Application of Spectral Polarimetry to a Hailstorm at Low Elevation Angle

APRIL 2019 W A N G E T A L . 567

YADONG WANG Department of Electrical and Computer Engineering, Southern Illinois University at Edwardsville, Edwardsville, Illinois

TIAN-YOU YU School of Electrical and Computer Engineering, and Advanced Radar Research Center, and School of Meteorology, University of Oklahoma, Norman, Oklahoma

ALEXANDER V. RYZHKOV Advanced Radar Research Center, and School of Meteorology, and Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

MATTHEW R. KUMJIAN Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

(Manuscript received 9 July 2018, in ﬁnal form 2 January 2019)

ABSTRACT

Spectral polarimetry has the potential to be used to study microphysical properties in relation to the dy- namics within a radar resolution volume by combining Doppler and polarimetric measurements. The past studies of spectral polarimetry have focused on using radar measurements from higher elevation angles, where both the size sorting from the hydrometeors’ terminal velocities and polarimetric characteristics are maintained. In this work, spectral polarimetry is applied to data from the 08 elevation angle, where polarimetric properties are maximized. Radar data collected by the C-band University of Oklahoma Polarimetric Radar for Innovations in Meteorology and Engineering (OU-PRIME) during a hailstorm event on 24 April 2011 are used in the analysis. The slope of the spectral differential reflectivity exhibits interesting variations across the hail core, which suggests the presence of size sorting of hydrometeors caused by vertical shear in a turbulent environment. A nearby S-band polarimetric Weather Surveillance Radar-1988 Doppler (KOUN) is also used to provide insights into this hailstorm. Moreover, a flexible numerical simulation is developed for this study, in which different types of hydrometeors such as rain and melting hail can be considered in- dividually or as a combination under different sheared and turbulent conditions. The impacts of particle size distribution, shear, turbulence, attenuation, and mixture of rain and melting hail on polarimetric spectral signatures are investigated with the simulated Doppler spectra and spectral differential reflectivity.

1. Introduction about hydrometeor’s size, shape, concentration, orientation, and type. Kinematic features such as convective Polarimetric weather radar has shown its capabilities in storm updrafts, vertical wind shear, and storm-relative improving quantitative precipitation estimation, hydro- helicity can be inferred from prominent polarimetric meteor classification, data quality control, and severe signatures such as the Z column and the Z arc (e.g., weather detection and forecasting (e.g., Meischner 2003; DR DR Kumjian and Ryzhkov 2008, 2009, 2012). In addition, the Ryzhkov et al. 2005; Park et al. 2009; Scharfenberg et al. kinematic information such as storm intensity, motion, 2005; Kumjian 2013a,b; Chandrasekar et al. 2013; Wang and turbulence can also be derived from Doppler mea- and Yu 2015; Hwang et al. 2017). Polarimetric variables surements of mean radial velocity and spectrum width. such as differential reflectivity Z , correlation coefficient DR These polarimetric and Doppler variables contain either r , and differential phase f can provide information HV DP the dynamical or the microphysical properties that are integrated over the radar resolution volume. Spectral po- Corresponding author: Yadong Wang, [email protected] larimetry, on the other hand, can unveil the polarimetric

DOI: 10.1175/JTECH-D-18-0115.1 Ó 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 10/05/21 11:15 AM UTC 568 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 36 variables directly as a function of radial velocities within observed by one C-band polarimetric radar at 08 eleva- one resolution volume (e.g., Unal et al. 2001; Russchenberg tion angle, where no size sorting should be expected if et al. 2008; Yu et al. 2012), and this provides a unique only the terminal speed of hydrometeors and the uniform opportunity to relate storm microphysical properties to background wind are considered. However, it was ob- kinematics. As shown in Doviak and Zrnic´ (1993),the served that spectral differential reflectivity exhibits dif- Doppler spectrum describes the returned power as a ferent patterns in different regions of the storm, which function of radial velocities, and the power at each ve- suggests the presence of a different ongoing size sorting locity bin is contributed by all scatterers with the same mechanisms. It is hypothesized that the variation of radial velocity within the radar resolution volume. Simi- the observed spectral differential reflectivity is a result of larly, spectral differential reflectivity represents a spec- shear-induced size sorting from raindrops and melting trum of differential reflectivity, and the value at each hailstones in a turbulent environment. The idea of a hy- bin is determined by the ratio of returned power be- drometeor’s trajectory in a sheared environment was first tween horizontal (H) and vertical (V) polarizations from proposed by Marshall (1953), and its velocity is derived to all the scatterers with the same radial velocity. Addi- be a function of its terminal velocity and environmental tionally, spectral correlation coefficient and spectral dif- vertical shear (e.g., Brussaard 1974). Therefore, we hy- ferential phase can be understood in the same manner. pothesize that when the vertical shear is present, the size- Therefore, if the relation between hydrometeor sizes and dependent velocity of hydrometeors can be revealed in radial velocities can be established (i.e., size sorting is polarimetric spectra even at 08 elevation angle. This work present), spectral polarimetry can reveal the polarimetric therefore focuses on the interplay of microphysics and properties as a function of hydrometeor sizes. Note kinematics within the resolution volume. The hypothesis that polarimetric variables can still be obtained from will be qualitatively supported from the analysis of radar these polarimetric spectra through integrating the whole observations, mesonet data, and storm reports. The resolution volume (Yu et al. 2012). impact of shear, turbulence, particle size distributions Since hydrometeors from different species and sizes of rain and melting hail, and attenuation on the slope of possess different terminal velocities (Atlas et al. 1973), spectral differential reflectivity will be investigated past studies using spectral polarimetry mainly relied on using simulations. size sorting originating from hydrometeors’ different In this paper, observations of a hailstorm from nearby terminal velocities. For vertically pointing radar obser- S-band and C-band polarimetric radars are first pre- vations, this size sorting is maximized; however, hydro- sented in section 2. In addition, the variation of spectral meteors (such as raindrops) viewed from below produce differential reflectivity revealed from the application of

