## 1. Introduction

An interpretation of radar observations of ice precipitation is complicated by the existence of a variety of ice hydrometeor types that differ in shape, size, and density (Magono and Lee 1966). A typical ice precipitation event can consist of a mixture of several different particle types, such as pristine ice particles, aggregates, and graupel. There are a number of known radar reflectivity ice water content relations presented in the literature (see, e.g., Sekhon and Srivastava 1970; Smith 1984; Matrosov 1992); however, because of the complexity of the problem there is no unique one.

Matrosov (1998) and Liao et al. (2005) have investigated the use of dual-wavelength radar for estimation of snow parameters. They have shown that by using measurements taken at two wavelengths, where at least one of them is located in a non-Rayleigh region, the parameters of a particle size distribution can be accurately estimated. In these studies, it is assumed that there is only one type of particle present in the observation volume.

Matrosov et al. (1996) have shown that by using dual-polarization radar measurements taken at several elevation angles it is possible to discriminate between various types of ice particles, such as planar crystals, columnar crystals, and aggregates in a homogeneous cloud. Furthermore, by using these measurements it is also possible to study their shape.

The use of very high-frequency (VHF) profiler measurements for retrieval of ice particle size distribution above the freezing level in the stratiform region of a tropical squall line was demonstrated by Rajopadhyaya et al. (1994). In this study it was shown that vertical-pointing profiler Doppler observations of precipitation can be converted to ice particle size distributions using a velocity–diameter relation (Langleben 1954; Locatelli and Hobbs 1974). The main limitation of the method is a limited sensitivity to small-size particles.

Recently, an application of dual-polarization spectral analysis is demonstrated for ice precipitation studies (Moisseev et al. 2004). It was shown that a combination of Doppler measurements taken at a 45° elevation angle and dual-polarization observations can be used to distinguish between different types of ice hydrometeors within a radar volume. In this paper we further develop this idea. The purpose of the study is to show that dual-polarization spectral measurements taken at a high elevation angle can be used for the retrieval of ice particle size distribution parameters. Furthermore, this approach can be used to estimate size distribution parameters for mixtures of different ice hydrometeor types.

The developed approach is applied to the ice precipitation above the melting layer of a stratiform rain case. The measurements were taken by the transportable atmospheric radar (TARA) on 19 September 2001 at Cabauw, Netherlands.

This paper is structured as follows. Section 2 gives an overview of the microphysical properties of ice particles, to be used in the model. Section 3 introduces the radar observables used for measuring ice particles. Section 4 describes the retrieval method for the particle size distributions of the two dominant types of ice hydrometeors. This retrieval procedure is applied on TARA measurements in section 5 and conclusions are presented in section 6.

## 2. Microphysical model of ice particles

Magono and Lee (1966) have demonstrated that there are more than 60 different types of ice crystals. The different ice particle types can be subdivided in three different groups, depending on their size. The most important ice particle types that represent the different groups are plates, dendrites, and columns for small pristine ice particles; rimed particles and aggregates for the intermediate particles; and graupel and hail for the large particles.

The developed microphysical model of this paper is based on the actual knowledge of ice hydrometeors found in the literature. Size, shape, canting angle, density, relative permittivity, and velocity are considered in the model. The types of ice particles are plates, dendrites, aggregates, graupel, and hail. Rimed particles are treated as aggregates because it is not possible to separate them from other type of aggregates (combinations of pristine ice particles) by radar only. The same microphysical properties, size, and shape, are expected for rimed particles and aggregates.

The occurrence of the different pristine ice particles strongly depends on temperature, pressure, and humidity of the atmosphere. Our observations are taken above the melting layer of a stratiform rain event, where the temperature is about −3°C (radiosonde measurement). Based on the measured temperature, it is not likely that columns are present above the melting layer during these observations (Fukuta and Takahashi 1999). However, in a future version, columns have to be implemented in the model.

A summary of the ice particle types that are modeled is given in Table 1. The given sizes of the different particles are typical sizes found in the literature (Mitchell et al. 1990; Mitchell 1996; Pruppacher and Klett 1978). It has to be taken into account that the boundaries are not strict in the sense that possibly particles may exist outside these boundaries.

### a. Shape of ice crystals

*w*is the smallest dimension of the spheroid, and

*D*is the largest dimension. Values for

*ξ*and

*ζ*in the case of plates, dendrites, and aggregates can be found in Matrosov et al. (1996) and Auer and Veal (1970). Bringi and Chandrasekar (2001) give the parameter values for hail. The values of

*ξ*and

*ζ*for the different particle types are summarized in Table 2.

Based on Eq. (1), the axis ratio, defined as the ratio of smallest over largest particle dimension, can be determined. In Fig. 1, the axis ratio is given as a function of maximum particle dimension. The axis ratio of hail and aggregates is constant over diameter. In reality these particles have very irregular shapes. On average they can be modeled as spheroids close to spheres, to simplify calculations (Bringi and Chandrasekar 2001). From the figure, it is also noticed that plates and dendrites have a more oblate shape than aggregates and hail.

Graupels are assumed to be conical. The axis ratio of conical graupel is calculated using Wang’s relations (Wang 1982), and a theoretical value of 0.95 is obtained. Measurements of mean axial ratios of conical graupels show values ranging from 0.75 to 0.9 for sizes in excess of 1 mm (Heymsfield 1978).

