## Introduction

The estimation of radar reflectivity from measured snow particle size spectra is a difficult task. This information, however, is of great interest for the interpretation of the results of conventional weather radars in winter, especially in alpine regions, where hydrometeors in the pulse volume usually consist of snow. In this note, a method is presented based on the works of Marshall and Gunn (1952) and Smith (1984). Rain and snow size spectra measured with the Particle Size and Velocity (PARSIVEL) optical disdrometer are used to compare estimated radar reflectivities with data from a C-band Doppler radar.

## Measuring system

The PARSIVEL disdrometer consists of an optical sensor that produces a horizontal sheet of light (30 mm wide, 1 mm high, 180 mm long). This is done by a 780-nm laser diode with a power of 3 mW. The light sheet is focused on a single photo diode in the receiver. Both the transmitter and the receiver are mounted in tunnel-like housings for protection. In the absence of particles, the receiver produces a 5-V signal from the sensor output. Particles passing through the light sheet cause a short decrease of the output voltage (Δ*U*) by light extinction. The Δ*U* depends linearly on the fraction of light sheet blocked by the particle. The amplitude of the voltage signal induced is a measure of particle size, and the duration of the signal (Δ*t*) allows for an estimate of the particle velocity. With this information, an automatic classification of the precipitation type can be made.

In the case of snow, the particles are assumed to be spherical, with a diameter *D*_{k} corresponding to the maximum width of the blocked area. The fall velocity is calculated from *υ* = (*D*_{k} + 1 mm)/Δ*t* (where 1 mm is the height of the laser beam). Obviously, this assumption is not true for real irregularly shaped snowflakes, but because of the wide variety in shape and density as well as the lack of proper empirical or theoretical relations, this assumption is currently used as a primary estimation. The influence of the irregular shapes can be reduced by measuring size and velocity distributions over long averaging times rather than looking at each individual particle. An averaging time of 5 min was chosen in this study.

In the case of water drops, a relation between *U* amplitude and volume equivalent diameter *D*_{V}, which was obtained empirically in a fall-tunnel experiment, is used to estimate *D*_{V}. From this information, the velocity is calculated using the empirical values measured by Gunn and Kinzer (1949). This calibration is necessary because rain drops with *D*_{V} larger than 1 mm are mainly oblate, not spherical. Therefore, the use of *υ* = (*D*_{k} + 1 mm)/Δ*t* would overestimate the fall velocity. Here, *D*_{k} is the measured horizontal diameter.

PARSIVEL measures size and velocity distributions and also rain rate, radar reflectivity, and other secondary information. For a detailed description of the instrument, see Löffler-Mang and Joss (2000).

## Radar reflectivity in the case of rain

*η,*which is the total backscatter cross section of all scatterers divided by the pulse volume

*V*

_{P}. If the Rayleigh approximation is valid and the individual particles are assumed to be spherical with a diameter

*D*

_{k},

*η*can be written as

*σ*is the individual backscatter cross section of the

_{k}*k*th particle,

*λ*is the wavelength and |

*K*|

^{2}= |(ε

_{r}− 1)/(ε

_{r}+ 2)|

^{2}is the dielectric factor (ε

_{r}=

*n*

^{2}, ε

_{r}is relative permittivity, and

*n*is complex index of refraction). The equivalent radar reflectivity factor

*Z*

_{e}is defined as

*K*

_{w}|

^{2}is the dielectric factor for water, indicating that the scatterers are expected to be water spheres.

*K*in (1) is identified by

*K*

_{w}. Therefore, the estimate of the radar reflectivity factor from measured drop size distributions (

*Z*

_{M}) can be expressed as the sixth moment of the drop size distribution with respect to the volume equivalent diameter,

*N*(

*D*

_{V}):

*D*

_{V}larger than 1 mm are oblate rather than spherical, the backscatter cross sections of these particles differ from those of spheres with the same volume. The difference depends on the departure from the spherical shape and the spatial distribution of their orientation. The error in estimating

*Z*

_{e}with (3) (when assuming the drops are spherical, with a diameter equal to

*D*

_{V}) is not considered to be a predominant source of error in comparison with other uncertainties when comparing radar-measured

*Z*

_{e}to estimates from ground measurements of drop size distributions. Therefore, (3) can be accepted to be a good estimate of

*Z*

_{e}.

