Freezing-Level Estimation with Polarimetric Radar

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EDWARD A. BRANDES AND KYOKO IKEDA

ABSTRACT
A simple empirical procedure for determining freezing levels with polarimetric radar measurements is described. The algorithm takes advantage of the strong melting-layer signatures and the redundancy provided by the suite of polarimetric radar measurements--in particular, radar reflectivity, linear depolarization ratio, and cross-correlation coefficient. Freezing-level designations can be made with all volumetric scanning strategies. Application to uniform (stratiform) precipitation within 60 km of the radar and with brightband reflectivity maxima of greater than 25 dBZ suggests an accuracy of 100-200 m.

1. INTRODUCTION
    An automated method for estimating freezing-level heights would have several benefits.(FN1) Freezing-level heights are required for designating potential icing layers in storms and for verifying their prediction with numerical forecast models. Knowing the height of the freezing level is important for determining whether precipitation observed by radar is rain or snow and is necessary for its quantification. Alerts could be provided for those engaged in snow removal and aircraft-deicing operations. The freezing-level height is also a component of polarimetric radar-based algorithms for general hydrometeor classification (e.g., Vivekanandan et al. 1999).
    The long-recognized association between melting hydrometeors and bright bands in vertical profiles of radar reflectivity measurements can be exploited to estimate 0°C heights (e.g., Mittermaier and Illingworth 2003; Gourley and Calvert 2003). Retrievals should improve if supplemental information is considered. White et al. (2002) describe a procedure using profiler measurements of radar reflectivity and vertical velocity for determining the 0°C level. The increase in hydrometeor terminal velocity that accompanies melting is used to confirm designations based on reflectivity and may permit freezing-level designations in situations in which strong reflectivity signatures are not observed.
    Here we examine the utility of polarimetric measurements for designating freezing levels. Measurements of differential reflectivity, linear depolarization ratio, cross-correlation coefficient, and differential propagation phase are particularly sensitive to the presence of large, wetted particles characteristic of melting layers. In comparison with radar reflectivity, they can give more precise definition of melting layers and may exhibit melting-layer signatures in convective precipitation.
    Although the problems are closely related, interest here is in freezing-level designation rather than melting-layer determination. In many respects the latter problem is more difficult. Radar-derived melting-layer characteristics depend on the habits and size of the frozen particles present. Melting-layer upper boundaries are often inferred by a change in curvature of the reflectivity profile (e.g., Fabry and Zawadzki 1995). However, reflectivity increases can also arise from particle growth and aggregation. It is not unusual for reflectivity increases to begin hundreds of meters above the freezing level and well before melting-layer signatures occur in other polarimetric measurements. Although lower boundaries of melting layers appear to be sharply defined in radar measurements, the boundary is probably indistinct. Large ice particles associated with stratiform precipitation can fall to temperatures of 5°C and higher before melting is complete (e.g., Willis and Heymsfield 1989). However, hydrometeors in the final stages of melting are virtually indistinguishable from raindrops. Radar depictions of melting layers are strongly influenced by beam smoothing (Sánchez-Diezma et al. 2000), and their precise determination with operational radars would require a deconvolution of the spatially distributed measurements.
    Here the problem of freezing-level detection is reduced to finding the height at which signatures for the various polarimetric parameters are maximized and then using predetermined statistical relationships between those heights and the freezing level. Retrieval is facilitated by redundancy among polarimetric measurements and their sensitivity to mixed-phase hydrometeors. There is a potential problem. Upper portions of melting layers may exhibit isothermal layers with dry-bulb temperatures close to 0°C (Stewart et al. 1984; Willis and Heymsfield 1989). The total distance over which melting occurs and the statistical relation between radar signatures and the freezing level could be affected. This situation was not observed in the data examined here.
    We begin by describing the properties of polarimetric radar measurements and examine typical profiles. Model parameter profiles are then presented and used to estimate freezing levels for several examples. Estimates are verified with soundings and aircraft observations.

