Versus Ice-Phase Clouds Using Dual-Polarization Radar with Application to Detection of Aircraft Icing Regions*

920 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

Discrimination of Mixed- versus Ice-Phase Clouds Using Dual-Polarization Radar with Application to Detection of Aircraft Icing Regions*

DAVID M. PLUMMER Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

SABINE GO¨ KE Department of Physics, University of Helsinki, Helsinki, Finland

ROBERT M. RAUBER AND LARRY DI GIROLAMO Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

(Manuscript received 29 April 2009, in ﬁnal form 30 October 2009)

ABSTRACT

Dual-polarization radar measurements and in situ measurements of supercooled liquid water and ice particles within orographic cloud systems are used to develop probabilistic criteria for identifying mixed- phase versus ice-phase regions of sub-08C clouds. The motivation for this study is the development of quantitative criteria for identiﬁcation of potential aircraft icing conditions in clouds using polarization radar. The measurements were obtained during the Mesoscale Alpine Programme (MAP) with the National Center for Atmospheric Research S-band dual-polarization Doppler radar (S-Pol) and Electra aircraft. The comparison of the radar and aircraft measurements required the development of an automated algorithm to match radar and aircraft observations in time and space. This algorithm is described, and evaluations are presented to verify its accuracy. Three polarization radar parameters, the radar reﬂectivity factor at horizontal polarization

(ZH), the differential reflectivity (ZDR), and the specific differential phase (KDP), are first separately shown to be statistically distinguishable between conditions in mixed- and ice-phase clouds, even when an estimate of measurement uncertainty is included. Probability distributions for discrimination of mixed-phase versus ice- phase clouds are then developed using the matched radar and aircraft measurements. The probability distributions correspond well to a basic physical understanding of ice particle growth by riming and vapor deposition, both of which may occur in mixed-phase conditions. To the extent that the probability distributions derived for the MAP orographic clouds can be applied to other cloud systems, they provide a simple tool for warning aircraft of the likelihood that supercooled water may be encountered in regions of clouds.

1. Introduction coalescence through the supercooled ‘‘warm rain’’ process (Huffman and Norman 1988; Pobanz et al. 1994; Supercooled liquid water (SLW) is an important Cober et al. 1996; Rauber et al. 2000). Supercooled component of cloud systems and precipitation de- water’s potential to enhance ice growth through the velopment. Ice crystals, in the presence of SLW, can accrete supercooled droplets and grow more rapidly into Bergeron–Findeisen process and accretion has made it precipitation-size particles. When ice particles are not an important topic of research for weather modiﬁcation, present, supercooled drops may grow by collision and which has led to a large number of publications char- acterizing the SLW distribution in orographic and stratiform cloud systems (e.g., Hill 1980; Heggli et al. * Supplemental material related to this paper is available at the 1983; Rauber et al. 1986; Rauber and Grant 1986, 1987; Journals Online Web site: http://dx.doi.org/10.1175/2009JAMC2267.s1. Heggli and Rauber 1988; Rauber 1992; Sassen and Zhao 1993). Most modern research related to SLW is con- Corresponding author address: David M. Plummer, Depart- cerned with particle growth processes in mixed-phase ment of Atmospheric Sciences, University of Illinois at Urbana– Champaign, 105 S. Gregory St., Urbana, IL 61801. clouds (e.g., Young et al. 2000; Korolev and Mazin 2003; E-mail: [email protected] McFarquhar and Cober 2004; Korolev and Isaac 2006).

DOI: 10.1175/2009JAMC2267.1

Ó 2010 American Meteorological Society MAY 2010 P L U M M E R E T A L . 921

Supercooled liquid water has a practical importance Hudak et al. (2002) showed scatterplots of ZDR and KDP to the aviation community. Supercooled droplets can measurements in low- and high-reflectivity regions of freeze on contact with exposed airframe surfaces, re- winter stratiform clouds over southeastern Canada. They ducing an aircraft’s aerodynamic properties and degrad- concluded that these parameters had potential for dis- ing its mechanical controls. Detection of icing conditions tinguishing mixed-phase from glaciated conditions. Wolde is of significant importance to aviation because of this et al. (2003) computed radar polarimetric parameters danger. Numerous studies have been devoted to the from in situ measurements in mixed-phase and ice characterization and identification of cloud systems for clouds and compared those with radar measurements of which icing is a hazard (e.g., Sand et al. 1984; Rasmussen ZH, ZDR, and KDP. They found that ZH computed from et al. 1992; Pobanz et al. 1994; Politovitch and Bernstein in situ data agreed with the radar data, but ZDR showed 1995; Cober et al. 2001b). agreement only in selected regions of the clouds. Field

