Shape of the rain drop size distributions and classification of ... - NOPR

Indian Journal of Radio & Space Physics Vol. 35, October 2006, pp. 360-367

Shape of the rain drop size distributions and classification of rain type at Gadanki Mahen Konwar, Diganta Kumar Sarma, Jyotirmoy Das1 & Sanjay Sharma Kohima Science College, Jotsoma, Kohima, Nagaland 797 002, India Electronics and Communication Sciences Unit, Indian Statistical Institute, Kolkata 700 108, India [e-mail: [email protected]]

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Received 26 August 2005; revised 22 May 2006; accepted 3 July 2006 Three different rain drop size distribution (RDSD) models namely exponential, lognormal and gamma distribution are fitted to RDSD as spectra observed from Joss-Waldvodgel Disdrometer (JWD) at Gadanki (13.8ºN, 79.18ºE). Gamma distribution shows overall good agreement with observed RDSD for all ranges of rainfall rate. Rainfall rate calculated from gamma drop size distribution is found to have minimum root mean square error and biasing compared to exponential or lognormal distribution. The intrinsic shape of RDSD is found out from normalized RDSD which follows an “S” shape for both low rain ≤ 10 mm h-1 and high rain > 10 mm h-1. The convective and stratiform rains are separated for an event. The equations of separation for convective and stratiform rains have been derived assuming power law for log10 (R) - Dm, log10 (N0*) - log10 (R) and log10 (N0*) - Dm. The coefficients and exponent for these equations are α DR = 0.145 , α RN = 103 , α DN = 0.85 × 103.15 , and β DR = 4.70 , β RN = −0.15 , β DN = −0.94 , respectively. Keywords: Rain drop size distribution (RDSD), Exponential DSD, Lognormal DSD, Gamma DSD, Convective rain, Stratiform rain PACS No: 92.60.Jq IPC Code: G01S13/95; G06T1/40

1 Introduction Rain drop size distribution (RDSD) is one of the most widely used parameters for better understanding and complete description of rain phenomenon. Different RDSD models namely exponential1, lognormal2, gamma3 and Weibull4 are being used to study rain characteristics. Marshall and Palmer1 parameterized the RDSD and found that it follows an exponential distribution of the following form

deviation from an exponential distribution in most cases. They concluded that the lognormal representation is suitable for a broad range of applications and can facilitate interpretation of the physical processes which control the shape of the distribution. Its parameters have a simple geometrical interpretation. The lognormal distribution has the following expression2

N ( D) = N 0 exp(−ΛD)

N ( D) =

... (1)

where, N(D) (m-3 mm-1) is the concentration of raindrops per diameter interval ∆D (mm), D (mm) the rain drop diameter, N0 the intercept parameter with a fixed value of 8 × 103 mm-1 m-3, Λ (mm-1) the slope parameter with a power law relation Λ = 4.1 R -0.21 mm-1, and R the rainfall intensity in mm h-1. It is found that exponential distribution under predicts (over predicts) the upper tail of the distribution in very light (heavy) rainfall5. Due to the departure of RDSD’s from exponentiality, many authors have preferred threeparameter models to describe rain characteristics2. Feingold and Levin2 fitted three-parameter lognormal model to frontal convective clouds and found

Nt × exp[− Ln 2 ( D / Dg ) / 2 Ln 2 σ] (2π) ( Lnσ) D … (2) 0.5

where N t (m-3) is the total number of drops, Dg (mm) the geometric mean of the drop diameter (or median size diameter), and σ the standard geometrical deviation of D. Ulbrich and Atlas6 has shown that accuracy in deducing rainfall rates from RDSD can be improved if it is assumed to be a gamma distribution. The gamma distribution is a three-parameter distribution. The following relation gives the modified gamma drop size distribution

N ( D) = N 0 D μ exp(−ΛD)

… (3)