2. POLARIMETRIC MEASUREMENTS
    Polarimetric measurements with strong melting-layer signatures include radar reflectivity Z, differential reflectivity Z[subDR], linear depolarization ratio (LDR), cross-correlation coefficient rho[subHV], and different propagation phase Phi[subDP]. [For detailed descriptions of these parameters, their usage, and typical values for different hydrometeor types, see Doviak and Zrnic (1993, chapter 8)]. Radar reflectivity at horizontal polarization (H) and vertical polarization (V) for a unit volume (mm[sup6] m[sup-3]) are defined as
    Z[subH,V] = lambda[sup4] / pi[sup5] |K|[sup2] [integral][supD[submax]][sub0] sigma[subH,V](D)N(D) dD,
    where lambda is the radar wavelength, K is the dielectric factor of water, sigma[subH,V](D) are the radar cross sections of scatterers at horizontal and vertical polarization, N(D) is the size distribution, and D is the particle equivalent diameter. Reflectivity is most often expressed in "dBZ" (10 × logZ[subH,V]).
    Differential reflectivity (dB) is computed from
    Z[subDR] = 10 × log(Z[subH]/Z[subV]).
    Differential reflectivity is positive (negative) for particles whose major axes are close to horizontal (vertical). Differential reflectivity is zero for particles that are spherical or for aspherical particles with a random distribution of orientations. Raindrops tend to flatten and orient themselves with their major axes close to horizontal. For rain, Z[subDR] ranges from 0.3 to 3 dB. Pristine ice crystals have small axis ratios (vertical dimension divided by horizontal dimension) and high bulk density, and they fall with their major axes near horizontal. Depending on crystal type, Z[subDR] can be 2-5 dB. Aggregates have large axis ratios and low bulk density; Z[subDR] for these particles is small (<0.5 dB).
    Hydrometeors whose principal axes are not aligned with the electrical field of the transmitted radiation cause a small amount of the energy to be depolarized and to appear in the orthogonal direction. The depolarized (cross polar) signal stems primarily from nonspheroidal particles that wobble or tumble as they fall, creating a distribution of orientations (canting angles). Signals are enhanced for wetted and melting particles. LDR (dB) is defined as the logarithm of the ratio of cross-polar and copolar signals; that is,
    LDR = 10 × log(Z[subVH]/Z[subH]),
    where Z[subVH] is the signal received at vertical polarization (cross-polar return) for a transmitted horizontally polarized wave. LDR is small (on the order of -34 to -25 dB, depending on antenna isolation) for rain and dry snow. Wet and melting snow can have an LDR from -15 to -20 dB.
    The cross-correlation coefficient rho[subHV] is computed from reflectivity at horizontal and vertical polarization. This parameter is sensitive to the distribution of particle sizes, axis ratios, and shapes. Theoretical values are ˜0.99 for raindrops and ice crystals. For melting aggregates, rho[subHV] can be less than 0.90. Because rho[subHV] and LDR are both sensitive to the presence of large wetted particles, their melting-layer responses are similar--but of opposite sign.
    The above polarimetric parameters are derived from signal measurements that depend upon backscattering properties of illuminated particles. Radar waves are also subject to attenuation and phase shifts. For an anisotropic medium like raindrops or pristine ice crystals, propagation constants for horizontally and vertically polarized waves differ. Horizontally polarized waves "see" a larger particle cross section and consequently propagate more slowly than do vertically polarized waves. Signals returned to the receiver for the two polarizations exhibit different accumulative phase (time) shifts depending on hydrometeor size, shape, orientation, quantity, and distance from the radar. The differential phase shift Phi[subDP] (°) actually has two components--a propagative component related to the difference in forward-scattering amplitudes and a component related to a backscatter differential phase. The backscatter phase shift can become significant when Mie scatterers (e.g., large melting aggregates) are present. As a consequence, Phi[subDP] measurements can also be used to detect melting layers.
    Radar measurements utilized in this study were obtained with the National Center for Atmospheric Research's S-band dual-polarization radar (S-Pol). To make the LDR measurement, the radar alternately transmits horizontally and vertically polarized electromagnetic waves and simultaneously receives scattered signals at both polarizations (Randall et al. 1997). Operational considerations dictate that the Weather Surveillance Radar-1988 Doppler (WSR-88D) modified for polarimetric measurements will transmit at 45° (slant) polarization and simultaneously receive returned signals at horizontal and vertical polarization (Doviak et al. 2000). The planned configuration precludes the LDR measurement.