Remote detection of SLW is important both for mi- et al. (2004) examined the ZH and ZDR signatures of crophysical studies and for aviation safety. Numerous mixed-phase and ice clouds for three aircraft flights, research efforts have been made to detect SLW remotely. finding that large ZDR values (.2 dB) above the melting Rauber et al. (1986) and Heggli and Rauber (1988), for layer were coincident with SLW, provided the particles example, used a microwave radiometer to detect varia- were growing by vapor deposition. However, ZDR de- tions in SLW content in orographic storms over northern creased when ice particles grew by aggregation, al- Colorado and the Sierra Nevada. Vivekanandan et al. though SLW sometimes was still present. (1999a) used differences in reflectivity and attenuation As suggested by this previous work, improved un- between dual-wavelength radar observations to identify derstanding of SLW detection with polarization radars water content for both liquid and mixed-phase clouds. may be achieved by employing combined aircraft and Zawadzki et al. (2001) used a vertically pointing X-band radar measurements. Orographic cloud systems, in par- Doppler radar to infer the presence of supercooled drop- ticular, provide an excellent opportunity to make these lets from observed bimodal Doppler spectra. Hogan et al. observations. Topographic features provide a constant (2003) combined lidar and radar observations to identify obstacle to lower-tropospheric flow, with the potential concentrations of supercooled droplets even when radar for long-term, consistent lifting of air. This vertical forc- observations were dominated by larger frozen particles. ing can result in persistent SLW production as lifted air Several field campaigns [Winter Icing and Storms Proj- cools and saturates (e.g., Hill 1980; Rauber et al. 1986; ect (WISP), Mount Washington Icing Sensors Project Heggli and Rauber 1988). Orographic storms provide (MWISP), and the Alliance Icing Research Studies I and the opportunity to make SLW observations over lengthy II (AIRS I and II); see Rasmussen et al. 1992; Ryerson time periods. However, even with large-scale consis- et al. 2000; Isaac et al. 2001; Isaac et al. 2005] have also tency between cases, orographic cloud systems do vary used short-wavelength (W, Ka, and X band) radars in structure and precipitation intensity. Observations to detect icing conditions (e.g., Reinking et al. 1997; from the Mediterranean Alps, the American Cascades, Vivekanandan et al. 2001; Reinking et al. 2002; Wolde the Sierra Nevada, and the Rocky Mountains, among et al. 2006; Williams and Vivekanandan 2007). others, have shown variations in precipitation develop- All of these studies employ instrumentation that is not ment and SLW production associated with local to- generally available for operational use. Polarization- pography and airflow (e.g., Heggli and Rauber 1988; capable radar systems, however, will provide a poten- Rasmussen et al. 1995; Medina and Houze 2003; Houze tially useful source of data for remote SLW identification and Medina 2005; Woods et al. 2005; Ikeda et al. 2007). as they become more widely available when the Weather In this paper, dual-polarization radar measurements Surveillance Radar-1988 Doppler (WSR-88D) is upgraded and in situ measurements of SLW and ice particles within (Ryzhkov et al. 2005). Various automated hydrome- orographic cloud systems are used to develop probabi- teor classification algorithms have been developed to listic criteria for identifying mixed-phase versus ice-phase use the variables measured by polarization radar sys- regions of sub-08C clouds. We first present an algorithm tems (e.g., Vivekanandan et al. 1999b; Straka et al. 2000; to obtain ‘‘matched’’ observations for direct comparison Liu and Chandrasekar 2000). Although some classifi- of aircraft and radar measurements. We then show that cation schemes include a category for SLW, the polari- polarization radar data can be used to statistically dis- zation signatures currently used to classify this particle tinguish between mixed-phase clouds (which, by defi- type feature significant ambiguities, and in situ verifi- nition, contain SLW) and ice-phase clouds. Finally, we cation of SLW with S-band radar has been very limited. develop probabilistic criteria for SLW detection that are The only studies we are aware of are those of Hudak consistent with a basic physical understanding of cloud et al. (2002), Wolde et al. (2003), and Field et al. (2004). microphysical processes. 922 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

