Evaluation of Some Raindrop Size Distribution Models for Different Rain Rates

Evaluation of Some Raindrop Size Distribution Models for Different Rain Rates

Joseph Mom

Department of Electrical and Electronics Engineering Joseph Sarwuan Tarka University, Makurdi, Nigeria

Soo Tyokighir

Department of Electrical and Electronics Engineering Joseph Sarwuan Tarka University, Makurdi, Nigeria

Gabriel Igwue

Department of Electrical and Electronics Engineering Joseph Sarwuan Tarka University, Makurdi, Nigeria

Abstract Results from the performance of the Negative exponential and Gamma function distribution model in a tropical location is presented in this work. The inputs to the drop size distribution (DSD) models have been obtained from Heipang, Nigeria (9.04N, 7.5E) using the pointing Micro Rain Radar (MRR). The effect of the raindrop diameter on the drop size distribution models (DSD) is investigated over the operating frequency range of 4 GHz to 40 GHz. The estimated rain rate for 99.99% availability for Heipang was used as input to the DSD models. From the results obtained, we observe that the highest drop size distribution for the diameter range 3.5 mm D 5.0 mm was 24.44 for a rainfall rate of 120 mm/h considering the Gamma function distribution. While that of the negative exponential distribution model was 88.87 for the range 3.0 mm D 5.0 mm. Results further suggest that a raindrop size of 0.7 mm can cause link outage for communication links operating with a frequency of 40 GHz. This is because of a wavelength large factor that ranged between 10.714 to 1.875. This study also shows that increasing rain rates of the convective rain types arise from the combined increase of the raindrop diameters across all frequencies. The results obtained from this study are useful for adequate network design and planning for achieving a higher quality of service across radio communication links.

KeywordsRaindrop size, Wireless communication, Gamma function, Radio communication, Micro rain radar

1. INTRODUCTION

   Modeling raindrop size distributions (DSD) is fundamental for connecting remote sensing observations with reliable precipitation products for hydrological applications [1]-[3]. To date, various standard probability distributions have been proposed to build DSD models [4], [5]. Microstructural parameters necessary for the prediction of attenuation due to rainfall includes the rainfall rate and raindrop size distribution (DSD). The DSD directly contributes to the determination of rainfall attenuation. The determination of attenuation due to rain is carried out by short-term or long-term studies of rainfall rates for the location of interest which depends on the availability of rainfall data. Rainfall being a complex phenomenon is mostly random, especially when considering the active natural variables that induce it. Studies from [6]-[9] show the practicability of using rainfall rates for the prediction of attenuation in different parts of the world. In this study, we test the performance of the negative exponential distribution model proposed by [10] and the Gamma function distribution model proposed by [11]. A generalized function of which the

   negative exponential is a special case, the gamma function was first proposed as a raindrop size distribution by [12] in the form:

   N(D) = N0Dp exp(Dq) (1)

   The methods from [10], [11] were used to estimate DSD values for the tropical location with frequency range 4 GHz to 40 GHz using the following rainfall rates as input: 40 mm/h, 60 mm/h, 80 mm/h and 120 mm/h. Since rainfall rate has been exhaustively used to predict attenuation in different parts of the world, the DSD models are used to determine their suitability in a tropical location. Internet service providers in a bid to improve the quality of service to customers, try to ensure the availability of services by compensating for errors due to attenuation resulting from rainfall through several corrective schemes such as base station power control algorithms. However, to adequately determine the amount of power to be added, proper estimations of attenuation values and fade margins are needed. To this end, the effect of raindrop diameters on DSD models becomes necessary. The effect of rain attenuation in microwave and satellite communication becomes increasingly disturbing at frequencies above 10 GHz [8], [13]-[15]. Since higher frequencies are characterized with smaller wavelengths, the effects from the rain droplets in the rainy medium often lead to signal absorption and scattering, and therefore, signal outage and deterioration [16]. Several statistical models have been used for the drop size distribution procedure. Popular among these statistical models include Marshall-Palmer negative exponential model, modified gamma model, and Weibull raindrop size distribution model [17]. The availability of rainfall DSD measurements provides a much more qualitative insight for radio engineers since it takes into reckoning the mechanics of rainfall microstructure [16]. Efforts have been made in the past to identify appropriate statistical models to represent rainfall DSD [18]. This is noticeable in the array and robustness of such models currently in use. Signal outages reduce bandwidth efficiency and spectrum utilization, which comes at a great cost to providers of network services [19], [20]. According to [21]-[24] the most deleterious factor to signal propagation in satellite/terrestrial link design is rainfall. Over the years, demand for utilization of higher frequencies has become paramount hence studies on drop size distribution models have become necessary.