*n*

_{i}is the number of measured drops in class

*i*during time

*t, D*

_{i}is the mean diameter in class

*i, F*is the measuring area, and

*υ*

_{i}is the mean velocity of drops in class

*i.*The denominator

*tFυ*

_{i}is necessary because the drops are counted by area and time and have to be transferred to a volume distribution. Here,

*t*was chosen to be 30 s.

Reflectivity data (dB*Z*) are presented for the early hours of 6 May 1999 (Figs. 1a,b) derived from (i) drop size distributions obtained by the PARSIVEL optical disdrometer and (ii) radar data. PARSIVEL was mounted on a platform in the Forschungszentrum Karlsruhe, 20 m to the side and 15 m below the antenna of the C-band-radar (*λ* = 5.4 cm). Unfortunately, it was not possible to measure farther away from the radar at a better site for comparison, because PARSIVEL requires a direct personal computer connection to record the data. (The next generation of the instrument will be stand alone.) For a reasonable comparison, the radar signals were averaged over a complete azimuthal scan on the lowest elevation (0.4°) in the nearest range gate (1.5–2 km) using a procedure in which a volume scan was performed every 5 min and then the spatial average and standard deviation of the innermost scan circle in the lowest elevation were calculated. This average was taken as an estimate of the radar-measured, 5-min-averaged reflectivity at the radar site. The spatial standard deviation provides an indication of the uncertainty of that estimate.

From the drop size distribution measured with PARSIVEL, *Z*_{M} was calculated every 30 s with (4) and then averaged over 5 min.

During the 8 hours of measurement, *Z*_{e} (radar) attained values between 5 and 40 dB*Z.* The spatial standard deviation of the radar data (error bars), which also serves as a measure for the spatial homogeneity of the rainfall, ranged from 10 to 40 dB*Z* (minimum at 0030 CEST, maximum at 0615). After 0330 CEST, its level increased from 15 dB*Z* to the maximum and then came to rest at a level of approximately 25 dB*Z.* The compared data agree reasonably well when taking the error bars into account, because most differences between radar means and PARSIVEL estimates range from 0 to ±5 dB*Z* and do not exceed 10 dB*Z.* Note that point measurements are being compared with volume data.

## Method of estimating *Z*_{e} for snow measurements

In this study, all types of ice particles (pristine ice crystals and soft hail as well as large and fragile agglomerates) are referred to as snow. Generally, snow particles are neither spherical nor homogenous in density. Furthermore, all these particles may have smooth, dry, rimed, or wet surfaces. They show a large variety of shapes, densities, and scattering properties.

Smith (1984) presented how to estimate *Z*_{e} for dry snow and ice particles based on previous studies of Marshall and Gunn (1952) and a theory by Debye (1929) for dielectric properties of a homogenous mixture of two dielectrics. In essence, it is assumed that the backscatter cross section of an irregularly shaped ice particle (composed of a weak dielectric such as ice and a nondielectric such as air) is the same as that of an ice sphere of the same mass. This assumption is valid for particles that are (i) small enough so that Rayleigh scattering theory can be applied, (ii) dry, (iii) of nearly homogeneous density, and (iv) not too far away from a spherical shape.

*Z*

_{e}can be estimated from snow size measurements by

*K*

_{i}|

^{2}/|

*K*

_{w}|

^{2}is the dielectric factor for ice/water,

*D*

_{M,k}is the ice mass equivalent diameter of particle

*k,*

*D*

_{k}is the measured diameter of particle

*k,*

*C*

_{K}= |

*K*

_{i}|

^{2}/|

*K*

_{w}|

^{2}, and

*C*

_{M,k}= (

*D*

_{M,k}/

*D*

_{k})

^{6}, where the sum has to be taken over all

*N*particles in

*V*

_{P}.

The correction factor *C*_{K} takes into account that the radar assumes water drops instead of ice particles, and *C*_{M,k} corrects the overestimation when using *D*_{k} instead of *D*_{M,k}.

