Estimation of Parameters for Lognormal Rainfall DSD Model for Various Rainfall Types in Durban

Estimation of Parameters for Lognormal Rainfall DSD Model for Various Rainfall Types in Durban Akintunde A. Alonge, Student Member, IEEE and Thomas J. Afullo, Senior Member, SAIEE School of Electrical, Electronics and Computer Engineering, University of KwaZulu-Natal, Durban, South Africa email: {2546050, afullot}@ukzn.ac.za Abstract This paper compares the behaviour of existing rainfall drop-size distribution models from 2-year disdrometer measurements in Durban (29 o 97 S, 0 o 95 E). The measurements were classified into four rainfall types: drizzle, widespread, shower and thunderstorm. Initial results indicate that the tropical 2-parameter lognormal model has the best fit for the location at different rainfall rates, especially at high rainfall rate. Consequently, the Method of Moment estimation technique is applied to derive the three input parameters of the lognormal model for rainfall types in Durban. The proposed model was seen to compare well with our measurements for various rainfall types. Index Terms Rainfall drop size distribution, rainfall types, method of moment, lognormal DSD model. I. INTRODUCTION Satellite and terrestrial communication systems deployed at microwave and millimetric frequencies (up to GHz and above) are prone to attenuation due to precipitation []. Precipitations (or hydrometeors) are known to be a major concern to link budget engineers particularly with respect to bandwidth availability and efficiency [2]. In past research involving studies of precipitation effects on signal transmission [] [8], rainfall has been identified as the active phenomena responsible for signal outages and losses during transmission. Instances of outages (either short or long) due to rainfall in packet-oriented networks often result in irreplaceable loss of time and resources. Hence, it is necessary to reduce these losses by embarking on corrective schemes which eliminate to a great extent the effects of rainfall on communication systems. The two well-known microstructural parameters used by researchers in the determination of rainfall attenuation are: rainfall rate and rain drop size distribution (or rainfall DSD) [2]. In this work, we approach rainfall attenuation studies for various rainfall types by modeling the rainfall DSD for Durban, evaluating their respective performances and deriving input parameters for the best model. The models under consideration include negative exponential model by Marshall and Palmer [], modified gamma model by Ulbrich [4] and tropical lognormal model by Ajayi et al. [5]. In this paper, we investigate rainfall DSD with the aid of Joss-Waldvogel impact disdrometer measurements available at the University of KwaZulu-Natal, Durban, South Africa between the period of January 2009 and December 20. II. THEORY OF RAIN DROPSIZE DISTRIBUTION Rainfall is made up of tiny droplets of spherical or oblateshaped particles whose number density contribute to attenuation in a microwave or satellite link [6]. The mechanism of rain drop attenuation are of two kinds: reflection mechanism and absorption mechanism. The two mechanisms are dependent on the transmission frequency and polarization sequence of a microwave link. Usually, rain droplets tend to produce reflection of transmitted signals at wavelengths larger than its diameter and absorption at smaller wavelengths [7]. In theory, the reflection encountered during rainfall account for the scattering mechanisms of signals due to rainfall this results in destructive coupling signals (noise) being added to the transmitted signal. The coefficients produced by scattering can range from real to complex values, which are very helpful, in the calculations of propagation coefficients. Mulangu et al. [7] and Odedina et al. [6] in their recent study determined the scattering parameters for locations in Botswana and South Africa respectively. The knowledge of these scattering parameters is helpful in the computation of the specific attenuation on signals due to rainfall droplets. Generally, the contribution of rain droplets to attenuation depends on other parameters such as rainfall rate, drop diameter, drop temperature, number of rain drops, fall velocity (or terminal velocity) of drops and drop diameter interval [2]. While the drop diameter and drop temperature influence the scattering coefficients, other parameters