significant negative *Z*_{dr} near the ground could be associated with three-body scattering (Hubbert and Bringi 1997).

Whether hail is dry or wet and spongy can also influence *Z*_{dr}. Longtin et al. (1987) found that wet, spongy ice spheroids exhibit increasing variation of *Z*_{dr} about zero with larger hail size, even for a fixed shape in the resonance scattering regime. Seliga and Bringi (1978) show that *Z*_{dr} for dry, oblate hailstones decreases with size and changes sign at *D* ≈ 50 mm, whereas tumbling hailstones tend to produce values of *Z*_{dr} ≈ 0.

Values of *Z*_{dr} for hail range from −2 < *Z*_{dr} < 0.5 dB in Table 2. For dry hail, −1 < *Z*_{dr} < 0.5 dB. For small wet hail, a bracket of −0.5 < *Z*_{dr} < 0.5 dB is indicated based on modeling studies and observations. The positive values account for small hailstones that might be horizontally oriented Rayleigh scatterers. Uncertainty in orientation is accounted for by straddling zero. The considerations above suggest that large and giant wet hail produce values of *Z*_{dr} that are possibly as large as 0.5 and 0 dB, respectively, but probably less than 0 dB (Balakrishnan and Zrnić 1990a,b; Carey and Rutledge 1998; Hubbert 1998). Because of uncertainties about the low threshold for appreciable hail sizes, we suggest values at least as low as −2 dB (Balakrishnan and Zrnić 1990a,b). For large and giant wet hail, resonance effects could produce values of *Z*_{dr} that are outside of the proposed ranges. Enhanced confidence is suggested for identification of hail, dry or wet hail, and, possibly, hail size using *Z*_{h}, *Z*_{dr} pairs (Fig. 2a; boundaries from values provided in Table 2). In practice, extrapolated curves associated with *Z*_{h}, *Z*_{dr} pairs should be terminated at *Z*_{h} ≈ 80 dB*Z* to avoid contamination by point scatterers such as airplanes.

#### Differential reflectivity hail signal

*H*

_{dr}(Aydin et al. 1986b, 1990) is defined as a function of pairs of

*Z*

_{h}and

*Z*

_{dr}such that

*H*

_{dr}

*Z*

_{h}

*f*

*Z*

_{dr}

*H*

_{dr}> 0 indicates ice and ice–liquid mixtures. There is some evidence that larger values correspond to larger hail. Above the melting level, use of

*H*

_{dr}to detect hail is limited, because other forms of ice also can be indicated. Leitao and Watson (1984) also plot a

*Z*

_{h},

*Z*

_{dr}curve that agrees with (Eq. 1) to delineate regions of rain and rain and ice mixtures. Use of

*H*

_{dr}is equivalent to partition of

*Z*

_{h},

*Z*

_{dr}pairs for frozen and mixed liquid–frozen precipitation (Fig. 2a). Note, though, that it has been suggested by Brandes et al. (1993) that use of

*H*

_{dr}alone might overestimate the coverage of hail.

#### Reflectivity difference

*Z*

_{dp}is below the line given by

*Z*

_{dp}

*Z*

_{hr}

*Z,*

*Z*

_{hr}is the

*Z*

_{h}(dB

*Z*) from rain (Golestani 1989). Other curves for

*Z*

_{dp},

*Z*

_{hr}pairs have been suggested, including

*Z*

_{dp}= 1.12(

*Z*

_{hr}−9) dB

*Z*by Conway and Zrnić (1993), and both

*Z*

_{hr}= 0.64

*Z*

_{dp}+ 14.8 dB

*Z*for 1 <

*Z*

_{dp}< 20 dB

*Z*and

*Z*

_{hr}= 0.88

*Z*

_{dp}+ 10 dB

*Z*for 20 <

*Z*

_{dp}< 52 dB

*Z*by Aydin and Giridhar (1992). Neither

*Z*

_{dp}nor a combination

*Z*

_{dp},

*Z*

_{h}have been used to classify hydrometeors, although the latter has discrimination potential. The suitable and accepted feature of

*Z*

_{dp}is the means to estimate the fraction of ice in the rain–hail mixture. Currently, we recommend that the ice fraction serve as the classification variable. To obtain its correct values, a

*Z*

_{dp}rain–hail line should be fit to actual data, so uncertainty in radar calibration is neutralized. For classification of pure hail, we suggest the fraction of ice

*F*

_{iz}from

*Z*

_{dp}(see section 4c) should be

*F*

_{iz}> 0.75 (Table 2).

#### Correlation coefficient

The thresholds in Table 2 for hail identification and size discrimination with |*ρ*_{hυ}(0)| are based on model results and observations by Balakrishnan and Zrnić (1990a,b), Liu et al. (1993), Bringi et al. (1996), Carey and Rutledge (1998), and Hubbert et al. (1998). Carey and Rutledge’s values are larger than Hubbert et al.’s even though they both studied the same storm. Balakrishnan and Zrnić show that |*ρ*_{hυ}(0)| decreases as 1) hail size increases, 2) hail protuberance-to-diameter ratio increases, 3) hail size distributions broaden, 4) hail is wetted or becomes spongy, or 5) hail mixes with other hydrometeors with different distributions and sizes. For wet/spongy hail, there is a marked reduction in |*ρ*_{hυ}(0)| at *D* ≈ 20 mm, with a more substantial reduction in |*ρ*_{hυ}(0)| at *D* > 50 mm because of resonance effects. Identification of hail is strengthened when both *Z*_{h} and |*ρ*_{hυ}(0)| in Table 2 are satisfied (Fig. 3). The hail curve at large *Z*_{h} in Fig. 3 is extrapolated from information in the cited references.

#### Specific differential phase

*K*

_{dp}insensitive to hail. First, the dielectric constant is smaller for ice than for liquid (Battan 1973). Nonetheless, on large, wet hail (

*D*> 20 mm), water coatings are thin (Rasmussen et al. 1984), and there may be only marginal increases in the dielectric constant. Next, hail is found in lower concentrations than other hydrometeor types (Pruppacher and Klett 1981). Last, propagation through tumbling hail (statistically isotropic) should be insensitive to polarization. As a result,

*K*

_{dp}≈ 0 is common in pure hail. These factors justify the thresholds in Table 2. By extrapolating 5-cm-wavelength results from Vivekanandan et al. (1993a) to 10-cm wavelengths, we obtain for the vertically oriented hail −1° <

*K*

_{dp}< 0° km

^{−1}for

*D*< 20 mm, and 0° <

*K*

_{dp}< 1° km

^{−1}for larger sizes. Large scatterers can produce

*δ,*which complicates

*K*

_{dp}estimations. Although regions of

*δ*can be recognized in radial profiles of

*ϕ*

_{dp}, methods to separate propagation contributions to

*ϕ*

_{dp}from

*δ*are prone to errors of about ±5° (Hubbert and Bringi 1995). Table 2 shows values of −0.5° <

*K*

_{dp}< 1° km

^{−1}for general hail identification, whereby the negative range of values is smaller than theoretical extrapolation from Vivekanandan et al. (1993a); we opt for the smaller range, because, in our data analysis, we have not encountered significant negative

*K*

_{dp}in hail. Smaller amplitude ranges about zero are associated with dry and/or smaller particles and larger ranges are associated with larger wet particles. Figure 4a shows that hail identification confidence can be increased with

*Z*

_{h},

*K*

_{dp}pairs. Similarly to

*H*

_{dr}, one can define

*K*

_{dr}as the distance (along a constant

*Z*

_{h}line) from the rain curve in the

*K*

_{dp},

*Z*

_{h}plane (Balakrishnan and Zrnić 1990a; Walsh 1993),

*K*

_{dr}

^{[0.125(Zh−49)]}

*K*

_{dp}

*K*

_{dr}should be greater than 1° km

^{−1}for hail, greater than 2° km

^{−1}for wet hail, and greater than 4° and 8° km

^{−1}for large and giant hail, respectively. Walsh (1993) found that

*K*

_{dr}helped to indicate the presence of ice particles with

*D*> 6 mm. Last, the fraction of ice

*F*

_{ik}from

*K*

_{dp}(section 4c) should be

*F*

_{ik}> 0.75 to classify hydrometeors in the pure hail category.

#### Backscatter differential phase

Balakrishnan and Zrnić (1990b) modeled values of *δ* for dry and wet hail with axis ratios of 0.8 and for wet, spongy hail with axis ratios of 0.6 and 0.8. They noted that dry hail produces values of *δ* ≈ 0 for *D* < 40 mm;hence, we set |*δ*| = 1 (Table 2). For wet hail with *D* ≈ 7–10 mm (oriented oblates, minor axis vertical), scattering models produce 0° > *δ* > −5°. At *D* ≈ 10 mm, there is a sharp transition, with *δ* ≈ 10°. Last, when *D* ≈ 50 mm, *δ* is substantially negative (−30° to −50°). Spongy hail, with an axis ratio of 0.8, behaves similarly to wet hail except that the sharp transition shifts to *D* ≈ 15 mm. Scattering by spongy hail with an axis ratio 0.6 produces similar results, but the magnitudes of *δ* are larger. An observational study by Zrnić et al. (1993b) suggests that large negative values of *δ* might indicate large hail in accord with the scattering-model results. They also noted that smaller negative values might indicate lower concentration of large hail and that positive values might indicate smaller hail. For large tumbling hailstones, *δ* ≈ 0, because they appear spherical in the mean. The thresholds in Table 2 are derived from the scattering models of oblate spheroids. Absolute values for *δ* in Table 2 accommodate both vertical and horizontal hydrometeor alignment because hail orientation generally is not known.