### b. Canting angles of ice crystals

Falling hydrometeors will be canted because of external forces like wind, turbulence, and draft. The canting angle is described by two orientation angles *δ* and *α*, where *δ* is the angle between the zenith and one symmetry axis of the ice particle, and *α* is the azimuth angle. For all ice particles, the azimuth angle *α* is uniformly distributed between 0 and 2*π*. For plates, dendrites, and aggregates, the orientation angle *δ* follows a Fisher probability function with the width parameter *κ* equal to 30 and a mean equal to zero according to Bringi and Chandrasekar (2001). Because of the conical shape, graupel has a different canting behavior for *δ* with respect to the other ice particles. Bringi and Chandrasekar (2001) describe this behavior with a Gaussian distribution with a mean equal to zero and a standard deviation of 30°. Hail particles have the tendency to tumble in the elevation plane. Therefore the distribution of the orientation *δ* is modeled by a uniform distribution between 0 and 2*π* (Bringi and Chandrasekar 2001).

### c. Density of ice crystals

*D*, where

*ρ*denotes the density of the ice crystals.

_{e}Pruppacher and Klett (1978) give values for the variables *k* and *l* for plates and dendrites (see Table 3). The density relation for conical graupel and hail is taken from El-Magd et al. (2000). Because aggregates occur in many different appearances, their density is difficult to model. Still, different density–diameter relations have been examined (Fabry and Szyrmer 1999). The relation *ρ _{e}* = 0.015

*D*

^{−1}from Mitchell et al. (1990) is chosen for the model because it falls roughly in the middle of the various relationships found in the literature (Fabry and Szyrmer 1999).

### d. Relative permittivity of ice crystals

_{eff}, ɛ

_{ice}, and ɛ

_{air}the permittivity of the ice particle, ice, and air, respectively;

*c*is the volume concentration of the inclusion of ice in air.

Ray (1972) gives a method to calculate the permittivity of ice over a broad spectral range.

### e. Velocity of ice crystals

*υ*of hydrometeors is taken from (Mitchell 1996): where

_{t}*A*is the area projected to the normal flow of the ice particle,

*ρ*is the density of air,

_{a}*m*is the mass of the particle,

*g*is the gravitational constant,

*D*is the largest dimension of the particle, and

*ν*is the kinematic viscosity of air.

*D*, Combining Eqs. (5)–(7), the terminal fall velocity of ice particles (cm s

^{−1}) is expressed as a power law of the maximum particle dimension

*D*(cm): The values of

*α*,

*β*,

*γ*, and

*σ*, for the different types of ice particles (except conical graupel), are summarized in Tables 4 and 5. They are reported by Mitchell (1996), whereas the values for conical graupel are given by Heymsfield and Kajikawa (1987). The values of

*a*and

*b*are derived in Khvorostyanov and Curry (2002).

The obtained fall velocities as a function of maximum dimension of the different particle types are shown in Fig. 2. From this plot, it is concluded that the different particles exist in different velocity regions. In the velocity region 0–2 m s^{−1}, only plates, dendrites, and aggregates exist. Graupel is observed in the region above 2 m s^{−1}, and hail exists above 10 m s^{−1}.

## 3. Radar observables to measure ice particles

*Z*

_{HH}and the differential reflectivity

*Z*

_{DR}. Their spectral representations—spectral reflectivity

*sZ*

_{HH}(

*υ*)

*dυ*and spectral differential reflectivity

*sZ*

_{DR}(

*υ*)

*dυ*—are given in Eqs. (9) and (10), respectively. Precipitation above the melting layer consists of multiple particle types, and the radar observables are therefore given by a summation over the

*n*types present in the radar volume: where the subscripts HH and VV denote, respectively, horizontal and vertical transmitting and receiving polarization modes of the radar;

*i*represents the particle type;

*N*(

*D*) is the particle size distribution; the Doppler velocity

*υ*is related to the terminal fall velocity; and

*σ*is the radar cross section.

With the given description of the properties of ice particles, the radar cross section of the different types of particles can be calculated. At S-band radar frequencies, the radar cross section of spheroidal hydrometeors can be determined using the Rayleigh scattering theory (Russchenberg 1992; Bringi and Chandrasekar 2001). With this theory, the radar cross sections of dendrites, plates, aggregates, and hail can be derived. The radar cross section of conical-shaped graupel can be calculated using the T-matrix method (Mishchenko et al. 2000). The radar cross section of the different types of ice hydrometeors is plotted versus velocity in Fig. 3.

Equation (9) represents the classical power Doppler spectrum measured with the horizontal polarization. The sum of *sZ*_{HH}(*υ*)*dυ* over all of the Doppler velocities gives the commonly used reflectivity value. Similarly, the spectral reflectivity *sZ*_{VV}(*υ*)*dυ* is measured. The ratio of these two power Doppler spectra measured with two different polarizations leads to the spectral differential reflectivity, given in (10), that represents the function of differential reflectivity versus Doppler velocities. It gives the possibility to separate a small particle type from a large particle type based on their different polarimetric properties. Spectral polarimetric parameters are defined in Unal and Moisseev (2004).

### a. Bulk observables of precipitation

Several bulk parameters are defined in this section. Their expressions are simple and assume a spherical shape of the ice particles, which is described by the equivolumetric diameter. These integral parameters will be used to test the retrieval algorithm. To obtain the total integral parameter from the retrieved particle size distributions, a summation is done over the bulk parameters obtained for the different types of ice hydrometeors.