help in the determination of the rainfall DSD and the resulting attenuation. The presence of these microphysical rainfall parameters can be used in the estimation of rainfall specific attenuation, and thus path attenuation, due to rain drops. The specific attenuation, A s, due to rainfall drops is given as: 4.4 - () where N(D) is the rainfall drop-size distribution in m - mm - and Q t (D) is the extinction cross section (ECS) of the arriving rain droplets in mm 2. As seen in (), rainfall DSD is an integral parameter in the attenuation function, hence, it is important to get a suitable model that fits the measurement. III. DETERMINATION OF RAINFALL DROPSIZE DISTRIBUTION An important process in the estimation of rainfall attenuation involves the determination of rainfall drop-size distribution. The disdrometer computation for rainfall DSD, N(D), is given as: (2) where n i represents the number of available rain drops per

bin, A represents the sampling area of the disdrometer (taken as 0.005 m 2 ), T is given as the sampling time of the disdrometer (taken as 60s), v(d i ) represents the corresponding fall velocity of the rain drops and D i is the diameter interval of the rain drops at the ith channel of the disdrometer. A. Rain Drop-size Distribution Statistical Functions Statistical functions for rainfall DSD have also been developed to represent N(D) for different rainfall regions and rainfall types. Popular among them are: Marshall- Palmer model [], modified gamma distribution by Ulbrich [4] and the tropical lognormal model by Ajayi et al. [5]. The modified gamma distribution is given by: expλ where N(D i ) has the same definition as (2), N o is the drop density per unit volume (or intercept parameter) in m - mm -, μ the shape parameter and Λ is the slope parameter. Typical values for the European region are obtained from Atlas et al. [8] where μ = 2, N o = 64500 R -0.5 m - mm - and Λ = 7.09 R -0.27. A special case of () occurs when the shape parameter (µ) equals zero where, N(D i ), becomes the Marshall-Palmer negative exponential model given by: and, expλ 4 Λ (5) where a and b are regression parameters of the rainfall variable, R. All other parameters retain the same definition as given in (). Table I gives the values for the parameters in (4) and (5) by Marshall and Palmer (M-P) and other values for drizzle, widespread and thunderstorm rain types suggested by Joss et al. [9]. While carrying out his work on four sites in South Africa in 2006, Owolawi [] obtained M-P parameters for Durban. He defined N o as a function of two parameters a and a 2 where No =. His parameters for Durban are given as : a = 500, a 2 = 0.26, a = 4. and b = -0.5. Lastly, the lognormal rainfall DSD model is given as: 2 exp 2 ln (6) where N T is the total number of drops per unit volume or concentration (in m - ), D i is the drop diameter in mm, µ is the mean of ln (D i ) and σ is the standard deviation. The input parameters for (6) are defined by the functions below: (7) ln (8) ln (9) TABLE I TYPICAL VALUES FOR N O AND Λ Parameters N o a Λ a General (M-P) 8000 4. b Joss-Drizzle 0000 5.7 b Joss-Widespread 7000 4. b Joss-Thunderstorm 400 a obtained from the studies of Marshall and Palmer [] b obtained from the studies of Joss et al.[9] TABLE II b AJAYI AND ADIMULA CONSTANTS FOR TROPICAL LOGNORMAL MODEL Rain type Drizzle Widespread Shower Thunderstorm N T µ σ 2 a o b o A µ B µ A σ B σ 78 0.99-0.505 0.28 0.08 0.0 264 0.22-0.47 0.74 0.6 0.08 7 0.70-0.44 0.24 0.22-0.04 6 0.49-0.78 0.95 0.209-0.00 Ajayi and Adimula [5] obtained values for various rainfall types in the tropical region by using the Method of Moment technique to determine all the unknown variables in equation (7-9). The result of their study is available in Table II. In this study, we intend to compare all the described rainfall DSD models with the measurements from the South African subtropical region to examine their suitability for our location. The model equations presented in ()-(9) will form the basis of our studies for Durban, South Africa. B. Data Acquisition and Measurement The measurements for this research work were undertaken at the Howard campus site of the University of KwaZulu- Natal, Durban. The one-minute rainfall data was acquired via the Joss-Waldvogel RD-80 impact disdrometer. It is installed at the School of Electrical, Electronics and Computer Engineering at latitude 0 º 58 E and longitude 29 º 52 S at a height of 