1

Combining Disdrometer Microscopic Photography and Cloud 1

Radar to Study Distributions of Hydrometeor Types Size and Fall 2

Velocity3

Xingcan Jia12 Yangang Liu2 Deping Ding3 Xincheng Ma3 Yichen Chen3 Bi 4

Kai3 Ping Tian3 Chunsong Lu4 Jiannong Quan15

1 Institute of Urban Meteorology Chinese Meteorological Administration Beijing 1000896

China 7

2 Brookhaven National Laboratory Upton NY 11973 US8

3 Beijing Weather Modification Office Beijing 100089 China9

4 Nanjing University of Information Science amp Technology Nanjing 210044 China10

11

Corresponding authors 12

Xingcan Jia xcjiaiumcn 13

Yangang Liu lygbnlgov 14

Deping Ding zytddpvipsinacom 15

Chunsong Lu clunuisteducn 16

17

Abstract Addressing solid precipitation poses additional challenges compared to warm 18

rain due to complex hydrometeor shapes involved including the dependence of fall 19

velocity on hydrometeor sizes hydrometeor size distributions and hydrometeor 20

classification This study is an extension of our previous work (Niu et al 2010) to 21

address these challenges by combining measurements from a PARSIVEL disdrometer 22

microscope photography and millimeter wavelength cloud radar The combined 23

BNL-211738-2019-JAAM

2

measurements are analyzed to classify the precipitation hydrometeor types examine 24

the dependence of fall velocity on hydrometeor sizes for different hydrometeor types 25

and determine the best distributions to describe the hydrometeor size distributions of 26

different hydrometeor types The results show (1) Hydrometeors can be classified to 27

four main types of raindrop graupel snowflake and mixed-phase according to the 28

dependence of terminal velocity on particle sizes corresponding microscope photos 29

and cloud radar observations (2) There are significant scatters in fall velocity for a given 30

hydrometer size velocities and the fall velocity spread for the solid hydrometeors 31

appear wider than that for raindrops across hydrometeor sizes with that for the 32

mixed-phase precipitation being largest suggesting that the effects of hydrometeor 33

shape on hydrometeor fall velocities (3) Hydrometeor size distributions for the four 34

types can all be well described by the Gamma or Weibull distribution Weibull (Gamma) 35

distribution performs better when skewness is less (larger) than 2 36

Key words hydrometeor fall velocity hydrometeor size distribution PARSIVEL 37

disdrometer microscope photography cloud radar 38

39

1 Introduction 40

Solid precipitation is important for weather and climate forecasting models since 41

predictions of precipitation amount location and duration depend greatly on how 42

precipitation particles are parameterized The last few decades have witnessed great 43

progress in both areas of parameterizing cold precipitation processes (Reisner et al 44

1998 Field et al 2007 Lin et al 2010 Agosta et al 2015) remote sensing (Tokay and 45

3

Short 1996 Souverijns et al 2017) and ground measurement (Chen et al 2011 46

Nurzyńska et al 2012 Ishizaka et al 2013 Huang et al 2017) of solid hydrometeors 47

Despite the great development solid precipitation measurement and parameterization 48

still suffer from large uncertainties and much work remains to be done Detailed solid 49

hydrometeor observations including size distribution fall velocity and shape of 50

hydrometeors are needed to improve microphysical parameterization in numerical 51

models and remote sensing 52

Hydrometeors properties (eg size concentration geometric shape and fall 53

velocity) are essential for further improving parameterizations of precipitation 54

processes and remote sensing (especially of polarized radar) In particular recent 55

developments in disdrometer and remote sensing techniques permit retrievals of more 56

hydrometeor size distribution (HSD) parameters and their vertical profiles over large 57

areas (Loumlffler-Mang and Blahak 2001 Matrosov 2007 Kneifel et al 2015) and 58

enhance our ability to monitor and investigate solid hydrometeor events and 59

microphysics At the same time more accurate assumptions regarding the spectral 60

shape of HSDs for different hydrometer types are needed which vary spatially and 61

temporally (Kikuchi et al 2013) Unfortunately our understanding of the hydrometeors 62

and direct measurements of solid precipitation is far from complete and more analyses 63

of in situ measurements are needed (Souverijns et al 2017) 64

Fall velocity is equally important and closely related to the HSD measurements 65

radar retrievals and parameterizations Fall velocity measurements of solid 66

hydrometeors can be traced to an empirical study by Locatelli and Hobbs (1974) which 67

4

is still utilized in microphysical parameterizations Later studies include those based on 68

fluid dynamics (Boumlhm 1989 Mitchell 1996 Khvorostyanov 2005 Heymsfield and 69

Westbrook 2010 Kubicek and Wang 2012) and using automated ground-based 70

disdrometers (Barthazy and Schefold 2006 Yuter et al 2006 Ishizaka et al 2013 Chen 71

et al 2011) 72

Most of these studies assume that the surrounding air is still rather than a 73

turbulent environment as in actual precipitating clouds Yuter et al (2006) obtained size 74

and fall velocity distributions within coexisting rain and wet snow (sleet) by using a 75

disdrometer but insufficient details of quantified results were provided The influence 76

of riming particle shape temperature and turbulence on the fallspeed of solid 77

precipitation in disdrometer measurements were further discussed (Barthazy and 78

Schefold 2006 Garrett and Yuter 2014 Geresdi et al 2014) However many factors 79

have influences on solid hydrometeor fall velocity and the complex effects have not 80

been yet adequately investigated 81

In a previous study (Niu et al 2010) we discussed the air density and other factors 82

(ie turbulence organized air motions break-up and measurement errors) that 83

potentially influence on distributions of raindrop sizes and fall velocities and called 84

attention to the turbulence induced the large velocity spread at given raindrop sizes 85

This work is a further extension of Niu et al (2010) to analyze measurements of size 86

and fall velocity distributions of solid hydrometeors collected during a recent field 87

experiment campaign conducted northeast of Beijing China to simultaneously 88

measure HSDs and fall velocities with a PARSIVEL disdrometer (see Section 2 for details) 89

5

This paper has two specific objectives (1) to distinguish and quantify hydrometeor 90

types and their fall velocities by combining disdrometer microscopic photography and 91

cloud radar observation (2) to characterize and compare the spectral shapes of HSDs 92

from different precipitation types 93

The rest of the paper is organized as follows Section 2 describes the experiment and 94

data Section 3 classifies hydrometeors and analyzes the size and fall velocity 95

distributions Section 4 examines the characters of HSDs and evaluates the distribution 96

function for describing HSDs The major findings are summarized in Section 5 97

2 Description of Experiment and Data 98

The observation site was on Haituo Mountain at a height of 1310 m located in 99

northwest of Beijing (40deg35primeN 115deg50primeE) China (Fig 1) The site is in the semiarid 100

temperate monsoon climate regime with mean winter precipitation amount for each 101

event is about 080 mm (2014-2015) (Ma et al 2017) All the observed events are stratiform 102

precipitation classified by the surface cloud radar China Weather Radar (Doppler radar) 103

and manual observations The entire radar reflectivity maximums are lower than 30 dBZ 104

which is chosen as the threshold reflectivity of convective precipitation (Zhang and Du 105

2000) 106

HSDs were measured with a PARSIVEL disdrometer and the measurement methods 107

are the same as employed in Niu et al (2010) Loumlffler-Mang and Joss (2000) and Tokay 108

et al (2014) provided detailed description of the PARSIVEL disdrometer The instrument 109

measures the maximum diameter of one-dimensional projection of the particle which is 110

smaller than or equal to the actual maximum diameter Snow particles are often not 111

6

horizontally symmetric and thus particle sizes for snow may underestimate actual maximum 112

particle diameter Battaglia et al (2010) pointed out that PARSIVELrsquos fall velocity 113

measurement may not be accurate for snowflakes due to the internally assumed relationship 114

between horizontal and vertical snow particle dimensions The uncertainty originates from 115

the shape-related factor which tends to depart more with increasing snowflake sizes and 116

can produce large errors When averaging over a large number of snowflakes the correction 117

factor is size dependent with a systematic tendency to underestimate the fall speed (but 118

never exceeding 20) The maximum error 20 of the empirical terminal velocities for 119

graupel and snowflake is used to estimate the instrument caused possible velocities such 120

as the dash lines in igure 2b In addition graupel is almost spherical hydrometeor and the 121

instrument error could not reach 20 The individual HSD sample interval was 10 seconds 122

The following criteria are used in choosing data for analysis (1) Particles smaller 123

than 025 mm are discarded (2) the total particle number of a HSD is over 10 counts 124

(every 10-sec sample) (Niu et al 2010) (3) precipitation lasted more than 30 minutes 125

are chosen (4) solid hydrometeor density is corrected with the equation ρs =126

017Dminus1 for solid hydrometeors (Boudala et al 2014) (5) Following Chen et al (2017) 127

raindrops outside +60 of the empirical terminal velocity of raindrop (Table 3) and -60 128

of empirical terminal velocity of densely rimed dendrites were excluded in the analysis 129

to minimize the effects of ldquomargin fallersrdquo winds and splashing Hydrometeors were 130

also collected with Formvar slides (76 cm long and 26 cm wide) which are exposed 131

outside for 5s to capture the hydrometeors with a sampling interval of 5 minutes The 132