0dB ZDR. Another extreme case is for 08 elevation spectral polarimetry to C-band data at 08 elevation angle angle, where ZDR magnitudes are maximized but size is shown. In section 3, a brief overview of size sorting in a sorting is totally diminished because of the same radial sheared environment is provided. To further investigate velocity (background wind) from all sizes of hydrome- the cause of these variations of spectral differential teors. Thus, spectral polarimetry typically has focused reﬂectivity, a simulation is developed in section 4,where on using measurements from higher elevation angles two cases of rain and a mixture of rain and melting hail where both the size sorting from the hydrometeors’ are examined. A summary and conclusions are provided terminal velocities and the polarimetric characteris- in section 5. tics are maintained. For example, Spek et al. (2008) re- trieved the particle size distributions of two mixed ice 2. Observations of a hailstorm on 24 April 2011 particles, background wind, and turbulence by applying spectral polarimetry to S-band data collected at 458 el- A total of 40 hailstorms were reported in Oklahoma evation angle. Moisseev et al. (2006) applied spectral on 24 April 2011, which cost approximately $65,000 polarimetry to estimate the shape parameter of raindrops (U.S. dollars) in property damage over 19 affected counties using data from the S-band CSU–CHILL at elevation of (http://www.ncdc.noaa.gov/stormevents). One of the 308 and the S-band Transportable Atmospheric Radar hailstorms is of special interest, because it was observed (TARA) at 458. simultaneously by two nearby polarimetric radars, the In addition to different terminal velocities, hydrometeor C-band University of Oklahoma Polarimetric Radar size sorting can be maintained by different mecha- for Innovations in Meteorology and Engineering nisms, including convective updrafts and nonzero storm- (OU-PRIME) (Palmer et al. 2011) and an S-band po- relative ﬂow associated with vertical wind shear (e.g., larimetric Weather Surveillance Radar-1988 Doppler Kumjian and Ryzhkov 2012; Dawson et al. 2014, 2015). In (WSR-88D) in Norman, Oklahoma (KOUN). The storm this work, spectral polarimetry was applied to a hailstorm was located within 60-km range from both radars during

Unauthenticated | Downloaded 10/05/21 11:15 AM UTC APRIL 2019 W A N G E T A L . 569 the time of interest. The level I in-phase and quadrature- data; therefore, they have the same origin to facilitate the phase (I and Q) signals from the OU-PRIME were comparison. collected, which allows the implementation of spectral Despite the generally similar Z and ZDR contours polarimetry. A more detailed description of the obser- from these two radars, enhanced reflectivity (.60 dBZ) vations is presented in the following subsection. and differential reflectivity (.4 dB) can be observed from OU-PRIME such as at approximately (240, 232), a. Dual-wavelength observations 2 2 ( 40, 0), and ( 20, 5) km (Figs. 1a,c). The rHV values at The C-band OU-PRIME is located approximately these regions (Fig. 1d) are as low as 0.8 from the C-band 6.78 km southeast of the S-band KOUN, and both of OU-PRIME, whereas relatively high values from the them are dual-polarization radars operating in simulta- S-band KOUN are still maintained (not shown). These neous transmission and reception mode. A detailed comparisons suggest the presence of resonance scatter- comparison of the specifications between OU-PRIME ing effects caused by large drops and small melting hail and KOUN is provided in Picca and Ryzhkov (2012). at C band (PiccaandRyzhkov2012). According to a Even though the scanning strategy and resolution are storm report from the National Oceanic and Atmospheric not the same for these two radars, valuable information Administration (http://www.ncdc.noaa.gov/stormevents), can be obtained by the simultaneous observations from hailstones of 1.9-cm size were observed at 1850 UTC in the two different wavelengths. For example, Borowska Rush Spring (about 50 km southwest from OU-PRIME; et al. (2011) studied the attenuation and differential at- annotated in Fig. 1), which can further support the fact tenuation from melting hail using these two different of hail. wavelengths radars. Picca and Ryzhkov (2012) used At 1858:21 UTC, OU-PRIME switched to RHI scan simultaneous OU-PRIME and KOUN data to study at azimuth angle of 233.38, and the fields from Z, ZDR, the hail size estimation and hail signatures near sur- and rHV are presented in Figs. 2a, 2c, and 2e. For com- face and aloft. In addition, Bodine et al. (2014) used parison purposes, data from the KOUN volume scan at these two radars to investigate tornado debris signa- 1856:07 UTC were interpolated to the same locations as tures from an EF4 (on the enhanced Fujita scale) in the OU-PRIME’s RHI scan using a nearest neigh- tornado. borhood method; the interpolated Z and ZDR are shown Between 1700:13 and 1959:49 UTC, the OU-PRIME on the right top two panels of Figs. 2b and 2d. Addi- was operated with alternating plan position indicator tionally, the HCA results obtained using those in- (PPI) and range–height indicator (RHI) scans. In the terpolated data are shown in Fig. 2f. The resonance PPI mode, a total of eight elevations were completed in effects at C band are still evident in the region at ranges approximately 2 min with a pulse repetition time (PRT) between 48 and 52 km and altitude below 1.5 km, which is of 0.8475 ms. In the RHI mode, OU-PRIME scanned manifested by enhanced Z (.55 dBZ)andZDR (.4dB), 8 8 8 , from 0 to 20 in elevation at the azimuth angle of 233.3 and reduced rHV ( 0.89) when compared to S-band data in approximately 52 s using the same PRT. The KOUN (Borowska et al. 2011; Picca and Ryzhkov 2012). radar was operated with volume coverage pattern (VCP) Differential attenuation at C band is evident after