#### Linear depolarization ratio

Many previous LDR_{υh} measurements associated with hail are at a 3-cm wavelength (e.g., Bringi et al. 1986b;Tuttle et al. 1989; Brandes et al. 1995). These measurements suggest values of LDR_{υh} > −25 dB for hail and values of −18 to −8 dB for unaligned or nonspherical hydrometeors that often are large, such as giant hail. Large values of LDR_{υh}, even at 3-cm wavelength, generally rule out smooth, smaller hail. Kennedy et al. (1997) suggest slightly smaller values of LDR_{υh} for hail discrimination at 10-cm wavelength. The observations and calculations of Frost et al. (1989, 1991) at 10-cm wavelength, however, indicate that the 3-cm wavelength values generally are applicable. In support of Frost et al.’s work, Vivekanandan et al.’s (1993b) scattering model results of LDR_{υh} include −28 < LDR_{υh} < −15 dB for dry hail and −24 < LDR_{υh} < −22 dB for wet hail, though they noted that ranges of these values are sensitive to assumptions about axis ratios, orientation, etc. Aydin and Zhao (1990) also carried out extensive modeling studies of hail and showed that hail can produce a wide range of LDR_{υh}, varying from less than −25 dB to greater than −15 dB (the larger values are caused by resonance effects). In addition, observational studies by Carey and Rutledge (1998) and Hubbert et al. (1998) at 10-cm wavelength with the CSU-CHILL radar include values of LDR_{υh} greater than −27 to −25 associated with hail and values of LDR_{υh} as large as −15 to −13 with wet, large/giant size hail. Based on the studies above, we suggest a threshold of LDR_{υh} greater than −26 dB to indicate hail in general, −26 to −18 dB for dry hail, greater than −24 dB for wet and small wet hail, greater than −20 dB for large wet and spongy hail, and greater than −16 dB for giant hail or large, water-coated hail. Confidence in discrimination of hail should be enhanced if the thresholds for *Z*_{dr} and LDR_{υh} (Table 2) are both valid (Fig. 5). Holler et al. (1994) also employed *Z*_{dr}, LDR_{υh} pairs at a 5-cm wavelength for hail discrimination; our values are similar to theirs. Last, Illingworth et al. (1986) found that a local increase of 6 dB in LDR_{υh} might be indicative of a hail shaft. This information also could help to discriminate hail.

### Graupel and/or small hail

Graupel (0.5 < *D* < 5 mm) and small hail (5 < *D* < 20 mm) often coexist and are indistinguishable. Even if graupel–small hail are physically distinguishable from each other, and one of the species is dominant, it may not be possible to determine which one it is. Thus, we are compelled to consider the two together. The density of graupel–small hail can range from 100 to 900 kg m^{−3}, size distributions can be represented by exponential or gamma functions, and number concentrations are on the order of 1–10^{3} m^{−3} (Pruppacher and Klett 1981). The shapes of graupel–small hail can be spherical or can be conical with axis ratios both larger and smaller than unity (Bringi et al. 1984; Aydin and Seliga 1984). In the study by Bringi et al. (1984), graupel with 2*a* less than 1 mm are assumed to be spherical, graupel with 1 ⩽ 2*a* ⩽ 4 mm are conical with *a*/*b* = 0.5, and graupel with 4 < 2*a* < 9 mm are conical with *a*/*b* = 0.75. Both smaller and larger particles might be spherical or irregular (e.g., lump graupel; highly irregular shaped rimed crystals and aggregates) in shape based on in situ observations. Low-density graupel sometimes is conical in shape, which might reveal its presence through distinct scattering properties related to aspect ratios (Aydin and Seliga 1984). In general, graupel–small hail tend to be relatively smooth in comparison with some large hailstones. The fall orientation of graupel–small hail is not known with great certainty, but some investigators hypothesize that the larger of these hydrometeors probably tumble, though conical graupel may have a preferential fall orientation (List and Schemenaur 1971; Pruppacher and Klett 1981). Some graupel may fall with their largest axis in the horizontal, whereas others may fall with their largest axis in the vertical. A summary of these details is presented in Table 3.

#### Reflectivity factor

During the Cooperative Convective Precipitation Experiment (CCOPE) in Montana (Bringi et al. 1984) and the May Polarization Experiments (MAYPOLE) in Colorado (Bringi et al. 1986a), graupel measurements by radar were compared with observations made with aircraft. In both experiments, graupel–small hail as large as *D* ≈ 9 mm were found in regions where measured *Z*_{h} was below 35 dB*Z.* In addition, *Z*_{h} computed from information obtained by precipitation probes ranged between 20 and 35 dB*Z* in the CCOPE storms. Similar *Z*_{h} values were measured in the MAYPOLE storms, but they were about 5 dB*Z* higher than was predicted by a scattering model. Bringi et al. attribute these higher values to the presence of wet graupel. They also might be associated with higher densities associated with these graupel particles. Walsh (1993) and Aydin et al. (1993) report similar results from an Oklahoma storm. Based on these findings, we associate dry, low-density graupel–small hail with *Z*_{h} < 35 dB*Z* (Table 4). In addition, wet, high-density graupel–small hail is associated with *Z*_{h} of 30–50 dB*Z* [e.g., Aydin and Seliga (1984) suggest *Z*_{h} < 45 dB*Z* for graupel]. Note that, according to Vivekanandan et al. (1990), *Z*_{h} probably is insensitive to whether graupel–small hail is spherical or conical.

#### Differential reflectivity

Graupel–small hail can be identified by negative *Z*_{dr} if they are elongated and fall in a vertically oriented manner (Aydin and Seliga 1984; Aydin et al. 1984). Thus, we set a lower threshold of −0.5 dB in Table 4. On the other hand, positive values of *Z*_{dr} could be produced by graupel–small hail if they are more oblate in shape and fall in a horizontally oriented manner (Bringi et al. 1984, 1986a). Aydin and Seliga (1984) show, using scattering-model results, that canting of 0°–30° reduces *Z*_{dr} by 0–1 dB for wet, high-density graupel and by 0–0.5 dB for dry, low-density graupel. Also, wetting of ice particles tends to enhance PR signatures because of dielectric constant effects and possible increases in particle density; thus, the upper threshold is 2 dB for wet, high-density graupel versus 1 dB for dry, low-density graupel. These results agree with the observations and computed values presented by Bringi et al. (1984, 1986a), who used particle probe data and assumed random canting to obtain *Z*_{dr} < 2.5 dB for graupel. It is noted, though, that Aydin and Zhao (1990) modeled values of *Z*_{dr} up to 3 dB for some shapes of graupel. Figure 2a shows the space where *Z*_{h}, *Z*_{dr} pairs should be used to enhance confidence in classification of graupel–small hail.

#### Reflectivity difference and hail signal

The fraction of ice from *Z*_{dp} should be *F*_{iz} > 0.75 for pure graupel–small hail. Differential reflectivity hail signal in Table 4 should be used as described for hail. Values of *H*_{dr} > 0 are suggested for dry graupel–small hail and of *H*_{dr} > 5 for wet, high-density particles.

#### Correlation coefficient

Because graupel–small hail are relatively smooth, we suggest a low threshold of |*ρ*_{hυ}(0)| > 0.95 (Table 4). This value is derived from measurements and scattering-model studies of small hail (Balakrishnan and Zrnić 1990b). Melting graupel/small hail mixed with rain could have a lower |*ρ*_{hυ}(0)| [0.92 in the model of Aydin and Zhao (1990)], but we classify such a combination of hydrometeors as rain–wet hail mixture (Table 7, presented later). Identification of graupel–small hail should be strengthened when the threshold ranges for both *Z*_{h} and |*ρ*_{hυ}(0)| are satisfied (Fig. 3).

#### Specific differential phase

Not much is known about *K*_{dp} in graupel, though signals in small hail should be similar to those discussed for hail. A lower threshold in Table 4 of −0.5° km^{−1} should accommodate vertically oriented wet graupel, and the upper threshold of 1.5° km^{−1} should be adequate for horizontally oriented small hail (oblate and wet). As described in section 3c on rain, however, if *Z*_{h} is greater than *Z*_{h}(*K*_{dp}), it is possible that an ice or an ice–rain mixture exists. The indication for graupel–small hail should be strengthened if both −0.5° < *K*_{dp} < 1.5° km^{−1} and 20 < *Z*_{h} < 50 dB*Z* (Fig. 4a) are satisfied. The fraction of ice from *K*_{dp} should be *F*_{ik} greater than 0.75 for pure graupel–small hail. Also, values of *K*_{dr}, which helps to discriminate between rain and ice or rain–ice mixtures, should be greater than 1° km^{−1} for graupel–small hail in general, and greater than 2° km^{−1} for wet, high-density particles.

#### Backscatter differential phase

Graupel particles are small in comparison with a 10-cm wavelength and generally are smooth. Thus, |*δ*| less than 1° can be associated with graupel.