2. Data sources The NCAR S-Pol radar (Keeler et al. 2000) was used during MAP. The radar primarily performed plan po- The radar and aircraft data used in this study were sition indicator (PPI) scans, but also performed RHI collected during the Mesoscale Alpine Programme (MAP), scans occasionally. Data for both scan types were used a multinational field campaign involving intensive ob- for this study. This 10-cm wavelength, dual polarization servations of cloud systems influenced by the topogra- radar system was situated in the Lago Maggiore region phy of the Mediterranean Alps (Bougeault et al. 2001). just south of the Italian Alps (see Fig. 1 of Medina and Cloud systems moving through this region have the po- Houze 2003). The radar transmits radiation alternately tential to be modified by the Alpine topography. Lower- between horizontal and vertical polarizations, allowing tropospheric flow often transports warm, moist air from measurement of several radar parameters including the the Mediterranean into this area, and enhanced pre- radar reflectivity factor at horizontal polarization (ZH), cipitation development can result if a significant upslope the Doppler velocity, the differential reflectivity (ZDR), flow component is present. A variety of weather systems the specific differential phase (KDP), and the correlation occurred during MAP. Widespread stratiform precip- coefficient between copolar horizontally and vertically itation was observed during the cases used here, with polarized echoes (rHV) [see Zrnic´ and Ryzhkov (1999), embedded convection also apparent in some cases de- Straka et al. (2000), or Bringi and Chandrasekar (2001) pending on the atmospheric stability and the orientation for definitions of these variables]. Because of a mis- of the lower-tropospheric flow relative to local topog- aligned feedhorn during the first part of the project, the raphy. ‘‘Unblocked’’ flow occurred in some cases when majority of the linear depolarization ratio (LDR) data the low-level flow was forced most directly over the to- were of low quality and were not used in this study pography, enhancing vertical forcing, which resulted in (http://www.eol.ucar.edu/rsf/MAP/SPOL/). the potential for significant and long-lasting SLW pro- The hydrometeor classification scheme developed by duction (Peterson et al. 1991; Medina and Houze 2003). Vivekanandan et al. (1999b) was operational with the Several cases during MAP resulted in significant pre- NCAR S-Pol during MAP. This scheme features a cat- cipitation developing over short time periods. This is egory for SLW, although the category is particularly likely the result of substantial riming in areas of enhanced ambiguous, as the polarization signatures used to iden- SLW production, as described by Medina and Houze tify this particle type significantly overlap with those (2003), Houze and Medina (2005), Lascaux et al. (2006), for several other particle categories. As described by and others. Vivekanandan et al. (1999b), the polarization signatures The data for this study were from MAP intensive ob- used to classify the ‘‘irregular ice crystal’’ type are par- servation periods (IOPs) that concentrated on orographic ticularly similar to those used to classify SLW. The data cloud systems that produced precipitation. The National used in this study provide an opportunity to further Center for Atmospheric Research (NCAR) Electra ob- understand SLW’s polarization signatures and decrease tained measurements within the S-band dual-polarization these ambiguities in its identification. Doppler radar (S-Pol) observation area for IOPs 2b, 3, 4, The NCAR Electra research aircraft transected many 6, 7, 9, and 15. IOPs 7 and 9 did not have Rosemount of the cloud systems observed by S-Pol. This study used probe data, which were required for the detection of SLW, measurements from several aircraft-based instruments. and were therefore rejected for this study. Data from all The output from the Rosemount Model 871 icing de- of the flights during the five remaining IOPs were used. tector was used as a binary measure of the presence of The synoptic characteristics of each IOP have been SLW. The LWC detection threshold of this instrument documented by the science directors involved in MAP, was calculated by Cober et al. (2001a) to be 0.007 6 with more detailed discussion of most IOPs in the pub- 0.01 g m23 for an aircraft traveling at 97 6 10 m s21. lished literature. IOP 2b has been extensively examined, Baumgardner and Rodi (1989) show that an LWC of for example, by Medina and Houze (2003), Rotunno and 0.01 g m23 is detectable over a 500-m distance, and an Ferretti (2003), and Chiao et al. (2004). IOP 3 has been LWC of 0.05 g m23 over a 100-m distance. For all radar described by Pujol et al. (2005) and Rotunno and Houze observations described below, the beamwidth exceeded (2007), and IOP 4 by Pradier et al. (2004) and Rotunno 500 m, so the detection thresholds were acceptable for and Houze (2007). IOP 6 has less documentation in the this study. Particle Measuring Systems two-dimensional formal literature, but details of the synoptic forcing can (2D-C, 2D-P) optical array probes (OAP; Knollenberg be found from Internet sources (see http://www.atmos. 1970) were used as a binary measure of the presence of washington.edu/gcg/MG/MAP/iop_summ.html). IOP 15 ice. The OAP images in all regions of the flights where has been described by Buzzi et al. (2003) and Rotunno the Rosemount probe was active were visually examined and Houze (2007). to identify predominant particle habits and to determine MAY 2010 P L U M M E R E T A L . 923 if droplets or distinctly rimed quasi-spherical ice particles signal retrieval was disrupted. Positional data were also were present. The binary measures (rather than concen- calculated using the Electra’s Inertial Reference System trations or size distribution parameters) were used to be (IRS) to improve accuracy. The IRS data were more consistent with the binary nature of SLW detection ex- continuous than the GPS measurements but could ac- pected to be used operationally with polarization radar. cumulate large errors over time. The two sets of posi- tions were filtered and combined, retaining the accuracy from the GPS but using the continuity of the IRS when 3. Automated aircraft–radar matching algorithm the GPS featured data spikes or dropouts (Miller and The purpose of this section is to describe the method Friesen 1987). All of these navigation corrections were developed to locate matched observations from the Elec- done by NCAR prior to release of the MAP datasets. tra aircraft and the S-Pol radar. This problem is nontrivial. The aircraft position in radar coordinates was obtained Radars observe large spatial volumes, while in situ ob- using the coordinate transformation described in the ap- servations are essentially point measurements. The rel- pendix. The uncertainty in the aircraft position in radar ative locations of both measurements must be taken into coordinates was determined by considering ‘‘skinpaints,’’ account before comparisons can be made between the which occur when an aircraft is within a radar pulse two. The simplest solution would be to use exactly col- volume. Skinpaints appear on a radar display as a few located observations, where the aircraft is located within pixels with large radar reflectivity factor values (Fig. 1). a radar pulse volume. Unfortunately, the radar obser- In this paper, a 40-dBZ minimum threshold was used. vations will be contaminated by the aircraft’s radar echo. Typically, the radar pixel containing the aircraft is the This contamination has been avoided (e.g., Plank et al. most prominent and features a larger reflectivity value 1980) by using radar measurements from one radar than those surrounding it. Sometimes neighboring pixels range gate away from the contaminated echo. A second also exhibit high reflectivity, a likely result of sidelobe issue, however, is the number of data points available for contamination. The positioning error was determined comparisons. Unless radar scan strategies and aircraft by considering 64 points in the MAP radar data where flight paths are specifically designed to sample the same skinpaints were clearly identifiable. volumes, collocated data points are rare. The absence of For each of these skinpaints, the difference in position sufficient data is a problem when in situ measurements between the center of the radar range gate and the air- are used to verify radar measurements (e.g., Ryzhkov craft was calculated in terms of range, elevation, azimuth, et al. 1998; Vivekanandan et al. 1999b). and absolute distance, as a fraction of the radar volume The method for locating matched data points pre- radius (Fig. 2). The mean and standard deviation of the sented here is an automated method applicable to large range differences were 278.9 6 81.2 m. The maximum datasets. The procedure transforms the aircraft’s posi- radar range resolution possible is 75 m for the S-Pol, half tion (initially in elevation, latitude, and longitude) into of its range gate length during MAP. The mean range the radar’s coordinate system (range, elevation, and azi- error is only slightly larger than this value, and the range muth). Mean particle trajectories are calculated forward error is still approximately within one range gate when and backward in time from the in situ measurements, the standard deviation is included, suggesting good ac- increasing the number of possible matches available be- curacy with respect to range. The elevation and azimuth tween the datasets. angle differences (Figs. 2b,c) were 20.20 6 0.368 and The S-Pol data are in spherical coordinates with lo- 0.07 6 0.458, both within the S-Pol’s 0.918 angular res- cations represented by range from the radar, as well as olution. The total distance between the aircraft and the azimuth and elevation angles with respect to 08 north radar pulse volume (Fig. 2d), as a fraction of the radar and 08 elevation, respectively. The Electra data also use beam’s radius, was 1.06 6 0.50. The mean difference was a spherical coordinate system, although locations are only slightly larger than half the radar volume’s radius, calculated with respect to the earth’s center. The air- again similar to the positional uncertainty inherent to craft’s location along its flight path was calculated from the radar data itself. The positional differences calculated Global Positioning System (GPS) satellites in terms of using skinpaints suggest that the coordinate transforma- altitude above ground level and longitude and latitude tion does not introduce significant error in its calcula- with respect to the earth’s center, with 1-s temporal res- tions beyond uncertainties inherent in the data. olution. For field projects that took place prior to May The next step in the matching algorithm was to cal- 2000 (as was the case for MAP), the GPS system fea- culate approximate forward and backward trajectories tured some accuracy issues, with location accuracy po- for the ensemble of particles observed at the position of tentially reduced to 100 m at times. The GPS data can the aircraft. The trajectories were calculated forward and also feature missing or erroneous data if the GPS satellite backward over a 300-s period using the aircraft-observed 924 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. 1. Example of ‘‘skinpaint’’ in PPI scan of radar reﬂectivity factor (dBZ) from MAP IOP 2b, 0612:42 UTC 20 Sep 1999. wind speed and direction and a fall velocity of 0.4 m s21 the maximum separation of the liquid and ice particles for every second that the aircraft was above the freezing would be 240 m. level. The fall velocity was an average of the negligible The matching algorithm calculated the distance be- fall speed for SLW droplets and an estimate of 0.8 m s21 tween the particle ensemble and the nearest radar pulse for ice crystals (e.g., Cronce et al. 2007), with an assumed volume at each point along these trajectories. For this negligibly weak updraft. At 300 s for these velocities, distance calculation, simple geometry shows that the unit