12 (4.5-min update time) during that period. Compari- 52 km, where OU-PRIME ZDR falls below 23.5 dB at sons of PPI scans from OU-PRIME at 08 elevation angle ranges of approximately 54 to 57 km. The ZDR from (left panels) and KOUN at 0.58 elevation angle (right S-band KOUN is approximately 2–3 dB at similar ranges. panels) both at 1856:07 UTC are shown in Fig. 1. The top Such anomalous attenuation at the C band is likely a re- (Figs. 1a,b)andmiddlerows(Figs. 1c,d)arethefields sult of large raindrops and melting hail (Meischner et al. of reflectivity Z and ZDR,andthefieldofrHV from 1991; Ryzhkov et al. 2007; Anderson et al. 2011; Ryzhkov OU-PRIME is shown in Fig. 1e. The result of hydro- et al. 2013). Moreover, HA was identified by KOUN at meteor classification algorithm (HCA) (Park et al. 2009) ranges around 50 to 55 km and at altitudes between from KOUN is shown in Fig. 1f, where ground clutter 1 and 4 km. Note that the mixture of hail and rain was (GC), biological scatterers (BI), dry snow (DS), wet not directly identified at lower elevations, which may snow (WS), crystals (CR), graupel (GR), big drops be a result of the time difference between these two (BD), light/moderate rain (RA), heavy rain (HR), observations or errors in the hydrometeor classification mixture of rain and hail (HA), and unknown (UK) are algorithm. It is speculated that the synthesized KOUN shown (Park et al. 2009). The locations of both radars radial at 1856:07 UTC only samples the edge of the are denoted by white asterisks. The azimuthal direction of storm core, as suggested by the HCA results in Fig. 1.In OU-PRIME’sRHIscanisalsoincludedanddenotedbya summary, the analysis of the observations from the two solid black line. Note that all the KOUN data have been wavelengths strongly suggests the presence of a mixture transformed into the same coordinates as OU-PRIME of hail and rain and large drops in the lowest elevation

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FIG. 1. The hailstorm observed by the PPI scan from (left) the C-band OU-PRIME and (right) the S-band KOUN both at 1856 UTC. The locations of OU-PRIME and KOUN are indicated with two white stars. The azimuthal direction of OU-PRIME’s RHI scan (as in Fig. 2) is also included and denoted by a solid black line. of the OU-PRIME RHI scan at a range of approxi- polarimetry, where 960 complex-valued IQ samples were mately 50–55 km at 1858:21 UTC. available for H and V channels at each gate. Speciﬁcally, at each range gate, 32 IQ samples for each channel were b. Spectral polarimetry at 08 elevation angle ﬁrst zero padded to 64 points and then Fourier trans- In this work, the OU-PRIME’s RHI data at the formed using a Hanning window. The power spectral lowest elevation angle (08) were used to calculate spectral density (PSD) for each of the H and V channels as well

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FIG. 2. The hailstorm observed by the RHI scan from (left) the C-band OU-PRIME at 1858 UTC and (right) the interpolated S-band KOUN at 1856 UTC. as the cross-spectral density (CSD) between the two S-band data (Borowska et al. 2011), can be clearly channels were computed. Note that PSD is also referred observed after 52.5 km. to as Doppler spectrum (Doviak and Zrnic´ 1993)or The Doppler spectrum from H channel (SH ) and sZDR spectral reflectivity after calibration. Subsequently, from five selected locations, indicated by the upward 30 PSDs and CSDs were averaged to reduce the var- arrows in Fig. 3, are shown in Fig. 4 from left to right. iance of the estimation (Yu et al. 2012). The averaged The range (in km), Z, and ZDR are also included at each PSDs for H and V channels are denoted by SH and panel as references. Note that both SH and sZDR were 21 SH, and spectral differential reflectivity is calculated dealiased to be within 225.4 to 9.4 m s for clearer as sZDR 5 SH /SV . presentation, based on the maximum unambiguous ve- 21 Before the discussion of sZDR, the range profiles of Z, locity of 17.7 m s . Although spectral polarimetry has 8 ZDR, and rHV along 233.3 between 43.75 and 58.625 km the potential for providing insights of hydrometeor from C-band OU-PRIME at 1858:21 UTC at 08 eleva- properties within the radar resolution volume, one of tion are shown in Fig. 3. The interpolated profiles from the challenges is that it is difficult to decide which part S-band KOUN at 1856:07 UTC are also included in of the spectra is reliable, owing to the high variance of

Fig. 3 as references. Relatively larger Z and ZDR and those estimators. Yu et al. (2012) devised a spectral SNR lower rHV from C-band OU-PRIME than those from threshold to meet a user-deﬁned quality of the spectral the S-band KOUN can be observed up to approxi- estimators for a given spectral averaging. Since 30 gates mately 50 km, which suggests the resonance effect. averaged PSDs and CSDs are used in this work, the

Additionally, the attenuation and differential attenu- derived thresholds for the bias of sZDR, the standard ation of OU-PRIME data, manifested by decreased deviation (SD) of sZDR, and the normalized spectral reflectivity and differential reflectivity compared to correlation coefficient were calculated as 0.04 dB, 0.707,

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FIG. 3. The range profiles of (a) reflectivity, (b) differential reflectivity, (c) correlation coefficient, and (d) slope of spectral differential reflectivity at 1858:21 UTC. The interpolated reflectivity, differential reflectivity, and correlation coefficient from KOUN are included with dashed lines. The five selected locations for Doppler spectrum and spectral differential reflectivity shown in Fig. 4 are also labeled. and 0.01, respectively (Yu et al. 2012). The portion of the spectrum (Doviak and Zrnic´ 1993). The presence of