#### Linear depolarization ratio

Bringi et al.’s (1986a) 3-cm wavelength scattering-model values of LDR_{υh} include −25 <LDR_{υh} <−20 dB for conical graupel and between −26 < LDR_{υh} < −24 dB for melting graupel. In addition, Vivekanandan et al. (1990) show that larger LDR_{υh} is possible for conical graupel than for spheroidal graupel. The LDR_{υh} enhancement probably is due to irregularly shaped or tumbling particles. Wetting of ice surfaces also increases LDR_{υh}. Computations of LDR_{υh} in dry graupel at a 10-cm wavelength (Frost et al. 1989, 1991) are −26 < LDR_{υh} < −22 dB and agree with the Rayleigh–Gans scattering model of randomly oriented oblate spheroids. Slightly higher LDR_{υh} values for graupel–small hail are expected for the 3-cm wavelength because of resonance effects. The thresholds in Table 4 are derived from the model and observation studies by Bringi et al. (1986a) and Vivekanandan et al. (1990) and are in accord with measurements and computations of Frost et al. (1989, 1991). Figure 5 shows the parameter space where *Z*_{dr}, LDR_{υh} pairs should strengthen PR classification confidence of graupel/small hail. This space is similar to that proposed by Holler et al. (1994) for PR variables at a 5-cm wavelength.

### Rain

*D*

_{e}(Pruppacher and Klett 1981); several relations exist, including

*a*

*b*

*D*

_{e}

*D*

_{e}is in millimeters,

*a*is the horizontal axis, and

*b*is the vertical axis (Pruppacher and Beard 1970; Pruppacher and Pitter 1971). Studies by Jones (1959), Jameson and Beard (1982), and Goddard et al. (1982) show, with concern, that in heavier rain events a large range in axis ratios might be expected with even prolates possible (though the latter are likely transient oscillations). After nearly two decades, raindrop axis ratio and its functional form, as well as the importance of drop oscillations, are under renewed scrutiny (e.g., Beard and Feng 1991; Beard et al. 1991; Tokay and Beard 1996; Bringi et al. 1998). Nevertheless, Eq. (4) and others’ similar relations specify that, for

*D*greater than 1 mm, drops become increasingly oblate with size. Raindrops generally fall with their minor axis oriented in the vertical, though a rare few drops might be temporarily elongated in the vertical (Takahashi and Kuhara 1993), possibly because of oscillations, collisions, or both. A theoretical study by Beard and Jameson (1983) and radar observations by Hendry et al. (1987) suggest that mean raindrop canting typically is less than 5°–10°. Size distributions can be approximated by exponential (Marshall and Palmer 1948) or gamma functions (Ulbrich 1983) for mean droplet spectra, but extreme local variations from these are observed (e.g., Rauber et al. 1991;Young 1993; Joss and Zawadski 1997). For the largest rain drops,

*D*≈ 3–5 mm; however, values as large as

*D*≈ 6–8 mm are documented (e.g., Rauber et al. 1991). Total number concentrations of rain are on the order of 10

^{3}to 10

^{4}m

^{−3}, but very large drops are usually present in low concentrations (<10

^{4}m

^{−3}). Details about the physical attributes of raindrops are presented in Table 5.

The National Weather Service classifies rain in six categories that are related to the reflectivity factor (Lipschutz et al. 1986; Bluestein 1992) including light (*Z*_{h} < 30 dB*Z*), moderate (30 < *Z*_{h} < 40 dB*Z*), heavy (40 < *Z*_{h} < 45 dB*Z*), very heavy (45 < *Z*_{h} < 50 dB*Z*), intense (50 < *Z*_{h} < 57 dB*Z*), and extreme (*Z*_{h} > 57 dB*Z*). For applications in numerical modeling, sizes of drops are important because they influence residence time in a storm and microphysical interactions. Polarimetric radar data are suitable for determining distribution and median sizes of raindrops. (Pertinent equations are presented in section 4.) Even if the median sizes cannot be determined (e.g., if *Z*_{dr} is not available, or hail is contaminating PR signals), it still may be possible to categorize rain into a few classes according to the median diameter *D*_{o}. We prescribe three categories in Table 6: small (*D*_{o} < 1 mm), medium (1 < *D*_{o} < 2 mm), and large (*D*_{o} > 2 mm).

#### Reflectivity factor

*Z*

_{h}thresholds in Table 6 are chosen to be compatible with

*Z*

_{dr}by substituting [Eq. 6, section 3c2] into Eq. (5), where

*Z*

_{hr}is found by equating the Marshall–Palmer relation (

*Z*

_{h}= 200

*R*

^{1.6}) to a relation

*R*(

*Z*

_{h},

*Z*

_{υ}), where

*R*is precipitation rate, proposed by Sachidananda and Zrnić (1987):

*Z*

_{hr}

*Z*

_{h}

*Z*

_{υ}

^{12.96}

*Z*are mm

^{6}m

^{−3}. Maximum values of

*Z*

_{h}in pure rain are ≈ 60 dB

*Z.*

#### Differential reflectivity

*Z*

_{dr}are uniquely related to size, but, for a distribution, that relation depends on the parameters of the distribution (e.g., Seliga and Bringi 1976; Hall et al. 1980; Bringi et al. 1991) and drop behavior (highlighted below). [Jameson (1985) and Zrnić and Doviak (1989) found that small canting of raindrops has a negligible influence on

*Z*

_{dr}observations.] A relation between

*D*

_{o}and

*Z*

_{dr}for rain is generated with a least squares fit of a second-order polynomial to a set of

*D*

_{o},

*Z*

_{dr}pairs, assuming a gamma drop size distribution and Eq. (4). The parameters of the distribution are varied to obtain

*D*

_{o}

*Z*

_{dr}

*Z*

^{2}

_{dr}

*D*

_{o}is in millimeters and

*Z*

_{dr}is in decibels. This equation is similar to the one proposed by Illingworth and Caylor (1989) and produces results comparable to those described by Al-Khatib et. al. (1979). The behavior of Eq. (6) is such that a larger

*D*

_{o}produces a larger

*Z*

_{dr}. The

*Z*

_{dr}thresholds in Table 6 recommended to discriminate rain satisfy Eqs. (4), (5), and (6).

As expected, there is a general trend for larger values of *Z*_{dr} to be found with larger values of *Z*_{h}. This is not always the case, however. For example, Illingworth et al. (1987) and Bringi et al. (1991, 1993, 1997, 1998) found that very large *Z*_{dr} (1.5–5.5 dB) with small *Z*_{h} (5–40 dB*Z*) probably indicate low concentrations of very large drops with *D* ≈ 6–8 mm. Aircraft observations described by Bringi et al. (1991, 1997, 1998) and others strongly support this theory. Rauber et al. (1991) also have observed very large raindrops in low concentrations in shallow Hawaiian precipitation systems. Based on these observations, we make an accommodation in Table 6 for small concentrations of large raindrops (and paucity of small drops).

Often, a column of large values of *Z*_{dr} extends from below the melting level to temperatures as low as −10°C or so. These “*Z*_{dr} columns” are associated with larger raindrops in many deep convective storms, where the drops probably are growing by coalescence. (We should note there could be other explanations for such columns that are beyond the scope of this paper.) A large source of hail embryos may be provided via freezing of drops in these columns (Conway and Zrnić 1993, Askelson et al. 1997, 1998; Carey and Rutledge 1998; Hubbert et al. 1998). Typically, the *Z*_{dr} column is a region with values of 30 < *Z*_{h} < 45 dB*Z,* 1 < *Z*_{dr} < 3 dB, *K*_{dp} ≈ 0° km^{−1}, reduced values of |*ρ*_{hυ}(0)|, and larger values of LDR_{υh} than usually are found with rain (Conway and Zrnić 1993; Hubbert et al. 1998). Values of the PR variables in the *Z*_{dr} column also are included in Table 6.

The *Z*_{h}, *Z*_{dr} pairs in Fig. 2a that define values associated with rain are a composite of observations (e.g., Aydin et al. 1986a,b; Leitao and Watson 1984; Illingworth 1988; Balakrishnan and Zrnić 1990a) and some of our recent simulations. We also have added subdelineations to the rain region to show small, medium, and large median drop sizes. In addition, we have included a parameter space for *Z*_{h}, *Z*_{dr} pairs to indicate increased classification confidence of large drops in low number concentrations in Fig. 2b. Last, the parameter space for *Z*_{h}, *Z*_{dr} pairs that define the *Z*_{dr} column is in Fig. 2b. Most of these *Z*_{h}, *Z*_{dr} regions are defined from the *Z*_{h} and *Z*_{dr} thresholds in Table 6. The largest values of *Z*_{dr} in Figs. 2a,b is 5 dB, which approximately corresponds to the largest *Z*_{dr} measured with PR and the largest drop sizes observed in situ with aircraft (though larger values for both certainly are possible).

#### Reflectivity difference and hail signal

As noted by Golestani (1989), pure liquid-phase hydrometeors may be present if values of *Z*_{dp} fall along the line given by Eq. (2). In addition, the fraction of ice from *Z*_{dp} should be *F*_{iz} < 0.25 for pure rain (Table 6). Values of *H*_{dr} less than 0 can signify rain (Table 6). Above the freezing level, *H*_{dr} < 0 can indicate other hydrometeor types as well; therefore, *H*_{dr} should be used only at *T* > 0°C for rain.