*C*

_{K}, the appropriate complex refraction indices for ice (

*n*

_{i}) and water (

*n*

_{w}) are required. For

*λ*= 4.7 cm, which is nearly equal to the wavelength of the C-band radar, Herman and Battan (1961) gave

*n*

_{i}= 1.78 − 0.0024

*i*and

*n*

_{w}= 7.95 − 2.202

*i*so that

*D*

^{3}

_{M,k}

*m*

_{k}/

*πρ*

_{i}follows

*ρ*

_{i}= 0.92 kg m

^{−3}is the density of ice and

*m*

_{k}is the mass of particle

*k.*

*Z*

_{M}in snow only differs from (4) in the presence of the two correction factors (6) and (7):

The measured mean fall velocity in the diameter class *i* requires additional discussion. It is obtained by assuming the snow particles are spherical and therefore uses their measured horizontal diameter to calculate the velocity from *υ* = (*D*_{k} + 1 mm)/Δ*t.* The average over all particles in each diameter class is then taken. This leads to an overestimation of the fall velocity, because snow particles settle with a preferred horizontal orientation of their main axis. Because snow particles are not symmetrical, this overestimation is partly reduced. This reduction is because, statistically, the measured diameter is smaller than the maximum horizontal diameter. When the distributions of the geometry and the orientations of the main axis are known, it is possible to calculate an estimation of the magnitude of the overestimation on a statistical basis.

## Mass–size relations

*C*

_{M,k}must be discussed in more detail and requires knowledge of the mass of each particle, which must be estimated from the measured size. Different authors offer mostly empirical relations, such as

*m*

*aD*

^{b}

*a*and

*b*in (9) are determined by the type of snow.

In this study, three different crystal types were chosen from Locatelli and Hobbs (1974): dense lump graupel with *m* = 0.078*D*^{2.8}, very loose rimed dendrites with *m* = 0.037*D*^{1.9}, and graupel-like snow of medium density represented by *m* = 0.059*D*^{2.1} (size *D*: mm and mass *m*: mg). These relations were developed by using the maximum possible diameter that fitted into the particles. With PARSIVEL, the maximum diameter of a projection is measured, which is smaller than or equal to the overall maximum diameter. Depending on the distribution of the orientation and the geometry of the particles, the mass of each measured particle is more or less underestimated. Together with the overestimation of the mean fall velocity in each diameter class, it is expected that *Z*_{e} is underestimated.

In Fig. 2, *C*^{*}_{M}*C*^{*}_{M}_{10}*C*_{M}). It can be seen that lump graupel requires a correction of approximately −20 dB over the whole size range, and for rimed dendrites the value is in the range between −27 and −44 dB. The choice of the mass–size relation has a strong influence on the magnitude of *C*_{M} because it allows differences in the estimated reflectivity of up to 25 dB. This variability must be remembered when discussing the following examples.

## Examples for snow

*C*

_{M,k}:

*D*

_{k}instead of the overall maximum particle diameter in the numerator generally yields an underestimation of the correction factor, as described in the previous section.

Two snow events (4 and 7 December 1998) are shown in Figs. 3a and 3b. The data processing was the same as that used for the rain measurements described in section 3 except that (8) was used instead of (4) in the estimation of radar reflectivity. Both case studies show a reasonably good agreement of the two compared datasets, especially when taking into account the spatial and temporal variability of snow crystal types, which were all treated with the same mass–size relation.

For 4 December 1998, the agreement is excellent (Fig. 3a). The maximum reflectivity of 29 dB*Z* in the radar data was reached at 1800 CET. Between 1750 and 1825 CET (the period of heaviest snow fall), the mean spatial standard deviation *σ*_{s} of the radar data was 6 dB*Z*; during the preceding and following period, it reached up to 23 dB*Z* (1855). The relatively small standard deviations during the period of snowfall suggest that the snowfall was spatially very homogeneous (stratiform). The time series of the PARSIVEL data looks to be shifted by 5 min, probably due to an incorrect clock synchronization. PARSIVEL underestimated the radar data by just a few decibels. This result indicates that the particles were nearly of the graupel-like snow type (in terms of reflectivity) described by the applied mass–size relation. This result is also consistent with the expected general underestimation (see previous section).

On 7 December 1998 (Fig. 3b), the event was less homogeneous (convective; *σ*_{s} up to 40 dB*Z* at 1750), and the snowflakes sometimes may have been less dense than graupel-like, as suggested by a tendency to overestimate the radar-measured reflectivity factors. The variability in the difference of the two instruments (from −6 dB at 1610 to 16 dB at 1915 CET) is due to the natural variability in the snow crystal types and the fact that a volume average is compared with a point measurement.