#### 1) Equivalent reflectivity

*K*is related to the relative permittivity of ice and

*K*corresponds to the relative permittivity of water. Here

_{r}*Z*,

_{e}*D*, and the particle size distribution

*N*(

*D*) are expressed, respectively, in mm

^{6}m

^{−3}, mm, and mm

^{−1}m

^{−3}.

#### 2) Ice water content

^{−3}), and

*m*the mass of the particle (g). The mass–diameter relation is given by (6).

#### 3) Number of particles

^{−3}), or total number of particles, is

### b. Sensitivity of radar observables to particle types

The proposed retrieval algorithm, which will be discussed in the next section, can retrieve the microphysical properties of two types of ice particles. Prior to the retrieval algorithm, a selection of types of particles, based on environmental conditions (measured using radiosondes), is carried out to reduce the number of types of ice particles. If necessary, a second step in the selection is performed to choose the types of measurable particles.

A slant measurement of ice precipitation above the melting layer is considered in this paper (see section 5) to illustrate the Doppler polarimetric retrieval algorithm. From section 3b, the selection of the types of ice hydrometeors, the model simulations, and the retrieval algorithm are related to this study case.

The selection of types of ice particles is carried out as follows. One type of ice particles, that is, the columns, is discarded because of the temperature value (radiosonde measurement). When the radar is pointing to the zenith, large fall velocities are not present, which is discarding hail and graupel (environmental condition). The possible types of particles are then the aggregates, the plates, and the dendrites. In case of a population of plates and dendrites with similar concentration, the radar observables are dominated by plates. The radar cross section of dendrites is smaller than the radar cross section of plates (Fig. 3). The plates are then predominant in the region of small Doppler velocities of the radar observables. When dendrites and plates have similar radar cross sections (small diameter), the other microphysical properties (e.g., axis ratio, density) are similar, and then there is no possibility to differentiate between them. This is different for aggregates and plates because, in that case, their axis ratios (Fig. 1) and thus their shapes (10) significantly differ and are related to different intervals of Doppler velocities. Therefore, the two particle types kept for modeling are aggregates and plates. The remaining type of ice particles, the dendrites, which is probably present in the radar volume, is eliminated (not measurable).

## 4. Retrieval of microphysical parameters

An algorithm is developed here that extracts microphysical properties of plates and aggregates from spectral radar measurements above the melting layer in stratiform precipitation. The model described above is extended to a model that produces spectral equivalent reflectivity and spectral differential reflectivity, using Eqs. (9) and (10). A schematic of the model is given in Fig. 4. The simulated spectra are fitted to the measurements using a nonlinear least squares optimization.

A form of particle size distribution (PSD) has to be selected. The gamma distribution and the exponential distribution are commonly used in the literature. Here, we choose the exponential distribution. Two parameters in the exponential distribution have to be retrieved instead of three parameters for the gamma distribution. The influence of this choice of particle size distribution is discussed in section 5b.

*N*the intercept parameter (mm

_{w}^{−1}m

^{−3}) and

*D*

_{0}the median volume diameter (mm).

This leads to two parameters to be retrieved for the aggregates (*N*^{agg}_{w}, *D*^{agg}_{0}) and two parameters for the plates (*N*^{pla}_{w}, *D*^{pla}_{0}). To obtain a realistic model of spectral radar observables, two extra parameters, the spectral broadening *σ*_{0} and the ambient wind velocity *υ*_{0}, are added to the model.

*σ*

_{0}the width of the broadening spectrum (m s

^{−1}).

*υ*

_{fall}the terminal fall velocity of the particle,

*α*the radar elevation angle, and

*υ*

_{0}the ambient wind velocity. The term

*υ*

_{0}consists of the horizontal wind velocity and the vertical wind velocity projected on the looking direction of the radar. The air densities

*ρ*and

_{a}*ρ*are taken at sea level and at the altitude of the observed radar volume, respectively.

### a. Dependence on PSD parameters of plates and aggregates

Before the parameters can be derived from spectral radar measurements, it is necessary to verify if the parameters have a significant effect on the two spectral radar observables. If a change in one parameter has no effect on the spectral observables, that parameter cannot be determined correctly. By changing the six parameters of the model one by one, while keeping the other five parameters constant, a good insight is obtained on the dependence of the spectral radar observables on the different parameters of the model.

In Figs. 5 –9, the plots are shown for changing the different parameters of the spectral model over a realistic range. The plot on the dependence of the spectral observables on the ambient wind *υ*_{0} is omitted because the ambient wind only generates a shift of the spectrum in velocity. For Figs. 5 –9, *υ*_{0} equals 0 m s^{−1} and the measurement convention, the terminal fall velocities are negative, is chosen for the Doppler velocities. Based on the results in Figs. 5 –9, the following conclusions are made:

- An increase in
*N*for aggregates leads to an increase in_{w}*sZ*_{HH}and a decrease in*sZ*_{DR}. The more aggregates there are, the more the total*sZ*_{DR}tends to the spectral differential reflectivity of aggregates, which is close to 0 dB, because of the near spherical shape of aggregates (see Fig. 1). - An increase in
*D*_{0}for aggregates leads to an increase and a wider spectrum for*sZ*_{HH};*sZ*_{DR}decreases for the same reason as explained for an increase in*N*, the contribution of aggregates to the total spectrum thus becomes dominant._{w} - An increase of
*N*for plates slightly affects the observed_{w}*sZ*_{HH}. This is due to the fact that the radar cross section for plates is smaller with respect to the cross section of aggregates (see Fig. 3). On the other hand,*sZ*_{DR}increases with increasing*N*. This is resulting from the high_{w}*sZ*_{DR}for plates due to their oblate shape (see Fig. 1). With an increasing concentration of plates, the observed spectral differential reflectivity tends toward the spectral differential reflectivity of plates in the region of small radial velocities. - An increase in
*D*_{0}generates a similar effect as an increase of*N*for plates. Note that the scale in decibels for the spectral reflectivity is large (60 dB), which makes small changes in_{w}*sZ*_{HH}not visible when the particle size distribution of plates is significantly changed. With*sZ*_{DR}we are looking at small variations of the spectra*sZ*_{HH}and*sZ*_{VV}. The scale of*sZ*_{DR}is only 1.5 dB. - The effect of spectral broadening on the horizontal spectral reflectivity is that the maximum of the spectrum becomes lower and the spectrum becomes wider and more symmetric as well. Next to that, an increase of spectral broadening flattens out the spectral differential reflectivity.

Summarizing, the spectral reflectivity depends strongly on the particle size distribution of aggregates, the spectral broadening factor, and the ambient wind velocity. The spectral reflectivity depends to a lesser extent on the particle size distribution of plates. The spectral differential reflectivity depends on all six parameters. All six parameters have a significant effect on the spectral radar observables, which makes their retrieval possible.

### b. Dependence of spectral observables on the model relationships

Parameters are used to express the microphysical properties of plates and aggregates in function of their size using a power law. The values of these parameters given in Tables 2 –5 may deviate from the actual values encountered in the measurement. In this section, the sensitivity of the spectral horizontal reflectivity and the spectral differential reflectivity with respect to the used parameters is discussed.

The values of the parameters are changed by plus and minus 10% in the sensitivity study. The spectra are simulated using the following example of particle size distribution: *D*^{agg}_{0} = 3.77 mm, *N*^{agg}_{w} = 2500 mm^{−1} m^{−3}, *D*^{pla}_{0} = 0.477 mm, *N*^{pla}_{w} = 5800 mm^{−1} m^{−3}, and *σ*_{0} = *υ*_{0} = 0 m s^{−1}. Based upon the obtained spectra for horizontal and differential reflectivity, conclusions are drawn on the sensitivity.

#### 1) Sensitivity of radar observables to the mass– and area–diameter relationships

With respect to the magnitude of the changes in the spectral radar observables, the modeled spectra of horizontal and differential reflectivity are most sensitive to the parameters of the mass and area relationships. A change in the mass and area parameters has an effect on the obtained fall velocities. An increase of the mass of particles (*α* larger and the exponent *β* smaller) and a decrease in the area of the particles (*γ* smaller and the exponent *σ* larger) both lead to an increase of fall velocities. An increase in velocity for aggregates and a decrease in velocity for plates will result in broadening of the spectral horizontal reflectivity. This change in velocity also results in different velocity regions for plates and aggregates, enhancing the possibility to separate the retrievals of plates and aggregates using the spectral differential reflectivity. In that case, the range of *sZ*_{DR} values is increased.

#### 2) Sensitivity of radar observables to the density– and shape–diameter relationships

The change in the density–diameter relation for plates and aggregates only has a significant effect on the spectral differential reflectivity. The magnitude of this effect is comparable to the effect on a change in the shape–diameter relation of aggregates. Here *sZ*_{DR} varies slightly with the shape–diameter relation of plates. A change in the shape–diameter relation or in the density–diameter relation results in a variation in the radar cross section of plates and aggregates. Then the ratio of the contribution of the spectral differential reflectivity of plates and aggregates to the total observed spectral differential reflectivity changes.

#### 3) Sensitivity of radar observables to the shape parameter of the PSD

The influence of the shape parameter *μ* in the particle size distribution on the modeled *sZ*_{HH} and *sZ*_{DR} is also examined. The values of *μ* are the same for plates and aggregates and equal 0, 2, 4, 6, and 8. An increase in *μ* results in a narrower spectral horizontal reflectivity and strongly affects the spectral differential reflectivity, which increases with the shape parameter. Narrower spectra for plates and aggregates result in a better separation of the contribution of both plates and aggregates in the spectral differential reflectivity.

### c. Retrieval of the particle size distributions, the spectral broadening, and the ambient wind velocity

*D*

^{pla}

_{0},

*D*

^{agg}

_{0},

*σ*

_{0}, and

*υ*

_{0}have to be known.

A second simplification of Eqs. (19) and (20) is done on the estimation of the ambient wind velocity *υ*_{0}. The ambient wind velocity creates a shift of the measured spectral horizontal reflectivity with respect to the modeled one. Assuming the other five parameters of the model are known, the shift between the modeled and the measured spectrum can be obtained by determining the lag of the cross correlation of the measured *sZ*^{meas}_{HH}(*υ*)*d**υ* and the modeled spectrum *sZ*^{mod}_{HH}(*υ*, *N*^{agg}_{w}, *D*^{agg}_{0}, *N*^{pla}_{w}, *D*^{pla}_{0}, *σ*_{0})*d**υ*.