    The radar observations were obtained during field programs conducted in various geographical areas. The freezing-level detection algorithm to be described (section 4) has been tested with both warm-season and cold-season precipitation. Experience shows that bright bands can often be detected to radar ranges of 40-60 km and more, depending on precipitation intensity. With increasing distance, beam smoothing causes the thickness of the melting layer to be increasingly overestimated and the magnitude of melting-layer signature extremes to be underestimated (e.g., Sánchez-Diezma et al. 2000). Freezing-level detections may be prevented altogether by signal loss in weak precipitation.
    Freezing-level designations are possible with all volumetric scanning strategies. Vertical cross-sectional scans at constant azimuthal angle are well suited for diagnosis because detailed height information is obtained along the various rays that make up the scan. Designations are also readily made from vertically pointing data, but vertical resolution (gate length) can be an issue with operational radars. Vertical cross sections and vertical scans are not permitted with the WSR-88D, which currently makes measurements in a series of 360° scans consisting of 9 or 14 elevation angles from 0.5° to 19.5°. Sánchez-Diezma et al. (2000) examined the dependence of reconstructed reflectivity profiles on the distribution of antenna elevation scans typical of operational radars. Melting-layer signature maxima can be displaced upward or downward, depending on the distribution of antenna elevation angles relative to the actual freezing level. However, averages of height estimates should be unbiased. An alternate approach is to estimate the freezing level from the distribution of measurements along radar rays at antenna elevation angles that pass through the melting layer.

3. POLARIMETRIC MELTING-LAYER SIGNATURES
    Profiles of radar reflectivity, linear depolarization ratio, cross-correlation coefficient, differential reflectivity, and differential propagation phase are shown in Fig. 1 for a winter storm observed at 1631 UTC 28 November 2001 as part of the Improvement of Microphysical Parameterization through Observational Verification Experiment (IMPROVE) field program conducted in the Oregon Cascade mountains (Stoelinga et al. 2003). [Previous descriptions of polarimetric measurements in melting layers include Bader et al. (1987), Hall et al. (1984), Herzegh and Jameson (1992), and Zrnic et al. (1993).] The profiles were computed at a radar range of 10 km. The data, from orographically forced stratiform precipitation, were collected in vertical cross sections with 1° azimuthal and 0.6° vertical spacing. The measurements were averaged over a radius of 2 km in the horizontal direction and over 0.2 km in the vertical direction. The 3-dB beamwidth (˜0.2 km at 10 km) also contributes to the smoothing. Smoothing artifically increases minima, reduces maxima, and deepens the radar-indicated melting layer. Advection in nonuniform precipitation can cause radar reflectivity profiles to differ considerably from reflectivity along particle trajectories (Fabry and Zawadzki 1995, their Fig. 2). However, the height at which the strongest melting-layer signatures appear should be relatively unaffected. Profiles of LDR and rho[subHV] are insensitive to advection.
    Radar reflectivity in the upper "ice only" precipitation layer (3-7 km) increases toward the ground at a rate of more than 4 dB km[sup-1] (in situ hydrometeor observations are not available for this event). Hydrometeor growth in this layer is thought to be due to vapor deposition and aggregation of small crystals. A rapid increase in radar reflectivity begins at 3.2 km--well above the melting level of 2.47 km determined from three soundings released within 60 km of the radar and having an average release time of 1612 UTC.(FN2) The break point in the reflectivity profile is considerably higher than that for LDR and rho[subHV] (2.7 and 2.6 km, respectively).(FN3) Differences from the measured freezing level respond in part to radar beam and analysis smoothing. However, in this case the higher altitude of the upper break point in the reflectivity trace is thought to be due primarily to a rapid increase in aggregation. Hobbs (1973) presents data showing accelerated ice particle growth by deposition and aggregation as ambient temperatures rise above -5°C and particle stickiness increases. Hence, the profile is illustrative of a potential problem when enhanced reflectivity is used as a melting indicator.
    Differential reflectivity in the ice layer is small and declines from 0.35 dB at 5.7 km to a minimum of 0.13 dB at 2.8 km (well above the 0°C level determined from soundings). Hydrometeors in the uppermost regions of the ice layer either had a greater degree of orientation with respect to the horizon or a greater eccentricity than the particles near the melting layer. At other times (not shown) Z[subDR] was less than 0.15 dB throughout the ice layer. Small Z[subDR] values are due in part to the lower dielectric factor for ice particles and their low bulk density. LDR in the ice layer increases toward the storm top. It is possible that particles at higher elevations had a broader distribution of canting angles. That the signal is not due to noise is supported by the cross-correlation coefficient, which does not decrease significantly with height. Differential propagation phase increases above the melting layer, particularly above approximately 5 km. The increase is also an indication of particle alignment.