#### Correlation coefficient

In theory, values of |*ρ*_{hυ}(0)| in rain are generally close to unity. Slight departures from near unity are due to a continual change in shape (a “breadth of the axis ratios”) through typical rain size distributions. Jameson and Dave (1988) imply that drop oscillations also can reduce |*ρ*_{hυ}(0)| from unity because the Δ*Z*_{h} and Δ*Z*_{υ} are not identical for the same increment in the drop size. Moreover, some reduction in correlation can be expected from effects such as drop oscillations, coalescence, and breakup, all of which can occur on the scale of the radar dwell time (Doviak and Zrnić 1993). For pure rain, the theoretical value for |*ρ*_{hυ}(0)| is 0.98, which is consistent with observations that values of |*ρ*_{hυ}(0)| are greater than 0.97 (Balakrishnan and Zrnić 1990b), 0.99 (Illingworth and Caylor 1991), and 0.975 in small drops (Liu et al. 1993). A threshold of 0.97 is assigned for smaller drops, and a value of 0.95 for larger sizes accommodates the effects of oscillations and canting. Pairs of *Z*_{h}, |*ρ*_{hυ}(0)| in Fig. 3 indicate enhanced confidence in rain discrimination over what would be observed by either variable. There is recent observational and theoretical evidence of decreased |*ρ*_{hυ}(0)| near the top of *Z*_{dr} columns and in association with frozen and/or freezing drops in the“LDR_{υh} cap” (Hubbert et al. 1995, 1998 Jameson et al. 1996). In these regions, |*ρ*_{hυ}(0)| may be as low as 0.96 and 0.92, respectively. It should be remembered that these observations are not of pure rain in a strict sense.

#### Specific differential phase

*K*

_{dp}are produced by statistically anisotropic hydrometeors such as rain when drops are larger than about 1 mm. Therefore,

*K*

_{dp}can be used to isolate the presence of rain from statistically isotropic hydrometeors such as tumbling hail. Whereas

*K*

_{dp}is a good indicator of rain if rainfall rates are relatively large (>20 mm h

^{−1}; section 4b), it is not uniquely related to median size. (Rather, it is more closely related to content.) We choose thresholds of

*K*

_{dp}in Table 6 to be nearly compatible with values of

*Z*

_{h}, which, in turn, are compatible with

*Z*

_{dr}. This choosing is done by matching the rain rates in the Marshall–Palmer

*Z, R*relation with the

*K*

_{dp},

*R*relation (Sachidananda and Zrnić 1987). Because

*K*

_{dp}depends on mass contents and the presence of oriented hydrometeors and is relatively insensitive to details of size distributions, the above procedure generally is very accurate except, perhaps, in regions where rain and mixed-phase hydrometeors coexist. It has been shown by Balakrishnan and Zrnić (1990a), for example, that ice and mixed-phase hydrometeors might occur to the left, and pure rain to the right, of the curve given by

*Z*

_{h}

*K*

_{dp}

*K*

_{dp}

*Z*

In practice, the observed value of *Z*_{h} can be compared with that from Eq. (7) [used to obtain Eq. (3)]. Pure rain is likely if *Z*_{h} < *Z*_{h}(*K*_{dp}) from the *Z*_{h}, *K*_{dp} curve given by Eq. (7). A pure hail or a rain/hail mixture is possible if *Z*_{h} > *Z*_{h}(*K*_{dp}). The rain region in the *Z*_{h}, *K*_{dp} parameter space (Fig. 4a) is in general accord with the work presented by Ryzhkov and Zrnić (1996) and Eq. (7). Last, the fraction of ice from *K*_{dp} should be *F*_{ik} < 0.25 for pure rain (Table 6).

Recently, Hubbert et al. (1998) found an indication of water drops with sizes as large as 2 mm and a mode of about *D* ≈ 1 mm in regions near the freezing level and close to the updraft of a severe hailstorm. Loney et al. (1999) confirmed this signature using radar and in situ aircraft data. Based on research by Rasmussen (1984) and Rasmussen and Heymsfield (1987), Hubbert et al. (1998) hypothesize that these drops are shed from melting or growing hailstones in a column near the updraft. Such regions are associated with enhanced values of *K*_{dp} (0.5° to 1.5° km^{−1}), thus the term “*K*_{dp} column.” Values of other PR variables in the *K*_{dp} column include 50 < *Z*_{h} < 60 dB*Z,* 0 < *Z*_{dr} < 2 dB, −22 < LDR_{υh} < −16 dB, and 0.94 < |*ρ*_{hυ}(0)| < 0.96 (Hubbert et al. 1998). The values for PR variables defining the *K*_{dp} column are included in Table 6, and the parameter space associated with *Z*_{h}, *K*_{dp} pairs is shown in Fig. 4b. Values of *Z*_{h}, *Z*_{dr} pairs associated with the *K*_{dp} column are presented in Fig. 2b.

#### Backscatter differential phase

For 10-cm wavelength radar, maximum raindrop sizes (≈8 mm) should produce barely measurable *δ* (Balakrishnan and Zrnić 1990b). The values of |*δ*| < 1° in Table 6 accommodate small statistical uncertainty. In contrast, it has been suggested recently that frozen or freezing drops might be associated with |*δ*| > 1° (Hubbert et al. 1998).

#### Linear depolarization ratio

The lowest values of LDR_{υh} (<−32 dB) can indicate small, nearly spherical raindrops (*D* < 1 mm). Slightly larger values of LDR_{υh} (−27 to −34 dB) are found in more intense rain, whereas values of LDR_{υh} > −27 dB might be produced by large, deformed, or canted drops (Bringi et al. 1986a). The largest values of LDR_{υh} in pure rain should be about −24 dB (Bringi et al. 1986a). These values are based upon observations with 3-cm-wavelength radar. Interestingly, they basically agree with observations presented by Hendry et al. (1987), valid at a 1.82-cm wavelength, and work presented by Frost et al. (1989, 1991) for 10-cm-wavelength radar. Based on previous work described above, that done by Frost et al. (1989, 1991) at a 10-cm wavelength, and recent observations with the CSU-CHILL radar (e.g., Carey and Rutledge 1998; Hubbert et al. 1998), we suggest LDR_{υh} < −25 dB in Table 6 as the upper bound for rain. Recall that our thresholds are not “hard” but can be exceeded, albeit with confidence less than 0.5. The confidence for pure rain identification should be increased if the threshold ranges for *Z*_{h}, LDR_{υh} pairs are both satisfied (Fig. 5).

### Rain–wet hail mixtures and mixed-phase hydrometeors

Thresholds for some of the variables in Table 7 for rain–wet hail mixture are largely compatible with the thresholds for rain (Table 6) and/or hail (Table 2). The rain–small wet hail mixture should also be considered to be valid for rain–melting graupel mixtures common in the U.S. High Plains. Note that thresholds for *Z*_{h}, LDR_{υh}, *H*_{dr}, *K*_{dr}, and *δ* for rain–wet hail are nearly identical to thresholds for hail.

#### Reflectivity

The values for *Z*_{h} are consistent with those for wet hail.

#### Differential reflectivity

In general, mixtures of larger raindrops and small, water-coated hailstones contribute to positive *Z*_{dr}, whereas the presence of larger hail tends to reduce *Z*_{dr} (Table 7). For small hail mixed with rain, we set the upper threshold to 6 dB, which is about 7 dB lower than the maximum values for a model of a melting hailstone 12 mm in diameter (Aydin and Zhao 1990) and is about 0.5 dB higher than similar scattering-model results of Vivekanandan et al. (1990) for a 10-mm hailstone. Extreme *Z*_{dr} of 5 and 6 dB might indicate very large water drops with ice cores [ice particles that are melting or experiencing wet growth as defined by Pruppacher and Klett (1981) or Young (1993)], because often a torus of liquid water forms about the equator of a small hailstone (*D* < 9 mm), making it more oblate [and stabilizing tumbling motions (Rasmussen et al. 1984; Rasmussen and Heymsfield 1987)]. Large melting or wet hailstones (*D* > 20 mm) tend to have smaller *Z*_{dr}, because they cannot support a torus of water around their equators. We, therefore, reduce the upper thresholds by 2 or 3 dB per hail size category (Table 7) and adjust the lower thresholds slightly upward from values for pure hail. Figure 2a shows the *Z*_{h}, *Z*_{dr} pairs associated with rain–wet hail mixtures, with larger values for *Z*_{dr} than those for pure hail. The values for rain–wet hail mixtures are in general agreement with observations and measurements described by Carey and Rutledge (1998) and Hubbert et al. (1998), though our upper thresholds for *Z*_{dr} are larger than those of Carey and Rutledge. The accommodation for melting graupel mixed with rain (incorporated in the rain–small wet hail category) is based on recent observations presented by Hubbert et al. (1998).

#### Reflectivity difference and hail signal

The fraction of ice from *Z*_{dp} should range from 0.25 < *F*_{iz} < 0.75 (Table 7). The values for differential reflectivity hail signal are consistent with those for wet hail. Use of *H*_{dr} for rain–wet hail is similar to that for wet hail.

