## Introduction

*Z*

_{e}measured at the different wavelength pairs using the dual-wavelength ratio (DWR), which is defined as

*l*denotes the lower-frequency radar band and

*h*denotes the higher-frequency radar band. Equivalent radar reflectivity is given by

*Z*

_{e}is used instead of

*Z*to indicate the possibility of non-Rayleigh scattering and has units of mm

^{6}m

^{−3}. Note that the dynamic range of the observed reflectivity values is large so reflectivity is often expressed as dB

*Z*

_{e}, where dB

*Z*

_{e}= 10 log(

*Z*

_{e}). The backscatter efficiency

*ξ*

_{b}is a function of particle diameter

*D*[mm]; radar wavelength

*λ*[m]; particle density

*ρ*[g cm

^{−3}]; and particle shape, which is discussed in a later section. Here,

*N*[mm

^{−1}m

^{−3}] is the particle number density. The dimensionless quantity

*K*is given as

*K*= {[

*n*(

*λ, ρ*)

^{2}− 1]/[

*n*(

*λ, ρ*)

^{2}+ 2]}, where

*n*is the complex index of refraction. Because the true index of refraction for the media observed is typically unknown, |

*K*|

^{2}is assumed to be that for liquid water |

*K*

_{w}|

^{2}at 0°C, which is typically 0.93 at low frequencies. Values of |

*K*|

^{2}for water and solid ice are given in Table 1 for the frequencies of interest in this study. Because |

*K*

_{w}|

^{2}is not constant with frequency DWR, as defined in Eq. (1), we will have slightly negative values for small diameter particles. DWR is defined in this manner to make clear the assumptions that

*Z*

_{e}is calculated with respect to a specific dielectric constant.

*Z*

_{e}has a more complex dependence upon particle size, number density, and dielectric constant or density. Below we assume that

*N*(

*D*) is the gamma distribution

*N*(

*D*) =

*N*

_{o}

*D*

^{μ}exp[−(3.67 +

*μ*)

*D*/

*D*

_{o}], as described by Ulbrich (1983). Using this relationship, (2) and (1) reduce to

*Z*

_{e}is a function of several variables, including

*N*

_{o}[mm

^{−1−μ}m

^{−3}]. Aside from the wavelength pair selected, DWR depends upon

*D*

_{o}[mm],

*μ,*and

*ρ*[g cm

^{−3}]. We will later show that DWR is fairly insensitive to density when

*ρ*decreases with

*D.*We also show that DWR

_{Ka,W}and DWR

_{S,W}are insensitive to

*μ*for

*D*

_{o}< 2 mm and that DWR

_{S,Ka}is fairly insensitive to

*μ*for values of

*D*

_{o}up to 10 mm. Therefore,

*D*

_{o}is obtained from DWR. If reasonable assumptions are made for

*ρ*(

*D*) and

*μ*(see section 3), then

*N*

_{o}can also be determined from

*Z*

_{e}.

Multifrequency particle sizing works when the higher-frequency experiences non-Rayleigh or Mie scattering. For cloud particle sizing, the higher frequency must exhibit non-Rayleigh scattering from small hydrometeors. For ice crystals this occurs at millimeter wavelengths where radar systems typically operate in one of several atmospheric transmission windows centered near 35, 95, 140, and 215 GHz. Radar–lidar and other techniques can detect smaller particle diameters in cirrus clouds and are superior to multifrequency radar in cold cirrus clouds where median diameters can be tens of microns. However, lidar signals are attenuated by low-level liquid clouds and are of limited use in optically thick clouds or during precipitation.

Although the minimum size that can be inferred from DWR is determined by the wavelength of the higher-frequency signal, atmospheric extinction increases with frequency and scattering losses from ice aggregates at 140 and 215 GHz can be substantial, even for dry ice particles (Nemarich et al. 1984). The choice of lower frequency is a trade-off between the maximum particle size that can be detected and system portability. As wavelength increases, the antenna diameter must also increase to maintain spatial resolution and sensitivity to clouds.

Multifrequency observations of precipitation and ice clouds measured by the University of Massachusetts—Amherst 33/95-GHz Cloud Profiling Radar System (CPRS) are reported by Sekelsky and McIntosh (1996). However, attenuation at these frequencies is still problematic in cases where the effects of non-Rayleigh scattering and attenuation cannot be separated with a high degree of confidence, such as when precipitation and melting particles are present in the propagation path. By combining CPRS measurements and those collected with the nonattenuating National Oceanic and Atmospheric Administration (NOAA) Aeronomy Laboratory S-band (2.8 GHz) radar, these uncertainties are better resolved and more accurate estimates of particle size distributions are possible. Table 2 compares extinction rates for the three frequencies considered in rain and snow, and Table 3 lists typical absorption rates for the humid tropical atmosphere. We see that S-band attenuation can be ignored for the zenith measurements considered in this paper.

The following sections describe the University of Massachusetts—Amherst (UMass) and NOAA radar hardware, scattering models for dry ice crystals and aggregates, and the neural network methodology that processes the multifrequency measurements. Finally, neural network outputs are presented for data collected during the Maritime Continent Thunderstorm Experiment (MCTEX).

## Radar hardware

### UMass Cloud Profiling Radar System

CPRS consists of 94.92 (W band) and 33.12 GHz (Ka band) subsystems, a programmable pedestal that facilitates various scanning modes and a high-speed VXI-bus-based data acquisition and digital signal processing (DSP) system (Sekelsky and McIntosh 1996;Lohmeier et al. 1997). Both the 33- and 95-GHz subsystems transmit and receive through a single 1-m-diameter dielectric lens antenna. The use of a single aperture eliminates pointing errors between the two frequencies and the lens acts as a radome protecting the hardware from precipitation. Other system characteristics are listed in Table 4.

Although CPRS is normally mounted on a truck, it was repackaged for operation from the standard 20-ft-length sea container shown in Fig. 1a, which is convenient for shipping and serves as a field laboratory. The microwave hardware is modular, and the antenna and radar front-ends sit outside the container atop an elevation-over-azimuth pedestal. Radar control and data acquisition for both CPRS and the NOAA radar, described below, are contained inside the air-conditioned sea container.