Formvar samples were examined and photographed with a microscope-camera system 133

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

3

Short 1996 Souverijns et al 2017) and ground measurement (Chen et al 2011 46

Nurzyńska et al 2012 Ishizaka et al 2013 Huang et al 2017) of solid hydrometeors 47

Despite the great development solid precipitation measurement and parameterization 48

still suffer from large uncertainties and much work remains to be done Detailed solid 49

hydrometeor observations including size distribution fall velocity and shape of 50

hydrometeors are needed to improve microphysical parameterization in numerical 51

models and remote sensing 52

Hydrometeors properties (eg size concentration geometric shape and fall 53

velocity) are essential for further improving parameterizations of precipitation 54

processes and remote sensing (especially of polarized radar) In particular recent 55

developments in disdrometer and remote sensing techniques permit retrievals of more 56

hydrometeor size distribution (HSD) parameters and their vertical profiles over large 57

areas (Loumlffler-Mang and Blahak 2001 Matrosov 2007 Kneifel et al 2015) and 58

enhance our ability to monitor and investigate solid hydrometeor events and 59

microphysics At the same time more accurate assumptions regarding the spectral 60

shape of HSDs for different hydrometer types are needed which vary spatially and 61

temporally (Kikuchi et al 2013) Unfortunately our understanding of the hydrometeors 62

and direct measurements of solid precipitation is far from complete and more analyses 63

of in situ measurements are needed (Souverijns et al 2017) 64

Fall velocity is equally important and closely related to the HSD measurements 65

radar retrievals and parameterizations Fall velocity measurements of solid 66

hydrometeors can be traced to an empirical study by Locatelli and Hobbs (1974) which 67

4

is still utilized in microphysical parameterizations Later studies include those based on 68

fluid dynamics (Boumlhm 1989 Mitchell 1996 Khvorostyanov 2005 Heymsfield and 69

Westbrook 2010 Kubicek and Wang 2012) and using automated ground-based 70

disdrometers (Barthazy and Schefold 2006 Yuter et al 2006 Ishizaka et al 2013 Chen 71

et al 2011) 72

Most of these studies assume that the surrounding air is still rather than a 73

turbulent environment as in actual precipitating clouds Yuter et al (2006) obtained size 74

and fall velocity distributions within coexisting rain and wet snow (sleet) by using a 75

disdrometer but insufficient details of quantified results were provided The influence 76

of riming particle shape temperature and turbulence on the fallspeed of solid 77

precipitation in disdrometer measurements were further discussed (Barthazy and 78

Schefold 2006 Garrett and Yuter 2014 Geresdi et al 2014) However many factors 79

have influences on solid hydrometeor fall velocity and the complex effects have not 80

been yet adequately investigated 81

In a previous study (Niu et al 2010) we discussed the air density and other factors 82

(ie turbulence organized air motions break-up and measurement errors) that 83

potentially influence on distributions of raindrop sizes and fall velocities and called 84

attention to the turbulence induced the large velocity spread at given raindrop sizes 85

This work is a further extension of Niu et al (2010) to analyze measurements of size 86

and fall velocity distributions of solid hydrometeors collected during a recent field 87

experiment campaign conducted northeast of Beijing China to simultaneously 88

measure HSDs and fall velocities with a PARSIVEL disdrometer (see Section 2 for details) 89

5

This paper has two specific objectives (1) to distinguish and quantify hydrometeor 90

types and their fall velocities by combining disdrometer microscopic photography and 91

cloud radar observation (2) to characterize and compare the spectral shapes of HSDs 92

from different precipitation types 93

The rest of the paper is organized as follows Section 2 describes the experiment and 94

data Section 3 classifies hydrometeors and analyzes the size and fall velocity 95

distributions Section 4 examines the characters of HSDs and evaluates the distribution 96

function for describing HSDs The major findings are summarized in Section 5 97

2 Description of Experiment and Data 98

The observation site was on Haituo Mountain at a height of 1310 m located in 99

northwest of Beijing (40deg35primeN 115deg50primeE) China (Fig 1) The site is in the semiarid 100

temperate monsoon climate regime with mean winter precipitation amount for each 101

event is about 080 mm (2014-2015) (Ma et al 2017) All the observed events are stratiform 102

precipitation classified by the surface cloud radar China Weather Radar (Doppler radar) 103

and manual observations The entire radar reflectivity maximums are lower than 30 dBZ 104

which is chosen as the threshold reflectivity of convective precipitation (Zhang and Du 105

2000) 106

HSDs were measured with a PARSIVEL disdrometer and the measurement methods 107

are the same as employed in Niu et al (2010) Loumlffler-Mang and Joss (2000) and Tokay 108

et al (2014) provided detailed description of the PARSIVEL disdrometer The instrument 109

measures the maximum diameter of one-dimensional projection of the particle which is 110

smaller than or equal to the actual maximum diameter Snow particles are often not 111

6

horizontally symmetric and thus particle sizes for snow may underestimate actual maximum 112

particle diameter Battaglia et al (2010) pointed out that PARSIVELrsquos fall velocity 113

measurement may not be accurate for snowflakes due to the internally assumed relationship 114

between horizontal and vertical snow particle dimensions The uncertainty originates from 115

the shape-related factor which tends to depart more with increasing snowflake sizes and 116

can produce large errors When averaging over a large number of snowflakes the correction 117

factor is size dependent with a systematic tendency to underestimate the fall speed (but 118

never exceeding 20) The maximum error 20 of the empirical terminal velocities for 119

graupel and snowflake is used to estimate the instrument caused possible velocities such 120

as the dash lines in igure 2b In addition graupel is almost spherical hydrometeor and the 121

instrument error could not reach 20 The individual HSD sample interval was 10 seconds 122

The following criteria are used in choosing data for analysis (1) Particles smaller 123

than 025 mm are discarded (2) the total particle number of a HSD is over 10 counts 124

(every 10-sec sample) (Niu et al 2010) (3) precipitation lasted more than 30 minutes 125

are chosen (4) solid hydrometeor density is corrected with the equation ρs =126

017Dminus1 for solid hydrometeors (Boudala et al 2014) (5) Following Chen et al (2017) 127

raindrops outside +60 of the empirical terminal velocity of raindrop (Table 3) and -60 128

of empirical terminal velocity of densely rimed dendrites were excluded in the analysis 129

to minimize the effects of ldquomargin fallersrdquo winds and splashing Hydrometeors were 130

also collected with Formvar slides (76 cm long and 26 cm wide) which are exposed 131

outside for 5s to capture the hydrometeors with a sampling interval of 5 minutes The 132

Formvar samples were examined and photographed with a microscope-camera system 133

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

4

is still utilized in microphysical parameterizations Later studies include those based on 68

fluid dynamics (Boumlhm 1989 Mitchell 1996 Khvorostyanov 2005 Heymsfield and 69

Westbrook 2010 Kubicek and Wang 2012) and using automated ground-based 70

disdrometers (Barthazy and Schefold 2006 Yuter et al 2006 Ishizaka et al 2013 Chen 71

et al 2011) 72

Most of these studies assume that the surrounding air is still rather than a 73

turbulent environment as in actual precipitating clouds Yuter et al (2006) obtained size 74

and fall velocity distributions within coexisting rain and wet snow (sleet) by using a 75

disdrometer but insufficient details of quantified results were provided The influence 76

of riming particle shape temperature and turbulence on the fallspeed of solid 77

precipitation in disdrometer measurements were further discussed (Barthazy and 78

Schefold 2006 Garrett and Yuter 2014 Geresdi et al 2014) However many factors 79

have influences on solid hydrometeor fall velocity and the complex effects have not 80

been yet adequately investigated 81

In a previous study (Niu et al 2010) we discussed the air density and other factors 82

(ie turbulence organized air motions break-up and measurement errors) that 83

potentially influence on distributions of raindrop sizes and fall velocities and called 84

attention to the turbulence induced the large velocity spread at given raindrop sizes 85

This work is a further extension of Niu et al (2010) to analyze measurements of size 86

and fall velocity distributions of solid hydrometeors collected during a recent field 87

experiment campaign conducted northeast of Beijing China to simultaneously 88

measure HSDs and fall velocities with a PARSIVEL disdrometer (see Section 2 for details) 89