The separate estimation of the intercept parameters and the ambient wind velocity results in a three-parameter (*D*^{pla}_{0}, *D*^{agg}_{0}, and *σ*_{0)} nonlinear least squares problem. Because of multiple minima in the cost functions, an iterative cascaded retrieval algorithm is preferred to obtain them. This algorithm is described in the appendix and illustrated in Fig. 10.

### d. Quality of retrieval technique

To get insight in the quality of the optimization procedure, the optimization is applied on simulated Doppler spectra. By comparing the input parameters used to create a simulated spectrum with the parameters obtained with the retrieval algorithm, conclusions can be drawn on their errors.

The simulated spectra are created using Eqs. (17) and (18). To generate signals with real statistical properties, noise is added according to Chandrasekar et al. (1986). The values of the parameters are selected randomly from the depicted intervals, given in Table 6. In addition to the constraints on the input parameters of the model, the retrieval algorithm is only applied on the simulated spectra when the spectral horizontal reflectivity *sZ*_{HH}(*υ*)*d**υ* exceeds −10 dB and the maximum spectral differential reflectivity exceeds 0.5 dB. The first threshold is to ensure that the spectrum has a sufficient signal-to-noise ratio to perform the optimization, and the second threshold is to ensure that the amount of plates is detectable.

It has been shown by simulation that the error on the median volume diameter of aggregates and plates increases rapidly for values of the spectral differential reflectivity maximum smaller than 0.5 dB.

The root-mean-square error of each parameter is given in Table 7. The error on both intercept parameters is large. Because of the layered structure of the retrieval algorithm, an error on the median volume diameter will be corrected by the estimated value of the intercept parameter to obtain the correct spectral reflectivity. Because the reflectivity is related to the sixth moment of the diameter and proportional to the intercept parameter, the error on the median volume diameter will have a large effect on the intercept parameter.

The same exercise is carried out on integral parameters, the equivalent reflectivity, the IWC, and the number of particles. The same dataset is used to obtain the errors on the particle size distributions and on the integral parameters. The results are given in Table 8. The root-mean-square error of the equivalent reflectivity is very small because the errors on the median volume diameter and intercept parameter cancel out in the estimate of the reflectivity.

The root-mean-square errors in Tables 7 –8 are estimated errors and do not include the uncertainties of the microphysical relationships of the model.

## 5. Application to radar data

The developed retrieval technique is applied to real radar measurements. The data is collected by the radar TARA (Heijnen et al. 2000) during a moderate stratiform rain event in Cabauw, Netherlands (Russchenberg et al. 2005). The reflectivity values of rain vary between 20 and 35 dB*Z*. The elevation angle of the radar is 45°. A sequence of five measurements (VV, HV, HH, and two offset beams) is collected in a data block of 5 ms. The Doppler spectrum is calculated from a time series of 512 samples (2.56 s). Ten Doppler spectra are averaged to obtain the Doppler spectrum that will be input for the inversion algorithm (25.6 s). The range and the Doppler resolution are, respectively, 15 m and 1.8 cm s^{−1}. With this high Doppler resolution, the Doppler spectrum of precipitating cloud consists of a large number of points.

Examples of measured spectral horizontal reflectivity and spectral differential reflectivity are shown in Figs. 11 –12. They are measured above the melting layer. A target approaching the radar has a Doppler velocity negative (definition). The positive Doppler velocities indicate the presence of the ambient wind velocity.

### a. Retrieval algorithm results

The Doppler spectra, used as input for the retrieval algorithm, are selected above the top of the melting layer (2000 m). Next, the constraints on the values of the spectral reflectivity and spectral differential reflectivity, which are considered in the simulation (section 4d), are applied. The chosen spectra are clipped at 10 dB below the maximum value of the spectral horizontal reflectivity. This clipping provides the interval of Doppler velocities (*υ*_{min}, *υ*_{max}). For spectral reflectivity values below the clipping level, the spectral differential reflectivity is severely affected by noise. Examples at two neighbor heights of the spectral horizontal reflectivity and spectral differential reflectivity data with their obtained fits as well as the obtained six parameters are given in Figs. 13 and 14.

Regarding the obtained values of the particle size distributions as well as the retrieved values of spectral broadening and ambient wind velocity, the outputs of the inversion algorithm for plates and aggregates are consistent for small variations in height and time.

### b. Influence of the shape parameter on the retrieval of the PSD

The most common PSD used in the literature is the gamma distribution, which consists of three parameters: *N _{w}*,

*D*

_{0}, and the shape parameter

*μ*. The exponential distribution used for the model is a simplification of the gamma distribution (

*μ*= 0). There is no good reason to fix the shape parameter to zero rather than some other value. The assumption of an exponential distribution was necessary to reduce the number of free variables in the model of spectral radar observables of plates and aggregates. To investigate the dependence of the output of the inversion algorithm on the value of the shape parameter, the particle size distribution parameters of plates and aggregates are also obtained from the radar dataset by fixing the values of the shape parameter to two, four, and six. The PSD now follows a gamma distribution and the same shape parameter is used for plates and aggregates.

The median volume diameters obtained for both aggregates and plates vary with the chosen value of the shape parameter. The maximum variation in the obtained value of the median volume diameter is 1.4 and 0.09 mm for aggregates and plates, respectively. The obtained differences are significantly larger than the root-mean-square errors: 0.6 mm for aggregates and 0.067 mm for plates (Table 7). Therefore, as expected, a change in the shape parameter leads to a change of the retrieved median volume diameter for both plates and aggregates. The intercept parameter will also vary with the value of the shape parameter. Therefore, the particle size distribution retrievals depend on the value of the shape parameter.