### NOAA S-band profiler

The NOAA S-band radar shown in Fig. 1b operates at 2.835 GHz and uses a vertical-pointing shrouded parabolic reflector antenna, a simple and reliable solid-state transmitter, and much of the same hardware and software that is used in Aeronomy Laboratory 915-MHz profilers (Carter et al. 1995; Ecklund et al. 1999). The solid-state transmitter and associated electronics are mounted on the outside of the antenna shroud to minimize feed line loss. For the data presented in this paper, 130 Doppler spectra (128 points) were averaged from each of 60 heights during each 30-s time interval. The average spectra provide vertical Doppler velocity, signal-to-noise ratio, noise level, and spectral width. The signal-to-noise ratio and noise level are used to compute equivalent reflectivity (dB*Z*_{e}) at each height. The system parameters are given in the last column of Table 4.

### Sample volume matching

*W*

_{r}, shown in Fig. 2a and in time (horizontally) with a weighting function,

*W*

_{t}(

*t, R*), which is a function of the S-band antenna pattern. Because the target is not stationary, horizontal cloud motions and radar averaging time must also be factored into the sample volume correction. The term

*W*

_{t}(

*t, R*) is the convolution of a time-averaging function,

*W*

_{TA}

*W*

_{ant}(

*t, R*), which are defined below. Here

*W*

_{TA}

*T*

_{A}seconds about time

*t*

_{o}. The antenna-weighting function depends on the angle with respect to the antenna boresight

*θ*and can be reparameterized as a function of time and range:

*V*

_{h}(

*R*) is the horizontal wind speed at height

*R.*

Figure 3 plots *W*_{t}(*t, R*) for *T*_{A} = 30 s at *R* = 2 km and *R* = 8 km. If the magnitude of horizontal wind is uniform with height, then *W*_{t} becomes wider with height, and the horizontal resolution degrades. The effect is analogous to that for a scanning radar where the effective beam is wider than that of a stationary antenna. However, in this case the radar is fixed and the target is moving across the beam. When the horizontal wind speed increases with height, *W*_{t} changes less with height. This situation often occurs, tending to reduce the horizontal resolution dependence on height. Figure 4 illustrates the performance of the volume matching scheme by comparing S- and Ka-band measurements of a sharp reflectivity gradient.

## Particle size estimation withmultifrequency radar

To understand how DWR is affected by density and size distribution shape we evaluate several particle scattering models. First, a comparison of Mie and Rayleigh models for a monodisperse distribution of liquid and frozen spheres of constant density shows the minimum size for which non-Rayleigh scattering is observed at W band (95 GHz). Next, we consider a more realistic ice-cloud model where the particle density decreases with *D* and the number concentration is modeled by a gamma size distribution. This generalized model describes DWR for both aggregates and spherical approximations of crystals. In the following section this model generates training data for a neural network that combines DWR_{Ka,W}, DWR_{S,W}, and DWR_{S,Ka} to form estimates of size distribution parameters for measured radar data. Distributions of columns and plates are also modeled to show how ice crystal shape and orientation influence DWR. Scattering for these particles is calculated using the the Discrete Dipole Approximation (DDA). Results show that DWR for planar crystals is highly dependent upon orientation, which agrees with previous investigations. Finally, there is a discussion of polarization effects, which have little influence on the zenith-pointing radar data presented.

### Non-Rayleigh scattering from spheres

At millimeter wavelengths, differences between measured backscatter and that predicted by the Rayleigh approximation are observed for precipitation and cloud particles (Lhermitte 1988). Figures 5a,b show how W-band scattering departs from the Rayleigh approximation and illustrates how Mie scattering is dependent upon hydrometeor phase and density. Figure 5a plots the ratio of Rayleigh to Mie backscatter at 94.92 GHz for monodisperse distributions of liquid and frozen water spheres with diameters up to 10 mm. Figure 5b shows details of the same curves for diameters less than 1 mm. Rayleigh backscatter efficiency is calculated with *ξ*_{b} = (4*π*^{4}*λ*^{−4})*D*^{4}|*K*|^{2} and Mie backscatter efficiency with a formulation of the Mie coefficients described by Deirmendjian (1969). Figure 5b shows that the degree of non-Rayleigh scattering for water spheres is somewhat sensitive to temperature. Non-Rayleigh scattering for ice spheres varies little with temperature but is sensitive to density. Variations of |*K*|^{2} with temperature are so small that the 0°C backscatter data for ice plotted in Fig. 5 are indistinguishable from the same quantities calculated at −30°C (not shown).

### Generalized DWR model for aggregatesand crystals

*N*

*D*

*N*

_{o}

*D*

^{μ}

*μ*

*D*

*D*

_{o}

*N*

_{o}[mm

^{−(μ+1)}m

^{−3}] is the peak number concentration,

*D*

_{o}[mm] the median volume diameter, and

*μ*is the dimensionless order of the distribution that determines its shape. The shape is concave upward for

*μ*< 1 and concave downward for

*μ*> 1. The gamma distribution is often used to model cirrus particle distributions (Kosarev and Mazin 1991; Matrosov et al. 1992;Matrosov 1993). Distributions of aggregates or snow tend to follow an exponential size distribution (Sekhon and Srivastava 1970; Thomason et al. 1995), which is also represented by (7) with

*μ*= 0. For the generalized model we apply (7) to distributions of both crystals and aggregates.

*D*

_{o}, while reflectivity depends upon

*D*

_{o},

*N*

_{o},

*μ,*and

*ρ*(

*D*). The retrieval problem is underconstrained, having three radar measurements and four variables. Therefore, density must be properly modeled in order to retrieve

*N*

_{o}from reflectivity data. Recent publications by Atlas et al. (1995), Brown and Francis (1995), and Thomason et al. (1995) discuss the requirement for improved bulk density parameterization. Density is often treated as a constant, independent of size. However, for aggregates and most crystal habits small particles tend be more dense than larger ones. Most size–density relationships follow the form

*ρ*is in grams per cubic centimeter. The variable

*A*is a positive constant and

*B*is negative, typically close to −1 (Locatelli and Hobbs 1974; Magono and Nakamura 1965; Mitchell et al. 1990).