    Melting and resultant changes in dielectric factor, particle fall speed, particle shape and size, and aggregation are factors that cause reflectivity bright bands (Battan 1973; Fabry and Zawadzki 1995). The total increase in radar reflectivity from the maximum curvature point in the reflectivity profile (23 dBZ) to the brightband maximum (40 dBZ, 2.3 km) is 17 dB. Part of the increase comes from changes in the density and dielectric factor of the hydrometeors (6.5 dB, Fabry and Zawadzki 1995). Enhanced aggregation is also believed to be a significant contributor to the increase in this case.
    Melting also affects the other polarimetric measurements. LDR, rho[subHV], and Phi[subDP] values just above the melting layer are -28 dB, 0.99, and 9.9°, respectively. Rapid changes occur as melting ensues, with LDR increasing to a maximum of -18 dB, rho[subHV] decreasing to a minimum of 0.91 (a melting-layer signature maximum), and Phi[subDP] increasing to 10.7°. Melting signatures for these three variables peak at 2.1 km. The decrease in rho[subHV] is likely a consequence of a greater variety of shapes and axis ratios associated with partly melted particles and the introduction of raindrops. Large LDR implies a distribution of mixed-phase particles with a broad distribution of canting angles, perhaps a result of increased hydrometeor fluttering and spinning. The increase in Phi[subDP] may be due primarily to melting and particle growth through a backscatter differential phase shift. The indicated lower boundary of the melting layer is 1.7 km for reflectivity, 1.6 km for LDR and Phi[subDP], and 1.5 km for rho[subHV]. For this event, the melting-layer signature for Phi[subDP] closely matches that for Z[subH], LDR, and rho[subHV]. In general, however, the measurement is noisy and responds to local gradients of reflectivity (Ryzhkov and Zrnic 1998), and the distribution of measurements through the melting layer may exhibit large fluctuations (e.g., Zrnic et al. 1993). S-Pol measurements on occasion show two maxima in the melting layer with a minimum between them at the height of the reflectivity maximum. Because Phi[subDP] depends on the total pathlength, the global maximum is usually at the storm top or at far distances for measurements at low antenna elevation angles.
    As temperatures rise and melting proceeds, the hydrometeors eventually collapse into raindrops. Particle terminal velocities increase, and faster-falling particles are removed from the region. The reflectivity decrease may also respond to the breakup of partly melted hydrometeors. Reflectivity at the top of the rain layer is ˜28 dBZ. LDR and rho[subHV] take on values close to those in the ice layer.
    Differential reflectivity maxima associated with melting usually occur at lower levels than do reflectivity maxima. This fact suggests that the maximum eccentricity of melting hydrometeors occurs at a lower level in the atmosphere than does their maximum size, assuming that the largest particles are located close to the reflectivity maximum. The height depression of the Z[subDR] maximum is a function of reflectivity magnitude. Mean depressions were typically 0.2 km for bright bands with a reflectivity maximum of 25 dBZ and 0.5-0.6 km for a maximum of 45 dBZ. In comparison, depressions for LDR and rho[subHV] are smaller and more narrowly distributed (more discussion in section 4). S-Pol beams for horizontal and vertical polarization have a small mismatch that can cause spurious Z[subDR] values in regions of strong reflectivity gradient. The minimum in the Z[subDR] trace at 2.8 km is thought to be a manifestation of this problem. Differential reflectivity spikes, like that at 1.9 km, appear in several datasets. At times, as in Fig. 1, the spike is close to the bottom of the melting layer just when other parameters are losing their melting-layer signatures. It is not clear whether the spike is an artifact or representative of the melting process. On occasion, the uppermost regions of storms exhibit Z[subDR] values that are greater than 2 dB, indicative of pristine ice crystals. Such measurements are often transient in our datasets and, like Phi[subDP] measurements in the melting layer, are not easily modeled. Also, there is no Z[subDR] or Phi[subDP] signature at zenith. (This mode is used for "calibrating" the Z[subDR] measurement and determining the error level in Phi[subDP].) Reflectivity is essentially independent of the antenna elevation angle. Simulations conducted by Vivekanandan et al. (1993) show that LDR and rho[subHV] have an elevation-angle dependence for precipitation in which ice crystals and aggregates contribute equally to reflectivity. Assumed axis ratios were 0.2 and 0.8, respectively. However, observed profiles obtained from low-elevation and vertically pointing scans show little difference, an indication that the observed particles were more spherical on average. Although Z[subDR] and Phi[subDP] may provide useful information in some situations, the task of identifying those events is made unnecessary by robust signatures for Z[subH], LDR, and rho[subHV]. As a consequence, only the latter parameters are used for designating freezing levels in this study.