9.7m. The disdrometer has 20 dedicated bins with average bin diameter ranging from 0.59 mm to 5.7 mm. It has a sampling time of oneminute and measures in real-time, quantities such as rain rate, rain drop-size, rain accumulation and rain reflectivity. Between January 2009 and December 20, a total of rain events (622 rainfall samples) were recorded with a few instances of equipment outages during the period. For the purpose of this study, the following procedure was undertaken to process the data: A maximum duration of five minutes was assumed as the interval between two independent rain events and samples with total sum of drops less than were discarded. The samples were then categorized, using similar classifications in [2], [5] and [] according to the following rainfall regimes in: drizzle (0 < R < 5 mm/h), widespread (5 R < mm/h), shower ( R < 40 mm/h) and thunderstorm (>40 mm/h). Table III shows the summary of the data collected for the stated period while Fig. shows the time series of a typical rain event. The event occurred on April 25, 2009 between 62 hours and 722 hrs; a maximum rainfall rate of 7.5 mm/h was recorded. In this particular event, it is observed that all the various rainfall types were present. The rainfall started as a drizzle rain type, and later transited into widespread and shower types. On attaining its peak as thunderstorm rainfall type at 69 hrs, it gradually decayed.

Rainfall Rate (mm/h) 20 80 60 40 20 Fig.. Time series of thunderstorm event in Durban on April 25, 2009 between 62 hours and 722 hours. TABLE III SUMMARY OF RAIN EVENTS RECORDED FROM 2009 TO 20 Rain Types Drizzle (0-5mm/h) Widespread (5-mm/h) Shower (-40mm/h) Thunderstorm (>40 mm/h) Number of events Total number of raindrops Number of Samples 0 0 20 0 40 50 4 2,569,75 42,50 75,5, 98 69 2,5,044 9556 6 48,0 54 Therefore, it can be said that thunderstorm rainfall types are bound at the edges by the combination of other rainfall types. C. Computational Procedure and Modeling The main thrust of this study is to investigate, among other things, the behaviour of statistical models for rain DSD and obtain important rainfall relationships for various rain types. The rainfall DSD statistical models such as Marshall-Palmer model, Joss model, tropical lognormal model and modified gamma model are compared with measured rainfall DSD. The Method of Moment (MM) estimation technique is applied afterwards to estimate the input parameters of the lognormal model for various rainfall types in Durban. By adopting a similar approach used for lognormal model in [5], [] [], we define the kth-moment generator for a lognormal DSD model as: exp 2 () The parameters of N T, µ and σ 2 have described in (6). The measured moment to be acquired from the data is given in [4] by:, () Time (Mins) where D i is the mid-class diameter of a drop class and D i is the diameter interval between the drop-sizes of different classes. The solutions of () for the third, fourth and sixth moments which correspond to known rainfall indices are provided in the studies of Kozu et al. [] and Timothy et al. [] and are given by: exp 24 27 6 (2).5.5 2 () (4) where L, L 4 and L 6 represent the natural logarithms of the measured moments M, M 4 and M 6. The solutions N T, µ and σ 2 given in (2) (4) are fitted statistically with rainfall rate, using regression technique, to derive empirical relationships for Durban. IV RESULTS AND DISCUSSIONS A. Comparison of Rain DSD with Statistical Rain DSD Models in Durban Applying equations (2)-(9), we obtained the rainfall DSD for various types as given in Figures 2-5. For each of the statistical models, the input parameters closer to the South African subtropical region were considered. Durban is classified as coastal savannah, with rainfall pattern closer to that of the tropics, albeit with lower rainfall accumulation [5]. Therefore, the typical values for negative exponential rainfall DSD (Marshal-Palmer and Joss) were obtained from Table I; Owolawi parameters for Durban was also applied. Ajayi et al. values in Table II for various rainfall types in tropical regions were used as input parameters in the lognormal rain DSD model and finally, the Atlas and Ulbrich values were used in the case of the modified gamma DSD model. The results of the rainfall DSD for the four rainfall types used in this study were at different rainfall rates: 4.99 mm/h (drizzle), 7.4 mm/h (widespread), 7.67 mm/m (shower) and 7.5 