*Z*

_{e}is explicitly related to density for Rayleigh scatterers. First, we note that for

*ρ*[g cm

^{−3}], (|

*K*|/

*ρ*)

^{2}= 0.208 is nearly constant with ice density (Battan 1973; Atlas et al. 1995) and virtually independent of frequency and temperature for the wavelengths considered. By substituting

*ξ*

_{b}= (4

*π*

^{4}/

*λ*

^{4})

*D*

^{4}|

*K*|

^{2}, equation (8) and the above constant for (|

*K*|/

*ρ*)

^{2}into (3) we obtain

*D*<

*A*/0.916. However, the error incurred in neglecting these terms in (9) is small. Using a relationship given by Ulbrich (1983), (9) is rewritten in a nonintegral form:

*Z*

_{e}, density, and the three gamma distribution parameters. Given that

*B*≈ −1, Eq. (10) can be further reduced:

*μ*by calculating DWR for the Mitchell et al. (1990) and Brown and Francis (1995) size–density functions using Deirmendjian’s formulation of the Mie coefficients (Deirmendjian 1969) and the Weiner density function described in the appendix. Brown’s density function for cirrus particles has the following form:

*ρ*is in grams per cubic centimeter and

*D*is in millimeters. For

*D*< 0.1 mm density is assumed to be that for solid ice. Snowflake density derived from recent measurements by Mitchell et al. (1990) has the following size dependence:

*ρ*[g cm

^{−3}] is density and

*D*[mm] is the maximum particle dimension. These two functions were chosen since they represent vastly different ice particle types.

DWR models plotted in Figs. 6a,b show little difference between the two density functions. The shaded regions represent DWR for a gamma size distribution with −1 < *μ* < +1. The dashed boundary of each family of curves corresponds to *μ* = −1 and the solid boundary to *μ* = 1. DWR for the two density functions do not substantially differ even though absolute reflectivity values for the Mitchell density function are higher than those corresponding to the Brown density function. Figure 6 indicates that DWR_{Ka,W} and DWR_{S,W} become sensitive to particle size distribution shape for *D*_{o} greater than approximately 2 mm. DWR_{S,Ka} shows much less sensitivity to size distribution shape, suggesting that DWR_{S,Ka} is better suited to estimate *D*_{o} for large snowflakes.

### DDA models of DWR for crystals

To assess the validity of our generalized scattering model we compare the above results to realistic models of columns and hexagonal plates. These results are calculated using the DDA with details given in the appendix. Previous models of DWR developed for millimeter-wave frequencies present scattering for spheroids (Matrosov 1993) and various realistic ice crystal geometries and orientations common to cirrus clouds (Tang and Aydin 1995; Aydin and Tang 1997). However, these works emphasize the use of frequency pairs with the highest frequency above 95 GHz, and they do not consider aggregated particles.

Figures 7a–e plot DWR for gamma distributions of aggregates and crystals with *D*_{o} < 2 mm. DWR for aggregates, previously plotted in Fig. 6, are replotted for the smaller size range in Figs. 7a,b for comparison with the DDA results for crystals.

The orientation of plates has a substantial impact on radar backscatter and DWR, and size ambiguities exist for these and other planar crystals because of uncertainties in their orientation. Plates normally lie with the position of their major axes distributed about the horizontal plane (Platt 1978; Hendry et al. 1976), and measurements of the tilt angle distribution suggest a Gaussian shape (Sassen 1987b). The tilt angle is defined as the angle between the major axes of the hexagon and the horizontal plane. Figures 7c,d show DWR for hexagonal plates with a Gaussian distribution of tilt angles having standard deviations of 2.5° and 10°, respectively. These model results agree qualitatively with previous results for DWR calculated using 95 and 220 GHz (Tang and Aydin 1995).

Figure 7e plots DWR for solid columns. Here the columns lie with their major (long) axis parallel to the horizontal plane and the direction of the major axis randomly oriented in the horizontal plane. These results are similar to those in Figs. 7a,b, although DWR for columns appears to be more sensitive to distribution shape than do spheres.

### Polarization effects

Up to this point there has been no discussion of polarization effects, which can influence *Z*_{e} and DWR. The radar cross section of nonsymmetric particles, such as the columns and hexagonal plates described above, depends heavily on the polarization of the incident electromagnetic field and the orientation of the particle relative to the observing radar. However, at zenith incidence these effects are minimal except in cases where crystals are oriented in a preferred direction by electric fields (Metcalf 1995; Galloway et al. 1997).

Previous computational studies of ice crystals at millimeter wavelengths show no difference between horizontal- and vertical-polarized backscatter at zenith incidence. For linear polarizations, differential reflectivity is defined as the ratio of H- to V-polarized reflectivity, *Z*_{DR} = 10 log(*Z*_{H}/*Z*_{V}), expressed in decibels. Models of differential reflectivity for bullet rosettes, columns, and hexagonal plates at 35, 94, and 220 GHz show substantial variation with radar elevation angle (Aydin and Walsh 1996). However, at zenith incidence *Z*_{DR} = 0 for each crystal type.