KONWAR et al.: RAIN DROP SIZE DISTRIBUTION & CLASSIFICATION

where, N0 (mm-1-μ m-3) is the intercept parameter, μ the shape parameter and Λ (mm-1) the slope parameter of the distribution.3, 7 Method of moments approach can be used to calculate the parameters8 N0, μ and Λ. Different representations of RDSDs such as exponential, lognormal and gamma have their own limitations. For example the intercept parameter of gamma distribution is not well defined and not considered as a physical quantity9. Hence normalization of N(D) is considered as one of the options in studying the shape of RDSD. Sekhon and Srivastava10, 11 and Willis7 suggested a normalization of the rain drop diameters and of the drop size distribution in order to deal directly with the whole set of the spectra on a unique plot, where the parameters of the RDSD were fitted more robustly and independently of R. Normalization of N(D) gives rise to the intrinsic shape of RDSD that compares the shape of the two spectra, not having same liquid water content (LWC) and/or median volume drop diameter (Dm). Testud et al.9 developed a mathematical technique to normalize rain spectra and found it to follow an “S” shape. For any rain event the classification of convective and stratiform rain is very important due to its different nature of contribution of latent heat released to climate. Tokay and Short8 observed a significant change of gamma parameter, i.e. interceptor N0, during the transition from convective to stratiform rain. Interceptor N0 of exponential distribution also demonstrates a similar nature during the transition period5. Atlas et al.12 classified rain events as convective, transition and stratiform type rather than convective and stratiform type. They identified rain event as initially convective, if rain rises sharply to peak in excess, about 10-15 mm h-1 while D0 (mass weighted mean diameter) does not vary greatly. When D0 and R decrease simultaneously following convective period, the rain is classified as transition. The stratiform rain is characterized by approximately steady rain having R ≤ 10 mm h-1 and usually with higher values of D0 (mm). In recent times many works have been carried out to find suitable relationships between the gamma parameters and rainfall type. Ulbrich3 reported that the gamma RDSD parameters such as N0, μ and Λ display a systematic dependence on one another between different rainfall types, as well from moment-to-moment within a given rainfall type. Tokay and Short8 found that it is possible to classify

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convective and stratiform types of rain from N0-R and Λ-R relations. Maki et al.13 studied the shape of modified gamma RDSD during convective and stratiform regions. They found distinct characteristic RDSDs for both occasions. Testud et al.9 also separated stratiform and convective rain types by normalizing the spectra. They presented scatter plots of log10 (R) versus Dm, log10 (R) versus log10 (N0*), and log10 (N0*) versus Dm and found separate clusters of convective and stratiform types of rain. In India numerous researchers are studying different RDSD models to fit them to rain phenomenon. For Indian climate Jassal et al.14, Verma and Jha15 studied lognormal drop size distribution and proposed lognormal RDSD model over Dehradun. Suresh and Bhatnagar (personal communication) found lognormal distribution to fit well with observed DSD in rain spectra at Cuddalore (11.46ºN, 79.46ºE) during north-east monsoon. However they observed some deviation in the rain rate, 10-50 mm h-1 during pre-monsoon and south-west monsoon season. A similar study was made by Reddy and Kozu16, who examined the seasonal variation of gamma parameters at Gadanki. Over Gadanki, Rao et al.17 separated precipitating systems using Doppler spectra of VHF/UHF wind profilers and established their associated Z-R relationships. They also discussed the respective RDSDs during convective, transition and stratiform periods. In this paper an effort is made to present a study of the characteristics of the RDSD and different types of rain with the help of Joss-Waldvodgel Disdrometer (JWD)18 observations. The study has been carried out with the following objectives: (i) To examine the fitting of the three RDSD models, i.e. exponential, lognormal and gamma, to the observed RDSD spectra for different rainfall intensity (ii) To examine the intrinsic shape of RDSD rain spectra (iii) To separate convective and stratiform rain types from RDSD characteristics This paper is organized in the following manner. Section 1 presents introduction. Section 2 describes the observational system and data analysis methodology. Sections 3 and 4 describe the observations. Results and conclusions are presented in Section 5. 2 System description and data analysis The JWD is one of the most widely used