5

This paper has two specific objectives (1) to distinguish and quantify hydrometeor 90

types and their fall velocities by combining disdrometer microscopic photography and 91

cloud radar observation (2) to characterize and compare the spectral shapes of HSDs 92

from different precipitation types 93

The rest of the paper is organized as follows Section 2 describes the experiment and 94

data Section 3 classifies hydrometeors and analyzes the size and fall velocity 95

distributions Section 4 examines the characters of HSDs and evaluates the distribution 96

function for describing HSDs The major findings are summarized in Section 5 97

2 Description of Experiment and Data 98

The observation site was on Haituo Mountain at a height of 1310 m located in 99

northwest of Beijing (40deg35primeN 115deg50primeE) China (Fig 1) The site is in the semiarid 100

temperate monsoon climate regime with mean winter precipitation amount for each 101

event is about 080 mm (2014-2015) (Ma et al 2017) All the observed events are stratiform 102

precipitation classified by the surface cloud radar China Weather Radar (Doppler radar) 103

and manual observations The entire radar reflectivity maximums are lower than 30 dBZ 104

which is chosen as the threshold reflectivity of convective precipitation (Zhang and Du 105

2000) 106

HSDs were measured with a PARSIVEL disdrometer and the measurement methods 107

are the same as employed in Niu et al (2010) Loumlffler-Mang and Joss (2000) and Tokay 108

et al (2014) provided detailed description of the PARSIVEL disdrometer The instrument 109

measures the maximum diameter of one-dimensional projection of the particle which is 110

smaller than or equal to the actual maximum diameter Snow particles are often not 111

6

horizontally symmetric and thus particle sizes for snow may underestimate actual maximum 112

particle diameter Battaglia et al (2010) pointed out that PARSIVELrsquos fall velocity 113

measurement may not be accurate for snowflakes due to the internally assumed relationship 114

between horizontal and vertical snow particle dimensions The uncertainty originates from 115

the shape-related factor which tends to depart more with increasing snowflake sizes and 116

can produce large errors When averaging over a large number of snowflakes the correction 117

factor is size dependent with a systematic tendency to underestimate the fall speed (but 118

never exceeding 20) The maximum error 20 of the empirical terminal velocities for 119

graupel and snowflake is used to estimate the instrument caused possible velocities such 120

as the dash lines in igure 2b In addition graupel is almost spherical hydrometeor and the 121

instrument error could not reach 20 The individual HSD sample interval was 10 seconds 122

The following criteria are used in choosing data for analysis (1) Particles smaller 123

than 025 mm are discarded (2) the total particle number of a HSD is over 10 counts 124

(every 10-sec sample) (Niu et al 2010) (3) precipitation lasted more than 30 minutes 125

are chosen (4) solid hydrometeor density is corrected with the equation ρs =126

017Dminus1 for solid hydrometeors (Boudala et al 2014) (5) Following Chen et al (2017) 127

raindrops outside +60 of the empirical terminal velocity of raindrop (Table 3) and -60 128

of empirical terminal velocity of densely rimed dendrites were excluded in the analysis 129

to minimize the effects of ldquomargin fallersrdquo winds and splashing Hydrometeors were 130

also collected with Formvar slides (76 cm long and 26 cm wide) which are exposed 131

outside for 5s to capture the hydrometeors with a sampling interval of 5 minutes The 132

Formvar samples were examined and photographed with a microscope-camera system 133

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

5

This paper has two specific objectives (1) to distinguish and quantify hydrometeor 90

types and their fall velocities by combining disdrometer microscopic photography and 91

cloud radar observation (2) to characterize and compare the spectral shapes of HSDs 92

from different precipitation types 93

The rest of the paper is organized as follows Section 2 describes the experiment and 94

data Section 3 classifies hydrometeors and analyzes the size and fall velocity 95

distributions Section 4 examines the characters of HSDs and evaluates the distribution 96

function for describing HSDs The major findings are summarized in Section 5 97

2 Description of Experiment and Data 98

The observation site was on Haituo Mountain at a height of 1310 m located in 99

northwest of Beijing (40deg35primeN 115deg50primeE) China (Fig 1) The site is in the semiarid 100

temperate monsoon climate regime with mean winter precipitation amount for each 101

event is about 080 mm (2014-2015) (Ma et al 2017) All the observed events are stratiform 102

precipitation classified by the surface cloud radar China Weather Radar (Doppler radar) 103

and manual observations The entire radar reflectivity maximums are lower than 30 dBZ 104

which is chosen as the threshold reflectivity of convective precipitation (Zhang and Du 105

2000) 106

HSDs were measured with a PARSIVEL disdrometer and the measurement methods 107

are the same as employed in Niu et al (2010) Loumlffler-Mang and Joss (2000) and Tokay 108

et al (2014) provided detailed description of the PARSIVEL disdrometer The instrument 109

measures the maximum diameter of one-dimensional projection of the particle which is 110

smaller than or equal to the actual maximum diameter Snow particles are often not 111

6

horizontally symmetric and thus particle sizes for snow may underestimate actual maximum 112

particle diameter Battaglia et al (2010) pointed out that PARSIVELrsquos fall velocity 113

measurement may not be accurate for snowflakes due to the internally assumed relationship 114

between horizontal and vertical snow particle dimensions The uncertainty originates from 115

the shape-related factor which tends to depart more with increasing snowflake sizes and 116

can produce large errors When averaging over a large number of snowflakes the correction 117

factor is size dependent with a systematic tendency to underestimate the fall speed (but 118

never exceeding 20) The maximum error 20 of the empirical terminal velocities for 119

graupel and snowflake is used to estimate the instrument caused possible velocities such 120

as the dash lines in igure 2b In addition graupel is almost spherical hydrometeor and the 121

instrument error could not reach 20 The individual HSD sample interval was 10 seconds 122

The following criteria are used in choosing data for analysis (1) Particles smaller 123

than 025 mm are discarded (2) the total particle number of a HSD is over 10 counts 124

(every 10-sec sample) (Niu et al 2010) (3) precipitation lasted more than 30 minutes 125

are chosen (4) solid hydrometeor density is corrected with the equation ρs =126

017Dminus1 for solid hydrometeors (Boudala et al 2014) (5) Following Chen et al (2017) 127

raindrops outside +60 of the empirical terminal velocity of raindrop (Table 3) and -60 128

of empirical terminal velocity of densely rimed dendrites were excluded in the analysis 129

to minimize the effects of ldquomargin fallersrdquo winds and splashing Hydrometeors were 130

also collected with Formvar slides (76 cm long and 26 cm wide) which are exposed 131

outside for 5s to capture the hydrometeors with a sampling interval of 5 minutes The 132

Formvar samples were examined and photographed with a microscope-camera system 133

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

6

horizontally symmetric and thus particle sizes for snow may underestimate actual maximum 112

particle diameter Battaglia et al (2010) pointed out that PARSIVELrsquos fall velocity 113

measurement may not be accurate for snowflakes due to the internally assumed relationship 114

between horizontal and vertical snow particle dimensions The uncertainty originates from 115

the shape-related factor which tends to depart more with increasing snowflake sizes and 116

can produce large errors When averaging over a large number of snowflakes the correction 117

factor is size dependent with a systematic tendency to underestimate the fall speed (but 118

never exceeding 20) The maximum error 20 of the empirical terminal velocities for 119

graupel and snowflake is used to estimate the instrument caused possible velocities such 120

as the dash lines in igure 2b In addition graupel is almost spherical hydrometeor and the 121

instrument error could not reach 20 The individual HSD sample interval was 10 seconds 122

The following criteria are used in choosing data for analysis (1) Particles smaller 123

than 025 mm are discarded (2) the total particle number of a HSD is over 10 counts 124

(every 10-sec sample) (Niu et al 2010) (3) precipitation lasted more than 30 minutes 125

are chosen (4) solid hydrometeor density is corrected with the equation ρs =126

017Dminus1 for solid hydrometeors (Boudala et al 2014) (5) Following Chen et al (2017) 127

raindrops outside +60 of the empirical terminal velocity of raindrop (Table 3) and -60 128

of empirical terminal velocity of densely rimed dendrites were excluded in the analysis 129

to minimize the effects of ldquomargin fallersrdquo winds and splashing Hydrometeors were 130

also collected with Formvar slides (76 cm long and 26 cm wide) which are exposed 131

outside for 5s to capture the hydrometeors with a sampling interval of 5 minutes The 132

Formvar samples were examined and photographed with a microscope-camera system 133

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

7

following the method described by Schaefer (1956) The two-dimensional photo of 134

hydrometeors allows a careful determination of the hydrometeor shape and size The 135

Formvar data were used to distinguish dominant hydrometeor types eg raindrop 136

graupel or snowflake The hydrometeor type with a percentage of occurrence gt60 is 137

used to represent the solid precipitation type for convenience of analysis 138

A Ka-band millimeter wavelength cloud radar (wavelength is 8 mm) was used to 139

probe the vertical structure of clouds by vertical scanning from 11202016 which can 140

be used to determine cloud properties at 1 min temporal and 30 m vertical resolution 141

A total of 11098 HSD samples were collected from 12 precipitation events (Table 1) 142

Four representative hydrometeor types from the precipitation events are identified 143

raindrop graupel snowflake and mixed-phase precipitation The identification of the 144

hydrometeor types is mainly based on hydrometeors velocity observations and Formvar 145

images which will be detailed in Section 4 Mixed-phased precipitation contains 146

raindrops graupels and snowflakes simultaneously and none is dominant over another 147