## Neural network particle sizedistribution estimator

Although DWR for any single wavelength pair can estimate particle size distribution parameters, quantities derived from a combination of DWR_{Ka,W}, DWR_{S,W}, and DWR_{S,Ka} can be more precise and noise tolerant. A fundamental question is how to weigh the different DWRs to produce a single estimate of size distribution parameters, especially *D*_{o}. In the previous section we noted that DWR_{Ka,W} and DWR_{S,W} show greater sensitivity to change in size than does DWR_{S,Ka} for small *D*_{o}. However, DWR_{S,Ka} is less sensitive to *μ* when *D*_{o} is large. Therefore, DWR_{Ka,W} and DWR_{S,W} are best suited to measure small particles and should be weighed more heavily than DWR_{S,Ka} in the smaller particle regime. DWR_{S,Ka} should be weighted more heavily than DWR_{Ka,W} and DWR_{S,W} for distributions of larger particles because DWR_{S,Ka} is less sensitive to *μ.*

Rather than arbitrarily assign weights to the different DWR values, a neural network was created to estimate median diameter, *D*_{o} [mm], and maximum particle concentration, *N*_{o} [mm^{−2} m^{−3}], for the first-order gamma size distribution, which is given by (7) with *μ* = 1. This distribution satisfies *N*(0) = 0 and is a compromise between the exponential distribution describing aggregates and the gamma distribution for cirrus clouds described by Kosarev and Mazin (1991), who indicate that 0 < *μ* < 2. Particle density is assumed to follow Brown’s function given by Eq. (12). This assumption affects *N*_{o}. Here, *D*_{o} is much less sensitive to density, as shown in the previous section.

Neural networks have been used to classify cloud type by Bankert (1994) and Lohmeier et al. (1997), to improve *Z–R*_{r} relationships in rain by Xiao and Chandrasekar (1997) and in numerous other engineering and remote sensing applications. For our purposes a neural network approximates the function relating the three radar reflectivity measurements or DWRs to *N*_{o} and *D*_{o}. Multivariate regression was attempted but found to be numerically unstable and difficult to modify. In contrast, the neural network solution described below is easily modified and more stable over a wide range of input values.

Figure 8 shows the geometry for a fully connected feed-forward multilayered perceptron (MLP) neural network that is used to retrieve particle size distributions. The specific geometry was selected through trial and error such that global errors are minimized, and there are no unused nodes. According to Haykin (1994), the universal approximation theorem holds that a three-layer (one hidden) feed-foreword MLP can model a function with arbitrary precision, given that a sufficient number of nodes exists. However, our experience is that a four-layer network (two hidden) train more quickly. Additional practical justification for using two hidden layers is given by Xiao and Chandrasekar (1997).

**X**

_{S,W}[dB], DWR

_{Ka,W}[dB], and W-band reflectivity in dB

*Z*

_{e}:

_{S,W}, DWR

_{Ka,W}, DWR

_{S,Ka}], 0.01[dB

*Z*

_{e,S}, dB

*Z*

_{e,Ka}, dB

*Z*

_{e,W}], etc. All contain the same information and may be used instead of (14). Scaled logarithmic values are input to reduce the dynamic range of values that must be modeled by the neural network. The output vector

**Y**

**Y**

*N*

_{o}

*D*

_{o}

**X**

**Y**

**Y**

**Ŷ**are minimized. This global difference is determined using the sum of the squared errors (SSE):

Our neural network was implemented using the Stuttgart Neural Network Simulator version 4.1 (Zell et al. 1995). Standard backpropagation was used to train the neural network with a training dataset that consisted of 12 800 sets of input and output vectors. The network was iterated many times until an acceptable error was achieved with respect to the training data. Here, 120 000 iterations resulted in maximum global error of less than 2%.

To evaluate the performance of the neural network’s ability to model its training data, it was tested with 10 000 pairs of input/output vectors that are similar but not identical to the training dataset. Testing with the training dataset can produce inaccurate results. Figure 9a plots the mean error in *D*_{o} averaged over all values of *N*_{o}, and Fig. 9b plots the mean error of *N*_{o}. These figures show a rapid increase in the magnitude of the output errors for *D*_{o} less than approximately 0.2 mm. This is an important ramification of the the neural network’s inability to build input–output relationships for distributions of small particles.

## Results

This section presents neural network estimates of particle size distribution parameters and derived quantities using measurements collected during MCTEX, described by Keenan et al. (1995) and Ackerman et al. (1996). MCTEX was conducted on Melville and Bathurst Islands, which lie between 11.3° and 11.9°S, and 130.0° and 130.6°E, approximately 50 km north of Darwin, Australia. Timing of the experiment coincided with the transition from dry to wet season during November and December 1995. During this period, daily convective activity produced numerous thunderstorms and a variety of cloud types were observed by the radars.

Neural network outputs are presented for the following three cloud cases: 1) a precipitating stratiform cloud, 2) a decaying convective cloud producing light drizzle, and 3) the leading edge of a massive cirrus anvil. As described in the previous section the neural network was trained using the Brown density function. While this assumption affects number concentration estimates, it has much less impact on size estimates. Ice mass content (IMC) is also derived from radar reflectivity and the neural network size estimates. Comparison with numerous published *Z*–IMC relationships shows that the additional size information derived from multifrequency radar data reduces uncertainties in IMC.

### Stratiform cloud case

Measurements presented for the stratiform cloud observed on 28 November were collected beginning about 2.5 h after intense convection. Radar Doppler measurements indicate no substantial updrafts, and precipitation ceases approximately halfway through the dataset, shown in Fig. 10. To correct for attenuation effects, cloud-top reflectivities were matched at the different frequencies, taking into account the differences in index of refraction between ice and water given in Table 1. S-band reflectivity values were used as a reference since the S-band signal experiences negligible attenuation.

*D*

_{o}is insensitive to the shape of both the size distribution and size–density functions and to particle shape. However, these limits are not approached monotonically for planar crystals. This means that our calibration scheme is insensitive to cloud microphysical parameters provided that the cloud particles are small.

Even though the smallest ice particles are often found near cloud tops, the last range gate showing a signal may not be the best choice for aligning reflectivity data because it may be comprised of volumes sampled above and below the cloud top. In addition, attenuation and weak cloud reflectivity may prevent one or more of the radar systems from detecting the cloud top. Instead of using the last range gate, a simple algorithm that correlates reflectivity features over small scales is used to determine what range gate close to the cloud top will be used for matching reflectivity values.