The same exercise is carried out on the ice water content and the number of particles; see Figs. 15 and 16. The maximum difference in the retrieved ice water content as a function of the shape parameter is close to 0.04 g m^{−3}, which is the root-mean-square error on the ice water content (Table 8). The IWC is then not significantly dependent on the chosen value of the shape parameter of the PSD of plates and aggregates. Concerning the number of particles, the maximum difference, 4000 m^{−3}, is larger than the root-mean-square error given in Table 8 (1940 m^{−3}).

## 6. Conclusions

The combination of dual-polarization and Doppler spectral analysis can be used to discriminate between different types of ice particles present in a radar observation volume. The potentiality of this technique is discussed and illustrated on a slant measurement of ice precipitation above the melting layer. For this measurement, separation is performed between two types of ice particles: plates and aggregates. A monomodal Doppler spectrum and the differential reflectivity for each class of Doppler velocity (spectral differential reflectivity) make this separation possible. The presence of aggregates influences the spectral differential reflectivity in the large Doppler velocities area. The plates, when their concentration is significant, influence the spectral differential reflectivity in the small Doppler velocities area.

A microphysical model is developed for dendrites, plates, aggregates, graupel, and hail. Columns will be included in the near future. Its output is the spectral reflectivity and the spectral differential reflectivity. This model of spectral radar observables depends on six parameters: the particle size distribution parameters of two types of ice particles, the spectral broadening, and the ambient wind velocity. Because only two types of ice hydrometeors can be retrieved, a selection of the two dominant types of ice particles is carried out based on environmental conditions. For the considered measurement of ice precipitation, dendrites, plates, and aggregates are most likely present in the radar resolution volumes. The type *dendrites* was eliminated because of its low radar cross section and similar polarimetric properties compared to plates.

The retrieval algorithm uses a nonlinear least squares approach to fit the modeled spectral radar observables to spectral radar measurements, by varying the six input parameters. We have shown that the six-parameter minimization problem can be reduced to a three-parameter minimization problem. The three-parameter minimization problem is solved by an iterative cascaded algorithm to avoid multiple minima problems.

The error analysis of the proposed retrieval algorithm is carried out on radar signal simulations. The relative root-mean-square errors of the median volume diameter for plates and aggregates are 15% and 17%, respectively. The retrieval of the intercept parameter of the particle size distribution is far less accurate. For those root-mean-square errors, the uncertainties on the microphysical relations of the model are not taken into account. Simulations have shown that, among the diverse microphysical relations, the parameters of the mass–diameter and area–diameter relationships affect the most the modeled spectra of horizontal reflectivity and differential reflectivity. The shape parameter of the particle size distribution also significantly affects the modeled spectra. The retrieved results of the particle size distributions of plates and aggregates depend on the choice of the shape parameter. First results indicate that the retrieved ice water content depends weakly on the shape parameter.

The retrieval algorithm is finally applied to radar data observations of stratiform precipitation. The outputs of the inversion algorithm for plates and aggregates are consistent for small variations in height and time. Based on the retrieved particle size distributions, time-dependent profiles of ice water content can be estimated: one for plates and one for aggregates. The retrieved ice water contents and the particle size distributions require validation with in situ data in dedicated experiments. The first experiment took place during the Convective and Orographically-induced Precipitation Study (COPS) campaign in the summer of 2007.

## Acknowledgments

The authors from the Delft University of Technology acknowledge the support of Climate for Spatial Planning and Earth Research Centre Delft. D. Moisseev and V. Chandrasekar were supported by the National Science Foundation (ATM-0313881).

## REFERENCES

Auer, A. H., , and Veal D. L. , 1970: The dimension of ice crystals in natural clouds.

,*J. Atmos. Sci.***27****,**919–926.Bringi, V. N., , and Chandrasekar V. , 2001:

*Polarimetric Doppler Weather Radar: Principles and Applications*. Cambridge University Press, 636 pp.Chandrasekar, V., , Bringi V. N. , , and Brockwell P. J. , 1986: Statistical properties of dual-polarized radar signals. Preprints,

*23rd Conf. on Radar Meteorology,*Snowmass, CO, Amer. Meteor. Soc., 193–196.Doviak, R. J., , and Zrnic D. S. , 1993:

*Doppler Radar and Weather Observations*. Academic Press, 562 pp.El-Magd, A., , Chandrasekar V. , , Bringi V. N. , , and Strapp W. , 2000: Multiparameter radar and in situ aircraft observation of graupel and hail.

,*IEEE Trans. Geosci. Remote Sens.***38****,**570–578.Fabry, F., , and Szyrmer W. , 1999: Modeling of the melting layer. Part II: Electromagnetic.

,*J. Atmos. Sci.***56****,**3593–3600.Fukuta, N., , and Takahashi T. , 1999: The growth of atmospheric ice crystals: A summary of findings in vertical supercooled cloud tunnel studies.

,*J. Atmos. Sci.***56****,**1963–1979.Heijnen, S. H., , Lighthart L. P. , , and Russchenberg H. W. J. , 2000: First measurements with TARA; An S-band transportable atmospheric radar.