2. RESEARCH METHOD

   Raindrop-size distribution

   (m^-3 mm^-1)

   Raindrop-size distribution

   (m^-3 mm^-1)

   The equipment used for the collection of data is the Micro Rain Radar equipment located at Heipang, Nigeria (9.04N, 7.5E). The radar equipment operates at a frequency of 26GHz and is a frequency modulated continuous wave. The determination of the rain cell statistics is carried out by a conversion of measured Doppler spectra into drop diameters which enables the realization of the radar reflectivity (Z), liquid water content (W), and rain rates. The spectral volume reflectivity with depth as realized by [22] is given as follows:

   2

   2

   (r, f)df = p(r, f)df c r t1(r) (2)

   r

   Where: p(r, f) = spectral power, f = doppler frequency in Hz, C = radar constant.

   2

   2

   Mean diameter of raindrops (mm)

   Mean diameter of raindrops (mm)

   The DSD can be estimated from the volume reflectivity (D) as follows:

3. RESULTS AND DISCUSSION

   10000

   10000

   1000

   100

   10

   1

   1000

   100

   10

   1

   4

   4

   6

   6

   Performance of the negative exponential and Gamma function distribution models are presented in Figures (1a)-(1d), (2a)- (2d). The figures show that as the mean diameter of raindrops increase, raindrop size distribution decreases.

   Negative exponential distribution @

   Negative exponential distribution @

   40 mm/h

   0

   40 mm/h

   0

   N(D) = (D)

   D

   (3)

   Fig. 1a. Drop size distribution @ 40 mm/h

   Negative exponential distribution @

   10000 60 mm/h

   1000

   100

   10

   1

   0 2 4 6

   Mean diameter of raindrops (mm)

   Negative exponential distribution @

   10000 60 mm/h

   1000

   100

   10

   1

   0 2 4 6

   Mean diameter of raindrops (mm)

   Raindrop size distribution

   (m^-3 mm^-1)

   Raindrop size distribution

   (m^-3 mm^-1)

   Where: N(D) = number of drops with size D to D+D in m/s. Mean fall velocity Vm is given as:

   P f df

   P f df

   0

   0

   m

   m

   V = f.P(f)df

   2 ( )

   0

   (4)

   Where: P(f) = spectral power, = wavelength.

   The DSD model is given by [10] and has the general form:

   N(D) = N0 exp(D) (5)

   Where: N(D) = concentration of raindrops per diameter (mm), D = rain drop diameter, N0 = intercept parameter given as 8 × 103 mm-1m-3, = slope parameter (mm)

   = 4.1R0.21mm1 (6)

   Where: R = rainfall rate (mm/h)

   The DSD model derived by [11] has the general form:

   N(D) = N0Dp exp(Dq) (7)

   Where: N0 = 1.76 × 106 R-1.2, p = 2, = 10R-0.31, q = 1

   To estimate the terminal velocity v(D), we can apply the following equation:

   v(D) = 9.65 10.30 exp(0.6D) + 0.65 exp(7D)

   (8)

   Fig. 1b. Drop size distribution @ 60 mm/h

   Negative exponential distribution @

   10000 80 mm/h

   1000

   100

   10

   1

   0 2 4 6

   Mean diameter of raindrops (mm)

   Negative exponential distribution @

   10000 80 mm/h

   1000

   100

   10

   1

   0 2 4 6

   Mean diameter of raindrops (mm)

   Raindrop-size distribution

   (m^-3 mm^-1)

   Raindrop-size distribution

   (m^-3 mm^-1)

   Fig. 1c. Drop size distribution @ 60 mm/h

   Gamma function distribution @ 80

   Negative exponential distribution @

   Negative exponential distribution @

   Raindrop-size distribution

   (m^-3 mm^-1)

   Raindrop-size distribution

   (m^-3 mm^-1)

   1000

   10000

   120 mm/h

   10000

   120 mm/h

   1000

   100

   10

   1

   1000

   100

   10

   1

   Rain drop-size distribution

   (m^-3 mm^-1)

   Rain drop-size distribution

   (m^-3 mm^-1)

   100

   10

   1

   mm/h

   0 2 4 6

   0

   0

   2

   2

   4

   4

   6

   6

   0.1

   Mean diameter of raindrops (mm)

   Mean diamter of raindrops (mm)

   Mean diamter of raindrops (mm)