*D*

_{o}< 0.5 mm. Figures 11a,b show results of the cloud-top matching for a particular profile. Figure 11a plots ice equivalent reflectivity at the measured cloud top for S, Ka, and W bands. Ice equivalent reflectivity is calculated using data from Table 1 and the following relationship:

*Z*

_{e}values created when

*K*

_{w}is assumed instead of

*K*

_{i}so that the overlap in reflectivity is clearly seen. Figure 11b plots DWR

_{Ka,W}. Imperfections in the volume matching scheme limit the accuracy of DWR

_{S,W}and DWR

_{S,Ka}.

Figures 12a–d show outputs of the neural network estimator for the stratiform cloud presented in Fig. 10. The cloud data used to create these figures were edited such that precipitation and the melting layer are not considered. The first range gate shown corresponds to the freezing level, approximately 4.2 km above ground level, indicated by the dashed line (labeled ML) in Fig. 10. Estimates of median particle diameter, shown as a histogram in Fig. 12a and a time–height plot in Fig. 12b, indicate that *D*_{o} monotonically decreases with height. The particle size trend is consistent with previous studies showing aggregate size increasing with temperature (Hobbs et al. 1974). The peak particle concentration shown in Figs. 12c,d increases with height.

### Convective cloud case

The convective cloud case observed on 4 December is shown in Figs. 13 and 14b–d. It occurred a few minutes after a near miss by a convective cell, but only weak precipitation (*R*_{r} < 1 mm h^{−1}) fell on the radars. Because the DWR near the cloud-top indicates non-Rayleigh scattering, the cloud-top matching scheme used to remove attenuation for the stratiform case cannot be applied here. Instead, a small correction for attenuation was applied using radar-derived rain rates, assuming Marshall–Palmer drop size distribution (Marshall and Palmer 1948) and combined radiosonde/dual-channel microwave radiometer estimates of water vapor absorption profiles.

Figures 14a,b show that, unlike the stratiform case, there is no particle size trend with height. On smaller timescales, Fig. 14b reveals slanted vertical features centered at approximately 3 and 22 min into the dataset. These correspond to high downward Doppler velocities and become the tops of wind-sheared rain shafts observed below the melting layer. Particle size estimates and radar Doppler data suggest that these features contain large particles falling through a background of smaller, lower terminal velocity particles. Note that since the neural network considers only radar reflectivity measurements, downdrafts do not influence its particle size estimates as they do when Doppler spectrum measurements are used (Gossard 1994).

### Cirrus cloud case

The last case analyzed is the leading edge of a large thunderstorm anvil, measured on 27 November, which is shown in Fig. 15. Although Ka-band and W-band radar measurements are well correlated, the cloud is very dynamic and the sample volume matching technique previously described could not adequately adjust for the spatial averaging effects of the S-band radar’s wide beamwidth antenna. Fortunately, attenuation in cirrus clouds is small and the S-band measurements are not required to eliminate cloud attenuation. Therefore, the particle size distribution parameters shown in Figs. 16b–d were produced using only CPRS Ka-band and W-band reflectivities. Here we see a trend similar to that for the stratiform case where particle size decreases with height, which is also in agreement with previous in situ measurements of anvils reported by McFarquhar and Heymsfield (1996).

### Calculation of ice mass content

*ρ*are in grams per cubic meter and D is in millimeters.

Equation (21) shows that knowledge of the particle size distribution and density are required to accurately determine IMC. It can be shown that for a gamma size distribution IMC is a function of *D*_{o}. Given that *μ* = 1 and *B* = −1 are reasonable assumptions, then for Rayleigh scatterers the relationship between *Z*_{e} and IMC depends only upon *A *and *D*_{o}. Here, *D*_{o} is obtained from DWR measurements.

*D*

_{o}by combining (10) and (22):

*μ*and

*B.*However, if

*B*= −1, then the bracketed quantity varies less than 15% for −1 <

*μ*< 3 and less than 6% for 0 <

*μ*< 3. Thus, IMC is fairly insensitive to

*μ.*For

*μ*= 1 and

*B*= −1 the bracketed term is 0.485 and (23) reduces to

*A,*which must be assumed. While this uncertainty in

*A*affects IMC, the variation in IMC is less than the spread among previous

*Z*

_{i}–IMC relationships for clouds with no corresponding in situ data. Below, profiles of IMC are presented for a realistic range of

*A*coefficients and compared to the

*Z*

_{i}–IMC relationships listed in Table 5.

Figures 17–22 show horizontal and vertical profiles of radar reflectivity, *D*_{o}, and IMC for the cloud cases analyzed. Here, IMC calculated using Eq. (25) is compared to values calculated using the *Z*_{i}–IMC relationships summarized in Table 5. The hatched region of the IMC plots is bounded by the Brown (*A* = 0.0706, upper bound) and Mitchell (*A* = 0.17, lower bound) density functions and represents the spread of values anticipated for particles ranging from cirrus particles to snowflakes.

Figures 17 and 18 plot profiles corresponding to the dotted lines in the reflectivity image for the stratiform cloud case shown in Fig. 10. The profiles show how IMC calculated with Eq. (25) react to variations in both *Z* and *D*_{o}*Z,* IMC increases with decreasing *D*_{o}. The *Z*_{i}–IMC relationships are only affected by *Z.*

Similarly, the convective case is shown in Figs. 19 and 20. Figure 20 shows interesting discrepancies between Eq. (25) and the *Z*_{i}–IMC relationships in the precipitating features seen in Fig. 13 where DWR indicates large particles.

Profiles for the cirrus case plotted in Figs. 21 and 22 show that IMC estimates are degraded for *D*_{o} less than approximately 0.2 mm. This is not surprising since *D*_{o} = 0.2 mm is the minimum size for which DWR, calculated with the radar wavelengths available, is sensitive to size.