,*Phys. Chem. Earth***25****,**995–998.Heymsfield, A. J., 1978: The characteristics of graupel particles in northeastern Colorado cumulus congestus clouds.

,*J. Atmos. Sci.***35****,**284–295.Heymsfield, A. J., , and Kajikawa M. , 1987: An improved approach to calculating terminal velocities of plate-like crystals and graupel.

,*J. Atmos. Sci.***44****,**1088–1099.Khvorostyanov, V. I., , and Curry J. A. , 2002: Terminal velocities of droplets and crystals: Power laws with continuous parameters over the size spectrum.

,*J. Atmos. Sci.***59****,**1872–1884.Langleben, M. P., 1954: The terminal velocity of snowflakes.

,*Quart. J. Roy. Meteor. Soc.***80****,**174–181.Liao, L., , Meneghini R. , , Iguchi T. , , and Detwiler A. , 2005: Use of dual-wavelength radar for snow parameter estimates.

,*J. Atmos. Oceanic Technol.***22****,**1494–1506.Locatelli, J. D., , and Hobbs P. V. , 1974: Fall speeds and masses of solid precipitation particles.

,*J. Geophys. Res.***79****,**2185–2197.Magono, C., , and Lee C. W. , 1966: Meteorological classification of natural snow crystals.

,*J. Fac. Sci., Hokkaido Univ.***2****,**321–335.Matrosov, S. Y., 1998: A dual-wavelength radar method to measure snowfall rate.

,*J. Appl. Meteor.***37****,**1510–1521.Matrosov, S. Y., 1992: Radar reflectivity in snowfall.

,*IEEE Trans. Geosci. Remote Sens.***30****,**454–461.Matrosov, S. Y., , Reinking R. F. , , Kropfli R. A. , , and Bartram B. W. , 1996: Estimation of ice hydrometeor types and shapes from radar polarization measurements.

,*J. Atmos. Oceanic Technol.***13****,**85–96.Mishchenko, M. I., , Hovenier J. W. , , and Travis L. D. , 2000:

*: Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications*. Academic Press, 690 pp.Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities.

,*J. Atmos. Sci.***53****,**1710–1723.Mitchell, D. L., , Zhang R. , , and Pitter R. L. , 1990: Mass-dimensional relationships for ice particles and the influence of riming on snowfall rates.

,*J. Appl. Meteor.***29****,**153–163.Moisseev, D. N., , Unal C. M. H. , , Russchenberg H. W. J. , , and Chandrasekar V. , 2004: Radar observations of snow above the melting layer. Preprints,

*Third European Conf. on Radar Meteorology and Hydrology (ERAD),*Visby, Sweden, 407–411.Pruppacher, H. R., , and Klett J. D. , 1978:

*Microphysics of Clouds and Precipitation*. Reidel, 954 pp.Rajopadhyaya, D. K., , May P. T. , , and Vincent R. A. , 1994: The retrieval of ice particle size information from VHF wind profiler Doppler spectra.

,*J. Atmos. Oceanic Technol.***11****,**1559–1568.Ray, P. S., 1972: Broadband complex refractive indices of ice and water.

,*Appl. Opt.***11****,**1836–1844.Russchenberg, H. W. J., 1992:

*Ground Based Remote Sensing of Precipitation Using Multi-Polarized FM-CW Doppler Radar*. Delft University Press, 206 pp.Russchenberg, H. W. J., and Coauthors, 2005: Ground-based atmospheric remote sensing in the Netherlands: European outlook.

,*IEICE Trans. Commun.***88****,**2252–2258.Rust, B. W., 2002: Fitting nature’s basic functions. Part III: Exponentials, sinusoids, and nonlinear least squares.

,*Comput. Sci. Eng.***4****,**72–77.Rust, B. W., 2003: Fitting nature’s basic functions. Part IV: The variable projection algorithm.

,*Comput. Sci. Eng.***5****,**74–79.Sekhon, R. S., , and Srivastava R. C. , 1970: Snow size spectra and radar reflectivity.

,*J. Atmos. Sci.***27****,**299–307.Smith, P. L., 1984: Equivalent radar reflectivity factors for snow and ice particles.

,*J. Climate Appl. Meteor.***23****,**1258–1260.Unal, C. M. H., , and Moisseev D. N. , 2004: Combined Doppler and polarimetric radar measurements: Correction for spectrum aliasing and nonsimultaneous polarimetric measurements.

,*J. Atmos. Oceanic Technol.***21****,**443–456.Wang, P. K., 1982: Mathematical description of the shape of conical hydrometeors.

,*J. Atmos. Sci.***39****,**2615–2622.

## APPENDIX

### Iterative Cascaded Retrieval Algorithm

A three-parameter nonlinear least squares optimization is carried out on the median volume diameters *D*^{pla}_{0}, *D*^{agg}_{0}, and the spectral broadening *σ*_{0} using a cascaded approach. The iterative optimization procedure is divided in five stages (see Fig. 10).

#### Iterative selection procedure (step 1)

The values of *D*^{pla}_{0}, *D*^{agg}_{0}, and *σ*_{0} are selected for consecutive optimizations. The values of *D*^{pla}_{0}, *D*^{agg}_{0}, and *σ*_{0} are bounded. The parameter *D*^{pla}_{0} varies from 0.01 to 0.6 mm (*N _{p}* values),

*D*

^{agg}

_{0}from 0.5 to 6 mm (

*N*values), and

_{a}*σ*

_{0}from 0 to 1 m s

^{−1}(

*N*values). For the start, the first values of

_{σ}*D*

^{pla}

_{0}(0.01 mm),

*D*

^{agg}

_{0}(0.5 mm), and

*σ*

_{0}(0 m s

^{−1}) are chosen.