   Fig. 1d. Drop size distribution @ 120 mm/h

   Rain drop-size distribution

   m^3 mm^-1

   Rain drop-size distribution

   m^3 mm^-1

   From the radar data obtained from Heipang, Nigeria, rainfall lasted for the period ranging from 1:08 to 2:23AM (18th August 2020) with the highest recorded rain rate and liquid water content (LWC) of 120mm/h and 28.55 respectively which occurred around 1:22 AM. The contribution formed by raindrop diameter intervals to the overall raindrop size distribution for different rain rates is illustrated in Figures (1a), (1b), (1c), (1d), (2a), (2b), (2c), (2d). The contribution of larger drop diameters as observed in the results show smaller raindrop size distributions and hence lesser attenuation values. For instance, in Figure (2d), the highest contribution in the diameter range 3.5 mm D 5.0 mm is 24.44 for a rainfall rate of 120mm/h considering the Gamma function distribution.

   Gamma function distribution @ 40

   10000 mm/h

   1000

   100

   10

   1

   Gamma function distribution @ 40

   10000 mm/h

   1000

   100

   10

   1

   0.1

   0

   2

   4

   6

   0.1

   0

   2

   4

   6

   0.01

   Mean diameter of raindrops (mm)

   0.01

   Mean diameter of raindrops (mm)

   Fig. 2a Gamma drop size distribution @ 40 mm/h

   Fig. 2c. Gamma drop size distribution @ 80 mm/h

   Gamma function distribution @ 120

   1000 mm/h

   Gamma function distribution @ 120

   1000 mm/h

   100

   100

   10

   10

   1

   1

   Mean diameter of raindrops (mm)

   6

   Mean diameter of raindrops (mm)

   6

   0

   0

   2

   2

   4

   4

   Raindrop-size distribution

   m^3 mm^-1

   Raindrop-size distribution

   m^3 mm^-1

   Fig. 2d. Gamma drop size distribution @ 120 mm/h

   Considering the negative exponential distribution proposed by

   [10] for the range 3.0 mm D 5.0 mm we obtained maximum raindrop size distribution of 88.87 for a rainfall rate of 120mm/h as shown in Figure (1d). Figures (1a), (1b), (1c), (1d), (2a), (2b), (2c) and (2d) show that for drop diameters ranging 0.5 mm D 3.0 mm we realize high drop size distributions of 452.38 and 55.87 for drop diameters 0.5mm and 3mm respectively. This implies that the largest contributions to attenuation are due to drop diameters not exceeding 3.5 mm for all rainfall regimes at all frequencies and this confirms the results obtained in other tropical locations such as Malaysia and Singapore. Hence, the diameter ranges 0.5 mm D 3 mm are critical to attenuation in Nigeria being a tropical region characterized by convective type rainfall.

   Raindrop-size distribution

   (m^-3 mm^-1)

   Raindrop-size distribution

   (m^-3 mm^-1)

   1000

   100

   10

   1

   0.1

   Gamma function distribution @ 60 mm/h

   160

   140

   120

   100

   80

   60

   40

   20

   0

   160

   140

   120

   100

   80

   60

   40

   20

   0

   C band

   C band

   0 2 4 6

   Raindrop size (mm)

   0 2 4 6

   Raindrop size (mm)

   Wavelength (mm)

   Wavelength (mm)

   0 2 4 6

   Mean diameter of raindrops (mm)

   Fig. 2b Gamma drop size distribution @ 60 mm/h

   Fig. 3a. Wavelength large factor for C band

   60

   50

   60

   50

   Ku band

   Ku band

   Wavelength (mm)

   Wavelength (mm)

   The wavelength large factors shown in Figures (3a), (3b), (3c), and (3d) are obtained by dividing the wavelength range for the respective bands (C band, Ku band, Ka-band, and V band) and their corresponding raindrop size. The wavelength large factor is denoted by lambda ( ) and is given as follows:

   40

   30

   20

   10

   0

   40

   30

   20

   10

   0

   wavelength range (mm)

   wavelength large factor () =

   raindrop size (mm)

   (9)

   0

   2

   4

   6

   0

   2

   4

   6

   Raindrop size (mm)

   Raindrop size (mm)

   Wavelength (mm)

   Wavelength (mm)

   Fig. 3b. Wavelength large factor for Ku band

   Ka band

   35

   30

   25

   20

   15

   10

   5

   0

   Ka band

   35

   30

   25

   20

   15

   10

   5

   0

   0 2 4 6

   Raindrop size (mm)

   0 2 4 6

   Raindrop size (mm)

   Wavelength (mm)