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instruments around the globe for analyzing the rain RDSD and rain characteristics. The data used in this study are collected from the JWD, located at National Laboratory for Atmospheric Research (NLAR) at Gadanki (13.8ºN, 79.18ºE), India. The JWD estimates the diameter of the drops by sensing the voltage induced from the downward displacement of a 50 cm2 styrofoam cone, once it is hit by rain drops. The output voltage relates to the diameter of the raindrop falling at terminal velocity. The standard output of the JWD is utilized in this study, which is the number of drops that are sorted into 20 size intervals ranging from 0.3 to about 5.0-5.5 mm for a one-minute integration time. However, one disadvantage of JWD is its inability to detect the lower drops during heavy precipitation. For reducing this error, dead error correction method has been applied19. Various rain parameters such as rainfall rate, liquid water content, radar reflectivity factor, kinetic energy of the falling drops can be measured with the help of JWD. The third moment of the RDSD gives the rain fall rate R in mm h-1, expressed by the following relation R = (π/6) (3.6/10 ) (1/AT) ∑ (ni D ) 6

3

… (4)

where A is the collecting area of the disdrometer, T the integration time, ni and D the number of drops and drop diameter of the ith channel of JWD, respectively. The intrinsic shape of the RDSD is obtained by normalizing the number density N(D) by a term N0*, thus eliminating the effect of rain rate R on the shape of the distributions. The normalized theoretical or observed raindrop spectra are obtained by the following expressions9 N ( D) = N 0* F ( D / Dm )

… (5)

The scaling parameter for concentration N 0* is calculated by the relation

N 0* =

44 LWC πρ w Dm4

… (6)

The mass weighted mean diameter Dm, which is equal to the ratio of fourth moment to the third moment of RDSD spectra, given by Dm =

M4 M3

… (7)

where M3, M4 are the 3rd and 4th moment of RDSD spectra. There is a relationship between Λ and μ of

gamma distribution expression3 Λ=

given

by

the

following

μ+4 Dm

… (8)

where the shape parameter μ is obtained from

μ=

11G − 8 + [G (G + 8)]1/ 2 2(1 − G )

with G =

M 43 M 32 M 6

… (9)

… (10)

where M6 is the 6th moment of RDSD spectra. For the present study, analysis is carried out with 21000 RDSD spectra collected during the period from 1998 to 2001 at Gadanki. RDSD spectrum of rainfall intensity greater than 0.1 mm h-1 is considered as a rainy event. 3 Observations and results The average RDSDs for rainfall intensity of 5, 25, 50 and 75 mm h-1 are shown in Figs 1(a), (b), (c) and (d), respectively. The averaged RDSDs are obtained by considering the various RDSD spectra of nearly equal R. In case of 5 mm h-1, the average spectrum is obtained by averaging 1170 spectra. For 25 mm h-1, 87 spectra in the range from 24 to 26 mm h-1 are considered. In case of 50 mm h-1, 11 spectra are averaged and lastly for 75 mm h-1, 6 spectra are considered. The exponential, lognormal and gamma RDSD model parameters are calculated from moments method2,5,8. The model parameters corresponding to each RDSD model for the four rain intensity regimes are given in Table 1. It is seen that for rain rate of 5 mm h-1, the three models are showing good fit to the observed RDSD [Fig. 1(a)]. At rain rate of 25 mm h-1 and 50 mm h-1, the exponential RDSD overestimates the observed RDSD at the smaller drop diameter and along the middle and higher diameter ranges it is showing good agreement with the observed RDSD. At rain rate of 50 mm h-1, the gamma and lognormal RDSD shows good agreement with the observed RDSD. At 75 mm h-1 the lognormal RDSD underestimates at the smaller diameter ranges [Fig. 1(d)]. On the other hand the exponential RDSD overestimates at the lower and higher drop diameter and underestimates at the middle drop diameter ranges. The gamma RDSD shows a

KONWAR et al.: RAIN DROP SIZE DISTRIBUTION & CLASSIFICATION

fairly good agreement between the model RDSD and observed RDSD at very heavy rain in all drop diameter ranges. Value of R (mm h-1) calculated from exponential, lognormal and gamma RDSD are compared with those observed from the JWD, and root mean square errors are found out. Figures 2(a), (b) and (c) show the frequency distribution of the root mean square error at various ranges in increment of 1.25 for exponential, lognormal and gamma RDSD model, respectively. The corresponding maximum range of error is found to be 15.0, 12.5 and 10.0 for these three models. The biasing factor in case of exponential RDSD is – 0.4600 while the root mean square error (RMSE) 1.1247. For lognormal RDSD, the biasing factor and the RMSE are found to be + 0.3172 and 0.9199, respectively. In case of gamma RDSD, the biasing factor comes out to be 0.0697 and RMSE