The duration sample number precipitation rate and dominant type of hydrometeor 148

are shown in Table 1 The precipitation rate is calculated from HSD using the method 149

presented in Boudala et al (2014) 150

3 Hydrometeor Classification and Size and Fall Velocity 151

Distributions 152

a General Feature 153

Fig 2 illustrates the observed mean number concentration as a function of the 154

maximum dimension and the fall velocity for the four types of hydrometeors 155

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

8

respectively Also shown as a reference are the seven solid curves representing 156

different types of hydrometeor terminal velocities obtained from the laboratory 157

measurements calibrated by coefficient 50

0

since the site altitude is 1310 m 158

average pressure is 868 hpa and the average temperature is -5 ordmC (Niu et al2010) 159

(equations are shown in Table 3) Here ρ is the actual air density and ρ0 is the 160

standard atmospheric density The updrafts downdrafts and turbulence often 161

accompany falling hydrometeors inducing deviations in the velocity and trajectory of 162

falling hydrometeors from those in still air (Donnadieu 1980) Hence in this field 163

observation fall velocities of hydrometeors distribute along the terminal velocity curves 164

with large spread for each given size Similar features have been previously reported in 165

some situ measurements of raindrop fall velocities (Niu et al 2010 Rasmussen et al 166

2012) 167

In the previous study we discussed the air density and potential influencing 168

factors (eg turbulence organized air motions break-up and measurement errors) for 169

raindrop (Niu et al 2010) For precipitation in winter the hydrometeor shape is vitally 170

important for fall velocity The hydrometeor shape influence on fall velocities physical 171

mechanisms and instrument error underlying the large spread in the measured fall 172

velocities of different types are discussed in detail in the next two sub-sections 173

b Effect of Hydrometeor Type on Fall Velocity 174

In Fig 2a the hydrometeor fall velocities agree with the typical raindrop terminal 175

velocity curve which is obviously larger than the terminal velocities of graupel and 176

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

9

snowflake at given maximum diameters From the fall velocity distribution we can infer 177

that these hydrometeors are mainly raindrops The microscope photos obtained on the 178

ground confirm this However unlike the raindrop size and velocity distribution in 179

summer in Figure 8a of Niu et al (2010) there are some small size hydrometeors 180

(05~35mm) at very low speeds (05~2 ms) distributing among the terminal velocity 181

curves for graupel and snowflake This result implies that beside liquid raindrops 182

(including small raindrops or droplets from drizzle) some small snowflakes also exist in 183

the measurement The microscope photos also support the finding that although 184

raindrops are dominant some small plane- and column- shaped snowflakes co-exist 185

Fig 2b shows the joint hydrometeor size and fall velocity distribution during the 186

graupel event The hydrometeors are mainly symmetrically distributed around the ideal 187

terminal velocity of graupel (Locatelli and Hobbs 1974) supported by the photo-based 188

hydrometeor classification On the other hand it is found that certain small droplets 189

with the maximum dimensions among 03 to 10 mm are distributed around the liquid 190

raindrop curve suggesting that there are some raindrops and droplets coexisting in the 191

graupel dominated precipitation It is recognized that graupels grow by snow crystals 192

collecting cloud droplets and small raindrops (Pruppacher and Klett 1998) high cloud 193

top more liquid water content and presence of small raindrops (as shown in 194

microscope photography) provide favorable conditions for graupel The microscope 195

photos also display that graupelsrsquo surfaces are lumpy because of riming super cooled 196

drops during growing 197

The fall velocities of hydrometeors in Fig 2c are much smaller than the terminal 198

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

10

velocities of raindrop and graupel but are close to the terminal velocity curves of snow 199

of hexagonal densely rimed dendrites aggregates of dendrites Synthesizing the size 200

data the hydrometeors can be inferred as snowflakes which are further confirmed by 201

surface microscope photos From the surface observation we found many 202

rimedunrimed or aggregated dendrites or hexagonal snowflakes Although riming and 203

aggregation microphysical processes have some influence on terminal velocity 204

(Barthazy and Schefold 2006 Garrett and Yuter 2014 Heymsfield and Westbrook 205

2010) but the influences are too small to distinguish only from observed fall velocity 206

since it is a combined result of many factors When maximum dimensions are among 207

03 to 10 mm there are some small hydrometeors with fall velocities larger than 208

snowflakes which maybe graupels raindrops and droplets 209

Fig 2d is the hydrometeor size and fall velocity distribution during a mixed-phased 210

precipitation The hydrometeor fall velocities exhibit the largest spread for example 211

the fall velocities for hydrometeors with maximum dimension equaling to 3 mm 212

change from 05 to 5 ms and do not distribute along any of the ideal terminal velocity 213

curves This result suggests that different hydrometeor types might coexist during the 214

same sampling time To substantiate supercooled raindrops graupels and different 215

kinds of snowflakes are observed from the surface microscope simultaneously Similar 216

changes in particle size and fall velocity distribution documented by the disdrometer as 217

the storm transitions from rain to mixed-phase to snow were found at Marshall Field 218

site (Rasmussen et al 2012) 219

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

11

c Other Influencing Factors 220

The different dependence on particle size of terminal velocities for different types 221

of hydrometeors can explain some systematic distribution of the PARSIVEL-measured 222

fall velocities but the large spreads in the measurement of the instant particle fall 223

velocities await further inspection 224

According to our previous work the fall velocity (V) of a hydrometeor measured by 225

the PARSIVEL can be regarded as a combination of the terminal velocity in still air (Vt) 226

and a component Vrsquo that results from all other potential influencing factors eg type 227

turbulence organized air motions break-up and measurement errors (Niu et al 228

2010) 229

119881 = 119881119905 + 119881 prime (1) 230

The large spread of the measurements at both sides of the terminal velocity curves 231

shown in figure 2 seems compatible with the notion of nearly random collections of 232

downdraft and updraft A wider spread for fall velocities implies a wider range of 233

variation in vertical motions when hydrometeors are liquid raindrops (Niu et al 2010) 234

Solid nonspherical hydrometeors are much more complex than liquid drops and the 235

large spread of velocities may be caused by several other factors First different types 236

of hydrometeors (columns bullets plates lump graupels and hexagonal snowflakes et 237

al) have different fall speeds It is long recognized that microphysical processes in cold 238

precipitation are more complex and consequently generate more types of solid 239

hydrometeors comparing to the generation of raindrops The averaged concentration 240

distributions of large number of samples shown in Figure 2 also represents the possible 241

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

12

distributions of hydrometeors If there is only one kind particle the high concentration 242

would along its ideal terminal velocity very well just like the raindrop possible 243

distribution (Niu et al 2010) The concentration of mixed precipitation in Fig 2d does 244

not along any singe ideal terminal velocity further improved there are more than one 245

kind of hydrometers Coexistence of different types of hydrometeors induces larger 246

spread of fall velocities Second different degrees of riming and aggregating may affect 247

the results Densely rimed particles fall faster than unrimed particles with the same 248

maximum dimension aggregated snowflakes generally fall faster than their component 249

crystals (Barthazy and Schefold 2006 Garrett and Yuter 2014 Locatelli and Hobbs 250

1974) Third solid hydrometeors more likely occur breakup aggregation collision and 251

coalescence processes because of the large range of speeds According to prior 252

research of raindrop breakup and coalescence would lead to the ldquohigher-than-terminal 253

velocityrdquo fall velocity for small drops but ldquolower-than-terminal velocityrdquo fall velocity for 254

large drops (Niu et al 2010 Montero-Martiacutenez et al 2009 Larsen et al 2014) Similar 255

results are expected for the fall velocity of solid hydrometeors Last but not the least 256

the disdrometer may suffer from instrumental errors due to the internally assumed 257

relationship between horizontal and vertical particle dimensions likely resulting in a 258

wider spreads of fall velocities for the graupel and snowflakes as compared to the 259

raindrop counterpart as discussed in Section 2 Although the instrument error is an 260

important factor that should be considered in velocity spread as shown in Fig2b the 261

instrument error could not cause that large velocity spread The possible distributions 262

of fall velocities are the combine effects of physical and instrument principle 263

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

13

Further support for the possible instrumental uncertainty is from the cloud radar 264

observations Fig 3 shows the cloud reflectivity and linear depolarization ratio (LDR) in 265

lower level (less than 08 km) for the graupel snowflake and mixed-phased 266

precipitations respectively The LDR is a valuable observable in identifying the 267

hydrometeor types (Oue et al 2015) As shown in Fig 3a the cloud top height of 268

graupel precipitation was up to 7km during 1900 to 2050 Beijng Time (BT) which was 269

highest among the three events The high cloud top implies stronger updraft and 270

turbulence which further induces a larger fall velocity spread The LDR value was the 271

smallest among the three events indicating that the hydrometeors were close to 272

spherical consistent to the surface observations The cloud top height of snowflake 273

precipitation was up to 35 km (Fig 3c) which was the lowest among the three types of 274

precipitations implying weakest updraft and turbulence Notably although the 275

snowflake types are more complex than those of graupels the velocity spread was not 276

as large as that of the graupel type likely because of the weakest updraft and 277

turbulence The LDR was relatively larger consistent with the surface snowflake 278

observations (Fig 3d) The cloud top height of mixed-phase precipitation was up to 45 279

km (Fig 3e) and the LDR in lower level was largest among the three types of 280

precipitations owing to coexistence of raindrops graupels and snowflakes (Fig 3f) 281