#### Correlation coefficient

Balakrishnan and Zrnić (1990b) suggest that |*ρ*_{hυ}(0)| can be as low as or lower than 0.90 in rain–hail mixtures. For two hydrometeor types in a resolution volume, |*ρ*_{hυ}(0)| is a minimum when the contribution to the echo power by one type is close to the contribution by the other type (Jameson 1989; Balakrishnan and Zrnić 1990b; Aydin and Zhao 1990). In rain–hail mixtures, |*ρ*_{hυ}(0)| decreases toward ground because of increasing rain amounts and, therefore, hydrometeor diversity below the melting level. This trend is nearly monotonic when the size and number of hailstones is significant; when not the case, |*ρ*_{hυ}(0)| reaches a minimum and starts to increase as hailstones melt into rain. Zrnić et al. (1993a) found that |*ρ*_{hυ}(0)| < 0.94 and *Z*_{dr} < −0.5 for rain–hail mixtures, with hail sizes of 20 < *D* < 50 mm. These findings, as well as recent measurements with both the Cimarron and CSU-CHILL radars (Hubbert et al. 1998), are used to define the thresholds in Table 7. Regions where the *Z*_{h}, |*ρ*_{hυ}(0)| pairs correspond to rain–wet hail mixtures are depicted in Fig. 3.

#### Specific differential phase

As described in the section for hail, the presence of hail should have a negligible influence on *K*_{dp}. In heavy rain, *K*_{dp} can be appreciably larger than zero, as seen in Table 4 and Fig. 4a. Therefore, these thresholds are also entered in Table 7. A small ice core (*D* < 9 mm) surrounded by a liquid torus might appear as a large water drop. Thus, large populations of these mixed-phase particles can produce large values of *K*_{dp}. Figure 4a shows *Z*_{h}, *K*_{dp} pairs for rain–wet hail mixtures to indicate enhanced classification confidence. The fraction of ice from *K*_{dp} should be 0.25 < *F*_{ik} < 0.75 (Table 7).

#### Backscatter differential phase

Values of *δ* for rain–wet hail mixtures are consistent with those for wet hail as described in the section on hail.

#### Linear depolarization ratio

For rain–wet hail mixtures, LDR_{υh} should be similar to or slightly smaller than that for pure wet hail, with larger values of LDR_{υh} for larger hail sizes. Carey and Rutledge (1998) indicate LDR_{υh} values for rain–hail mixtures that are slightly smaller than those for pure hail at 10-cm wavelength, as does Holler et al. (1994) for 5-cm wavelength. We employ this trend in Table 7 and Fig. 5.

Some recent, interesting observations of LDR_{υh} include caps (−25 < LDR_{υh} < −19 dB) on top of *Z*_{dr} columns. In these columns, *Z*_{dr} is large (up to about 5 dB), and |*ρ*_{hυ}(0)| is smaller than expected in pure rain. The LDR_{υh} caps also are associated with reduced values of |*ρ*_{hυ}(0)|, and *K*_{dp} typically is between 0° and 1° km^{−1}. Jameson et al. (1996) and Hubbert et al. (1995, 1998) attribute these sets of observations to zones of freezing or frozen drops. Hubbert et al. (1995, 1998) provide modeling support for this hypothesis. The threshold values for the PR variables from Hubbert et al. (1998) for LDR_{υh} caps are provided in Table 6. The parameter space where *Z*_{dr}, LDR_{υh} pairs satisfy the thresholds (Fig. 5) strengthens the confidence in the identification of LDR_{υh} caps. Values of *Z*_{h}, *K*_{dp} pairs as well as values of *Z*_{h}, *Z*_{dr} pairs associated with LDR_{υh} caps also are plotted in Fig. 2b and Fig. 4b, respectively, to show when increased confidence in these measurements might be valid. More modeling studies and in situ observations are needed to help to verify these values.

### Snow crystals and aggregates

Whereas PR data and, possibly, the temperature of the environmental air can be used to identify snow aggregates, classification of crystal types at 10-cm-wavelength radar is still in need of much research. One of the problems is that ice crystal types are difficult to discern because the dielectric constant for ice crystals and snow aggregates is dependent on particle density (which can be strongly size dependent); therefore, low-density particles produce lower contrasts among the PR variables. Small crystal sizes also can reduce magnitudes of PR signatures. Furthermore, ice crystals have a large variety of shapes that are difficult to model. Nevertheless, much recent work has been done, particularly for 3-cm-and-shorter-wavelength radar for Rayleigh scatterers that is of relevance (e.g., Matrosov 1992;Aydin and Tang 1997; Vivekanandan and Adams 1993;Vivekanandan et al. 1993a, 1994; Atlas et al. 1995; Gosset et al. 1995; Matrosov et al. 1996; Reinking et al. 1997). There also has been some work done at 10-cm wavelength (e.g., Thomason et al. 1995; Ryzhkov and Zrnić 1998a; Ryzhkov et al. 1998). Of particular interest is that low-density crystals that are within the limit of validity of the Rayleigh–Gans approximation exhibit little or no polarization dependence on *Z*_{h} even if they have very complex shapes (Matrosov 1992). Nevertheless, we constructed Table 8 on the basis of available model results and observational measurements with 10-cm-wavelength radar and modification of information of Rayleigh-regime results from shorter-wavelength radar (which would be at least qualitatively consistent with Rayleigh-regime information from longer-wavelength radar).

The density of snow crystals and aggregates varies as a function of habit from 50 to 900 kg m^{−3}, with higher values expected for solid ice structures and wetted particles. The size distributions of ice crystals and snow aggregates can be represented by exponential and gamma functions, and the total number of concentrations is on the order of 1–10^{4} m^{−3} for aggregates, 10–10^{9} m^{−3} for individual crystals at colder temperatures (*T* < −20°C), and often as high as 10^{4} m^{−3} at warmer temperatures (Pruppacher and Klett 1981). The size of large aggregates can be *D* ≈ 20–50 mm, whereas the size of large crystals can be *D* ≈ 1–5 mm. The shapes of aggregates are approximately spherical to extremely oblate, and the approximate shapes of crystals can vary from extreme prolates and oblates to essentially spheres (Pruppacher and Klett 1981). Most individual crystals tend to fall with their largest dimension horizontally oriented unless there are pronounced electric fields. Aggregates also can fall in a horizontally oriented manner or may tumble. A summary of crystal densities and size relations is presented in Table 9. Radar signatures of ice crystals are sensitive to their mean canting angle (Vivekanandan et al. 1993a).

#### Reflectivity factor

Measured *Z*_{h} (or *Z*_{υ}) of snow aggregates and crystals is generally smaller than for most other precipitating hydrometeors except, perhaps, drizzle largely because of low dielectric effects, regardless of any of the possible crystal shapes. Moreover, *Z*_{h} is highly dependent on particle density (densities of aggregates typically are much lower than those for crystals as shown in Table 9; also see references listed above). Boucher and Wieler (1985) report a range of 10–36 dB*Z.* Ohtake and Henmi (1970) computed *Z*_{h} from size distribution data and concluded that larger aggregates of dendrites produce the largest *Z*_{h}. The five crystal forms considered by Ohtake and Henmi give a maximum of *Z*_{h} < 35 dB*Z,* which agrees with the results of Ryzhkov et al. (1998) and Ryzhkov and Zrnić (1998a). Thus, we set an upper threshold for dry snow at 35 dB*Z.* Wetting could increase the reflectivity by 5–10 dB*Z* or so, from increased particle density and increased dielectric constant as suggested by observations and models (Zrnić et al. 1993a;Vivekanandan et al. 1993; Ryzhkov et al. 1998; Ryzhkov and Zrnić 1998a; among others). In addition, *Z*_{h} can be enhanced for wet snow, because these particles can grow to larger sizes through more efficient aggregation in association with wet ice surfaces. Moreover, PR signatures of aggregates can be enhanced as they collapse into raindrops. All of these aspects associated with aggregates can lead to melting-level brightband signatures in the *Z*_{h} fields (as well as in other PR fields). Hence, the upper threshold of *Z*_{h} ≈ 45 dB*Z* is valid for wet aggregates. Vertical gradients in *Z*_{h} at and below the melting level can be useful in discriminating between aggregates and graupel. The lack of a prominent bright band (gradient yet to be specified) can be useful in indicating graupel when there are warm surface temperatures. Moreover, the lack of any bright band can be useful in indicating snow when temperatures through a column are all below the freezing point of water.