mm/h (thunderstorm). These rainfall rates are selected spectral within their respective rainfall types, however, they can provide an insight into the behaviour of the existing statistical models. For drizzle rainfall type (0 < R< 5 mm/h), as shown in Fig. 2, we considered R = 4.99 m/h from the rainfall samples. By comparing the measured rainfall DSD with the other statistical models, it appears the M-P (general) model performs better particularly at this low rainfall rate value. The modified gamma model and M-P (Joss) appear to under-estimate the measured rain DSD, while the tropical lognormal model over-estimates particularly at middiameter bounds. It could be said from our observation that the distribution is better represented by the Marshall-Palmer model. For widespread rainfall type (5 mm/h R < mm/h) in Fig., it was observed that all the statistical models underestimates the measured rain DSD to certain variation at R = 7.4 mm/h; it also appears that M-P (Joss) and M-P (general) coincide within this rainfall regime. The lognormal model performs better in this regard because it takes the shape of our measured DSD although it under-estimates it. In Fig. 4, the results for shower rainfall type ( mm/h R < 40 mm/h) indicate that the tropical lognormal model follows the path of the measured rainfall DSD at R = 7.67 mm/h. M-P (Joss) slightly under-estimates the measurement between.2 mm and 2 mm diameter bounds. Other statistical models are observed to perform fairly better,

N(D) (m - mm - ) Fig. 2. Rainfall DSD at Durban for drizzle rainfall types at 4.99 mm/h. N(D) (m - mm - ) Fig.. Rainfall DSD at Durban for widespread rainfall types at 7.4 mm/h. N(D) (m - mm - ) 0 0. 0 2 4 0 0. M-P (General) M-P (Joss) Lognormal (Ajayi et al.) 0.0 0 2 4 5 6 00 M-P (general) M-P (Joss) Lognormal (Ajayi et al.) 0.0 0 2 4 5 6 Fig. 4. Rainfall DSD at Durban for shower rainfall type at 7.67 mm/h. N (D) (m - mm - ) 0 0. 0 M-P (general) M-P (Joss) Lognormal (Ajayi et. al.) 0. 0 2 4 5 6 M-P (General) M-P Joss lognormal (Ajayi) Fig. 5. Rainfall DSD at Durban for thunderstorm rainfall types at 7.5 mm/h. albeit, with over-estimation at lower drop diameter bound. Lastly, for thunderstorm rainfall type (40 mm/h R < 20 mm/h), the tropical lognormal model again fits the measured rainfall DSD shape. Although, it appears the modified gamma model has a good fit also, the lognormal parameters provided by Ajayi et al. under-estimate the measured DSD in Durban. Generally, it is observed that the larger diameter drops (at lower drop densities) appear more prominent at showery and thunderstorm rain events, while smaller diameter drops (at higher drop densities) appear at drizzle and widespread rain events. The results from Figures 2-5 have shown that only the tropical lognormal model by Ajayi and Adimula, though with cases of DSD under-estimation, to a large extent, is closer to the measured DSD in Durban. This is closely followed by the gamma model based on the fact that it caters for a spectrum of rain drop diameters below 2 mm. The negative exponential models (Marshall-Palmer and Joss) grossly performs poorly most especially at higher rainfall rates which is of interest to microwave studies. B. Parameters of Lognormal Rainfall DSD for Durban The case for modeling parameters for lognormal model in Durban has been shown in the results from Figures 2 5. It is important to note that only lognormal and gamma models cater for rain drops in the lower diameter region of any rain DSD. It is on this premise that we estimate our parameters for various rainfall types in Durban. From the modeling results, it is observed that the solution of µ ranges from.02 to.57. On the hand, σ 2 ranges from 0 to 0.6. There is also a confirmation that a positive correlation exists for the parameters of µ and σ 2 with respect to the measured rainfall rate for our location. Results from our model for the parameters of lognormal model in Durban are presented in Table IV. Our proposed model is examined at rainfall rates of 4 mm/h, 9 mm/h, 26 mm/h and 76.4 mm/h; these values represent the drop-size distribution spectral of the various rainfall types for the proposed model. Our graphical results representing the listed rainfall rates are shown in Figures 6-9. It should be noted that even though the proposed lognormal model for Durban fitted the rainfall DSD at these rainfall rates, there are still deviations in our values. This is attributed