Fig. 1—Averaged number density spectra (solid line), exponential RDSD (dash line), lognormal RDSD (long short dash line) and gamma RDSD (dot line) for (a) 5 mm h-1, (b) 25 mm h-1, (c) 50 mm h-1 and (d) 75 mm h-1 rainfall intensities

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0.3662. Rainfall rate calculated from exponential RDSD underestimates while lognormal RDSD overestimates towards higher rain rate compared to observed values. Gamma RDSD exhibits a good agreement between calculated and observed rainfall intensity, where the RMSE and biasing factor are found to be minimum compared to the other two RDSD models. Hence from this comparative study it is revealed that the Gamma RDSD is performing better compared to the other two RDSD models. Figures 3(a), (b), (c), (d) and (e), present the frequency distribution of R, Dm, µ, Λ and log10 (N0*) for rainfall intensity ≤ 10 mm h-1. Frequency distribution of R is shown in Fig. 3(a), where mean rainfall intensity is 1.56 mm h-1 with standard deviation (SD) of 1.96 mm h-1. The mean value of Dm is found to be 1.21 mm and SD of 0.42 mm. As shown in Fig. 3(c), the mean value of gamma shape parameter µ is found to be 10.60 and SD of 8.19. In case of Λ, the mean value is found to be 13.84 mm-1 while the SD of 9.94 mm-1. The mean value of scaling parameter log10 (N0*) is 3.42 and SD 0.535. Rain events having R greater than 10 mm h-1 are referred here as high rain and belong mostly to convective regime. High rains consist of nearly 10.0% of total rainfall observations. Frequency distributions of the various parameters for high rain are presented in Figs 3(f), (g), (h), (i) and (j). Mean rainfall intensity is 28.60 mm h-1 with a SD of 22.13 mm h-1 [Fig. 3(f)].

Fig. 2—Bar diagram of errors obtained from the observed rain rate and estimated rainfall intensity from (a) exponential, (b) lognormal and (c) gamma RDSD

Table 1—Model parameters for various RDSD models Rainrate R mm h-1 5 25 50 75

Exponential N0 Λ m-3 mm-1 mm-1 4.49×103 4.82×103 4.30×103 1.50×104

2.69 1.93 1.62 1.98

Nt m-3 607 713 806 1133

Rain Drop size distribution Lognormal Gamma N0 µ Dg σ mm mm mm-1 mm-1-μ m-3 0.209 0.177 0.176 0.110

0.355 0.110 0.282 0.399

6.235 ×103 7.511×103 4.777×103 5.593×104

0.53 1.93 2.34 6.71

Λ

3.01 2.73 2.43 4.74

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Figure 3(g) shows the frequency diagram of Dm, with the mean Dm found to be 2.09 mm with a SD of 0.626 mm. The mean of μ is 9.47 and SD value of 6.22 [Fig. 3(h)]. The mean Λ is found to be 6.90 mm-1 with SD of 3.43 mm-1 [Fig. 4(i)]. Frequency distribution of log10 (N0*) is shown in Fig. 3(j). Its mean and SD values are 3.70 and 0.49, respectively. It is observed

that though there is a large variability of R for both low (stratiform) and high (convective) rainfall intensity categories, there is nearly same mean value of μ and log10 (N0*) for both types of rain. The intrinsic shape for low rain ≤ 10 mm h-1 is shown in Fig. 4(a). It is obtained by plotting N(D)/N0* versus D/Dm. This shape is different from any of

Fig. 3—Frequency distribution of (a) R, (b) Dm, (c) μ, (d) Λ, (e) log10(N0*) for R ≤ 10 mm h-1 and frequency distribution of (f) R, (g) Dm, (h) μ, (i) Λ, (j) log10(N0*) for R >10 mm h-1