4 Hydrometeor Size Distributions 282

a Comparison of HSDs between Different Types 283

There were four representative types of precipitations during this field experiment 284

providing us a unique opportunity to examine the HSD differences between the four 285

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

14

types of hydrometeors The sample numbers mean maximum dimensions and number 286

concentrations for each kind of HSDs are given in Table 2 Figure 4 is the averaged 287

distribution of all measured HSDs of 4 kinds As shown in Figure 4 on average the 288

mixed-phased precipitation tends to have most small size hydrometeors with D lt 15 289

mm while snowflake precipitation has the least small hydrometeors among the four 290

types The snowflake precipitation has more large size hydrometeors ie 3 mm lt D lt 11 291

mm Graupel size distribution tends to have two peeks around 13 mm and 26 mm The 292

mean maximum dimension is largest for snowflake (126 mm) and smallest for the 293

mixed HSD due to many small particles Similar results were found in precipitation 294

occurring in the Cascade and Rocky Mountains (Yuter et al 2006) The PARSIVEL 295

disdrometer doesnrsquot provide reliable data for small drops lower than 05 mm and tends to 296

underestimate small drops so the relatively low concentrations of small drops likely resulted 297

from the instrumentrsquos shortcomings (Tokay et al 2013 and Chen et al 2017) 298

b Evaluation of Distribution Functions for Describing HSDs 299

Over the last few decades great efforts have been devoted to finding the 300

appropriate analytical functions for describing the HSD because of its wide applications 301

in many areas including remote sensing and parameterization of precipitation (Chen et 302

al 2017) The two distribution functions mostly used up to now are the Lognormal 303

(Feingold and Levin 1986 Rosenfeld and Ulbrich 2003) and the Gamma (Ulbrich 1983) 304

distributions The Gamma distribution has been proposed as a first order generalization 305

of the Exponential distribution when μ equals 0 (Table 4) Weibull distribution is 306

another function used to describe size distributions of raindrops and cloud droplets (Liu 307

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

15

and Liu 1994 Liu et al 1995) Nevertheless the question as to which functions are 308

more adequate to describe HSDs of solid hydrometeors have been rarely investigated 309

systematically and quantitatively Thus the objective of this section is to examine the 310

most adequate distribution function used to describe HSDs 311

Considering size distributions of raindrops cloud droplets and aerosol particles are 312

the end results of many (stochastic) complex processes Liu (1992 1993 1994 and 313

1995) proposed a statistical method based on the relationships between the skewness 314

(S) and kurtosis (K) derived from an ensemble of particle size distribution 315

measurements and from those commonly used size distribution functions Cugerone 316

and Michele (2017) also used a similar approach based on the S-K relationships to study 317

raindrop size distributions Unlike the conventional curve-fitting of individual size 318

distributions this approach applies to an ensemble of size distribution measurements 319

and identifies the most appropriate statistical distribution pattern Here we apply this 320

approach to investigate if the statistical pattern of the HSDs of four precipitation types 321

follows the Exponential Gaussian Gamma Weibull and Lognormal distributions (Table 322

4) and if there are any pattern differences between them Briefly S and K of a HSD are 323

calculated from the corresponding HSD with the following two equations 324

23

2

3

t

iii

t

iii

dDN

)(Dn)D(D

dDN

)(Dn)D(D

S

minus

minus

=

(2) 325

32

2

4

minus

minus

minus

=

dDN

)(Dn)D(D

dDN

)(Dn)D(D

K

t

iii

t

iii

(3) 326

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

16

where Di is the central diameter of the i-th bin ni is the number concentration of the 327

i-th bin and Nt is total number concentration Note that each individual HSD can be 328

used to obtain a pair of S and K which is shown as a dot in Figure 5 Also noteworthy is 329

that the SndashK relationships for the families of Gamma Weibull and Lognormal are 330

described by known analytical functions (Liu 1993 1994 and 1995 Cugerone and 331

Michele 2017) represented by theoretical curves in Figure 5 The Exponential and 332

Gaussian distributions are represented by single points on the S-K diagram (S=2 and K=6 333

for Exponential) and (S=0 and K=0 for Gaussian) respectively Thus by comparing the 334

measured S and K values (points) with the known theoretical curves one can determine 335

the most proper function used to describe HSDs 336

Figure 5 shows the scatter plots of observed SndashK pairs calculated from the HSD 337

measurements for raindrop graupel snowflake mixed-phase precipitations Also 338

shown as comparison are the theoretical curves corresponding to the analytical HSD 339

functions A few points are evident First despite some occasional departures most of 340

the points fall near the theoretical lines of Gamma and Weibull distributions suggesting 341

that the HSD patterns for all the four types of precipitations well follow the Gamma and 342

Weibull distributions statistically Second HSDs of raindrops and mixed hydrometeors 343

are better described by Gamma and Weibull distribution than graupels and snowflakes 344

Third the S and K do not cluster around points (2 6) and (0 0) suggesting that the 345

Gaussian and Exponential distribution are not appropriate for describing HSDs 346

especially Gaussian distribution 347

To further evaluate the overall performance of the different distribution functions 348

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

17

we employ the widely used technique of Taylor diagram (Taylor 2001) and ldquoRelative 349

Euclidean Distancerdquo (RED) (Wu et al 2012) as a supplement to the Taylor diagram The 350

Taylor diagram has been widely used to visualize the statistical differences between 351

model results and observations in evaluation of model performance against 352

measurements or other benchmarks (IPCC 2001) Correlation coefficient (r) standard 353

deviation (σ) and ldquocentered root-mean-square errorrdquo (RMSE hereafter) are used in 354

the Taylor diagram The RED is introduced to add the bias as well in a dimensionless 355

framework that allows for comparison between different physical quantities with 356

different units In this study we consider the S and K values derived from the 357

distribution functions as model results and evaluate them against those derived from 358

the HSD measurements Brieflythe expressions for calculating r and σ and RMSE are 359

shown below 360

119903 =1

119873sum (119872119899minus)(119874119899minus)119873119899=1

120590119872120590119900 (4) 361

119877119872119878119864 = 1

119873sum [(119872119899 minus ) minus (119874119899 minus )]2119873119899=1

12

(5) 362

120590119872 = [1

119873sum (119872119899 minus )119873119899=1 ]

12

(6) 363

1205900 = [1

119873sum (119874119899 minus )119873119899=1 ]

12

(7) 364

where M and O denote distribution function and observed variables The RED 365

expression is shown below 366

119877119864119863 = [(minus)

]2

+ [(120590119872minus120590119874)

120590119900]2

+ (1 minus 119903)212

( 8 ) 367

Equation (8) indicates that RED considers all the components of mean bias variations 368

and correlations in a dimensionless way the distribution function performance 369

degrades as RED increases from the perfect agreement with RED=0 370

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

18

In the Taylor diagrams shown in Fig 6 the cosine of azimuthal angle of each point 371

gives the r between the function-calculated and observed data The distance between 372

individual point and the reference point ldquoObsrdquo represents the RMSE normalized by the 373

amplitude of the observationally based variations As this distance approaches to zero 374

the function-calculated K and S approach to the observations Clearly the values of r 375

are around 097 for Gamma and Weibull r for Lognormal function is slightly smaller 376

(095) The normalized σ are much different for Lognormal Gamma and Weibull 377

distributions with averaged value being 057 0895 and 0785 respectively Therefore 378

the Lognormal distribution exhibits the worst performance for describing HSDs and 379

Gamma and Weibull distributions appear to perform much better (Fig 6a) 380

Noticing that S and K of Gamma and Weibull distributions are equal at the point of S 381

= 2 and K = 6 and have positive relationship (If S lt 2 then K lt 6 and if S gt 2 then k gt 6) In 382

this part we further separately examine the group of data with S lt 2 and K lt 6 (Fig 6b) 383

and compare its result with those of total data The averaged normalized σ are 103 and 384

109 and r are 091 and 090 for Weibull and Gamma distributions respectively 385

suggesting that Weibull distribution performs slightly better than Gamma distribution 386

for the four HSD types in the data group of S lt 2 and K lt 6 whereas the opposite is true 387

for all data This conclusion is further substantiated by comparison of RED values for the 388

four type of precipitation (Fig7) which shows that Lognormal has the largest RED 389

Gamma the smallest and Weibull in between for four types of precipitations in all data 390

whereas Weibull (Lognormal) has smallest (largest) RED values when S lt 2 and K lt 6 It 391

can be inferred for data group of S gt 2 and k gt 6 Gamma distribution performs better 392

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

19

Further comparison between different precipitation types shows these functions all 393

performs better for raindrop size distributions and mixed-phase than graupel and 394

snowflake size distributions The RED for snowflakes is largest among the four types 395