FIG. 2. Positional differences (aircraft 2 radar) in terms of (a) range in meters, (b) azimuth in degrees, (c) elevation in degrees, and (d) perpendicular distance as a fraction of corresponding radar volume radius. MAY 2010 P L U M M E R E T A L . 925

its closest to a radar pulse volume. If dm was within set thresholds (250 m vertically and 1000 m horizontally for this study), the observations were considered matched, allowing the in situ measurements to be compared directly to the radar measurements. These thresholds

are well below the thresholds recommended for dm of 500 m vertically and 3000 m horizontally by Hudak et al. (2004). Inherent in these calculations is the as- sumption that there is no structure on scales smaller than these thresholds, and that while some very thin SLW layers can occur (e.g., altocumulus clouds) these are unlikely to be important for icing. The mean particle trajectory calculations assume that measurements made by the aircraft instrumentation are valid for some distance temporally and spatially about the aircraft’s location. The distance over which in situ measurements are representative of local conditions depends on the inherent variability of the cloud system. The standard deviations of the aircraft-observed horizontal and vertical wind velocities were used as a measure of this local variability. From these, estimates of the maximum positional uncertainty of particles along their trajectories were calculated using the wind variability and a maximum time window of 300 s. These positional FIG. 3. (a) Geometry of radar beam, showing components of unit uncertainties were related to the width of the S-Pol radar vector c, in direction of beam: cx 5 sinuR cosuR, cy 5 cosuR cosuR, beam, which increases with range from the radar. The and cz 5 sinuR, given beam elevation uR, and azimuth fR. (b) Di- width of the radar beam is given by h 5 2r tan(Du /2), agram showing shortest distance d from air parcel and particle R ensemble at location t to radar beam p. where r is the range from the radar and DuR is the angular beamwidth, 0.918. Figure 4 shows the maximum positional uncertainty along the particle trajectories, vector c 5 (cx, cy, cz) in the direction of the radar beam is given by both as the total distance (Fig. 4a) and normalized by the width of the radar beam at the corresponding range of each matched data point (Fig. 4b). The positional un- cx 5 sinfR cosuR, (1) certainty of nearly all matched points was less than the

cy 5 cosfR cosuR, and (2) width of the radar beam. The automated algorithm was applied to ﬁve Electra