SH and sZDR meeting those requirements are super- vertical shear is inferred from the range profile of mean imposed and denoted by thick lines. It is evident radial velocity from the two lowest elevations (08 and that the pattern of sZDR at different ranges exhibits 0.58) of OU-PRIME. Vertical shear of more than 2 2 interesting and distinct variations. To quantify the 0.02 s 1 in the negative radial direction (i.e., ,20.02 s 1, variations, a linear fitting was performed using only where negative direction is toward the radar) can be those sZDR that meet the quality requirements. The observed up to range of 48.5 km, where the height dif- fitted lines are denoted by dashed lines in the sZDR plots ference of the two lowest beams are approximately 2 2 and the value of slope [dB (m s 1) 1] is also presented. 423.2 m (Doviak and Zrnic´ 1993). Beyond the range 21 The range profile of sZDR slope is also presented in of 48.5 km, an averaged shear of 20.01 s is still ob- Fig. 3d. tained. The strong vertical shear was also verified From Fig. 4, it is clear that the Doppler spectra are by the observation from a mesonet station close to fairly wide with non-Gaussian shapes. It has been re- Rush Spring, which locates at (241.2, 253.9) km from ported that similar broad and flattop spectra can be OU-PRIME. observed in strongly sheared environments (Yu et al. From Fig. 4, it can be observed that from location 2009). Additionally, turbulence can further broaden 1 to 2, the negative slope becomes steeper and both

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FIG. 4. (top) Doppler spectra and (bottom) spectral differential reflectivity from five specific locations indicated in Fig. 3. The data are from 08 elevation angle observed by OU-PRIME at 1858:21 UTC. The fitted lines are denoted by dashed lines in the sZDR plots and the 2 2 value of the slope [dB (m s 1) 1] is also included.

Z and ZDR increase. The largest negative slope of approximated trajectory of a hydrometeor and its in- 2 2 approximately 20.17 dB (m s 1) 1 and the maximum clination are related to vertical shear. The equation

ZDR value of 5.7 dB were obtained at location 2. At lo- of motion for falling hydrometeors in sheared envi- cation 3, the sZDR becomes ﬂattened with a slope of ronment was derived in many previous studies, such approximately zero, and both Z and ZDR reach their as Brussaard (1974, 1976), Bohne (1982),andBeard maximum values of 60.5 dBZ and 5.7 dB, respectively. and Jameson (1983), and is brieﬂy reviewed here.

Though similar ZDR values of approximately 5.7 dB are The primary forces on a single hydrometeor are the obtained at locations 2 and 3, distinctly different sZDR gravity force and drag force. In this work, only hori- slopes are observed. This suggests the dynamic and/or zontal ah and vertical az directions are considered, microphysics in these two radar volumes are different, in which positive az is upward. For a single hydrome- and spectral polarimetry has the potential to reveal it. teor of size D, when it falls through a sheared region,

Moreover, sZDR exhibits positive slope at locations 4 its velocity is V 5 ahVh 1 azVz. Initially, the hydro- and 5, while sZDR values at location 5 are generally meteor is located at height H and its velocity is equal lower than those at location 4 for most of radial veloci- to the environment wind velocity at H, U(H) 5 ahUh, ties. It is evident from Fig. 3 that the decreases of Z and where only the horizontal air motion is consid-

ZDR in location 5 are caused by attenuation and differ- ered. From Newton’s second law, the net force Fn ential attenuation. Given the analysis presented so far, it on the hydrometeor is represented in the following is hypothesized that the observed variations of sZDR equation: slope at 08 elevation angle is caused by shear-induced F 5 F 1 F , (1) size sorting of a mixture of rain and melting hail in a n g d turbulent environment. In section 4, this hypothesis will where F 5 mg is the gravity force, m is the mass, be tested, and the impacts of microphysical and dy- g g 5 a 0 1 a (2g), and g is the gravitational acceleration. namical characteristics on the sZ will be studied using h z DR The drag (or air resistance) force F is obtained using simulation. d the following equation (Bohne 1982): V 2 U 3. A brief overview of shear-induced size sorting F 52m , (2) d t Thetrajectoryofhydrometeors in a sheared environment was ﬁrst investigated in Marshall (1953). where the time constant t 5 Vt(D)/g and Vt(D)is Pinsky and Khain (2006) also showed that the the terminal speed of a hydrometeor with size D.