#### Differential reflectivity

In general, it is difficult at best to discriminate snow aggregates and crystal types with *Z*_{dr}, let alone any of the other PR variables. One reason is there are but very few known relations between physical particle attributes and the very complicated scattering characteristics of aggregates and crystals. Another is the unknown index of refraction, which depends on particle density and on which *Z*_{dr} depends. Nevertheless, it has been found in some instances that *Z*_{dr} might be used to identify aggregates, needles, columns, and plates under ideal conditions (e.g., Vivekanandan et al. 1993; Brandes et al. 1995). Individual crystals have large axis ratios and generally fall with a horizontal orientation, which would produce positive *Z*_{dr}. For 5-cm-wavelength radar, Meischner et al. (1991a,b) suggest that *Z*_{dr} of 2–5 dB might be associated with pristine dendrites or aggregates composed of 5–6 crystals, and that *Z*_{dr} of 0–0.5 might be associated with larger dendrites and aggregates. The lower values for dry aggregates take into account that they may be 1) small, 2) of low density, 3) associated with low dielectric constants, 4) nearly spherical, and/or 5) tumbling while falling. A scattering model of oriented oblate spheroids with densities of snow produces *Z*_{dr} less than 1 dB (Illingworth et al. 1987), which is the upper limit for dry snow aggregates in Table 8. For wet aggregates, measured values of *Z*_{dr} at the bottom of the melting layer range from 1 dB (Zrnić et al. 1993a; Moninger et al. 1984) to 2.5 dB (Hagen et al. 1993). Recent measurements of *Z*_{dr} up to 4 dB (CSU-CHILL radar) have been ascribed to snow aggregates. There are not many scattering model results or observations yet of scattering by columns (or thick plates), plates (or sectors and dendrites), or needles (or sheaths) at a 10-cm wavelength. Geometric considerations dictate that, of these three crystal forms, plates have the largest *Z*_{dr}, and needles have the lowest *Z*_{dr}. We use information available for a 10-cm wavelength (Vivekanandan et al. 1993; Vivekanandan 1994) and extrapolate results from an 8-mm wavelength (Evans and Vivekanandan 1990; Vivekanandan and Adams 1993) to estimate the range of thresholds provided in Table 8. The largest dimensions considered by Evans and Vivekanandan are 2 mm, which is in the resonance region of scattering and causes large excursions of *Z*_{dr}. The upper thresholds in Table 8 are smaller by about 2 dB from the model results, because at a 10-cm wavelength, resonance effects generally are not present in ice crystals.

In summary, 0.4 < *Z*_{dr} < 3.0 dB with 30 < *Z*_{h} < 45 dB*Z* may indicate wet snow at warmer temperatures, whereas 0.4 < *Z*_{dr} < 3.0 dB with 5 < *Z*_{h} < 30 dB*Z* may indicate pristine ice crystals or lightly aggregated crystals at colder temperatures. At cold temperatures, measurements of 0 < *Z*_{dr} < 0.2 dB with 5 < *Z*_{h} < 30 dB*Z* could be indicative of dry aggregates. In addition, values of *Z*_{dr} and *Z*_{h} that are similar to those for aggregates in the melting layers might be indicative of the rain–snow line as seen in some Oklahoma winter storms (Ryzhkov et al. 1998; Ryzhkov and Zrnić 1998a). These values are incorporated into Table 8 and in Fig. 2b. Below the melting level, as aggregates collapse into drops, measurements of *Z*_{dr} typically show a decrease. Further decreases in *Z*_{dr} from 1 to 2 km below the melting layer could be associated with drops breaking up into smaller particles from collisions with each other.

It should be noted that bulk interpretations of aggregates and crystals might be misleading. For example, Bader et al. (1987) found that a few large, low-density aggregates could produce low bulk values of *Z*_{dr} that could conceal high values of *Z*_{dr} possible from many numerous, small needles and columns when the aggregates and crystals are mixed.

#### Reflectivity difference and hail signal

A few investigators have used *Z*_{dp} implicitly to separate rain from frozen hydrometeors and for quantification of ice amounts (e.g., Golestani et al. 1989; Conway and Zrnić 1993; Ryzhkov et al. 1998). It has not been widely accepted for identification of specific ice hydrometeor types, however. The most detailed example possibly is that by Meischner et al. (1991a), who had a 5-cm-wavelength radar and in situ measurements. At this time, we are not confident that *Z*_{dp} (at 10-cm wavelength) can indicate crystal type. Also, *H*_{dr} has not been used to indicate crystals or aggregates.

#### Correlation coefficient

Values of |*ρ*_{hυ}(0)| in pure snow are similar to those for rain if there is little variation in the canting angles of snow crystals (e.g., Balakrishnan and Zrnić 1990b). Observations by Illingworth and Caylor (1989) indicate |*ρ*_{hυ}(0)| that are as low as 0.60 in the bright band associated with melting aggregates. Values this low also have been seen regularly with the Cimarron radar in Oklahoma (e.g., Ryzhkov et al. 1998). It also has been demonstrated observationally and theoretically that |*ρ*_{hυ}(0)| decreases substantially in regions of mixed-phase hydrometeors at the bottom of the melting zone where snow aggregates begin to collapse into raindrops (e.g., Zrnić et al. 1993a, 1994; Ryzhkov and Zrnić 1998a; Ryzhkov et al. 1998). These considerations lead us to suggest a range of 0.50 < |*ρ*_{hυ}(0)| < 0.90 for wet aggregates, a lower threshold of 0.95 for dry aggregates, and a lower threshold of 0.95 for other types of snow particles. Confidence in the wet snow aggregate category should be enhanced if both 30 < *Z*_{h} < 45 dB*Z* and 0.50 < |*ρ*_{hυ}(0)| < 0.90 are satisfied (Fig. 3; Ryzhkov et al. 1998; Ryzhkov and Zrnić 1998a).

#### Specific differential phase

Attempts have been made to use *K*_{dp} to identify crystal habits. Hendry et al. (1976) suggest that *K*_{dp} should be larger for dendrites than for aggregates and recorded 0.36° km^{−1} associated with snow. More recently Golestani et al. (1989) noted *K*_{dp} of 0.75° km^{−1} in oriented crystals that possibly were mixed with some aggregate. (The aggregates were not assumed to contribute significantly to *K*_{dp}.) In addition, *K*_{dp} measurements in a storm anvil ranged from 0.25° to 0.5° km^{−1} (Vivekanandan et al. 1994). These measurements were attributed to horizontally aligned crystals in a mixture with effectively spherical aggregates. Observations with CSU-CHILL radar and scattering-model results suggest that *K*_{dp} might increase when crystal density increases. These observations are consistent with values modeled by Vivekanandan et al. (1994) and Evans and Vivekanandan (1990), which, for example, predict *K*_{dp} ≈ 0.25° km^{−1} for snow. Vivekanandan and Adams (1993) show PR scattering-model results at an 8-mm wavelength for various elevation angles between 0° and 90° that suggest some habit discrimination with *K*_{dp}, as well as with other PR variables; however, this possibility needs to be tested at 10-cm wavelength. Recent observations with the Cimarron radar and an instrumented T-28 aircraft suggest the presence of horizontal crystals if 0.0° < *K*_{dp} < 0.6° km^{−1} and 5 < *Z*_{h} <30 dB*Z.* In addition, vertically oriented crystals are indicated if 0.0° > *K*_{dp} > −0.6° km^{−1} and 5 < *Z*_{h} < 30 dB*Z* in electric fields (Ryzhkov and Zrnić 1998a; Ryzhkov et al. 1998), as presented in Table 8 and Fig. 4b. These values are valid for *T* < 0°C. Carey and Rutledge (1998) also suggest *K*_{dp} < −0.25° km^{−1} and *Z*_{h} < 40 dB*Z* for vertically aligned crystals in electric fields, which concurs with Ryzhkov et al.’s values. Wet aggregates in the melting layer can produce *K*_{dp} ≈ 0.5°–1° km^{−1} (Zrnić et al. 1993a), which can help to discriminate dry from wet aggregates, with wet aggregates producing the large phase shift. Last, Ryzhkov and Zrnić (1998a) found that *K*_{dp} in snow is almost always larger at a fixed value of *Z*_{h} than that observed in rain. In Table 8, *K*_{dp} is not used to discriminate among columns, plates, and needles, as suggested by the 8-mm-wavelength results of Vivekanandan and Adams (1993). Extrapolation to 10-cm wavelength is possible because the absence of resonance effects (at 10-cm wavelength) can only reduce discriminating properties present at shorter wavelengths. The small positive thresholds (*K*_{dp} > 0.0° km^{−1}) indicate a general trend in *ϕ*_{dp} caused by the presence of numerous horizontally oriented particles. Vertically oriented crystals also are taken into account in Table 8 and Fig. 4b; the orientation is due to the presence of strong electric fields that can align small crystals. Whereas *K*_{dp} is sensitive to electric fields because many small crystals can be reoriented, *Z*_{dr} generally is not sensitive to electric fields because less numerous but larger crystals probably cannot be reoriented (Ryzhkov et al. 1998).

#### Backscatter differential phase

Of all the crystal and snow forms, only large wet aggregates are known to produce measurable *δ.* At a 10-cm wavelength, aggregates might produce *δ* when sizes exceed *D* ≈ 10 mm. The range −5° < *δ* < 10° in Table 8 is obtained from a scattering model and observational measurements (Zrnić et al. 1993a).