Fig. 4—Intrinsic shape for (a) R ≤ 10 mm h-1 and (b) R > 10 mm h-1

KONWAR et al.: RAIN DROP SIZE DISTRIBUTION & CLASSIFICATION

RDSD models namely exponential, lognormal or gamma distribution and it follows an “S” type shape. The intrinsic shape for high rainfall rate, i.e. greater than 10 mm h-1, also follows a similar “S” shape to that of low rain intensity [Fig. 4(b)]. It suggests that irrespective of rainfall rate, liquid water content, type of rain, median volume drop diameter and gamma RDSD parameters, the intrinsic shape of RDSD follows a universal “S” shape. This result is similar to the intrinsic shape as shown by Testud et al.9 It is encouraging to note that the intrinsic shape of normalized RDSD follows the same shape, i.e. “S” shape, irrespective of different geographical locations. It is to be noted that in general rain events having R > 10 mm h-1 mostly occurs during convective rain and low intensity rain having R ≤ 10 mm h-1 during stratiform rain period. Though different types of microphysical mechanisms are dominant during convective and stratiform rain, similar types of “S” shape are observed in both types of rain. 4 Classification of rain type In order to separate convective and stratiform type of rain from R, Dm and N0* parameters, a rain event on

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17-18 May 1999 is selected. The temporal variation of R, N0, log10 (N0*) and Dm are presented in Figs 5(a), (b), (c) and (d) respectively, from 2123 to 0332 hrs LT. Maximum rainfall intensity of 87 mm h-1 is observed from JWD observations, considered to belong to convective type of rain. The sudden jump down of N0 parameter is considered as the starting of stratiform rain5. As shown in Fig. 5(b), at the time of starting of stratiform rain, the gamma intercept parameter N0 jumps down from 2.6158 × 1016 to 335 mm-1-μ m-3 at around 2344 hrs LT. As the transition period is conventionally classified as a part of the decaying stage of convective rain (Atlas et al.)12, for simplicity we considered the data of transition phase as part of convective regime. In Fig. 5 this is marked by a solid line to specify the boundary line for both types of rain. The normalization parameter N0* is found to follow the rainfall intensity [Fig. 5(c)]. During stratiform rain, considerably large and nearly constant values of Dm is observed; although small rainfall with low rainfall intensity < 10 mm h-1 is observed for a long duration of time from 2344 to 0332 hrs LT [Fig. 5(d)]. Scatter plots of log10(R) - Dm, log10 (N0*) – log10(R) and log10 (N0*) - Dm are shown

Fig. 5—Temporal variation of (a) R, (b) N0, (c) log10 (N0*) and (d) Dm of the event on 17-18 May, 1999 from 2123 LT to 0333 LT

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R = 0.145 Dm4.70

… (11)

N 0* = 103 R −0.15

… (12)

N 0* = 0.85 × 103.15 Dm−0.94

… (13)

Similar observations are also reported by Testud et al 9. However their equations of separation are different from present analysis. They studied the JWD data collected from Tropical Ocean and Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE). They found the following as the equations of separations for stratiform and convective type of rain R = 1.64 Dm4.25 , N 0* = 107 R −0.1 and N 0* = 0.95 × 107 Dm−0.42 . Both the convective and stratiform type of rain are having well separated clusters exhibiting distinct behaviour of both types of rain.

Fig. 6—Scatter plot of (a) log10(R)- Dm, (b) log10(N0*)- log10(R) and (c) log10(N0*)-Dm [The convective rain is indicated by circle while stratiform by star marks.]

in Figs 6(a), (b) and (c), respectively. Two separate clusters of stratiform and convective type of rain are observed in all the plots. Power law equations are assumed between them β RN β DR ∗ of the form R = α DR Dm , N 0 = α RN R and N 0∗ = α DN DmβDN . The estimated equations that separate these two clusters are given as follows

5 Conclusions The gamma distribution is found to be a good representative of RDSD, showing an overall agreement with observed RDSD. The RMSE of estimated rain from gamma distribution is found to be minimum, which is better than both exponential and lognormal distributions. The mean values of μ and log10 (N0*) show more or less constant values for both types of low and high rainfall intensity rain categories, though they have large variability in R. The intrinsic shape of RDSD has been examined by the procedure followed by Testud et al.9 It is found that the “S” shape structure is prominent in convective as well as in stratiform rain. Invariability in universal “S” shape is observed despite having widely varying ranges of rainfall intensity, mass weighted drop diameter and gamma RDSD parameters. The robustness of the universal shape of RDSD also evokes that any presumption of the shape of RDSD is not required as it departs from exponential, gamma and lognormal RDSD. Rain type such as convective and stratiform are separated by scatter plots of log10(R) - Dm, log10 (N0*) - log10(R) and log10 (N0*) - Dm. Separate clusters of stratiform and convective rain are obtained in each scatter plot and corresponding equations for separation are derived. Acknowledgement Authors from Kohima Science College, Nagaland acknowledge gratefully the financial support from the