cz 5 sinuR, (3) flights, resulting in a set of matched S-Pol and aircraft- based observations. If the Rosemount probe indicated where uR and fR are the beam’s elevation and azimuth SLW across the width or greater in the matched radar (Fig. 3a). As shown in Fig. 3b, the projection p of the pixel, the matched point was classified as containing particles’ location (t) onto the radar beam (unit vector c) SLW. SLW was only assumed to be present if the voltage is given by increased nearly continuously across the distance of the radar pixel. Single voltage spikes were rejected if the p 5 t c (4) voltage returned to its original level after the spike. Otherwise, it was classified as ice only (IO). and the distance (d) between the particles and the radar Figure 5a shows the frequency distribution of the ra- beam at each point is found using dar range for each matched data point. The median qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi range was 62 km. Since the rotation rate of the radar d 5 jjt 2 jjp 2. (5) antenna during MAP was ;108 s21, the sweep velocity of the beam in the azimuthal direction at the median The matching algorithm compared these calculated range was 10.9 km s21. The aircraft’s true airspeed at 21 distances and located the smallest value, dm, for each the matched points was 140 6 9ms ,withanabso- trajectory. At this point, the assumed trajectory was at lute uncertainty of approximately 4 m s21. Since in our 926 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. 5. (a) Range from radar (km) for all matched data points. (b) Observed air temperature (8C) for all matched data points for SLW conditions. (c) Observed air temperature (8C) for all matched data points for IO conditions.

points. SNR was not recorded during MAP, but was calculated as the ratio of the raw power received over the average noise power (2115.9 dBm during MAP). Based on recommendations in Melnikov and Zrnic´

FIG. 4. Maximum positional uncertainty along particle ensemble (2007), a threshold was applied to remove data with trajectory, represented as (a) total distance in kilometers and (b) SNR , 15 dB. Also, based on recommendations from the a fraction of radar beamwidth at corresponding range. NCAR Earth Observing Laboratory (R. Rilling 2009, personal communication), a second threshold was ap- analysis, no more than one matched data point was re- plied, removing data with rHV , 0.92, since intrinsic rHV trieved per second of aircraft data, and the aircraft velocity values should be close to one for observations of mixed- and sweep velocity of the radar were vastly different, the phase clouds (Bringi and Chandrasekar 2001). Thirty- probability that more than one matched data point could one percent of the matched data points were removed occur at the same altitude per radar volume was van- using these filters. Figures 6b–e show the impact of each ishingly small. Thus, we are confident that the matched threshold on the number of observations as a function of data points are independent. SNR. Frequency distributions of each variable were con- The distribution of the temperatures at the matched structed separately for SLW and IO conditions (Fig. 7). data points for SLW and IO is shown in Figs. 5b and 5c. Figures 7a and 7b show the distribution of observed ZH Matched data points from individual flights tended to be for the matched data points. The range of values was concentrated in narrow temperature ranges. However, similar between the SLW and IO observations, with taken together, the observations from all flights featur- nearly all values less than 25 dBZ. The median ZH value ing SLW span the temperature range from 08 to 2228C, was 10.3 dBZ for SLW and 11.0 dBZ for IO. The IO while the IO range was from 08 to 2268C. histogram features a more distinct peak at larger ZH. Distributions for ZDR and KDP are shown in Figs. 7c,d and 7e,f, respectively. While the total range of observed 4. Choice of polarization signatures ZDR values was similar between the two distributions, The polarization radar parameters ZH, ZDR, KDP, and the median value for the SLW observations was 0.01 dB rHV were determined for each of the matched data as compared with 0.20 dB for the IO observations. The points. Figure 6a shows the distribution of rHV versus median KDP value for the SLW observations was 0.018 signal-to-noise ratio (SNR) for all the matched data km21, slightly smaller than the median of 0.068 km21 for MAY 2010 P L U M M E R E T A L . 927

FIG. 6. (a) Correlation coefficient vs SNR (dB) for all matched data points. Horizontal line shows 0.92 threshold for correlation coefficient, and vertical line shows 15-dB threshold for SNR. Frequency histograms of observed SNR values shown for (b) all matched data points, (c) matched data points after 0.92 correlation coefficient threshold applied, (d) matched data points after 15 dB SNR threshold applied, and (e) matched data points after both thresholds applied.