4. MODEL AND IMPLEMENTATION
    Model freezing-level profiles for reflectivity, linear depolarization ratio, and cross-correlation coefficient are shown in Fig. 2. The profiles are based on statistics gathered from several field programs conducted in different climatic regimes and are representative of average profiles obtained with operational radars at ranges of less than 60 km. The profiles are used to determine if the basic melting-layer patterns described in section 3 exist. The procedure is insensitive to the profile details; rather, it is the overall correspondence between the observed and model profiles and the presence of melting-layer signature maxima that is important. For illustration purposes, a freezing-level height of 4.5 km is assumed. For reflectivity, a constant rain-layer value is also assumed (27 dBZ). Within the lower portion of the bright band, reflectivity increases with height to 35 dBZ over a depth of 0.7 km and then decreases to 23 dBZ over the next 0.5 km. Above the upper break point (4.7 km), the reflectivity decreases at a rate of 6 dB km[sup-1]. The upper break point is taken to be above the 0°C level to allow for aggregation that begins before melting. The reflectivity maximum is shown to be 0.3 km below the 0°C level. Although a depression of 0.3 km is a good mean value, the actual offset depends on precipitation intensity. Profiles presented by Fabry and Zawadzki (1995, their Fig. 10) disclose that the brightband reflectivity maximum lowers and the melting layer thickens as the reflectivity increases in magnitude. The profiles were constructed from 350 h of measurements obtained with a vertically pointing radar at a resolution of 15 m. They assume that the upper break point in the reflectivity profile coincides with the 0°C level. Depressions for their profiles were determined and plotted versus maximum brightband reflectivity in Fig. 3a. A conservative fit to the depressions is
    d[sub1] = 0.0886 - 0.000 400Z[subH] + 0.000 112Z[sup2][subH], (1)
    where d[sub1] (km) is the depression of the reflectivity maximum from the estimated 0°C level and Z[subH] is in dBZ. Also plotted in Fig. 3a is a sampling of depressions from field experiments conducted in Florida and Brazil for which sounding information was available. The depressions are based on measurements from constant antenna-elevation-angle and constant azimuthal-angle scans in a ratio of 6:7. The depressions, which are shown regardless of correlation with the model, agree nicely in the mean with the measurements of Fabry and Zawadzki (1995), an indication that aggregation above the 0°C level does not dominate the relation.
    Modeled values of LDR and rho[subHV] are taken as constant and equal in the respective ice-only and rain-only layers. The assumed values are -25 dB and 0.993. Model melting-layer signatures for LDR and rho[subHV] are confined to the 3.5-4.5-km layer. Melting-layer signature extremes (-16 dB for LDR and 0.92 for rho[subHV]) are assumed to reside at the middle of the layer or 0.5 km below the 0°C level on average and 0.2 km below the reflectivity brightband maximum.
    Melting-layer depressions for LDR maxima and rho[subV] minima from the 0°C level are also functions of precipitation intensity. Height depressions tend to be larger than for radar reflectivity. This is illustrated in Fig. 3b, where the depression of LDR and rho[subHV] melting-layer extremes from the brightband reflectivity maximum are shown. Mean heights for LDR and rho[subHV] melting-layer extremes from constant-elevation scans are usually within 0.1 km; hence, the depressions were assumed to be equal. The relationship between the depression of the LDR and rho[subHV] melting-layer extremes from the reflectivity maximum is not quite linear. A polynomial fit was applied to the measurements (Fig. 3b), and the result was added to (1) to obtain
    d[sub2.3] = 0.121 + 0.000 445Z[subH] + 0.0002Z[sup2][subH], (2)
    where d[sub2] and d[sub3] are the depressions for the LDR and rho[subHV] melting-layer signature maxima from the 0°C level.