## Conclusions

Cloud reflectivity measurements collected by multiwavelength radars operating at one or more millimeter wavelengths can estimate median particle size, and other particle size distribution characteristics in clouds composed of dry ice particles for median diameters larger than approximately 0.2 mm. In contrast to techniques requiring lidar or IR radiometer measurements (Matrosov et al. 1992; Intrieri et al. 1993), the multiwavelength radar technique described is substantially less affected by atmospheric attenuation and works for clouds having large optical depths. However, combined lidar–radar and IR–radar techniques are superior to multiwavelength radar for measuring distributions of particles with *D*_{o} < 0.2 mm.

For measurements at two radar wavelengths, particle size information is contained in the DWR, which is the ratio of equivalent radar reflectivity measured at the longer wavelength to that at the shorter wavelength. DWR is typically positive although limiting values for small diameters are slightly negative due to the frequency dependent dielectric constant for water used to calculate DWR. Selection of the upper frequency is a trade-off between minimum detectable size and cloud extinction, while the lower frequency sets the maximum size that is detected and is a compromise between attenuation and system portability. Scattering models show that the range of dry particle diameters that can be unambiguously estimated using the Ka-band–W-band frequency pair is approximately 0.2–5 mm, while the S-band–W-band pair has the same minimum size limitation but can be used to retrieve much larger particle diameters. The S-band signal also experiences negligible attenuation. There are, however, trade-offs between spatial resolution and system portability for the lower frequency. For instance, the S-band radar requires a substantially larger antenna to match the spatial resolution of the CPRS Ka-band subsystem. Additionally, while measurements of horizontal winds can be used to correct for sample volume mismatches when a widebeam antenna is employed, the correction does not work well for highly dynamic clouds, such as the cirrus anvil measurements presented.

When measurements are available for more than two radar wavelengths, neural networks are a practical means for weighting the measurements and producing a single estimate of particle size distribution parameters. Using W-band (95 GHz), Ka-band (33 GHz), and S-band (2.8 GHz) radar reflectivity measurements collected during MCTEX, median particle size *D*_{o} and peak number concentration *N*_{o} are estimated, assuming a first-order gamma distribution for three distinct cloud cases:1) a precipitating stratiform cloud, 2) a convective cloud, and 3) a cirrus anvil. While histograms of *D*_{o} show a characteristic decrease with height in the stratiform cloud and cirrus anvil, no such trend is seen in convective cloud where large particles are detected at all altitudes.

Estimates of IMC are also calculated from the neural network–derived size distributions and compared to *Z*_{i}–IMC relationships for various cloud types. The *Z*_{i}–IMC relationships listed in Table 5 are statistical in nature and therefore applicable only to specific cloud types, whereas IMC estimates calculated using radar-derived *D*_{o} and Eq. (25) do not depend upon cloud type and react to the microphysics of the clouds observed.

## Acknowledgments

The radar measurements and data analysis performed by the University of Massachusetts—Amherst were supported by the Department of Energy Atmospheric Radiation Measurement (ARM) program under Grant 352095-AQ5 and Grant DEFG02-90ER61060, respectively. The NOAA Aeronomy Laboratory S-band work was supported in part by the National Science Foundation under Grant ATM-9214800 and by the Department of Energy as part of the ARM program. Special thanks to Dr. Thomas Ackerman of The Pennsylvania State University (PSU) for introducing the University of Massachusetts to MCTEX, to Jay Eshbaugh, Eric Knapp, Lihua Li, Michael Petronino, and Brian Roberts for preparing the sea container and radar hardware, and to Eugene Clothiaux (PSU), Paul Johnston (NOAA), and personnel from the Australian Bureau of Meteorology Research Centre (BMRC) for helping to maintain the CPRS and the NOAA radar on Melville Island.

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## APPENDIX

### Scattering Models for Ice Crystals and Aggregates

#### Index of refraction

*n,*for dry ice particles is related to bulk density using Wiener’s theorem (Oguchi 1983):

*n*

_{i}is the index of refraction for solid ice and

*n*

_{a}is the index of refraction for air. Here,

*n*

_{i}is obtained using data from Ray (1972) and

*n*

_{a}is assumed to have a constant value of unity. Additionally,

*f*

_{i}and

*f*

_{a}are the dimensionless volume fractions of ice and air, respectively, which are subject to the constraints

*f*

_{i}+

*f*

_{a}= 1 and

*ρ*

_{i}

*f*

_{i}+

*ρ*

_{a}

*f*

_{a}=

*ρ.*Here,

*ρ*

_{i}is the density of solid ice (

*ρ*

_{i}= 0.916 g cm

^{−3}),

*ρ*

_{a}is the density of air, and

*ρ*is particle bulk density. Combining these constraints the volume fraction of ice is obtained:

*ρ*

_{a}≪

*ρ*and

*ρ*

_{a}≪

*ρ*

_{i}this can be further -reduced to

#### Backscatter computations for ice crystals

Numerical modeling techniques can accurately determine backscatter for irregularly shaped hydrometeors with maximum dimensions larger than approximately *λ*/10, which is typically the maximum size for which the Rayleigh approximation is applied. Bohren and Singham (1991) reviews various methods for calculating radar backscatter and discusses inadequacies of classic approaches such as Rayleigh and Rayleigh–Gans theory. On the other hand, the feasibility of numerical techniques for short radar wavelengths, including Ka band and W band, has been demonstrated (O’Brien and Goedecke 1988; Evans and Vivekanandan 1990; Draine and Flatau 1994; Aydin and Tang 1997). The accuracy of the DDA, the finite difference time domain method, and other numerical methods is limited mainly by the availability of adequate computer resources and target microphysical characteristics. In situ data collected with balloons or aircraft are limited to small sampling volumes that may not represent the characteristics of adjacent regions. In situ data is often unavailable for mixed-phase cloud regions or intense storms, such as those measured during MCTEX.

Backscatter and extinction cross sections for the spheres and crystals described above are calculated using the DDA. DDSCAT (V5a) FORTRAN code (Draine and Flatau 1996) was run on a Silicon Graphics Power Challenge with four R10K processors. Spherical particle models were compared with those calculated using Mie cross sections to verify the accuracy of the DDA models.