IO observations. Additionally, the distributions show comparing the sum of observed rankings to the distribu- that larger KDP values appeared more frequently when tion of all possible rankings. A ﬁnite number of possible IO conditions were present. Finally, the frequency his- cumulative rankings exists, following a Gaussian distri- tograms for rHV feature a distinct maximum near one bution that is based solely on the total number of data for both the SLW and IO cases (Figs. 6g,h). points used. The observed ranking is transformed into The Mann–Whitney U test (Wilcoxon 1945; Mann a standard Z score using the mean and standard de- and Whitney 1947) uses the cumulative ranking of two viation of the Gaussian ranking distribution, allowing observed distributions to identify the likelihood that a p value to be obtained using common statistical pro- the distributions’ medians are statistically distinct, by cedures (e.g., Wilks 2006). The standardized Z score 928 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. 7. Frequency histograms of observed values of radar reflectivity factor (dBZ) for (a) SLW and (b) IO; differential reflectivity (dB) for (c) SLW and (d) IO; specific differential phase (8 km21) for (e) SLW and (f) IO; correlation coefficient for (g) SLW and (h) IO. MAY 2010 P L U M M E R E T A L . 929 indicates the point at which the observed result appears with the statistical mean estimated for 1000 bootstrap on a normalized Gaussian distribution, which has a mean samples of each variable, creating a distribution of es- of zero and a standard deviation of one. The p value timated population means for the SLW and IO ob- indicates the likelihood of the observed (or any more servations. Figures 8a–c show these distributions. The extreme) result appearing because of chance, and is used distributions of the likely means of ZDR and KDP are here to evaluate the null hypothesis that the medians completely distinct with no overlapping values, with the of the two observed distributions arise from statistically distributions’ means separated by 0.22 dB for ZDR and 21 indistinguishable populations, given the known cumu- 0.06 8 km for KDP. The distributions of the ZH mean lative ranking. The null hypothesis was rejected here for estimates have some overlap, with the distributions’ p values of 0.05 or less, indicating that the observed dis- means separated by 0.23 dBZ. The ZH distributions are tributions were statistically distinct with 95% or greater distinct at the 98% confidence level, lending confidence certainty. that the variables may be used to develop probabilistic The calculations used here follow Hollander and estimates of the presence of SLW and IO conditions. Wolfe (1999). The scores calculated for the SLW and IO The probabilistic application of these results requires an observations of ZH were jZj 5 2 and p 5 0.02, and for in-depth assessment using an independent dataset. This 25 ZDR and KDP were jZj . 19 and p , 10 , indicating that is necessary to determine the degree to which uncer- the medians of the SLW and IO distributions of the tainties in any given set of polarimetric radar measure- three variables are very likely to be statistically distinct. ments affect the usefulness of the algorithm developed. Significantly more matched data points were located for IO conditions than for SLW conditions. To ensure that 5. Supercooled water identification the U test was not biased because one of the distribu- The purpose of this section is to develop a three- tions contained many more data points than the other, dimensional lookup table that provides estimates of the the IO observations were randomly sampled and re- probability that SLW is present in a cloud, given a duced in size to match the number of SLW observations. measurement of Z , Z , and K . For this purpose, The Mann–Whitney U test was applied to these paired H DR DP the matched radar observations were sorted into bins distributions and the results were found to be consistent based on their Z , Z , and K values. The condi- with those using the full sample datasets for the Z and H DR DP DR tional probability of SLW’s occurrence given a known K measurements. The calculated p values for the Z DP H radar signature was calculated using measurements varied between 0.02 and 0.10 depending on the random sample of IO observations chosen. SLWA As a second statistical test of the distinctness of the P(BjA) 5 , (6) ALLA distributions, the bootstrap method (e.g., Efron and Gong 1983; Hesterberg et al. 2005) was used to determine the where P(BjA) is the probability that SLW is present (B) likelihood that the means of the distributions of ZH, given a radar measurement (A), SLWA is the number of ZDR, and KDP were statistically distinct. Bootstrapping measurements of SLW in the binned radar interval A, involves the creation of new sample distributions by and ALLA is the number of measurements of SLW and randomly resampling data from the original observa- IO in A. A variety of bin sizes was tested for these cal- tions. Each bootstrap sample features the same number culations. Larger bins allowed more observations to of data points as the original dataset but is sampled with be used in each probability calculation, increasing the replacement, meaning that any particular observation confidence in the results. However, larger bins also de- may appear more than once within the bootstrap sam- crease the resolution of the calculated probabilities. The ple. The original observations are considered to repre- final bin sizes used to sort the data represent a balance sent a sample from the polarization parameter’s true between these two concerns. Additionally, the results population. A large number of bootstrap samples are shown below are for bins containing 10 or more obser- created to supplement these original observations by vations. Probability values (P) were calculated for all functioning as additional sample estimates of the true bins, but a minimum of 10 observations was used as an population. A statistical parameter (e.g., the mean) can arbitrary threshold for reporting the probability calcu- be calculated for each bootstrap sample. Because the lations. The data are represented in Figs. 9–11. Tabular bootstrap samples function as estimates of the true pop- values of SLWA and ALLA for the data used to construct ulation, the most likely value of the mean and a 95% Figs. 10 and 11 are given in the electronic supplement confidence interval for the mean are easily estimated (http://dx.doi.org/10.1175/2009JAMC2267.s1). (Hesterberg et al. 2005). These bootstrap calculations Between 10 and 30 bin divisions were tested for the were applied to the ZH, ZDR, and KDP observations, ZH and ZDR observations (Figs. 9a–d). The use of fewer 930 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. 8. Bootstrap estimates of statistical mean of (a) radar reflectivity factor (dBZ), (b) differential reflectivity (dB), and (c) specific differential phase (8 km21). SLW shown in dark gray and IO in light gray.

than 15 bins for ZH or ZDR resulted in a very coarse most commonly for ZDR values below 0.3 dB. The ma- resolution for the calculated probabilities, although they jority of P values greater than 0.50 over this ZH range covered a wider range of values (Fig. 9a). Increasing the are associated with slightly negative ZDR values. These number of ZH and ZDR bins (Figs. 9b–d) provided better P values occur for ZDR between 0.0 and 20.3 dB when resolution but reduced the range of bins with at least 10 ZH is below 20 dBZ, but some larger P values are also observations. Based on this analysis, we chose 25 bins for associated with ZDR between 0.0 and 0.3 dB for larger ZH (1.6-dBZ binwidth) and 20 bins for ZDR (0.15-dB ZH values of 20–25 dBZ. With the exception of some binwidth). Additionally, between one and five bins were values associated with negative ZH, P generally decreases tested for KDP. Using three or fewer bins tended to from 0.25 toward zero as ZDR increases above 0.2–0.3 dB. cluster nearly all of the observations into one KDP bin, Figures 11a–c show P for the same ZH and ZDR bins effectively ignoring its contribution to the identification but with observed KDP values divided into four separate 21 criteria. Four bins of 0.1258 km width were used. bins. The first KDP bin, including KDP values less than Figure 10 shows the probability distribution for SLW 20.0758 km21, does not feature usable data after the 10 presence for all KDP. The P values between 0.25 and 0.90 observation per bin threshold was applied, although the occur over the entire range of observed ZH values, but number of observations is reported in tabular form in MAY 2010 P L U M M E R E T A L . 931