    For measurements collected as a stepped series of constant-elevation scans or as vertical cross sections, the algorithm constructs measurements profiles at 0.1 km resolution. This vertical spacing readily reproduces prominent profile features (e.g., Fig. 1). In general, all measurements within a 3-km horizontal radius of the desired profile location are averaged. The vertical averaging interval is 0.2 km. For vertically pointing data, the range spacing of the radar (usually 0.15 km) is used, and the radar data are averaged in time over one antenna rotation of 360°.
    Operational radars scan by rotating the antenna through full 360° sweeps, incrementing the elevation angle through a tilt series. Vertical resolution may be too coarse to construct a profile of radar parameters accurately. However, radar measurements from beams that pass through the melting level at high elevation angles in stratiform precipitation often show well-defined bright bands. Vertical profiles with resolutions of a few tens of meters can be constructed from these data by plotting the measurements as a function of height rather than distance.
    For Z[subH] and LDR all averaging is done in the logarithmic domain to reduce the impact of measurements with high reflectivity. Averaging for rho[subHV] is in linear space. Several thresholds are applied. Only those portions of profiles with radar reflectivity of more than 10 dBZ and only those profiles with a reflectivity maximum of more than 25 dBZ are retained for analysis. These constraints remove measurements associated with weak precipitation and those potentially influenced by system noise. Also, searches for freezing levels are conducted only over atmospheric depths at which they are climatologically likely. For winter storms in the Oregon Cascade mountains, the limits were 1-4 km. (A limit lower than 1 km was preferred but was prevented by an extended ground-clutter problem). For summer storms in Florida, the height limits were 3-6 km.
    The model profiles are used to verify that the expected melting-layer signatures are present through pattern matching. For example, the height of the radar-observed reflectivity maximum is determined and the model profile overlaid such that the height of the observed reflectivity maximum coincides with the model reflectivity maximum. A data file is then generated that consists of paired observed and model reflectivity values at heights for which measurements are available. The linear correlation coefficient rho[sub1] is computed from the paired reflectivities as a measure of how closely the observed profile matches the model. The procedure is repeated for LDR and rho[subHV] by again matching the heights of melting-layer signature extremes. For Fig. 1, rho[sub1](Z[subH]) = 0.91, rho[sub2](LDR) = 0.95, and rho[sub3](rho[subHV]) = 0.97. Observed profiles with a correlation of more than 0.7, that is, profiles with at least 50% of their variance explained by the model, are retained. The estimated freezing-level, height is simply
    h[subi] = htmax[subi] + d[subi],
    where i is the parameter index, htmax, is the height at which the signature maximum appears, and d[subi] is the estimated depression from the freezing level. For Fig. 1, h[sub1] = 2.55 km and h[sub2] = h[sub3] = 2.57 km. A consensus (weighted) height is determined with
    h[subfzlv] = [Graphic Character Omitted] h[subi]rho[sup2][subi] / [Graphic Character Omitted] rho[sup2][subi].
    The summation is for the n parameters that meet the correlation criterion. Profiles with higher correlations have greater weight, and the final precision of the consensus freezing-level estimate will, in general, be better than 100 m. For Fig. 1, h[subfzlv] = 2.56 km. As a measure of scatter, the standard deviation sigma[subh] is computed from
    sigma[subh] = [root]1 / n [Graphic Character Omitted] (h[subi] - h[subfzlv])[sup2].
    Usually, sigma[subh] is nonzero, and the parameter serves as a confidence factor for the retrieval.
    Note that the correlation is independent of systematic differences between the observed and model profiles. Further, correlations are relatively insensitive to profile details, in particular for LDR and rho[subHV], because the correlation magnitude is dictated largely by the matching of the signature maxima. By keying on profile extremes, the need to retrieve the details of the melting-layer signatures by a deconvolution of the smoothed radar measurements is avoided.