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The hydrometeor’s horizontal and vertical velocities can be derived as sV2(D) V (t) 5 U (t) 1 t , (3a) h h g 52 Vz(t) Vt(D), (3b) where s 5 dUh/dz is constant vertical shear. The canting angle of the hydrometeor (g) in shear is also obtained as tang 52sVt(D)/g (e.g., Brussaard 1974, 1976). The radar-observed radial velocity from the hydrometeor is the sum of the radial components from the two velocities as y 5 Vh cosu 1 Vz sinu, where u is the elevation angle. For a higher elevation angle, the size sorting can be contributed from both horizontal and vertical components of the falling hydrometeors. However, for the el- FIG. 5. (a) The profiles of OU-PRIME radial velocity from the 8 8 evation angle of 08, the presence of size sorting is elevation angles of 0 (solid line) and 0.5 (dashed line). (b) The simulated radar volume. The beamwidth is 0.58, and the distance contributed only from the shear term in the horizontal from this resolution volume to radar is 50 km. The background wind 2 component. For example, for a distribution of hydro- gradually changes from 18 m s 1 toward the radar at the bottom to 21 meteors with sizes ranging between Dmin and Dmax in 8ms forward from the radar at the top of the volume. a sheared environment without turbulence and background wind [Uh(t) 5 0], an interval of radial velocities 2 2 8 The method of simulating the sheared environment from sVt (Dmin)/g to sVt (Dmax)/g results from 0 eleva- is demonstrated in Fig. 5b, where the distance from tion angle, where Dmin and Dmax are the minimum and maximum sizes, respectively. As expected, no size OU-PRIME to the radar gate of interest is 50 km as in sorting will be observed at 08 elevation angle if the shear the observation (Fig. 4). The Gaussian shape beam is not present, and all the hydrometeors have the same patternisalsoincluded.AsshowninFig. 5a, the radial velocities at 08 and 0.58 elevation angles are 213 and radial velocity as the background wind. Note the sign of 2 217 m s 1 at location 1, respectively. Since the maximum shear will determine the sign of radial velocity, where 2 unambiguity velocity of OU-PRIME is 17.7 m s 1,the the convention of positive sign indicates outbound ve- 21 locity. Additionally, the background wind of U (t) will dealiasing radial velocity can reach around 17 m s at h 8 only shift the interval of radial velocity by the amount 0.5 elevation angle. Therefore, in the current work, the background wind at the bottom of the radar volume is set of Uh(t). 2 as 18 m s 1 toward to the radar, and gradually changes to 2 8ms 1 forward from the radar at the top of the radar 4. Simulation volume. Given the fact that OU-PRIME’s beamwidth is half degree, the simulated vertical shear is approximately a. Description of simulation 2 s 520.046 s 1 at 50 km, and that is similar to the ob- To test the hypothesis and investigate the impacts of servation from a mesonet station close to Rush Spring. particle size distributions on Z, ZDR, and the slope of Within this sheared environment, particles from different sZDR, a numerical simulation was developed in the heights within one radar resolution volume possess dif- current work. Spek et al. (2008) developed a model of ferent background velocities, and their radial velocities S-band Doppler spectrum and spectral differential re- are the combination of background wind and the veloci- flectivity for a mixture of two types of ice particles with ties derived from section 3. Such strong vertical shear can different particle size distribution functions. A similar generate a wide Doppler spectrum, as shown in Fig. 4. model was also developed in the particle size distribu- A flowchart of the simulation is provided in Fig. 6, tions retrieval for a mixture of rain and melting hail in a where the inputs and outputs are denoted by orange and turbulent environment (Wang 2010). In this work, the green boxes, respectively. The input parameters include simulation used in Wang (2010) is extended to include the particle size distributions from rain [Nr(Dr)] and both shear-induced size sorting and attenuation. The at- from hail [Nh(Dh)], shear s, elevation angle u, range r, tenuated Doppler spectrum and spectral differential re- and turbulence broadening sb. The idea is to first gen- flectivity can be generated from single or mixed types of erate Doppler spectra for both H and V channels from hydrometeors in a turbulent and sheared environment. individual hydrometeor type separately in a sheared

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hailstones) as defined by Ryzhkov et al. (2011). When hailstones start to melt, the mass water fractions vary with diameters and can be calculated with the model proposed by Rasmussen and Heymsfield (1987).As shown in Fig. 7b, if the hailstones with diameters less than 8 mm are assumed fully melted, the mass water fraction decreases with the increase of the hailstones’ size, and it reaches 0.13 when the diameter is 25 mm. This is consistent with the simulation results from Ryzhkov et al. (2013). Dielectric constants of melting hailstones are determined by the dielectric constants of water, ice, air, and water volume fraction (Ryzhkov et al. 2011). The aspect ratio of melting hail is calculated using FIG. 6. Simulation flowchart. Input and output variables are denoted by orange and green boxes, respectively. Inputs are par- the model proposed by Rasmussen et al. (1984). The size ticle distribution of hail [Nh(Dh)] and raindrops [Nr(Dr)], vertical dependencies of ZDR of melting hail are also shown in shear s, radar elevation angle u, distance r, and turbulence broad- Fig. 7d, where the mean canting angle is 08. The standard ening sb. deviations of canting angle are taken as a function of mass water content following (Ryzhkov et al. 2011): environment, and resulting Doppler spectra for H and s 5 s 1 f (s 2 s ), (6) V channels are the summations of individual spectra s mw r s from each hydrometeor type. Subsequently, spectra are where f is the volume water fraction, and s and s attenuated and further broadened by the turbulence. mw r s are the standard deviations of canting angle distribution In the current work, a gamma and an exponential for raindrops and frozen hydrometeors (Ryzhkov et al. distribution are used to simulate the particle distribu- 2011). Note that using water-coated particles (i.e., with tions of raindrops and hailstones, respectively (e.g., the two-layer T-matrix code) should produce more ac- Ulbrich 1983; Zhang et al. 2001): curate simulation results, especially for very large (Dh . 2L 25 mm) wet hail (Ryzhkov et al. 2011). However, since 5 m rDr Nr(Dr) Nr Dr e , (4) 0 the maximum hail size is limited to 25 mm in this study, 2L the uniform particles with Mishchenko (2000) T-matrix 5 hDh Nh(Dh) Nh e , (5) 0 code can produce reasonable results. With the obtained backscattering cross sections from where the diameters of raindrops D and hailstones D r h raindrops and hailstones, the S and sZ could be are set as 0.1–8 and 8–25 mm, respectively. The intercept H DR : 2 2 calculated with the spectral processing methods (e.g., N is 400L4 11 m 3 mm 1 as suggested by Ryzhkov et al. h0 h Unal et al. 2001; Russchenberg et al. 2008; Yu et al. (2013). The relations between N , m, and L used in this r0 r 2012). For an ideal case of maximum size sorting, where work fit the results discussed by Zhang et al. (2001). The each radial velocity associated with each size can be fully impacts of N , N , m, L , and L on the simulated S r0 h0 r h H resolved, sZ from raindrop will have the same shape and sZ will be discussed in section 4b. DR DR as in Fig. 7c. In practice, the shape of sZ will be The C-band radar backscattering cross section for DR smoothed because velocity resolution is limited by dwell each size and hydrometeor type is calculated from the time and some amounts of turbulence are likely to occur. complex backscattering amplitudes [f (p) and f (p)] a b When a mixture of rain and melting hail occurs, the using the T-matrix code (Mishchenko 2000). The aspect shape of sZ is difficult to visualize intuitively, and will ratio of raindrop as a function of its equivolume di- DR be determined from the amount of backscattered power ameter (Brandes et al. 2002) and the dielectric constant at each velocity bin from the H and V channels. The of water at 208C are inputs into the T-matrix code. The unattenuated Doppler spectrum (H channel) from a obtained size dependencies of differential reflectivity hydrometeor of size D at velocity y can be obtained Z of raindrops are shown in Fig. 7c, where the mean i DR using the following equation (Spek et al. 2008): canting angle is 08, and standard deviation of canting angle are 108. It is evident that the maximum value of n Sun(y)dy 5 å N (D fyg)s W(y), (7) differential reflectivity resulting from resonance effects H i i HH,i i51 is 6.32 dB at Dr 5 5:4 mm. The hailstones in this work are modeled as uniform where i 5 r for rain and i 5 h for hail, Ni(Di)isthe particles with various fractional water contents (spongy particle size distribution defined in Eq. (4) or (5),