to the sharp changes in the values of µ which influences the location of the modeled DSD, with σ and N T, both scaling the DSD. The fitting procedures for µ and σ alongside their irregular variation at different rainfall rates also contributed to some of the deviations noticed in our model. As expected for drizzle rainfall type in Durban in Fig. 6, it is observed that a large percentage of the rain drop-size fall below the mm diameter bound; this is also reflected in our model. It should be noted that for widespread rain types at 9 mm/h (Fig. 7), majority of the rain drops are well above mm and closer to the bound for 4 mm diameter mark. TABLE IV ESTIMATED CONSTANTS FOR LOGNORMAL MODEL IN DURBAN Rain type N T µ σ 2 a o b o A µ B µ A σ B σ Drizzle Widespread Shower Thunderstorm 20.2 22.4 258.28 4.9 0.4089 0.2 0.095 0.625-0.287-0.92 2 0.2989 0.7 0.249 0.248 0.06 0.079 0.082 0.078 0.06 0.07-0.00 0.005 0.022

N(D) (m-mm-) 0 0. Probability density function (mm-).6.4.2 0.8 0.6 0.4 0.2 Drizzle (4 mm/h) Widespread (9 mm/h) Shower (26 mm/h) Thunderstorm (76 mm/h) 0.0 0 2 4 5 6 0 0 2 4 5 6 Mean diameter of drops (mm) Fig. 6. Rainfall DSD at Durban for drizzle rainfall type at 4 mm/h. N(D) (m - mm - ) 0 0. 0.0 0 2 4 5 6 Fig. 7. Rainfall DSD at Durban for widespread rainfall type at 9 mm/h. N(D) (m - mm - ) 0 0. 0.0 0 2 4 5 6 Fig. 8. Rainfall DSD at Durban for Shower rainfall type at 26 mm/h. N(D) (m - mm - ) 00 0 0. 0.0 0 2 4 5 6 Fig. 9. Rainfall DSD at Durban for thunderstorm rainfall type at 76.4 mm/h. Showery rainfall type at 26 mm/h also has its rain drop diameter bound closer to 4 mm, which the proposed model generally estimated between 5 mm and 6 mm as seen in Fig. 8. For thunderstorm rainfall type at 76.4 mm/h in Fig. 9, our model shows a general estimation for all mean drop diameters present. In most cases of thunderstorm rain types in Durban, most of the disdrometer bins were active with the Fig.. Lognormal PDF of the mean diameters of the drops at different rainfall rates. preponderance of drops in the larger diameter category (between 4 mm and 5.7 mm). This is expected because at higher rainfall rates, agglomeration of smaller diameter rain drops do occur, forming larger diameter rain drops as suggested in [] [2]. This is due to the general microphysics of the rainfall atmosphere during rainfall conditions. Fig. shows the probability density functions (PDFs) for the different rainfall types. Based on the PDFs, it was noticed that for Durban, the peak probabilities decreased with increasing rainfall rates. The consequence of this confirms the fact that in Durban, drizzle rain types have the largest concentration of smaller diameter rain drops, while the thunderstorm rainfall types have the largest concentration of larger diameter rain drops. Therefore, it can be said that Durban experiences a higher percentage of drizzle rainfall events, than other rainfall events, for most part of the year. CONCLUSION In this study, we examined three statistical rainfall DSD models. It was established that the tropical lognormal DSD model, followed by gamma DSD model, are the most appropriate models for Durban based on their advantage of good estimation for rain drops at lower diameter regions. We employed the Method of Moment estimators to compute the two-parameter lognormal relationships for Durban for four rainfall types: drizzle, widespread, shower and thunderstorm. The proposed model for Durban is seen to compare well with measurements from our disdrometer. With these results, an overall rainfall DSD for South Africa can be developed. This will essentially be helpful in the development of rainfall attenuation models for Durban and other major cities in South Africa. REFERENCES [] R.K. Crane, Electromagnetic Wave Propagation Through Rain. New York: John Wiley, 996, pp 40. [2] G.O. Ajayi, S. Feng, S.M. Radicella, B.M. Reddy (Ed), Handbook on Radiopropagation Related to Satellite Communications in Tropical and Subtropical Countries, Trieste: ICTP, 996, pp. 7 4. [] J. S. Marshall and W. Palmer, The distributions of raindrop with size, Journal of Meteorology, 5, pp. 65 66, 948. [4] C.W. Ulbrich, Natural variation in the analytical form of the raindrop size distribution, J. of Climate and Applied Meteor., vol. 2, pp. 764 775, 98. [5] I.A Adimula and G.O. Ajayi, Variation in raindrop size distribution and specific attenuation due to rain in Nigeria, Ann. Telecom, vol.5, No. -2, pp. 87 9, 996.