FIG. 9. Probability of correct identification of SLW, using (a) 10, (b) 15, (c) 20, and (d) 25 bins each for radar reflectivity factor (dBZ) and differential reflectivity (dB), for all specific differential phase (8 km21) values. Probability values between 0 and 1 are indicated by color scale for bins featuring at least 10 total observations. the electronic supplement cited on the title page. A in Fig. 11c. Although the data coverage is less extensive in majority of the observations is contained in the second Fig. 11c, P values of zero (and one value of 0.07) occur for and third KDP bin divisions, which are from 20.0758 to ZH between 10 and 25 dBZ with ZDR up to 0.7 dB. Ad- 0.0508 km21 and from 0.0508 to 0.1758 km21, respectively ditionally, comparison of Figs. 11a–c reveals a trend as-

(Figs. 11a,b). In Fig. 11a, the largest P values (between sociated with KDP.NearlyallP values greater than 0.50 0.50 and 0.80) are most common for ZH between 5 and occur with near-zero KDP,andP typically decreases for 20 dBZ,withZDR between 20.3 and 0.3 dB. The P values the larger KDP categories. decrease below 0.30 as ZDR increases, particularly for Z . 0.2 dB and Z , 15 dBZ. In Fig. 11b, P values are DR H 6. Discussion signiﬁcantly lower than those shown in Fig. 11a, with only one value above 0.50 and most values below 0.25. The The P values and polarization signatures discussed in

P values above 0.25 are mainly associated with ZH be- the previous section conform to a microphysical un- tween 5 and 15 dBZ and near-zero ZDR. Again, P values derstanding of particle growth mechanisms in mixed- decrease as ZDR increases. For ZH between 5 and 20 dBZ, phase conditions. Particle development is dependent on P decreases from 0.25 to 0 as ZDR increases to ap- the relative concentrations of SLW and ice particles. proximately 0.7 dB. Similarly low P values are apparent Supercooled liquid water droplets alone are typically 932 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. 10. Probability of SLW identifications as in Fig. 9, but using 25 bins for radar reflectivity factor (dBZ) and 20 bins for differential reflectivity (dB). Corresponds to Tables 1 and 2 in the supplemental material cited on the title page. small and result in weak ZH signatures. Because ZDR and KDP are sensitive to particles’ size and mass characteristics in the horizontal and vertical dimensions, these droplets’ roundness results in near-zero ZDR and KDP values (e.g., Doviak and Zrnic´ 1993). However, mixtures of SLW and ice particles result in more distinctive polarization signatures. Ice particles are able to grow by riming when SLW is present in sufficient quantity. Rimed particles dominate the polarization signal compared to the smaller SLW droplets, with larger ZH values possible than for SLW alone. However, riming produces rela- tively rounded ice particles with similar size and mass in their horizontal and vertical dimensions, and so ZDR and KDP values near zero result from these statistically isotropic particles. When mixed-phase conditions exist, crystals grow at the expense of SLW droplets through the Bergeron–Findeisen process. Again, the larger crystals dominate the radar reflectivity, with larger ZH values possible depending on their size, concentration, and den- sity. Additionally, these crystals tend to develop a horizontal orientation relative to the wind, resulting in a larger ZDR and KDP than is evident for rimed particles FIG. 11. Probability of SLW identification as in Fig. 10 but for (a) (e.g., Straka et al. 2000). 21 21 21 20.0758 km # KDP , 0.0508 km , (b) 0.0508 km # KDP , These microphysical characteristics are apparent in the 21 21 0.1758 km ,and(c)KDP $ 0.1758 km .DataforKDP , observed polarization signatures and calculated P values 20.0758 km21 are not shown as all bins featured fewer than 10 described in the previous section. The signatures asso- observations. Corresponds to Tables 5–10 in the supplemental ma- ciated with rimed particle growth are evident in Figs. 10 terial cited on the title page. and 11 for the P values above 0.50. These occur over the range of observed ZH values but for near-zero ZDR,as when SLW is present in sufficient quantities to produce expected in mixed-phase conditions where SLW con- significant riming. Conditions associated primarily with tributes to ice particle riming. These P values are also ice-phase clouds are also evident in these figures. As more evident for near-zero KDP in Fig. 11a than for the ZDR increases in each figure, P values show a distinct larger KDP values in Figs. 11b and 11c, again as expected decrease. Also, P values significantly decrease as KDP MAY 2010 P L U M M E R E T A L . 933 becomes larger, as shown in Figs. 11a–c. The calculated P values and observed signatures are consistent with the characteristics expected for ice particle growth in the absence of significant riming. Overall, the observed P values and polarization signatures correspond well to known particle growth mechanisms.