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FIG. 7. (a) Terminal velocities of rain (dashed line) and hail (solid line) from different size. (b) The distribution of mass water fraction of melting hailstones along size. The hailstone with size less than 8 mm is assumed fully melted. (c) The differential reﬂectivity from raindrops with different sizes. The mean canting angle and the standard deviation are 08 and 108, respectively. (d) The differential reﬂectivity from melting hailstones with different sizes. The mean canting angle is 08, and the standard deviation is as a function of the water mass fraction.

sHH,i (sVV,i) is the backscattering cross section from dependence of Doppler spectrum on drop size can be horizontal (vertical) direction, and W(y) is the beam converted onto radial velocity as in Eq. (7). Specifically, pattern at velocity y,asshowninFig. 5. The superscript the radial velocity for each size of hydrometeor in a ‘‘un’’ means that the obtained Doppler spectrum is sheared environment is determined by the background not attenuated. wind (Fig. 5b) and its shear-induced radial velocity ob- un After SH (y) is calculated using Eq. (7), attenuation tained from Eqs. (3a) and (3b) at an elevation angle of u. effects are accounted for as In this work, the terminal velocity for raindrops and hail (Fig. 7a) is determined by using the formula in Atlas 5 un 2 3 et al. (1973) and Mitchell (1996), respectively. The SH(y) SH (y) Ah r, (8) spectra for each hydrometeor types and polarization are where r is the distance, and Ah is specific attenuation then broadened by turbulence by convolving with a 21 (dB km ) at horizontal polarization, which is calculated Gaussian kernel of width sb (e.g., Spek et al. 2008). The 5 : 3 23 as Ah 8 686 10 Im(hfa(0)i). The variable fa(0) is the resulting Doppler spectrum for the H channel SH is forward scattering amplitude at horizontal polarization the sum of the broadened spectra from rain and melting and is calculated using the T-matrix method as described hail for the H channel. above, and hfa(0)i is the average over the particle size distribution. Similarly, the attenuated Doppler spectrum b. Simulation results for the V channel SV could be calculated with hfb(0)i,and 1) CASE I: RAIN ONLY the attenuated sZDR can be therefore obtained. It should be noted that attenuation is a path and volume integrated, In this case, Doppler spectrum and spectral differ- and the same attenuation is applied for all the particles ential reflectivity are simulated with raindrops only, in the same simulated radar volume. Therefore, the and the simulated results are shown in Figs. 8 and 9, specific attenuation and specific differential attenuation respectively. Because of the size sorting, particle size will affect the obtained Z and ZDR (Jameson 1992; distribution shows clear vertical variations in one ra- Carey et al. 2000), but the effects on the shape of Sh and dar resolution volume, and these variations are simu- L sZDR can be ignored. lated through changing coefficients of Nr0, m,and r The next step is to determine the relationship be- in Eq. (4).InFig. 8, from the top to the bottom of L tween the drop size and radial velocity; therefore, the the resolution volume, Nr0 ( r) gradually changes

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FIG. 8. (a) Simulated Doppler spectrum, (b) spectral differential reflectivity, and (c) particle size distribution of raindrop. Only raindrops are used in this case; details about the particle size distribution can be found in text. Examples of particle size distributions from top, center, and bottom in the resolution volume are presented with solid, dotted, and dashed lines, respectively. The obtained Doppler spectrum, spectral differential reflectivity, slope of sZDR, reflectivity, and differential reflectivity are similar to the observations from location 1.

2 2 2 2 2 from 13 000 m 3 mm 1 (1.9 mm 1) to 8000 m 3 mm 1 more large drops are added as shown in Fig. 9c. These 2 (2.7 mm 1), and the value of m is set as 1 in all the alti- large raindrops mainly come from the shedding process tudes. All these values are within the reasonable ranges of melting hailstones (Ryzhkov et al. 2013). From the as indicated by Zhang et al. (2001). Examples of Nr from top of the resolution volume to the bottom, the values of top, center, and bottom of the radar resolution volume m gradually increase from 21.9 to 3.5, and the setting L are shown in Fig. 8c with solid, dotted, and dashed lines, for Nr0 and r are the same as in location 1. The ob- respectively. Because of the size sorting, the amount tained Z, ZDR, and slope of sZDR are 57 dBZ, 4.8 dB, 2 2 of small drops shows decreasing trend from the top and 20.16 dB (m s 1) 1. Comparing the results from to the bottom in the resolution volume, and small size locations 1 and 2, although they show similar Z and ZDR, raindrops are gradually absent when they fall down signiﬁcantly different slopes of sZDR can be found. It (Kumjian and Ryzhkov 2012). Therefore, raindrops should be noted, in the observation, the sZDR can reach 2 with all sizes (from 1 to 8 mm) could be found from the approximate 8 dB at velocity bin of 218 m s 1, but the top of the resolution volume, but only big drops could be maximum value as simulation is around 5.12 dB. These found from the bottom of the resolution volume. In large sZDR bins may come from big drops from fully or this work, we assume raindrops with diameter smaller partially melted hail. Because the aspect ratio for such than 5.1 mm are totally absent at the bottom of the large drops or small melting hailstones are still under resolution volume. The simulated Sh and sZDR are investigation (Thurai et al. 2013; Kumjian et al. 2018), shown in Figs. 8a and 8b, the obtained Z, ZDR, and slope we did not add such big drops in the simulation. As a 21 21 of sZDR are 57 dBZ, 3.9 dB, and 20.08 dB (m s ) , result of lacking such big drops, the simulated ZDR is respectively. These values are close to the observations 4.8 dB, which is smaller than the observed 5.7 dB. from location 1. 2) CASE II: MIXTURE OF RAIN AND MELTING HAIL Following the same procedure, the observed Doppler spectrum and spectral differential reﬂectivity from lo- In this case, a mixture of rain and melting hail is cation 2 are simulated, and the results are shown in investigated, and results are shown in Figs. 10 and 11. Figs. 9a and 9b. Different from location 1, a smaller Two scenarios are simulated with different amounts amount of small drops are used in the simulation, but of hail, and the particle size distributions of rain are the