#### Linear depolarization ratio

Values of LDR_{υh} between −29 and −26 dB have been observed in association with dry, moderately heavy snow at a wavelength of 1.82 cm (Hendry et al. 1987). Hendry et al. also noted an LDR_{υh} bright band near the melting level in precipitation. Similarly, Frost et al. (1991) measured LDR_{υh} > −22 dB in the bright band at a 10-cm wavelength. They report more moderate values of −6 < LDR_{υh} < −22 dB in dry ice regions, however. Scattering-model results of LDR_{υh} for aggregates at 10-cm wavelength are between −25 and −20 dB (Vivekanandan et al. 1993a), though much smaller values were produced by the scattering model of Frost et al. (1989). Matrosov (1991) suggests that there might be some crystal discrimination possibility with LDR_{υh} but that the signal differences are weak. His results showed that plates and needles produce the largest LDR_{υh} signals (LDR_{υh} ≈ −27 to −28 dB, respectively), and that thick plates, solid columns, and bullets produce the smallest (LDR_{υh} ≈ −30 to −31 dB, respectively). Recent wintertime observations with the CSU-CHILL and S-POL radars agree, in general, with the larger observed values described, especially near the melting level. The thresholds in Table 8 and Fig. 5 are approximately valid for a 10-cm wavelength and are based on some of the observations described above and on the scattering-model results from Matrosov (1991), Vivekanandan and Adams (1993), and Vivekanandan et al. (1993a, 1994). Note that electric fields that align small ice crystals into the vertical prior to lightning discharges have been shown to increase, reduce, and not affect LDR_{υh} signatures (Caylor and Chandrasekar 1996). Therefore, LDR_{υh} is not used to indicate vertically aligned crystals in electric fields. The possibility to study storm electrification and lightning location with dual-polarimetric radar has begun to gain acceptance in the meteorology community in the past five yr or so.

### Case where *Z*_{h} and *T* are the only variables available

If *Z*_{h} is the only radar variable available, then, with the addition of temperature, some general information for hydrometeor identification could be extracted by eliminating physically unrealistic possibilities (Table 10). Ambiguities are impossible to avoid with just *Z*_{h} and *T.*

## Estimating precipitation amounts

For as long as weather radar has existed, there have been attempts made to use *Z*_{h} to quantify how much precipitation is falling, often in conjunction with statistical procedures. Over the past two decades, PR has emerged as a very useful remote sensing instrument for estimating precipitation amounts, with most improvements for rain amount estimation.

For correct quantitative estimation of precipitation amounts and other parameters, the type of contributing hydrometeor needs to be determined, as described in section 3. Determining hydrometeor amounts is significantly more difficult than hydrometeor classification. In this section, we suggest some quantitative estimates for *D*_{o}, terminal velocity *V, R,* ice or liquid water content *M,* and total number concentration *N*_{t}. These estimates depend on assumptions about the size distributions, axis ratios *a*/*b,* particle densities *ρ,* fall orientations, shapes, dielectric constants for water, etc. The accuracy of these estimates depends on the type and amount of hydrometeors and on the availability, resolution, reliability, and accuracy of the PR data. The information for the quantitative estimation is put forward in Tables 1, 3, 5, and 9 for use in scattering and cloud modeling (with references provided for more detailed explanation). For brevity, the discussion in the remainder of this section is limited to estimation of precipitation rates and contents.

To transform the PR data into quantitative precipitation estimates, equations described in previously published studies are employed. Many of these equations were obtained from model simulations of realistic size distributions, whereas others were determined from in situ measurements. Some precipitation types, such as rain, can be quantified from two or more fundamentally different relations. Then, consistency among precipitation amounts can be used to check the estimate.

Quantification of ice hydrometeors is complicated by the presence of hydrometeors that are themselves mixed phase, as well as by mixed–hydrometeor phase populations. For most of the ice species, amount estimates rely heavily on the reflectivity factor. When ice is mixed with rain, the liquid contribution to *Z*_{h} often can be removed by careful use of some of the PR variables. Recently, some precipitation amount estimates for ice using *K*_{dp} have been developed (section 4e). For quantifying amounts for all ice habits, however, there is still much work to be done. We submit, though, that, at the very least, if the classification is correct, then the quantification based on the reflectivity factor alone, with all its uncertainties, might be more reliable. The quantitative precipitation estimate equations for the previously described variables, including hail, graupel, rain, and snow crystals are summarized in Tables 1, 3, 5, and 9, respectively. A brief discussion of each follows.

### Hail amounts

Estimates of hail content and size are crude at best (Table 1), and the errors could be significant, depending on how much the actual size distribution deviates from the assumed distribution. The Cheng and English (1983) hail size distribution has gained some acceptance and is suggested, at least, for the present. The exponent and concentration of the distribution are related, and therefore the exponent is a function of *Z*_{h} alone.

### Rain amounts

Whereas natural hydrometeor size distributions are highly variable, approximations with a three-parameter gamma function might be adequate for most applications (Ulbrich 1983); we urge caution, however, because exceptions commonly occur. If *Z*_{h} is the only variable, then the Marshall and Palmer (1948) distribution (Table 5) and semiempirical relations can be used to determine rainfall rates. In the WSR-88D system (Table 5), *Z*_{h} is truncated at 51 dB*Z* for arid and semiarid regions to prevent contamination by hail (Vieux and Bedient 1998). In maritime, tropical, and other precipitation systems, truncation should be at 55 dB*Z* (Vieux and Bedient 1998; Aydin et al. 1995). Care must be taken with *R*(*Z*) estimates, even at 10-cm wavelengths, for various reasons, including attenuation (Ryzhkov and Zrnić 1995c). However, *R*(*Z*) relations can be tuned to some extent with PR variables (Ryzhkov and Zrnić 1997). In addition, multidimensional *Z*_{h} properties (gradients, horizontal and vertical morphology, precipitation system type, etc.) might be used to improve *R*(*Z*) estimates and identify the *Z*_{h} “bright band” (Rosenfeld et al. 1995a,b) based on objective classification of precipitation system type.

Rain rate estimates using *Z*_{h} and *Z*_{dr} are based on several assumptions, including 1) drops fall with their minor axis vertically oriented, 2) axis ratios and size relations are known, and 3) drops exhibit little canting in the mean. In addition, the assumptions that the raindrop size distribution is exponential and that *D*_{max} is reasonably large are also made. The *R*(*Z*_{h}, *Z*_{dr}) for rainfall rate in Table 5 was developed by Sachidananda and Zrnić (1987). A similar relation for M is also in Table 5. Chandrasekar et al. (1990) note that *R*(*Z*_{h}, *Z*_{dr}) formulations outperform *R*(*Z*) relations. Other similar relations for *R* and *M* using *Z*_{h} and *Z*_{υ} have been proposed by Chandrasekar et al. (1990), Aydin and Giridhar (1992), and Gorgucci et al. (1993), among many others.

Sachidananda and Zrnić (1987) and Steinhorn and Zrnić (1988) proposed that *R* for rain can be estimated using *K*_{dp} from 10-cm-wavelength radar (Table 5). Similar *R*(*K*_{dp}) and *M*(*K*_{dp}) relations also are proposed by Aydin and Giridhar (1992), Chandrasekar et al. (1990), Gorgucci et al. (1993, 1995), Aydin et al. (1995), and Ryzhkov and Zrnić (1995, 1996a,b), and are partially summarized by Brandes et al. (1997). Note that *R* and *M* are nearly linearly dependent on *K*_{dp}. More recently, *R*(*Z*_{dr}, *K*_{dp}) relations have been proposed (Jameson 1991;Aydin et al. 1995; Ryzhkov and Zrnić 1995a, 1996b). A summary of a number of *R* and *M* relations for various PR variable combinations is presented in Table 5.

Studies by Sachidananda and Zrnić (1987), Balakrishnan and Zrnić (1990a), and Chandrasekar et al. (1990) suggest an *R*(*Z*) relation for *R* < 20 mm h^{−1}, *R*(*Z*_{h}, *Z*_{υ}) for 20 < *R* < 50 mm h^{−1}, and *R*(*K*_{dp}) for *R* > 50 mm h^{−1} (Table 5). Note that Chandrasekar et al. (1990) use a 70 mm h^{−1} threshold for *R*(*K*_{dp}). The smaller value we suggest capitalizes on the other advantageous properties of *K*_{dp} rather than just statistical errors of estimates; these values follow recommendations by Sachidananda and Zrnić (1987). More recently, Ryzhkov and Zrnić (1996) recommend one *R*(*K*_{dp}) relation for all rain-rate regimes but with longer range averaging intervals for smaller rain rates.

Errors resulting from drop size distribution variations and statistical uncertainty, under ideal conditions, [simulations with gamma distribution and Pruppacher and Beard (1970) axis ratios] are 30%–40% for *R* < 20 mm h^{−1}, 20%–30% for 20 < *R* < 60 mm h^{−1}, and better than 20% for *R* > 60 mm h^{−1} (Chandrasekar et al. 1990;Sachidananda and Zrnić 1987; Balakrishnan et al. 1989). Recent findings suggest accumulation errors with *R*(*K*_{dp}) between 10% and 20% (Aydin et al. 1995; Ryzhkov and Zrnić 1995a; Ryzhkov and Zrnić 1996b). Errors for *R*(*Z*_{dr}, *K*_{dp}) estimates are similar to, or slightly better than, those for *R*(*K*_{dp}) (Ryzhkov and Zrnić 1995a).