*d,*must be small compared to both the maximum particle dimension and radar wavelength and 2) the number of dipoles,

*N,*must be sufficiently large to describe the target shape. Using the constraints described by Draine and Flatau (1996), these conditions are satisfied when

*a*

_{eff}is the radius for a sphere of volume equal to that of the model particle and

*n*is the index of refraction. Given the maximum particle dimensions and particle density in Table A1, the model space dimensions are adjusted to ensure accuracy and optimize memory usage for each particle geometry. For spheres, a model space of 36 × 36 × 36 produces errors of only a few percent at W band for diameters up to 18 mm for the Mitchell density function. Errors are less than 2% for median diameters as large as 10 mm.

(a) Range-weighting functions and (b) two-way antenna patterns for CPRS and the NOAA S-band radar.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Range-weighting functions and (b) two-way antenna patterns for CPRS and the NOAA S-band radar.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Range-weighting functions and (b) two-way antenna patterns for CPRS and the NOAA S-band radar.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Examples of time (horizontal) weighting functions for S-band radar illustrating dependence upon horizontal winds. Averaging time, *T*_{A}, is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Examples of time (horizontal) weighting functions for S-band radar illustrating dependence upon horizontal winds. Averaging time, *T*_{A}, is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Examples of time (horizontal) weighting functions for S-band radar illustrating dependence upon horizontal winds. Averaging time, *T*_{A}, is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Results of sample volume matching for cloud edge with a sharp reflectivity gradient.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Results of sample volume matching for cloud edge with a sharp reflectivity gradient.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Results of sample volume matching for cloud edge with a sharp reflectivity gradient.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ratio of Rayleigh to Mie backscattering efficiencies for liquid water droplets and ice spheres. (b) Detail for small diameters illustrating when Mie scattering occurs.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ratio of Rayleigh to Mie backscattering efficiencies for liquid water droplets and ice spheres. (b) Detail for small diameters illustrating when Mie scattering occurs.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ratio of Rayleigh to Mie backscattering efficiencies for liquid water droplets and ice spheres. (b) Detail for small diameters illustrating when Mie scattering occurs.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs median particle diameter for (a) Brown and (b) Mitchell aggregate density functions. Here, *D* is in millimeters for the density relationships shown. A general gamma size distribution is assumed with −1 < *μ* < 1. The dashed boundary corresponds to *μ* = −1 and the solid boundary corresponds to *μ* = +1.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs median particle diameter for (a) Brown and (b) Mitchell aggregate density functions. Here, *D* is in millimeters for the density relationships shown. A general gamma size distribution is assumed with −1 < *μ* < 1. The dashed boundary corresponds to *μ* = −1 and the solid boundary corresponds to *μ* = +1.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs median particle diameter for (a) Brown and (b) Mitchell aggregate density functions. Here, *D* is in millimeters for the density relationships shown. A general gamma size distribution is assumed with −1 < *μ* < 1. The dashed boundary corresponds to *μ* = −1 and the solid boundary corresponds to *μ* = +1.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs *D*_{o} for several aggregate density functions and crystal types. A gamma distribution is assumed with 0 mm < *D*_{o} < 2 mm and −1 < *μ* < 1. Spheres with (a) Brown and (b) Mitchell size–density functions. Hexagonal plates with Gaussian distributions of tilt angles having *σ* equal to (c) 2.5° and (d) 10°. (e) Hollow columns with long axis randomly distributed in horizontal plane.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs *D*_{o} for several aggregate density functions and crystal types. A gamma distribution is assumed with 0 mm < *D*_{o} < 2 mm and −1 < *μ* < 1. Spheres with (a) Brown and (b) Mitchell size–density functions. Hexagonal plates with Gaussian distributions of tilt angles having *σ* equal to (c) 2.5° and (d) 10°. (e) Hollow columns with long axis randomly distributed in horizontal plane.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

DWR vs *D*_{o} for several aggregate density functions and crystal types. A gamma distribution is assumed with 0 mm < *D*_{o} < 2 mm and −1 < *μ* < 1. Spheres with (a) Brown and (b) Mitchell size–density functions. Hexagonal plates with Gaussian distributions of tilt angles having *σ* equal to (c) 2.5° and (d) 10°. (e) Hollow columns with long axis randomly distributed in horizontal plane.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Geometry of the MLP neural network used to estimate particle size distribution parameters.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Geometry of the MLP neural network used to estimate particle size distribution parameters.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Geometry of the MLP neural network used to estimate particle size distribution parameters.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Mean error in neural network approximation of (a) *D*_{o} and (b) *N*_{o} for the Brown density function. The curves were created using 10 000 pairs of input/output test vectors, spanning 0.025 mm ⩽ *D*_{o} 9 mm and 100 mm^{−1} m^{3} ⩽ *N*_{o} ⩽ 5 × 10^{6} mm^{−1} m^{3}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Mean error in neural network approximation of (a) *D*_{o} and (b) *N*_{o} for the Brown density function. The curves were created using 10 000 pairs of input/output test vectors, spanning 0.025 mm ⩽ *D*_{o} 9 mm and 100 mm^{−1} m^{3} ⩽ *N*_{o} ⩽ 5 × 10^{6} mm^{−1} m^{3}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Mean error in neural network approximation of (a) *D*_{o} and (b) *N*_{o} for the Brown density function. The curves were created using 10 000 pairs of input/output test vectors, spanning 0.025 mm ⩽ *D*_{o} 9 mm and 100 mm^{−1} m^{3} ⩽ *N*_{o} ⩽ 5 × 10^{6} mm^{−1} m^{3}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for a stratiform cloud case. Averaging time is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for a stratiform cloud case. Averaging time is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for a stratiform cloud case. Averaging time is 30 s.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ice equivalent reflectivity and (b) DWR_{Ka,W} at stratiform cloud top.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ice equivalent reflectivity and (b) DWR_{Ka,W} at stratiform cloud top.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