KONWAR et al.: RAIN DROP SIZE DISTRIBUTION & CLASSIFICATION

Indian Space Research Organization (ISRO), Bangalore to carry out this work under RESPOND program (10/4/362). Authors thank the Director, NMRF for providing the Disdrometer data. Active support from the Engineers of NMRF is thankfully acknowledged. The kind support from coordinator of UGC-SVU center of MST Radar application is gratefully acknowledged. Authors are indebted to the authorities of Kohima Science College for providing necessary facilities to carry out the research work. References 1 Marshall J S & Palmer W M, The distribution of raindrops with size, J Meteorol (USA), 5 (1948) 165. 2 Feingold G & Levin Zev, The lognormal fit to raindrop size spectra from frontal convective clouds in Israel, J Clim & Appl Meteorol (USA), 25 (1986) 1346. 3 Ulbrich C W, Natural variations in the analytical form of the raindrop size distribution, J Clim & Appl Meteorol (USA), 22 (1983) 1764. 4 Jiang H, Sano M & Sekine M, Weibull raindrop-size distribution and its application to rain attenuation, IEE Proc Microwave Antennas Propag (UK), 144 (1997) 197. 5 Waldvogel A, The N0 jump of raindrop spectra, J Atmos Sci (USA), 31 (1974) 1068. 6 Ulbrich C W & Atlas D, Assessment of the contribution of differential polarization to improved rainfall measurements, Radio Sci (USA), 19 (1984) 49. 7 Willis P T, Functional fits to some observed drop size distributions and parameterization of rain, J Appl Meteorol (USA), 41 (1984) 1648.

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8 Tokay A & Short D A, Evidence from tropical raindrop spectra of the origin of rain from stratiform versus convective clouds, J Appl Meteorol (USA), 35 (1996) 355. 9 Testud J, Oury S, Blake R A, Amayene P & Dou X, The concept of “Normalized” distribution to describe raindrop spectra: A tool for cloud physics and cloud remote sensing, J Appl Meteorol (USA), 40 (2001) 1118. 10 Sekhon R S & Srivastava R C, Snow size spectra and radar reflectivity, J Atmos Sci (USA), 27 (1970) 299. 11 Sekhon R S & Srivastava R C, Doppler radar observations of drop-size distributions in a thunderstorm, J Atmos Sci (USA), 28 (1971) 983. 12 Atlas D, Ulbrich C W, Marks Jr F D, Amitai E & Williams C R, Systematic variation of drop size and radar-rainfall relations, J Geophys Res (USA), 104 (1999) 6155. 13 Maki Masayuki, Keenan T D, Sasaki Y & Nakamura K, Characteristics of the raindrop distribution in tropical continental squall lines observed in Darwin, Australia, J Appl Meteorol (USA), 40 (2001) 1393. 14 Jassal B S, Verma A K & Singh L, Rain drop-size distribution and attenuation for Indian climate, Indian J Radio Space Phys, 23 (1994) 193. 15 Verma A K & Jha K K, Raindrop size distribution model for Indian climate, Indian J Radio Space Phys, 25 (1996) 15. 16 Reddy K K & Kozu T, Measurements of raindrop size distribution over Gadanki during south-west and north-east monsoon, Indian J Radio Space Phys, 32 (2003) 286. 17 Rao T N, Rao D N, Mohan K & Raghavan S, Classification of tropical precipitating systems and associated Z-R relationships, J Geophys Res (USA), 106 (2001) 699. 18 Joss J & Waldvogel A, Ein Spektrograph für Niedersclagstropfen mit automatischer Auswertung, Pure Appl Geophys (USA), 68 (1967) 240. 19 Sauvageot H & Laucaux J P, The shape of averages drop size distributions, J Atmos Sci (USA), 52 (1995) 1070.