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FIG.9.AsinFig. 8, but the obtained Doppler spectrum, spectral differential reflectivity, slope of sZDR, reflectivity, and differential reflectivity are similar to the observations from location 2.

2 2 same. From the top to the bottom of the resolution vol- positive slope of 0.09 dB (m s 1) 1, which is similar to L 23 21 ume, Nr0 ( ) gradually changes from 13 000 m mm the observation from location 4. From location 2 to lo- 21 23 21 21 (1.9 mm ) to 8000 m mm (2.7 mm )asin cation 4, Z and ZDR show similar values, but the slope of Fig. 10c, and the value of m is set as 2 in all the alti- sZDR changes from negative to positive. With the same tudes. This particle size distribution of rain is ap- background wind environment, the changes in the sZDR plied to both scenarios. In Fig. 10d, Nh0 is set as slope are simulated through adjusting the particle size 23 21 21 400 m mm ,andLh is set as 1.14 mm from all the distributions. The observed Z and ZDR is higher than altitudes in the resolution volume; the obtained Z, ZDR, simulation because of the attenuation, and the attenu- and slope of sZDR are 60 dBZ, 4.2 dB, and 20.02 dB ation effect is more obvious in location 5. 2 2 (m s 1) 1. The simulated Doppler spectrum and spec- 3) CASE III: RAIN ONLY FROM A REGION tral differential reﬂectivity are shown in Figs. 10a and WITHOUT VERTICAL SHEAR 10b, and they show similar features as the observations from location 3 in Fig. 4. It should be noted that the The Doppler spectrum and the spectral differential

Z and ZDR values from locations 1, 2, and 3 show reflectivity from regions away from the strong vertical similar values, but the slopes of sZDR show signifi- shear were also examined and simulated in the current cantly different values. In these three locations, sig- work. Three continuous locations (azimuthal angle: nificant changes in the particle size distribution and 2408; ranges: 44.375, 44.5, and 44.625 km) were selected hydrometeor species are simulated, but the integrated because 1) they are away from the shear region, and

Z and ZDR values are not sensitive to these changes. 2) they are associated with large signal-to-noise ratio. Therefore, the slope of sZDR couldbeusedasanin- The SH (Figs. 12a–c) and sZDR (Figs. 12d–f) are shown dicator of variations in the particle size distribution in Fig. 12. The linearly fitted sZDR is also shown and classes. in Figs. 12d–f with a dashed line. It should be noted that In Fig. 11, from the top to the bottom of the resolution only the high signal-to-noise ratio (SNR) portion with 21 volume, Lh gradually changes from 0.84 to 0.3 mm , SH(y) . 10 dB (thick solid line) is used in the fitting, and 23 21 and Nh0 is set as 400 m mm . Examples of Nh from the low SH(y) portion (think dashed line) is not used in top, center, and bottom of the radar resolution volume the fitting because they are associated with large fluc- are shown in Fig. 11c with solid, dotted, and dashed tuation caused by low SNR. It could be found that the lines, respectively. For the given particle size distri- SH exhibits a Gaussian-shaped spectrum, and the sZDR butions of rain and hail, the obtained sZDR shows shows a flattop-shape signature. Because there is no

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FIG. 10. (a) Simulated Doppler spectrum, (b) spectral differential reflectivity, (c) particle size distribution of raindrop, and (d) particle size distribution of hail. Details about the particle size distribution can be found in text. The obtained Doppler spectrum, spectral differential reflectivity, slope of sZDR, reflectivity, and differential reflectivity are similar to the observations from location 3.

strong vertical shear, both SH and sZDR are associated uniform particle size distribution is used in the whole with small spectral width. simulation volume with the following settings: Nr0 5 23 21 21 The simulated Doppler spectrum and the spectral 9000 m mm , m 5 1, and Lr 5 2mm . Because this differential reﬂectivity are presented in Fig. 13. Because region is away from the hail region, there are no hail- there is no size sorting caused by strong vertical shear, a stones added in the simulation. The background wind is

FIG. 11. As in Fig. 10, but the obtained Doppler spectrum, spectral differential reflectivity, slope of sZDR, reflectivity, and differential reflectivity are similar to the observations from location 4.

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FIG. 12. (a)–(c) Doppler spectrum and (d)–(f) spectral differential reﬂectivity observed from three locations away from the shear region. The azimuthal angle of these locations is 2408, and ranges are 44.375, 44.5, and 44.625 km. The linearly ﬁtted results are also included as dashed lines.