[6] M.O. Odedina and T.J. Afullo, Determination of rain attenuation from electromagnetic scattering by spherical raindrops: Theory and experiment, Radio Sci., vol. 45, 20. [7] C.T. Mulangu and T.J. Afullo, Variability of the propagation coefficients for microwave links in southern Africa, Radio Sci., vol. 44, 2009. [8] D. Atlas and C.W. Ulbrich, The physical basis for attenuationrainfall relationships and the measurement of rainfall parameters by combined attenuation and radar methods, J. Rech. Atmos., 8, pp. 275 298, 974. [9] J. Joss, J.C. Thams and A. Waldvogel, The Variation of raindropsize distribution at Locarno, Proc. of Inter. Conf. on Cloud Physics, pp. 69-7, 968. [] P.A. Owolawi, Rainfall rate and rain drop size distribution models for line-of-sight millimetric systems in South Africa. M.Sc thesis submitted to the University of KwaZulu-Natal, Durban, 2006. [] K.I. Timothy, J.T. Ong and E.B.L. Choo, Raindrop size distribution using method of moments for terrestrial and satellite communication applications in Singapore, IEEE Antennas Propagat., vol. 50, pp. 420 424, October 2002. [2] S. Das, A. Maitra and A.K. Shukla, Rain attenuation modeling in the - GHz frequency using drop size distributions for different climatic zones in tropical India, Progress in Electromagnetics Research, vol. 25, pp. 2 224, 20. [] T. Kozu and K. Nakamura, Rainfall parameter estimation from dualradar measurements combining reflectivity profile and path-integrated attenuation, J. of Atmos. and Oceanic tech., pp. 259 270, 99. [4] G.O. Ajayi and R.L. Olsen, Modeling of a tropical raindrop size distribution for microwave and millimeter wave applications, Radio Science, Vol. 20, number 2, pp. 9 202, Apr. 985. [5] M.O. Fashuyi, P.A. Owolawi and T.O. Afullo, Rainfall rate modelling for LoS radio systems in South Africa, Trans. of South African Inst. of Elect. Engineers (SAIEE), vol. 97, pp. 74 8, 2006. Akintunde A. Alonge received the BEng. (First Class Hon) degree from the Federal University of Technology, Akure, Nigeria (2007). He is presently undergoing his M.Sc. studies at the University of KwaZulu-Natal (UKZN), Durban, South Africa. His research interests include radio planning and budgeting, wireless communication systems and signal processing. Thomas J. Afullo received the BSc. (Hon) Electrical Engineering from the University of Nairobi, Kenya (979), the MSEE from West Virginia University, USA (98), and the Bijzondre License in Technology and Ph.D in Electrical Engineering from the Vrije Universiteit Brussel (VUB), Belgium (989). He is currently an Associate Professor, School of Electrical, Electronic & Computer Engineering, University of KwaZulu- Natal (UKZN), Durban, South Africa.