(a) Ice equivalent reflectivity and (b) DWR_{Ka,W} at stratiform cloud top.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Neural network outputs for stratiform cloud case using the Brown density function. Data are averaged to 30-s temporal resolution and histograms show number of occurrences for indicated bin sizes. (a) Histogram of *D*_{o}, (b) false color map of *D*_{o}, (c) histogram of *N*_{o}, and (d) false color map of *N*_{o}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Neural network outputs for stratiform cloud case using the Brown density function. Data are averaged to 30-s temporal resolution and histograms show number of occurrences for indicated bin sizes. (a) Histogram of *D*_{o}, (b) false color map of *D*_{o}, (c) histogram of *N*_{o}, and (d) false color map of *N*_{o}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Neural network outputs for stratiform cloud case using the Brown density function. Data are averaged to 30-s temporal resolution and histograms show number of occurrences for indicated bin sizes. (a) Histogram of *D*_{o}, (b) false color map of *D*_{o}, (c) histogram of *N*_{o}, and (d) false color map of *N*_{o}.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 12, except for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 12, except for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 12, except for convective cloud case.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements of a cirrus anvil cloud. Note the clear-air return below 6 km at 2.8 GHz, which is not seen by the millimeter-wave radars.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements of a cirrus anvil cloud. Note the clear-air return below 6 km at 2.8 GHz, which is not seen by the millimeter-wave radars.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Radar reflectivity measurements of a cirrus anvil cloud. Note the clear-air return below 6 km at 2.8 GHz, which is not seen by the millimeter-wave radars.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 12 and 14, except for cirrus cloud case. As discussed in the text these data are derived using Ka- and W-band data only.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 12 and 14, except for cirrus cloud case. As discussed in the text these data are derived using Ka- and W-band data only.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 12 and 14, except for cirrus cloud case. As discussed in the text these data are derived using Ka- and W-band data only.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Vertical profiles of radar measurements and derived quantities for the convective cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) corresponding to 0726:33 UTC. Here, (d), (e), and (f) correspond to 0810:22 UTC. The upper bound on hatch regions correspond Brown’s density function given by (8), and the lower bound is given by Mitchell’s relationship for dry snow.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Vertical profiles of radar measurements and derived quantities for the convective cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) corresponding to 0726:33 UTC. Here, (d), (e), and (f) correspond to 0810:22 UTC. The upper bound on hatch regions correspond Brown’s density function given by (8), and the lower bound is given by Mitchell’s relationship for dry snow.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Vertical profiles of radar measurements and derived quantities for the convective cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) corresponding to 0726:33 UTC. Here, (d), (e), and (f) correspond to 0810:22 UTC. The upper bound on hatch regions correspond Brown’s density function given by (8), and the lower bound is given by Mitchell’s relationship for dry snow.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Profiles of radar reflectivity and derived IMC vs time for the 28 Nov stratiform cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) at a height of 7.3 km. Here, (d), (e), and (f) correspond to height of 8.8 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Profiles of radar reflectivity and derived IMC vs time for the 28 Nov stratiform cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) at a height of 7.3 km. Here, (d), (e), and (f) correspond to height of 8.8 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Profiles of radar reflectivity and derived IMC vs time for the 28 Nov stratiform cloud case. (a) Radar reflectivity measurements, (b) neural network estimates of *D*_{o} assuming a first-order gamma distribution, and (c) comparison of neural network–derived IMC with published *Z*_{i}–IMC relationships (Sassen 1987) at a height of 7.3 km. Here, (d), (e), and (f) correspond to height of 8.8 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 17, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to 0508:33 UTC and (d), (e), and (f) correspond to 0532:58 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 17, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to 0508:33 UTC and (d), (e), and (f) correspond to 0532:58 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 17, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to 0508:33 UTC and (d), (e), and (f) correspond to 0532:58 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 18, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to a height of 9.55 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 18, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to a height of 9.55 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Fig. 18, except for 4 Dec convective cloud case. Here, (a), (b), and (c) correspond to a height of 9.55 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 17 and 19, except for 27 Nov cirrus anvil. Here, (a), (b), and (c) correspond to 0623:00 UTC and (d), (e), and (f) correspond to 0634:15 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 17 and 19, except for 27 Nov cirrus anvil. Here, (a), (b), and (c) correspond to 0623:00 UTC and (d), (e), and (f) correspond to 0634:15 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 17 and 19, except for 27 Nov cirrus anvil. Here, (a), (b), and (c) correspond to 0623:00 UTC and (d), (e), and (f) correspond to 0634:15 UTC.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 18 and 20 except for 27 Nov cirrus anvil. Here, (a), (b) and (c) correspond to a height of 12.3 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 18 and 20 except for 27 Nov cirrus anvil. Here, (a), (b) and (c) correspond to a height of 12.3 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Same as Figs. 18 and 20 except for 27 Nov cirrus anvil. Here, (a), (b) and (c) correspond to a height of 12.3 km and (d), (e), and (f) correspond to height of 12.5 km.

Citation: Journal of Applied Meteorology 38, 1; 10.1175/1520-0450(1999)038<0005:PSEIIP>2.0.CO;2

Here, |*K _{w}*|

^{2}for water at 0°C and |

*K*|

_{i}^{2}for solid ice are given for several radar bands. The last row shows the ratio of these two quantities that is used to convert between ice equivalent reflectivity and water equivalent reflectivity according to Eq. (20).

One-way extinction rates as a function of frequency calculated for rain using a Marshall–Palmer (1948) drop size distribution and an exponential distribution for snow described by Gunn and Marshall (1958).

Summary of MCTEX water vapor and oxygen absorption measurements from combined radiosonde, dual-channel radiometer, and surface observations. The mean surface humidity measured at Garden Point, Melville Island, is 19.9 g m^{−3} and the mean water vapor path is 4.98 cm. Zenith absorption is calculated by integrating the absorption rate from the surface to a height of 15 km.

Radar parameters.

*Z _{i}*–IMC relationships described in (Sassen 1987). IMC is in g m

^{3}and

*Z*is in mm

_{i}^{6}m

^{−3}.

Table A1. Density and dimensional relationships for several particle geometries corresponding to the DWR models in Fig. 7.