The performance of Gamma is better for describing solid HSDs of all data but Weibull 396

distribution is better when S lt 2 and K lt 6 397

5 Conclusion 398

The combined measurements collected during a field experiment conducted at the 399

Haituo Mountain site using PARSIVEL disdrometer millimeter wavelength cloud radar 400

and microscope photography are analyzed to classify the precipitation hydrometeor 401

types examine the dependence of fall velocity on hydrometeor size for different 402

hydrometeor types and determine the best distribution functions to describe the 403

hydrometeor size distributions of different types 404

Analysis of the PARSIVEL-measured fall velocities show that on average the 405

dependence of fall velocity on hydrometeor size largely follow their corresponding 406

terminal velocity curves for raindrops graupels and snowflakes respectively suggesting 407

that such measurements are useful to identify the hydrometeor types The associated 408

microscope photos and cloud radar observations support the classification of 409

hydrometeor types based on the PARSIVEL-observed fall velocities Furthermore there 410

are velocity spreads in fall velocity across hydrometeor sizes for all hydrometeors and 411

the spreads for the solid hydrometeors could cause by hydrometeor type turbulence or 412

updraftdowndraft and instrument measured principle The coexistence of various 413

types of hydrometeors likely induces additional spread of fall velocity for given 414

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

20

hydrometeor sizes such as mixed precipitation Meanwhile complex microphysical 415

processes in cold precipitation such as riming aggregation breakup and collision and 416

coalescence influence fall velocity which likely induce larger spread of fall velocity Last 417

but not the least the disdrometer may suffer from instrumental errors due to the 418

internally assumed relationship between horizontal and vertical particle dimensions 419

Comparison of the type-averaged hydrometeor size distributions shows that the 420

mixed precipitation has more small hydrometeors with D lt 15 mm than others 421

whereas the snowflake precipitation has the least number of small hydrometeors 422

among the four types On the contrary the snowflake precipitation has more large 423

hydrometeors with D between 3 mm and 11 mm 424

The commonly used size distribution functions (Gamma Weibull and Lognormal) 425

are further examined to determine the most adequate function to describe the size 426

distribution measurements for all the hydrometeor types by use of an approach based 427

on the relationship between skewness (S) and kurtosis (K) Taylor diagram and RED are 428

further introduced to assess the performance of different size distribution functions 429

against measurements It is shown that the HSDs for the four types can all be well 430

described statistically by the Gamma and Weibull distribution but Lognormal 431

Exponential and Gaussian are not fit to describe HSDs Gamma and Weibull distribution 432

describe raindrop and mixed-phase size distribution better than snowflake and graupel 433

size distribution and their performances are worst for snowflake It may due to graupel 434

and snowflake precipitation have more big particles When S less than 2 and K less than 435

6 Weibull distribution performs better than Gamma distribution to describe HSDs but 436

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

21

Gamma distribution performs better for total data 437

A few points are noteworthy First although this study demonstrates the great 438

potential of combining simultaneous disdrometer measurements of particle fall velocity 439

and size distributions microscopic photography and cloud radar to address the 440

challenges facing solid precipitation the data are limited More research is needed to 441

discern such effects on the results presented here Second as the observations show 442

that in addition to sizes there are many other factors (eg hydrometeor type 443

microphysical processes turbulence and measuring principle) that can potentially 444

influence particle fall velocity as well More comprehensive research is needed to 445

further discern and separate these factors Finally the combined use of S-K diagram 446

Taylor diagram and relative Euclidean distance appears to be a powerful tool for 447

identifying the best function used to describe the HSDs and quantify the differences 448

between different precipitation types More research is in order along this direction 449

Acknowledgements 450

This study is mainly supported by the Chinese National Science Foundation under Grant 451

No 41675138 and National Key RampD Program of China (2017YFC0209604) It is partly 452

supported by Beijing National Science Foundation (8172023) the US Department 453

Energys Atmospheric System Research (ASR) Program (Liu and Jia at BNL) Lu is 454

supported by the Natural Science Foundation of Jiangsu Province (BK20160041) The 455

observation data are archived at a specialized supercomputer at Beijing Meteorology 456

Information Center (ftp10224592) which are available from the corresponding 457

author upon request The authors also thank Beijing weather modification office for 458

their support during the field experiment 459

Reference 460

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

22

Agosta CC Fettweis XX Datta RR 2015 Evaluation of the CMIP5 models in the 461

aim of regional modelling of the Antarctic surface mass balance Cryosphere 9 2311ndash462

2321 httpdxdoiorg105194tc-9-2311-2015 463

Barthazy E and R Schefold 2006 Fall velocity of snowflakes of different riming degree 464

and crystal types Atmos Res 82(1) 391-398 httpsdoiorg101016jatmosres 465

200512009 466

Battaglia A Rustemeier E Tokay A et al 2010 PARSIVEL snow observations a critical 467

assessment J Atmos Oceanic Technol 27(2) 333-344 doi 1011752009jtecha13321 468

Boumlhm H P 1989 A General Equation for the Terminal Fall Speed of Solid Hydrometeors 469

J Atmos Sci 46(15) 2419-2427 httpsdoiorg1011751520-0469(1989)046 lt2419A 470

GEFTTgt20CO2 471

Boudala F S G A Isaac et al 2014 Comparisons of Snowfall Measurements in 472

Complex Terrain Made During the 2010 Winter Olympics in Vancouver Pure and 473

Applied Geophysics 171(1) 113-127 474

Chen B W Hu and J Pu 2011 Characteristics of the raindrop size distribution for 475

freezing precipitation observed in southern China J Geophys Res 116 D06201 476

doi1010292010JD015305 477

Chen B Z Hu L Liu and G Zhang (2017) Raindrop Size Distribution Measurements at 478

4500m on the Tibetan Plateau during TIPEX-III J Geophys Res 122 11092-11106 479

doiorg10100 22017JD027233 480

23

Cugerone K and C De Michele 2017 Investigating raindrop size distributions in the 481

(L-)skewness-(L-)kurtosis plane Quart J Roy Meteor Soc 143(704) 1303-1312 482

httpsdoiorg101002qj3005 483

Donnadieu G 1980 Comparison of results obtained with the VIDIAZ spectro 484

pluviometer and the JossndashWaldvogel rainfall disdrometer in a ldquorain of a thundery typerdquo 485

J Appl Meteor 19 593ndash597 486

Feingold G and Levin Z 1986 The lognormal fit to raindrop spectra from frontal 487

convective clouds in Israel J Clim Appl Meteorol 25 1346ndash1363 488

httpsdoiorg1011751520-0450(1986)025lt1346tlftrsgt20CO2 489

Field P R A J Heymsfield and A Bansemer 2007 Snow Size Distribution 490

Parameterization for Midlatitude and Tropical Ice Clouds J Atmos Sci 64(12) 491

4346-4365 httpsdoiorg1011752007JAS23441 492

Garrett T J and S E Yuter 2014 Observed influence of riming temperature and 493

turbulence on the fallspeed of solid precipitation Geophysical Research Letters 41(18) 494

6515ndash6522 495

Geresdi I N Sarkadi and G Thompson 2014 Effect of the accretion by water drops 496

on the melting of snowflakes Atmos Res 149 96-110 497

Gunn R and G D Kinzer 1949 The terminal velocity of fall for water drops in stagnant 498

air J Meteor 6 243-248 httpsdoiorg1011751520-0469(1949)006lt0243TTVOFFgt 499

20CO2 500

24

Heymsfield A J and C D Westbrook 2010 Advances in the Estimation of Ice Particle 501

Fall Speeds Using Laboratory and Field Measurements J Atmos Sci 67(8) 2469-2482 502

httpsdoiorg1011752010JAS33791 503

Huang G-J C Kleinkort V N Bringi and B M Notaroš 2017 Winter precipitation 504

particle size distribution measurement by Multi-Angle Snowflake Camera Atmos Res 505

198 81-96 506

Intergovernmental Panel on Climate Change (IPCC) 2001 Climate Change 2001 The 507

Scientific Basis Contribution of Working Group I to the Third Assessment Report of the 508

Intergovernmental Panel on Climate Change edited by J T Houghton et al 881 pp 509

Cambridge Univ Press New York 510

Ishizaka M H Motoyoshi S Nakai T Shiina T Kumakura and K-i Muramoto 2013 A 511

New Method for Identifying the Main Type of Solid Hydrometeors Contributing to 512

Snowfall from Measured Size-Fall Speed Relationship Journal of the Meteorological 513