2 set as 215 m s 1, and there is no vertical shear applied. C-band OU-PRIME. Broad and flattop spectra were We found that Gaussian-shaped SH with small spectral observed and the spectral differential reflectivity ex- width and flattop-shaped sZDR were obtained, and these hibited interesting and distinct variation of slopes at results are consistent with the observations shown in different ranges. It is hypothesized that those observed Fig. 12. features in spectra of reflectivity and differential reflectivity are caused by shear-induced size sorting of a mixture of rain and melting hail in a turbulent envi- 5. Summary and conclusions ronment. Additionally, the profiles of mean radial ve- Spectral polarimetry links the Doppler and polari- locities from the lowest two elevations and mesonet metric information within the radar resolution volume wind data further support the presence of strong vertical through spectral processing. For example, Doppler shear. By comparing the polarimetric variables from the spectrum and spectral differential reflectivity represent C-band OU-PRIME and the nearby S-band KOUN, the distributions of reflectivity and differential reflectivity resonance effects at C band caused by large raindrops and as a function of radial velocity. In the past, spectral po- melting hail are evident, which is manifested by enhanced larimetry has been applied to observations at higher reflectivity and differential reflectivity and decreased elevations where the size sorting of hydrometeors is correlation coefficient. dominated by their size-dependent terminal velocities. Numerical simulations were carried out to study the In this work, spectral polarimetry was applied to ob- impact of various microphysical and dynamical vari- servations of a hailstorm at 08 elevation angle with the ables on the slope of spectral differential reflectivity.

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FIG. 13. As in Fig. 10, but for (a) the obtained Doppler spectrum, and (b) spectral differential reﬂectivity according to the observations from Fig. 12.

In this work, Doppler spectrum and spectral differential Acknowledgments. Matthew R. Kumjian was funded reflectivity from four distinct locations are simulated by NSF Award AGS-1661679 and by an award from the using different particle size distributions and hydrome- Insurance Institute for Business and Home Safety (IBHS). teor species in a sheared environment. The reflectivity The authors thank Dr. Guifu Zhang for helpful discussion of and differential reflectivity from these four locations the simulation, especially the implementation of attenuation. show similar values, but the slopes of spectral differential reflectivity show significantly different values. REFERENCES Through adjusting the particle size distributions within a reasonable range, the observations from these four lo- Anderson, M. E., L. D. Carey, W. A. Petersen, and K. R. Knupp, cations could be regenerated. It should be noted that 2011: C-band dual-polarimetric radar signatures of hail. Electron. J. Oper. Meteor., 2011-EJ02, http://nwafiles.nwas.org/ only one set of particle size distributions are used to ej/pdf/2011-EJ2.pdf. simulate the Doppler spectrum and spectral differen- Atlas, D., R. C. Srivastava, and R. S. Sekhon, 1973: Doppler radar tial reflectivity for each scenario in the current work. characteristics of precipitation at vertical incidence. Rev. Geophys. Other sets of particle size distribution should be able to Space Phys., 11, 1–35, https://doi.org/10.1029/RG011i001p00001. generate similar results, and we do not enumerate all Beard, K. V., and A. R. Jameson, 1983: Raindrop canting. J. Atmos. Sci., 40, 448–453, https://doi.org/10.1175/1520-0469(1983)040,0448: of them. RC.2.0.CO;2. One of the limitations in spectral polarimetry for op- Bodine, D., R. D. Palmer, and G. Zhang, 2014: Dual-wavelength erational application is the requirement of spectral av- polarimetric radar analyses of tornadic debris signatures. eraging. For example, 30 spectra were averaged in this J. Appl. Meteor. Climatol., 53, 242–261, https://doi.org/10.1175/ work to reduce the statistical fluctuations in the spectral JAMC-D-13-0189.1. Bohne, A. R., 1982: Radar detection of turbulence in precipitation polarimetric variables. Even though spectral averaging environments. J. Atmos. Sci., 39, 1819–1837, https://doi.org/ can also be achieved in range, azimuthal, and frequency 10.1175/1520-0469(1982)039,1819:RDOTIP.2.0.CO;2. domain, the resolution in those domains is compro- Borowska, L., A. V. Ryzhkov, and D. S. Zrnic´, 2011: Attenuation mised. Fortunately, a resampling technique based on and differential attenuation of 5-cm-wavelength radiation in bootstrapping was recently developed to tackle this melting hail. J. Appl. Meteor. Climatol., 50, 59–76, https://doi.org/ 10.1175/2010JAMC2465.1. limitation to improve the quality of spectral polarimetric Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experi- variables with relatively small number of samples ments in rainfall estimation with a polarimetric radar in a (Umeyama 2016). It is envisioned that spectral polar- subtropical environment. J. Appl. Meteor., 41, 674–685, https:// imetry can be applied to other types of scenarios. For doi.org/10.1175/1520-0450(2002)041,0674:EIREWA.2.0.CO;2; example, the study of tornado debris is one of them, Corrigendum, 44, 186, https://doi.org/10.1175/1520-0450(2005) , . where polarimetric signatures of debris are evident and 44 186:C 2.0.CO;2. Brussaard, G., 1974: Rain-induced cross polarization and raindrop centrifugal force can provide the size sorting. In this canting. Electron. Lett., 10, 411–412. work, we focused on only one of the spectral polari- ——, 1976: A meteorological model for rain-induced cross polarimetric variables, spectral differential reflectivity, at the zation. IEEE Trans. Antennas Propag., 24,5–11,https://doi.org/ lowest elevation. Two other variables of spectral corre- 10.1109/TAP.1976.1141282. lation coefficient and spectral differential phase can Carey, L. D., S. A. Rutledge, D. A. Ahijevych, and T. D. Keenan, 2000: Correcting propagation effects in C-band polarimetric provide different aspects of microphysical properties. radar observations of tropical convection using differential Studies on these two variables will be reported in propagation phase. J. Appl. Meteor., 39, 1405–1433, https://doi.org/ subsequent work. 10.1175/1520-0450(2000)039,1405:CPEICB.2.0.CO;2.

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