7. Conclusions With the future upgrade of the WSR-88D network to dual-polarization capability on the horizon, it is important that the potential of these radars to detect hazard- ous conditions in clouds be explored. The motivation for this study was the development of quantitative criteria for identification of potential aircraft icing conditions in clouds using polarization radar. The data presented here provide a first step toward this effort. This study’s primary goal was to examine polarization radar signatures associated with SLW in orographic cloud systems and quantify their use as probabilistic re- FIG. A1. Aircraft position vector (xi, yi, zi) relative to Earth’s mote SLW identification criteria. Verification of SLW’s center (Ec), where xi 5 RA cosQA cosFA, yi 5 RA cosQA sinFA, and presence was accomplished with the use of in situ mi- zi 5 RA sinQA, given range from the radar RA, aircraft latitude QA, crophysical measurements. This verification necessitated and aircraft longitude FA. the development of an automated routine used to locate ‘‘matched’’ data between sets of radar and aircraft-based APPENDIX observations. This algorithm’s method was described, along with the evaluations used to verify that it did not introduce significant positional errors into the data. This Transformation of Aircraft Position into Radar automated routine was developed specifically to com- Coordinates pare radar- and aircraft-based observations for this study, The aircraft’s location in space is initially specified but it is also applicable for comparison of radar mea- using its latitude, longitude, and altitude. We can rede- surements with other airborne instrumentation. fine its position at time i in a Cartesian coordinate sys- Probability distributions for SLW detection were de- tem (CA) with respect to the earth (xi, yi, zi), where veloped using these matched dual-polarization radar and in situ observations. Three polarization radar pa- p xi 5 RA sin QA cosFA 5 RA cosQA cosFA, (A1) rameters, ZH, ZDR, and KDP, were separately shown to 2 be statistically distinguishable between SLW and IO p conditions, even when considering measurement un- y 5 R sin Q sinF 5 R cosQ sinF , and i A 2 A A A A A certainty. The matched dataset was used as the training (A2) set to develop probabilistic criteria for remote SLW identification, with the end result being a probability p z 5 R cos Q 5 R sinQ , (A3) distribution of the likelihood of SLW’s presence as a i A 2 A A A function of binned polarization variables. The probability distributions correspond well to a basic physical with RA is the earth’s effective radius [corrected for understanding of ice particle growth by riming and va- nonsphericity and beam propagation using the 4/3R ap- por deposition, both of which may occur in mixed-phase proximation (Doviak and Zrnic´ 1993, p. 21), where R is conditions. the earth’s actual radius] plus the aircraft’s altitude above sea level, QA is the aircraft’s latitude, and FA is its Acknowledgments. This material is based upon work longitude (Fig. A1). The unit vectors of this system are supported by the National Science Foundation under such that x points from the earth’s center toward the Award NSF-ATM-0438071. We thank Scott M. Ellis equator at 08 longitude, z points from the earth’s center and William A. Cooper of the National Center for At- north along its rotation axis, and y is orthogonal to x and mospheric Research for their contributions to this effort. z, pointing toward the equator at 908E longitude. 934 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 49

FIG. A2. (a) Translation of aircraft coordinates from CA to CR, where RA 2 RR 5 (Dx, Dy, Dz). (b) Rotation counterclockwise about z by FR. (c) Rotation clockwise about y9 by G5 (p/2) 2QR (illustration assumes x9 parallel to xearth for simplicity). (d) Rotation counterclockwise about z0 by (p/2). Aircraft and radar location RA and RR, radar longitude FR, and radar latitude QR, all with respect to the earth’s center.

The aircraft’s position (xi, yi, zi)inCA is translated to orienting y9 tangential to the earth’s surface at the radar’s its position in a Cartesian coordinate system (CR) cen- location (Fig. A2b). The coordinate system is then rotated tered at the radar location with axes parallel to CA using clockwise about y9 by an angle of G5(p/2) 2QR using

xi05xi9 cosG zi9 sinG5xi9 sinQR zi9 cosQR, (A10) Dxi 5 RA cosQA cosFA RR cosQR cosFR, (A4)

yi05yi9, and (A11) Dyi 5 RA cosQA sinFA RR cosQR sinFR, and (A5)

zi05xi9 sinG1zi9cosG5xi9 cosQR 1 zi9 sinQR (A12) Dzi 5 RA sinQA RR sinQR, (A6) to orient z0 perpendicular to the earth’s surface at the where RR is the earth’s effective radius plus the radar’s location of the radar (Fig. A2c). Finally, the coordinate altitude above sea level, and QR and FR are the radar’s system is rotated counterclockwise about z0 by an angle latitude and longitude (Fig. A2a). This coordinate sys- of (p/2): tem is still oriented with z parallel to the earth’s rotation axis. In the next step, the coordinate system is rotated xi09 5 yi05yi9, (A13) counterclockwise around z by an angle of FR, using yi09 5 xi05xi9 sinQR 1 zi9 cosQR, and (A14) 95D F 1D F xi xi cos R yi sin R, (A7) zi09 5 zi05xi9 cosQR 1 zi9 sinQR (A15) to align the coordinate system so that x999 points east, y999 yi95Dxi sinFR 1Dyi cosFR, and (A8) points north, and z999 points upward at the radar location

zi95Dzi, (A9) (Fig. A2d). MAY 2010 P L U M M E R E T A L . 935

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