Society of Japan Ser II 91(6) 747-762 httpsdoiorg102151jmsj2013-602 514

Khvorostyanov V 2005 Fall Velocities of Hydrometeors in the Atmosphere 515

Refinements to a Continuous Analytical Power Law J Atmos Sci 62 4343-4357 516

httpsdoiorg101175JAS36221 517

Kikuchi K T Kameda K Higuchi and A Yamashita 2013 A global classification of 518

snow crystals ice crystals and solid precipitation based on observations from middle 519

latitudes to polar regions Atmos Res 132-133 460-472 520

Kneifel S A von Lerber J Tiira D Moisseev P Kollias and J Leinonen 2015 521

Observed relations between snowfall microphysics and triple-frequency radar 522

25

measurements J Geophys Res Atmos 120(12) 6034-6055 523

httpsdoi1010022015JD023156 524

Kubicek A and P K Wang 2012 A numerical study of the flow fields around a typical 525

conical graupel falling at various inclination angles Atmos Res 118 15ndash26 526

httpsdoi101016jatmosres201206001 527

Larsen M L A B Kostinski and A R Jameson (2014) Further evidence for 528

superterminal raindrops Geophys Res Lett 41(19) 6914-6918 529

Lee JE SH Jung H M Park S Kwon P L Lin and G W Lee 2015 Classification of 530

Precipitation Types Using Fall Velocity Diameter Relationships from 2D-Video 531

Distrometer Measurements Advances in Atmospheric Sciences (09) 532

Lin Y L J Donner and B A Colle 2010 Parameterization of riming intensity and its 533

impact on ice fall speed using arm data Mon Weather Rev 139(3) 1036ndash1047 534

httpsdoi1011752010MWR32991 535

Liu Y G 1992 Skewness and kurtosis of measured raindrop size distributions Atmos 536

Environ 26A 2713-2716 httpsdoiorg1010160960-1686(92)90005-6 537

Liu Y G 1993 Statistical theory of the Marshall-Parmer distribution of raindrops 538

Atmos Environ 27A 15-19 httpsdoiorg1010160960-1686(93)90066-8 539

Liu Y G and F Liu 1994 On the description of aerosol particle size distribution Atmos 540

Res 31(1-3) 187-198 httpsdoiorg1010160169-8095(94)90043-4 541

Liu Y G L You W Yang F Liu 1995 On the size distribution of cloud droplets Atmos 542

Res 35 (2-4) 201-216 httpsdoiorg1010160169-8095(94)00019-A 543

Locatelli J D and P V Hobbs 1974 Fall speeds and masses of solid precipitation 544

26

particles J Geophys Res 79(15) 2185-2197 httpsdoi101029JC079i015p02185 545

Loumlffler-Mang M and J Joss 2000 An optical disdrometer for measuring size and 546

velocity of hydrometeors J Atmos Oceanic Technol 17 130-139 547

httpsdoiorg1011751520-0426(2000)017lt0130AODFMSgt20CO2 548

Loumlffler-Mang M and U Blahak 2001 Estimation of the Equivalent Radar Reflectivity 549

Factor from Measured Snow Size Spectra J Clim Appl Meteorol 40(4) 843-849 550

httpsdoiorg1011751520-0450(2001)040lt0843EOTERRgt20CO2 551

Matrosov S Y 2007 Modeling Backscatter Properties of Snowfall at Millimeter 552

Wavelengths J Atmos Sci 64(5) 1727-1736 httpsdoiorg101175JAS39041 553

Mitchell D L 1996 Use of mass- and area-dimensional power laws for determining 554

precipitation particle terminal velocities J Atmos Sci 53 1710-1723 555

httpsdoiorg1011751520-0469(1996)053lt1710UOMAADgt20CO2 556

Montero-Martiacutenez G A B Kostinski R A Shaw and F Garciacutea-Garciacutea 2009 Do all 557

raindrops fall at terminal speed Geophys Res Lett 36 L11818 558

httpsdoiorg1010292008GL037111 559

Niu S X Jia J Sang X Liu C Lu and Y Liu 2010 Distributions of raindrop sizes and 560

fall velocities in a semiarid plateau climate Convective vs stratiform rains J Appl 561

Meteorol Climatol 49 632ndash645 httpsdoi1011752009JAMC22081 562

Nurzyńska K M Kubo and K-i Muramoto 2012 Texture operator for snow particle 563

classification into snowflake and graupel Atmos Res 118 121-132 564

Oue M M R Kumjian Y Lu J Verlinde K Aydin and E E Clothiaux 2015 Linear 565

27

Depolarization Ratios of columnar ice crystals in a deep precipitating system over the 566

Arctic observed by Zenith-Pointing Ka-Band Doppler radar J Clim Appl Meteorol 567

54(5) 1060-1068 httpsdoiorg101175JAMC-D-15-00121 568

Pruppacher H R and J D Klett 1998 Microphysics of clouds and precipitation Kluwer 569

Academic 954 pp 570

Rasmussen R B Baker et al 2012 How well are we measuring snow the 571

NOAAFAANCAR winter precipitation test bed Bull Amer Meteor Soc 93(6) 811-829 572

httpsdoiorg101175BAMS-D-11-000521 573

Reisner J R M Rasmussen and R T Bruintjes 1998 Explicit forecasting of 574

supercooled liquid water in winter storms using the MM5 mesoscale model Q J R 575

Meteorol Soc 124 1071ndash1107 httpsdoi101256smsqj54803 576

Rosenfeld D and Ulbrich C W 2003 Cloud microphysical properties processes and 577

rainfall estimation opportunities Meteorol Monogr 30 237-237 578

httpsdoiorg101175 0065-9401(2003)030lt0237cmppargt20CO2 579

Schaefer V J 1956 The preparation of snow crystal replicas-VI Weatherwsie 9 580

132-135 581

Souverijns N A Gossart S Lhermitte I V Gorodetskaya S Kneifel M Maahn F L 582

Bliven and N P M van Lipzig (2017) Estimating radar reflectivity - Snowfall rate 583

relationships and their uncertainties over Antarctica by combining disdrometer and 584

radar observations Atmos Res 196 211-223 585

Taylor K E 2001 Summarizing multiple aspects of model performance in a single 586

diagram J Geophys Res 106 7183ndash7192 httpsdoi1010292000JD900719 587

28

Tokay A and D A Short 1996 Evidence from tropical raindrop spectra of the origin of 588

rain from stratiform versus convective clouds J Appl Meteor 35 355ndash371 589

Tokay A and D Atlas 1999 Rainfall microphysics and radar properties analysis 590

methods for drop size spectra J Appl Meteor 37 912-923 591

httpsdoiorg1011751520-0450(1998)037lt0912RMARPAgt20CO2 592

Tokay A D B Wolff et al 2014 Evaluation of the New version of the laser-optical 593

disdrometer OTT Parsivel2 J Atmos Oceanic Technol 31(6) 1276-1288 594

httpsdoiorg101175JTECH-D-13-001741 595

Ulbrich C W 1983 Natural variations in the analytical form of the raindrop size 596

distribution J Appl Meteor 5 1764-1775 597

Wen G H Xiao H Yang Y Bi and W Xu 2017 Characteristics of summer and winter 598

precipitation over northern China Atmos Res 197 390-406 599

Wu W Y Liu and A K Betts 2012 Observationally based evaluation of NWP 600

reanalyses in modeling cloud properties over the Southern Great Plains J Geophys 601

Res 117 D12202 httpsdoi1010292011JD016971 602

Yuter S E D E Kingsmill L B Nance and M Loumlffler-Mang 2006 Observations of 603

precipitation size and fall velocity characteristics within coexisting rain and wet snow J 604

Appl Meteor Climatol 45 1450ndash1464 doi httpsdoiorg101175JAM24061 605

Zhang P C and B Y Du 2000 Radar Meteorology China Meteorological Press 511pp 606

(in Chinese) 607

608

29

Table 1 Summary of observed precipitation events in winter 609

Events Duration

(BJT)

Number of

samples

Dominant

hydrometeor

type

Surface

temperature

()

Precipitation

rate (mmh)

2016116 1739-2359 1793 Snowflake -88 09

2016116 2010-2359 797

Raindrop

(2010-2200)

and snowflake

-091 04

2016117 0000-0104 280 Snowflake 043 054

20161110 0717-1145 1232 Snowflake -163 068

20161120 1820-2319 1350 Graupel -49 051

20161121 0005-0049

0247-1005 2062

Graupel

(0005-0049)

and mixed

-106 071

20161129 1510-1805 878 Snowflake -627 061

20161130 0000-0302 780 Snowflake -518 096

2016125 0421-0642 718 Mixed -545 05

20161225 2024-2359 317 Snowflake -305 028

20161226 0000-0015

0113-0243 112 Mixed -678 022

201717 0537-1500 726 Snowflake -421 029

610

30

611

31

Table 2 Averaged information for each HSD 612

HSD type Sample

number

Mean maximum

dimension (mm)

Number

concentration (m-3)