#### Estimation procedure (step 2)

Estimation of the intercept parameters *N*^{agg}_{w} and *N*^{pla}_{w} using the linear optimization (21) and (22) on *sZ*_{DR}|_{D0agg,D0pla,σ0} and *sZ*_{HH}|_{D0agg,D0pla,σ0} with *υ*_{0} (0 m s^{−1}).

Next, the ambient wind velocity *υ _{0}* is calculated as the lag of the cross correlation of the measured

*sZ*

^{meas}

_{HH}(

*υ*)

*d*

*υ*and the modeled spectrum

*sZ*

^{mod}

_{HH}(

*υ*,

*N*

^{agg}

_{w},

*D*

^{agg}

_{0},

*N*

^{pla}

_{w},

*D*

^{pla}

_{0},

*σ*

_{0})

*d*

*υ*.

#### Output of the model (step 3)

A set of six parameters is obtained with the corresponding modeled spectra.

#### Optimization procedure from cost function study (step 4)

##### 1) Optimization on *σ*_{0}

*N*times for each value of

_{σ}*σ*

_{0}. Here

*D*

^{pla}

_{0}and

*D*

^{agg}

_{0}are fixed. We obtain thus

*N*values of

_{σ}*N*

^{agg}

_{w},

*N*

^{pla}

_{w}, and

*υ*

_{0}, which depend on

*σ*

_{0}. The minimization of

*L*(

*σ*

_{0}) is carried out using the obtained

*N*values of

_{σ}*N*

^{agg}

_{w},

*N*

^{pla}

_{w}, and

*υ*

_{0}. The result is

*σ*

^{opt}

_{0},

*N*

^{agg}

_{w}(

*σ*

^{opt}

_{0}),

*N*

^{pla}

_{w}(

*σ*

^{opt}

_{0}), and

*υ*

_{0}(

*σ*

^{opt}

_{0}) for fixed

*D*

^{pla}

_{0}and

*D*

^{agg}

_{0}.

##### Optimization on D_{0}^{agg}

*D*

^{agg}

_{0}. Here

*D*

^{pla}

_{0}is fixed. We obtain thus a set of

*N*values of

_{a}*σ*

^{opt}

_{0}(

*D*

^{agg}

_{0}),

*N*

^{agg}

_{w}(

*σ*

^{opt}

_{0}),

*N*

^{pla}

_{w}(

*σ*

^{opt}

_{0}), and

*υ*

_{0}(

*σ*

^{opt}

_{0}). The cost function is minimized using the set of

*N*values. The result is

_{a}*D*

^{agg}

_{0 opt},

*σ*

^{opt}

_{0}(

*D*

^{agg}

_{0 opt}),

*N*

^{agg}

_{w}(

*σ*

^{opt}

_{0}),

*N*

^{pla}

_{w}(

*σ*

^{opt}

_{0}), and

*υ*

_{0}(

*σ*

^{opt}

_{0}) for fixed

*D*

^{pla}

_{0}.

##### Optimization on D_{0}^{pla}

*D*

^{pla}

_{0}. We obtain thus a set of

*N*values of

_{p}*D*

^{agg}

_{0 opt}(

*D*

^{pla}

_{0})

*σ*

^{opt}

_{0}(

*D*

^{agg}

_{0 opt}),

*N*

^{agg}

_{w}(

*σ*

^{opt}

_{0}),

*N*

^{pla}

_{w}(

*σ*

^{opt}

_{0}), and

*υ*

_{0}(

*σ*

^{opt}

_{0}). The cost function is minimized using the set of

*N*values.

_{p}The solution of the cascaded retrieval algorithm is *D*^{pla}_{0 opt}, *D*^{agg}_{0 opt} (*D*^{pla}_{0 opt}), *σ*^{opt}_{0} (*D*^{agg}_{0 opt}), *N*^{agg}_{w} (*σ*^{opt}_{0}), *N*^{pla}_{w} (*σ*^{opt}_{0}), and *υ*_{0} (*σ*^{opt}_{0}).

#### Final outcome (step 5)

The solution consisting of the optimized six parameters, and the final fit between measured and modeled radar observables are obtained.

Types and typical sizes of ice particles. The given diameter denotes the maximum particle dimension of the ice particles.

Parameters of the shape–diameter relation for different ice crystals given by (1). Data are taken from Matrosov et al. (1996), Auer and Veal (1970), and Bringi and Chandrasekar (2001).

Parameters of the density–diameter relation for different ice crystals given by (2). Data are taken from Pruppacher and Klett (1978), El-Magd et al. (2000), and Fabry and Szyrmer (1999).

Parameters of the mass–diameter relation for different ice crystals given by (6). Data are taken from Mitchell (1996) and Heymsfield and Kajikawa (1987).

Parameters of the area–diameter relation for different ice crystals given by (7). Data are taken from Mitchell (1996) and Heymsfield and Kajikawa (1987).

Regions of variables.

The rms errors of the six retrieved parameters.

The rms errors of the integral parameters.