   Wavelength (mm)

   Figure. 3c. Wavelength large factor for Ka band

   16

   14

   12

   10

   8

   6

   4

   2

   0

   V band

   16

   14

   12

   10

   8

   6

   4

   2

   0

   V band

   0 1 2 3 4

   Raindrop size (mm)

   0 1 2 3 4

   Raindrop size (mm)

   Fig. 3d Wavelength large factor for V band

   The wavelength range for the C band, Ku band, Ka-band, and V bands is 7.5 cm to 3.75 cm, 2.5 cm to 1.67 cm, 11.1 mm to

   1. mm, 500 nm to 700 nm respectively. Reference raindrop sizes ranged from 0.5 mm to 4.0 mm. From Figure (3a), we observe that the wavelength is large by a factor ranging between 150 to 75 for a raindrop size range of 0.5 mm to 1mm. Hence the effect of the raindrop size on the link can be seen as insignificant. For raindrop size ranging 1.1 mm to 1.6 mm a large factor of 68.1818 to 46.8750 is realized. For raindrop sizes ranging from 1.2 mm to 4.0 mm, large factor ranged between 62.5 to 18.75. Due to the large factors obtained as shown in Figure (3a), it can be said that the raindrop size effect on the C band is insignificant. Considering Figure (3b) large factors ranged between 50 to 10.869 for raindrop size of 0.5 mm to 2.3 mm. Figure (3b) further suggests hat for a raindrop size of 2.3 mm the effect of the raindrop begins to be significant on the link and could result in to link outage. For the raindrop size range of 2.4 mm to 4.0 mm, a continuous drop in large factor is realized which ranged between 10.4166 to 6.25. For the Ka-band link shown in Figure (3c), raindrop size begins to severely affect the link at a size of 1.5 mm. Drop sizes ranging between 1.5 mm to 4.0 mm produced a large factor ranging between 10 to 3.75. Figure (3d) suggests that the drop size effect on link outage for V bands starts at approximately 0.7 mm. This is because of a large factor that ranged between 10.714 to 1.875.

4. CONCLUSION

This study compares and discusses the effect of raindrop diameters for raindrop size distributions in Nigeria. The effect of raindrop size on radio links for the C, Ku, Ka, and V bands is investigated. Results suggest that the V band is prone to severe attenuation effect as raindrop size as small as 0.7 mm could result to link outage. Furthermore, results suggest that for the convective rain types, the maximum drop diameters lead to a reduced raindrop size distribution suggesting that increasing rain rates of the convective rains arise from the combined increases of the raindrop diameters across all frequencies. Results from this study provide a good understanding of raindrop diameters as it relates to drop size distribution and rainfall attenuation characteristics which can be helpful to radio engineers for proper network design and planning.

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BIOGRAPHIES OF AUTHORS

Joseph Mom received his BEng in Electrical and Electronics Engineering from Joseph Sarwuan Tarka University, Makurdi, Nigeria in 2004, he also received his MEng and Ph.D. Degree in Electronics Engineering from the University of Nigeria, Nsukka in 2009 and 2015 respectively. He is currently an associate Professor in the Department of Electrical and Electronics Engineering, Joseph Sarwuan Tarka University, Makurdi. He has published many articles in reputable journals and conference proceedings. He is a registered Engineer with the Council for the Regulation of Engineering in Nigeria (COREN) and also a member of the IEEE and ACM.

technologies.

technologies.

Soo Tyokighir holds BEng and MEng degree in Electrical and Electronics Engineering from the Joseph Sarwuan Tarka University, Makurdi in 2013 and 2019. He is currently a Ph.D. student and research assistant at Joseph Sarwuan Tarka University, Makurdi. He is a member of the Space & Satellite Professionals International, USA (SSPI), IEEE, Nigerian Society of Engineers (NSE) and COREN. His research interests are in engineering physics, radio communication and RF

Gabriel Igwue holds a B.Sc from the University of Lagos in 1973; M.Sc from the Massachusetts Institute of Technology, USA and a Ph.D from the North Caroline State University, USA, all in Electrical Engineering in 1975 and 1982 respectively. Professor Igwue has taught Electrical Engineerig courses at both polytechnic and University for over thirty years. He has held many management positions in the public and private sectors. He is a COREN registered Engineer and a member of the Nigerian Society of Engineers (NSE). He has written several books in Electrical Engineering including Circuit Theory; Electrical Engineering Materials; Mathematical Methods for Engineers; Principles of Communications and Basic Circuit Theory and Industrial Electronics for physicists. He is married with children.