Raindrop 378 101 7635

Graupel 1608 109 6508

Snowflake 6785 126 4488

Mixed 2634 076 16942

613

614

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

28

Tokay A and D A Short 1996 Evidence from tropical raindrop spectra of the origin of 588

rain from stratiform versus convective clouds J Appl Meteor 35 355ndash371 589

Tokay A and D Atlas 1999 Rainfall microphysics and radar properties analysis 590

methods for drop size spectra J Appl Meteor 37 912-923 591

httpsdoiorg1011751520-0450(1998)037lt0912RMARPAgt20CO2 592

Tokay A D B Wolff et al 2014 Evaluation of the New version of the laser-optical 593

disdrometer OTT Parsivel2 J Atmos Oceanic Technol 31(6) 1276-1288 594

httpsdoiorg101175JTECH-D-13-001741 595

Ulbrich C W 1983 Natural variations in the analytical form of the raindrop size 596

distribution J Appl Meteor 5 1764-1775 597

Wen G H Xiao H Yang Y Bi and W Xu 2017 Characteristics of summer and winter 598

precipitation over northern China Atmos Res 197 390-406 599

Wu W Y Liu and A K Betts 2012 Observationally based evaluation of NWP 600

reanalyses in modeling cloud properties over the Southern Great Plains J Geophys 601

Res 117 D12202 httpsdoi1010292011JD016971 602

Yuter S E D E Kingsmill L B Nance and M Loumlffler-Mang 2006 Observations of 603

precipitation size and fall velocity characteristics within coexisting rain and wet snow J 604

Appl Meteor Climatol 45 1450ndash1464 doi httpsdoiorg101175JAM24061 605

Zhang P C and B Y Du 2000 Radar Meteorology China Meteorological Press 511pp 606

(in Chinese) 607

608

29

Table 1 Summary of observed precipitation events in winter 609

Events Duration

(BJT)

Number of

samples

Dominant

hydrometeor

type

Surface

temperature

()

Precipitation

rate (mmh)

2016116 1739-2359 1793 Snowflake -88 09

2016116 2010-2359 797

Raindrop

(2010-2200)

and snowflake

-091 04

2016117 0000-0104 280 Snowflake 043 054

20161110 0717-1145 1232 Snowflake -163 068

20161120 1820-2319 1350 Graupel -49 051

20161121 0005-0049

0247-1005 2062

Graupel

(0005-0049)

and mixed

-106 071

20161129 1510-1805 878 Snowflake -627 061

20161130 0000-0302 780 Snowflake -518 096

2016125 0421-0642 718 Mixed -545 05

20161225 2024-2359 317 Snowflake -305 028

20161226 0000-0015

0113-0243 112 Mixed -678 022

201717 0537-1500 726 Snowflake -421 029

610

30

611

31

Table 2 Averaged information for each HSD 612

HSD type Sample

number

Mean maximum

dimension (mm)

Number

concentration (m-3)

Raindrop 378 101 7635

Graupel 1608 109 6508

Snowflake 6785 126 4488

Mixed 2634 076 16942

613

614

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

29

Table 1 Summary of observed precipitation events in winter 609

Events Duration

(BJT)

Number of

samples

Dominant

hydrometeor

type

Surface

temperature

()

Precipitation

rate (mmh)

2016116 1739-2359 1793 Snowflake -88 09

2016116 2010-2359 797

Raindrop

(2010-2200)

and snowflake

-091 04

2016117 0000-0104 280 Snowflake 043 054

20161110 0717-1145 1232 Snowflake -163 068

20161120 1820-2319 1350 Graupel -49 051

20161121 0005-0049

0247-1005 2062

Graupel

(0005-0049)

and mixed

-106 071

20161129 1510-1805 878 Snowflake -627 061

20161130 0000-0302 780 Snowflake -518 096

2016125 0421-0642 718 Mixed -545 05

20161225 2024-2359 317 Snowflake -305 028

20161226 0000-0015

0113-0243 112 Mixed -678 022

201717 0537-1500 726 Snowflake -421 029

610

30

611

31

Table 2 Averaged information for each HSD 612

HSD type Sample

number

Mean maximum

dimension (mm)

Number

concentration (m-3)

Raindrop 378 101 7635

Graupel 1608 109 6508

Snowflake 6785 126 4488

Mixed 2634 076 16942

613

614

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

30

611

31

Table 2 Averaged information for each HSD 612

HSD type Sample

number

Mean maximum

dimension (mm)

Number

concentration (m-3)

Raindrop 378 101 7635

Graupel 1608 109 6508

Snowflake 6785 126 4488

Mixed 2634 076 16942

613

614

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

31

Table 2 Averaged information for each HSD 612

HSD type Sample

number

Mean maximum

dimension (mm)

Number

concentration (m-3)

Raindrop 378 101 7635

Graupel 1608 109 6508

Snowflake 6785 126 4488

Mixed 2634 076 16942

613

614

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

32

615

Table 3 Relationships between max dimension and terminal velocity for different types of 616

hydrometeors (Gunn and Kinzer 1949 Locatelli and Hobbs 1974) calibrated by air density in Haituo 617

Mountain 618

Hydrometeor types Terminal velocity

Raindrop 119881119905 = 11405(965 minus 103119890minus06119863)

Graupel 119881119905 = 11405(13119863066)

Aggregates of unrimed radiating assemblages of

plates bullets and columns 119881119905 = 11405(069119863041)

Graupel like snow of hexagonal type 119881119905 = 11405(086119863025)

Aggregates of densely rimed radiating assemblages

of dendrites 119881119905 = 11405(079119863027)

Densely rimed dendrites 119881119905 = 11405(062119863033)

Aggregates of unrimed radiating assemblages of

dendrites 119881119905 = 11405(08119863016)

619

620

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

33

Table 4 Functions used to fit HSD 621

Name Function Parameters

Gamma 119899(119863) = 1198730119863120583119890119909119901(minus120582119863) No λ μ

Lognormal

119873 =119873119879

radic2120587119897119900119892120590

1

119863exp(minus

1198971199001198922(119863119863119898

)

21198971199001198922120590)

NT σ Dm

Weibull 119873 = 1198730119863119902minus1119890119909119901(minus120582119863119902) No λ q

Exponential 119899(119863) = 1198730119890119909119901(minus120582119863) No λ

Gaussian 119873 =119873119879

120590radic2120587

1

119863exp(minus

(119863 minus 119863119898)2

21205902) NT σ Dm

622

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

34

623

624

Figure 1 Topographic distribution of the experiment site The elevation of the site is 625

1310 m 626

627

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

35

628

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

36

Figure 2 Fall velocity as a function of particle maximum dimension (maximum diameter) for the four 629

hydrometeor types of (a) supercool raindrop (b) graupel (c) snowflake and (d) mixed The color 630

scheme denotes the particle number concentration The solid red curve denotes the typical 631

expression of terminal velocity of raindrops The black curve represents the terminal velocity of 632

lump graupels The blue curve represents the typical terminal velocity of aggregates of densely 633

rimed planes bullets and columns The light blue yellow green and orange curves represent 634

terminal velocities of hexagonal aggregates of densely rimed dendrites densely rimed dendrites 635

and aggregates of unrimed dendrites The dash lines provide the estimate of possible range of 636

instrumental errors for graupels and snowflakes (averaged velocity of these types) estimated by the 637

empirical terminal velocity plusmn20 The representative microscope photographs of the corresponding 638

different precipitation types observed during the same period are shown on the left panel 639

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

37

640

Figure 3 Height-time distribution of cloud radar parameters of graupel precipitation (a b) snowflake 641

precipitation (c d) mixed-phase (e f) reflectivity (dBz) (a c e) linear depolarization ratio (LDR) 642

under 08 km (dB) (b d f) 643

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

38

644

645

Figure 4 Averaged hydrometeor size distributions for the raindrop graupel snowflake 646

mixed-phase precipitations 647

648

0 1 2 3 4 5 6 7 8 9 10 11001

01

1

10

100

1000 raindrop

graupel

snowflake

mixed

N (m

-3m

m-1

)

Max dimension(mm)

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

39

649

650

Figure 5 S-K diagram comparing the relationships between observations (dots) and the 651

commonly used size distribution functions (solid lines) for raindrop (a) graupel (b) 652

snowflake (c) mixed-phase precipitation (d) The blue red and green lines represent 653

S-K relationships of Gamma Lognormal and Weibull distribution respectively The S 654

and K of Exponential and Gaussian distributions are the point (2 6) and (0 0) 655

respectively 656

657

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

-1 0 1 2 3 4 5 6

0

10

20

30

40

50

Raindrop

K

S

Gamma

Lognormal

Weibull

Graupel

K

S

Snowflake

K

S

MixedK

S

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

40

658

659

Figure 6 Taylor diagrams of K from (left) all the data and (right) data with S less than 2 660

and K less than 6 The numbers ldquo1rdquo ldquo2rdquo and ldquo3rdquo denote ldquoGammardquo ldquoLognormalrdquo and 661

ldquoWeibullrdquo distributions respectively 662

663

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

41

664

Figure 7 RED for all data (left) and data with S less than 2 and K less than 6 (right) The 665

shorter the distance the better distribution function performance is 666

raindrop graupel snowflake mixed00

05

10

15

20

25

30

35R

ela

tive

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull

raindrop graupel snowflake mixed0

1

2

3

4

5

6

Re

lative

Eu

clid

ea

n D

ista

nce

Gamma

Lognormal

Weibull