## Conclusions

A method for the estimation of the radar reflectivity factor from measured snow size spectra has been presented in this work. Some measurements of the PARSIVEL optical disdrometer during one rain and two snow events are compared with the results of a conventional C-band precipitation radar. It is shown that, in the case of snow, the applied mass–size relation has a large influence on the PARSIVEL results, because the correction factor *C*^{*}_{k}

The first snow event was rather homogeneous in space and time (intensity and snow type), and therefore the reflectivity factors obtained by a point measurement and a radar area average are reasonably comparable. With a possible time shift taken into account, PARSIVEL systematically underestimated the radar-measured *Z*_{e} by 3–5 dB. This underestimate could be due to non-graupel-like particles and/or the tendency of PARSlVEL to underestimate *Z*_{e}. The second event was convective, and therefore the comparison is more difficult. Most of the time, the difference shows an overestimation (up to 16 dB) by PARSIVEL. However, sometimes an underestimation of as much as −6 dB was observed. In addition to the problems in the comparison from the strong spatial and temporal inhomogeneity of the event, there may have been temporal variations of the snow type with the presence of snow particles for which the model is not valid (see section 4). These, however, are case studies;to prove the abilities of the method, more snowfalls must be examined.

It is hoped that the fall velocity information of the snowflakes (also available from PARSIVEL measurements) will improve the results in the future. Locatelli and Hobbs (1974) gave not only mass–size relations but also velocity–size relations for different crystal types. The velocity information could therefore help to determine the crystal types and then to choose the approriate mass–size relation for the correction factor *C*_{M,k}.

Another issue is the improvement of the size and velocity estimation of each measured snow particle itself. Because, in the case of the snow, the measured horizontal diameter [which is smaller than or equal to the overall maximum diameter required for the mass–size relations of Locatelli and Hobbs (1974) and depends on the geometry and orientation of the main particle axes] was used to calculate the fall velocity and the correction factor *C*_{M,k}, it is generally expected to underestimate the reflectivity factor. Information about the spatial distributions of the orientation and the geometry of the particles for different snow types could be used to estimate the magnitude of this underestimation on a statistical basis. This information, however, is not yet available as needed.

A third important issue is to examine how the particles that fall out of the region where the method is applicable (e.g., dendrites) can be treated.

## Acknowledgments

The authors thank Ronald Hannesen for the preparation of the radar data and Klaus Beheng and Jürg Joss for their very helpful comments on the subject. In addition, we express our appreciation for the helpful comments of three anonymous reviewers.

## REFERENCES

Debye, P., 1929:

*Polar Molecules.*Chemical Catalogue Co., 172 pp.Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for water drops in stagnant air.

*J. Meteor.,***6,**243–248.Herman, B. M., and L. J. Battan, 1961: Calculations of Mie back-scattering from melting ice spheres.

*J. Meteor.,***18,**468–478.Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles.

*J. Geophys. Res.,***79,**2185–2197.Löffler-Mang, M., and J. Joss, 2000: An optical disdrometer for measuring size and velocity of hydrometeors.

*J. Atmos. Oceanic Technol.,***17,**130–139.Marshall, J. S., and K. L. S. Gunn, 1952: Measurement of snow parameters by radar.

*J. Meteor.,***9,**322–327.Smith, P. L., 1984: Equivalent radar reflectivity factors for snow and ice particles.

*J. Climate Appl. Meteor.,***23,**1258–1260.

Correction factor *C*^{*}_{M}

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2

Correction factor *C*^{*}_{M}

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2

Correction factor *C*^{*}_{M}

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2

Same as Fig. 1 but for snow events (a) 1600–2000 CET 4 Dec 1998 and (b) 1600–2000 CET 7 Dec 1998

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2

Same as Fig. 1 but for snow events (a) 1600–2000 CET 4 Dec 1998 and (b) 1600–2000 CET 7 Dec 1998

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2

Same as Fig. 1 but for snow events (a) 1600–2000 CET 4 Dec 1998 and (b) 1600–2000 CET 7 Dec 1998

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0843:EOTERR>2.0.CO;2