5. RESULTS

A. EXAMPLES
    Freezing-level retrievals on a 5-km grid determined from vertical cross-sectional scans for the Cascades storm at 1538 UTC are shown in Fig. 4. The upper number to the right of the grid point is the consensus height estimate. The central number is the standard deviation of the estimates, and the lower number is the number of parameter profiles from which the consensus height is estimated. Freezing-level designations are made to a range of 30 km. The spread in consensus designations is narrow. The largest standard deviations occur in regions with ground clutter and weak precipitation. Inspection of the estimated heights reveals that the highest freezing levels are close to the radar and that the freezing level lowers to the east (right).
    Trends in the designated freezing levels for the three radar parameters are presented in Fig. 5. Individual data points represent either median or weighted-mean heights determined for the domain shown in Fig. 4. Weights were assigned as in section 4. The decrease in heights beginning at 1800 UTC coincides with passage of a cold front.
    Examination of the scatter about the mean trends shows that for this storm the most repeatable designations are from the LDR parameter. The range in heights at any particular time is about 0.1 km. The series for radar reflectivity shows a number of spurious designations at heights of 1.4-2.0 km. These arise from clutter-contaminated measurements and can easily be removed with a consistency check.
    Freezing-level heights determined from soundings released at Salem, Oregon, (60 km north-northwest of the radar) and at a special sounding site (55 km to the southwest) and from aircraft observations (25 km west of the radar) are also shown in Fig. 5. The comparison is excellent, suggesting that detailed information regarding the 0°C level can be retrieved and that errors are small. For individual parameter designations on 28 November, the percentage of profiles that did not meet the correlation criterion was 37% for reflectivity, 9% for linear depolarization ratio, and 54% for cross-correlation co-efficient.
    A retrieval for a summer stratiform rain event that formed off the east coast of Florida during the "PRECIP98" field program (Brandes et al. 2002) is presented in Fig. 6. The radar was operated in sector mode, and the antenna was stepped through a series of elevation scans. The storm was somewhat stronger than that in Fig. 4, permitting freezing-level designations to distances of 60 km. The actual freezing level, as observed by a research aircraft, is shown. In Fig. 6, two LDR profiles and four [subHV] profiles did not meet the correlation criterion.
    An example of melting-layer signatures for Z[subH], LDR, and rho[subHV] at 6.0° elevation for the dataset in Fig. 6 is presented in Fig. 7. The lower-right panel in Fig. 7 shows retrievals for constant elevation scans at 4.8°, 6.0°, 7.2°, and 9.6°. The data were averaged over azimuthal sectors of 10° at each range bin. Plotted data points are centered on the sector midpoint and radar range corresponding to the designated freezing-level height. For this analysis, one reflectivity and four linear depolarization ratio profiles were rejected. The melting-level pattern, with heights decreasing on average to the east, is similar to that in Fig. 6. There are some differences that stem from the way the data were processed. The input data in each case are the same. However, in Fig. 6 the data are interpolated to a rectangular grid to produce a column of measurements whose vertical resolution is dictated in large part by the spacing between successive elevation scans (1.2°). At a range of 40 km, the vertical distance between scans is about 0.8 km. Melting-layer signature maxima can be displaced up-ward or downward, depending on the distribution of antenna elevation angles relative to the actual freezing level (Sánchez-Diezma et al. 2000).
    The large number of designations that is possible for a precipitation area can be used for evaluation, much like the consistency among individual parameters at a particular location. Figure 8 shows a histogram of freezing-level designations for the individual radar parameters and the consensus estimates for yet another storm system. Estimated freezing-level heights are for the grid shown in Fig. 4 and include all designations within a 30-min window. Some of the variation is undoubtedly caused by meteorological factors. However, the number counts are concentrated in the 2.1-km-height category for each parameter. The dispersion in Fig. 8 can be used to eliminate suspected retrievals. The most consistent freezing-level designations in this case (after removing outliers) are with radar reflectivity.

B. VERIFICATION
    A comparison of algorithm freezing-level designations and in situ measurements made by research aircraft and local temperature soundings in presented in Fig. 9. The available verification data for IMPROVE are from soundings located 25-60 km from the radar and not in the region of volumetric sampling. Comparisons labeled as PRECIP98 are based on coincident aircraft and radar measurements. The dataset is small, but overall the agreement is good, with little or no bias. The mean absolute error (the departure from the 1:1 line) is 0.14 km, and the root-mean-square error is 0.17 km[sup2].