### Rain–wet hail mixture amounts

Because *K*_{dp} is almost unaffected by the presence of hail, it is probably the most suitable PR variable to estimate the amount of liquid water content in rain–ice mixtures (e.g., Aydin et al. 1995). After Balakrishnan and Zrnić (1990a), *Z*_{hr} = 24 800 *K*^{1.386}_{dp}*Z*_{hr} is the reflectivity (mm^{6} m^{−3}) from rain. Aydin and Giridhar (1992) also proposed a similar rain curve using *K*_{dp} at a 5-cm wavelength. The contribution of ice to the reflectivity is simply *Z*_{hi} = *Z* − *Z*_{hr}. Golestani et al. (1989) suggest that the reflectivity difference [(Eq. A4)] can be applied to separate the amounts of rain and hail in a mixture of the two. In pure rain, tightly correlated pairs of *Z*_{dp} and *Z*_{hr} lie along a well-defined line given by Eq. (2). Significant lateral displacement (2dB) of the data from this line is indicative of mixed phase or hail and is caused by the reflectivity of ice *Z*_{hi}.

The fraction of ice contribution *F*_{i} to *Z*_{h} in ice and liquid mixtures is computed as *F*_{im} = *Z*_{hi}(*Z*_{hi} + *Z*_{hr})^{−1}, where *Z*_{hi} and *Z*_{hr} are in units of mm^{6} m^{−3}, and the subscript *m* refers to the method of computing rain amount (*F*_{iz} is from *Z*_{dp} and *F*_{ik} is from *K*_{dp}). If *K*_{dp} is used to estimate the rain amounts in the mixture, the errors are comparable to those in pure rain. For estimates from *Z*_{dp}, errors are at least as large as in pure rain. If both *K*_{dp} and *Z*_{dp} are available, it probably is better to compute the liquid and frozen fractions from *K*_{dp} because the measurement of the liquid part is more robust (Balakrishnan and Zrnić 1990b).

### Graupel–small hail amounts

There is a paucity of data concerning quantitative relations for graupel–small hail. Part of the problem is a nonexistence of well-defined relations between PR variables and parameters of the size distribution. For lack of better estimates, we list Hauser and Amayenc’s (1986) formulation using *Z*_{h} for low-density (150 km m^{−3}) lump graupel (Table 3). We suggest that estimates for *R* and *M* for high-density graupel–small hail follow those for hail (e.g., Table 1). Errors in the graupel rate and content calculations are uncertain.

### Snow crystals and aggregate amounts

Radar estimates of ice water content of crystals and aggregates are greatly complicated by the multitude of crystal sizes and shapes, various crystal and aggregate densities, and dielectric constants, among others. Estimates of snow amounts have been made from *Z*_{h} [e.g., Sekhon and Srivastava 1970; Heymsfield 1977 (stratiform); Herzegh and Hobbs 1980; Smith 1984; Sassen 1987; Matrosov 1992; Detwiler et al. 1993 (crystals); Atlas et al. 1995 (cirrus); Aydin and Tang 1997; Thomason et al. 1995; Matrosov 1998]. Some of these relations are in Table 9. Vivekanandan et al. (1993a, 1994) relate ice contents for crystals as a function of density and shape to *K*_{dp} and *Z*_{dp}; these relations might be used for columns, plates, and needles (Table 9). More recently, Ryzhkov and Zrnić (1995) and Ryzhkov et al. (1998) used *K*_{dp} and combined *Z*_{dr} and *K*_{dp} to estimate ice water contents for crystals and lightly aggregated snowflakes (Table 9). Of the PR variables, *K*_{dp} and *Z*_{dp} might be the most useful, because they are more sensitive to number concentration (especially *K*_{dp}) than to shape. An ice water content from *Z*_{dr} (or *Z*_{dp}) and *K*_{dp} also might be of value, because it probably is insensitive to crystal shape and density. Of all the snow types, the determination of the amount of wet aggregates is probably the most difficult. Still needed are reliable estimates of errors in *R* and *M* for snow.

## Summary

The results of numerous modeling and observational studies have been synthesized into relations between PR variables and hydrometeor types. These relations are expressed as threshold boundaries in multidimensional PR variable spaces meant for deducing bulk hydrometeor types, which subsequently permits the quantification of hydrometeor amounts. On a PR variable threshold boundary or on the edge of a multidimensional PR variable space, our confidence or belief of a correct hydrometeor identification is suggested to be 0.5 on a scale from 0 to 1. Therefore, the boundaries and spaces are suitable for use in fuzzy classification and similar algorithms. Relations to determine the amounts and average sizes of classified hydrometeors are tabulated succinctly. At present, we consider these relations to be some of the more reliable estimates. A better understanding of PR variables through comparison with observations and models will improve these relations. A fuzzy classifier algorithm that also quantifies precipitation amounts will be described in a subsequent paper and is considered to be versatile in that it accepts any of the available PR variables. Researchers interested in the determination of bulk qualitative or quantitative hydrometeor properties throughout a PR sample volume might find the synthesized information herein useful either for testing similar or for developing alternative procedures.

## Acknowledgments

Partial funding for this research was provided by the National Science Foundation grants ATM-9120009, ATM-9311911, EAR-9512145, and ATM-9617318; the National Severe Storms Laboratory;the Cooperative Institute for Mesoscale Meteorological Studies; and the Graduate College of the University of Oklahoma (Dr. E. C. Smith). Conversations over the past five years with Drs. V. Bringi, A. Detwiler, R. Doviak, K. Emanuel, M. Fritsch, R. Gall, J. Keeler, S. Lasher-Trapp, D. Lilly, R. Maddox, S. Nelson, E. Rasmussen, P. Smith, J. Vivekanadan, and J. Wurman, and the support of Drs. K. Droegemeier, F. Carr, and P. Lamb are gratefully acknowledged. We thank Messrs. M. Askelson and B. Gordon and Ms. Y. Liu for fruitful discussions and for editing radar data. Ms. K. Kanak is thanked for the tremendous amount of time she invested in helping to critique, to organize, and to edit this paper. Ms. Nichole Peltier is acknowledged for helping with some of the literature search.

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## APPENDIX

### Polarimetric Radar Variables

*Z*

_{h}

*λ*

^{4}

*π*

^{4}

*K*

_{w}

^{2}

*s*

_{hh}

^{2}

*Z*

_{υ}

*λ*

^{4}

*π*

^{4}

*K*

_{w}

^{2}

*s*

_{υυ}

^{2}

*Z*

_{dr}

*s*

_{hh}

^{2}

*s*

_{υυ}

^{2}

*Z*

_{dp}

*Z*

_{h}

*Z*

_{υ}

*Z*

_{h}

*Z*

_{υ}

*ϕ*

_{dp}

*ϕ*

_{hh}

*ϕ*

_{υυ}

*K*

_{dp}

*ϕ*

_{dp}

*r*

_{2}

*ϕ*

_{dp}

*r*

_{1}

*r*

_{2}

*r*

_{1}

^{−1}

*ρ*

_{hυ}

*s*

_{υυ}

*s*

^{*}

_{hh}

*s*

^{2}

_{hh}

^{1/2}

*s*

^{2}

_{υυ}

^{1/2}

_{υh}

*s*

_{υh}

^{2}

*s*

_{hh}

^{2}

Two additional backscatter variables are the complex correlations between copolar and cross-polar voltages; however, they are not discussed here because it is not known how useful the information they provide is (Doviak and Zrnić 1993). In addition, attenuation and depolarization from propagation are difficult to measure, and their use is limited; thus they are not discussed. In this appendix, *s*_{ij} refers to an element of the backscattering matrix of a hydrometeor (Zrnić 1991). The first subscript *i* indicates the polarization of the backscattered field [horizontal *h* or vertical *υ*], and the second subscript *j* refers to the polarization of the incident field; *K*_{w} = (*ϵ*_{w} − 1)/(*ϵ*_{w} + 2) is a factor related to the dielectric constant of water; *ϵ*_{w} is the dielectric constant;*λ* is radar wavelength; *ϕ*_{hh} and *ϕ*_{υυ} are the phases of the horizontally polarized and vertically polarized waves; and *r*_{1} and *r*_{2} are the distances of measurements 1 and 2 from a radar. The brackets 〈 〉 indicate expectations expressed in terms of the distribution of mean hydrometeor properties such as size, shape, shape irregularities, fall orientation, canting angle, particle density, composition, dielectric constant, and others.

Equations and parameters for hail.

Thresholds for some variables to classify hail.

Equations and parameters for graupel–small hail.

Thresholds for some variables to classify graupel–small hail.

Equations and parameters for rain (and rain when mixed with wet hail).

Continued.

Thresholds for some variables to identify rain. Note: Large drops in low concentration, *Z*_{dr} column, *K*_{dp} column, and LDR* _{υh}* cap are not considered in the general category of rain.

Thresholds for some variables to classify rain–wet hail mixtures.

Thresholds for some variables to classify snow crystals and aggregates.

Equations and parameters for snow crystals and aggregates.

Continued.

Hydrometeor types using *Z _{h}* and

*T.*

^{1}

All discussion herein about hydrometeor classification with PR refers to bulk hydrometeor identification or identification of the hydrometeor type that dominates the various PR signatures in a radar volume. A word of caution: it is possible, because of physical attributes of a hydrometeor type such as particle structure, density, etc., that relatively few in number of one hydrometeor type (e.g., a few, low-density snow aggregates) might dominate PR signatures in a radar volume even when there are large numbers of another type in the same volume (e.g., many small needle and column crystals).