6. SUMMARY AND CONCLUSIONS
    A simple empirical algorithm for estimating freezing-level heights from polarimetric radar measurements of reflectivity, linear depolarization ratio, and cross-correlation coefficient is described. Designations are based on melting-layer signatures and are determined by matching observed profiles of the polarimetric variables with idealized profiles. Consensus freezing-level designations are made by weighing the goodness of fit for the individual parameters. The algorithm was designed for stratiform precipitation with brightband reflectivity of greater than 25 dBZ. Testing on a limited dataset reveals that the method allows for retrieval of fine detail in the freezing-level surface with an accuracy of 0.1-0.2 km at radar ranges of less than 60 km. Potential problems could arise with events characterized by deep isothermal 0°C layers. However, application to the reflectivity profiles from isothermal events described by Stewart et al. (1984) and Willis and Heymsfield (1989) suggests that the procedure should yield reasonable estimates for the base of the 0°C layer. The response of LDR and rho[subHV] in deep isothermal 0°C layers is yet to be determined. The current algorithm was not designed to determine multiple freezing levels routinely. However, designations can be made if there is sufficient separation between freezing levels and algorithm searches are restricted to heights that contain only one freezing level (e.g., Ikeda et al. 2004, manuscript submitted to J. Atmos. Sci.).
ADDED MATERIAL
    EDWARD A. BRANDES AND KYOKO IKEDA
    National Center for Atmospheric Research,* Boulder, Colorado
    * The National Center for Atmospheric Research is sponsored by the National Science Foundation.
    Corresponding author address: Dr. Edward A. Brandes, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.
    E-mail: brandes@ncar.ucar.edu
    Acknowledgments. We thank Robert Rilling and Jean Hurst for preparing the numerous radar data tapes used in the analysis and Drs. Marcia Politovich and Guifu Zhang for constructive comments on the manuscript. This research is in response to requirements of and funding from the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.
FIG. 1. Profiles of polarimetric measurements (Z[subH], LDR, rho[subHV], Z[subDR], and Phi[subDP]). The estimated 0°C level (2.47 km) is shown by a horizontal line. Heights are above mean sea level.
FIG. 2. Model profiles (Z[subH], LDR, and rho[subHV]). The assumed freezing level (4.5 km) is shown.
FIG. 3. Height depressions of (a) maximum brightband Z[subH] from the 0°C level and (b) maximum melting-layering signatures for LDR and rho[subHV] from the brightband reflectivity maximum.
FIG. 4. Retrieved 0°C heights. The grid spacing is 5 km. The upper (boldface) number is the consensus freezing-level height (km MSL), the middle number is rho[subh] (km), and the lower number is the number of parameters with rho > 0.7. The reflectivity pattern at 1.5° elevation is underlaid.
FIG. 5. Time series of freezing-level designations for radar reflectivity, linear depolarization ratio, and cross-correlation coefficient. Observed freezing levels deduced from soundings and aircraft are superimposed.
FIG. 6. Retrieved freezing-level heights, as in Fig. 4, but for a summer stratiform rain event. The grid spacing is 10 km; the reflectivity pattern at 1.2° elevation is underlaid. An aircraft measurement of the freezing-level height and its location (dot) are shown.
FIG. 7. Measurements of Z[subH], LDR, and [subHV] at 6° antenna elevation for the dataset in Fig. 6. The lower-right panel shows retrieved freezing-level heights, as in Fig. 6, except computed from measurements at 4.8°, 6.0°, 7.2°, and 9.6° antenna elevation.
FIG. 8. Histograms of freezing-level designations based on individual radar parameters and the consensus estimate.
FIG. 8. Algorithm-retrieved freezing levels for winter (IMPROVE) and summer (PRECIP98) storms plotted against in situ measurements.

FOOTNOTES
1 The freezing level is defined (Glickman 2000) as the lowest level in the atmosphere at which the temperature, measured by a thermometer exposed to the air, is 0°C. It is recognized that cloud droplets may supercool to much lower temperatures before freezing and that the melting of hydrometeors may begin at a slightly warmer air temperature because of the wet-bulb effect.
2 All heights are above mean sea level (MSL).
3 Break points correspond to changes in the profile curvature as used, for example, by Fabry and Zawadzki (1995) to delimit reflectivity bright bands.

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