Empirical Models For Estimation Of Rain Rate In The Fifteen ITU Rain Zones 85 Mathematical and Software Engineering, Vol. 2, No. 2 (2016), 85-92 Varεpsilon Ltd, http://varepsilon.com Empirical Models for Estimation of Rain Rate in the Fifteen ITU Rain Zones Orogun Avuvwakoghene Jonathan1, Kufre M. Udofia2, Constance Kalu3 1,2,3Department of Electrical/Electronic and Computer Engineering, University of Uyo, AkwaIbom, Nigeria. *Corresponding Author: 2kmudofiaa@yahoo.com Abstract Rain rate data are essential for the computation of rain attenuation that can be experienced by wireless signal passing through a given area. The International Telecommunication Union (ITU) divided the world into fifteen rain zones and for each rain zone ITU published rain rate data for just seven different link percentage availability, namely: 99%, 99.7%, 99.9%, 99.97%, 99.99%, 99.997% and 99.999%. In this paper, two empirical models are developed for estimating the rain rate for any given link percentage availability in all the fifteen ITU rain zones. The goodness of fit of the models are indicated in terms of coefficient of determination (otherwise called 2r ), root mean square error and prediction accuracy. In all, 90.3777% is the lowest prediction accuracy recorded for Model 1 for rain zone C and 91.6306% is the lowest prediction accuracy recorded for Model 2 for rain zone B. The best prediction accuracy recorded for Model 1 is 98.2456% for rain zone Q and best prediction accuracy recorded for Model 2 is 95.3553% for rain zone H. The models are useful for the estimation of rain rate and hence rain attenuation for any given link percentage availability in all the fifteen ITU rain zones. Keywords: Rain rate, rain attenuation, percentage availability, rain zone, empirical models. 1 Introduction When propagating through rain, radio waves suffer from power loss (attenuation) due to rain [1,2,3]. According to research findings, rain attenuation is the dominant propagation impairment at frequencies above about 10 GHz [4,5,6,7,8,9]. Furthermore, rain attenuation depends on the rainfall rate and the raindrop size distribution. Consequently, rain attenuation is computed with knowledge of rain rate for a given region [1,10,5,11,12]. However, the amount of rainfall an area receives depends on the geographic location and the climate of the area [13,14,15]. As such, in the International Telecommunication Union (ITU) recommendations, ITU-R P. 837-1 and ITU-R P838, the climatic map for the whole world is divided into 15 climatic zones on rain rate distribution or rain intensities at 1% to 0.001% probabilities [16,17,18,19,20,21]. However, for each of the fifteen rain zones, ITU published rain 86 rate data for just seven different link percentage availability, namely: 99%, 99.7%, 99.9%, 99.97%, 99.99%, 99.997% and 99.999%. In this paper, empirical models are developed and validated for the estimation of the rain rate for each of the fifteen rain zones and for any given percentage of time the rain rate is exceeded. Such models will make it possible for researchers and wireless network designers to effectively study the variation of rain rate and rain attenuation with percentage of time exceeded in any of the rain zones. The models will also enable researchers and wireless network designers to effectively study the relationship between rain rate, percentage of time exceeded and wireless communication link outages or link availability. The two empirical models are developed based on the actual rain rate data published by ITU for the fifteen rain zones. 2. Methodology In this paper, the available ITU rain rate data for the 15 ITU rain zones are analysed in order to select the appropriate empirical model for estimating the rain rate for any given percentage of time exceeded. Specifically, three different preliminary trend line curves are fitted on the plot of rain rate versus the percentage of time exceeded (Fig. 1) for rain zone F. The error analysis on the trend line equations shows that power trend line model has the highest Coefficients of Determination ( 2r ) of above 0.98 followed by logarithmic trend line model with 0.92 < 2r < 0.98 and lastly the exponential trend line model with 0.73< 2r < 0.92. Table 1. ITU Rain Rate and Percentage of Time Exceeded for Rain Zone E (Source: [17, 18] ) Percentage of Time Exceeded (%) Rain Rate (mm/h) 1 0.6 0.3 2.4 0.1 6 0.03 12 0.01 22 0.003 41 0.001 70 y = 0.872449x-0.703755 R² = 0.971276 Power Trend Line Model y = -6.598618ln(x) - 5.166157 R² = 0.866371 Logarithmic Trend Line Model y = 15.967293e-3.587894x R² = 0.814051 Exponential Trend Line Model -10 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 R a in R a te ( m m / h ) Percentage Of Time Exceeded (%) Figure 1. The Graph Of ITU Rain Rate and Percentage of Time Exceeded for Rain Zone E. 87 Consequently, in order to predict rain rate for the ITU rain zones, two empirical models are used, namely: MODEL 1: Y = cxeaX 2 (1) MODEL 2: Y = caX b + 1 (2) Model 1 is a product of power and exponential components, whereas Model 2 is the inverse of the sum of power and constant components. In the two models Y is the predicted rain rate in a particular rain zone, X is the percentage of time the rain rate is exceeded (otherwise called percentage of time exceeded), a, b and c are empirical constants determined for each model in each rain zone. (Note, x = 100% – link percentage availability. So, link percentage availability of 99% amounts to X = 1%). Statistical error analysis parameters such as Coefficients of Determination ( 2r ), Root mean square error (RMSE) and Prediction Accuracy (PA) for each of the models for each rain zone are determined to quantify the goodness of fit of the model for predicting the rain rate in the given rain zone. After selecting the two empirical models, Matlab program is used to find the values of the empirical constants a, b and c for each of the model. Afterwards, the nonlinear optimisation option in Microsoft Excel Solver is used to determine a constant (Kopt) that will be added to the model in order to minimise the prediction error. Essentially, the effective models used for estimating the rain rate for each of the rain zones are: MODEL 1: Y = cxeaX 2 + Kopt (3) MODEL 2: Y = caX b + 1 + Kopt (4) 3. Goodness of Fit Measures The Coefficients of Determination ( 2r ): If n is the number of data items x and y, the Correlation Coefficient (r) is given as: ( ) ( )( )( ) ( ) ( )[ ] ( ) ( )[ ]      −− − = ∑∑∑∑ ∑∑∑ 2222 yynxxn yxxyn r (5) Also, the Coefficients of Determination ( 2r ) is given as: ( ) ( )( )( ) ( ) ( )[ ] ( ) ( )[ ] 2 2222         −− − = ∑∑∑∑ ∑∑∑ yynxxn yxxyn r (6) Prediction Accuracy: The prediction accuracy (PA in %) is calculated as follows: %100* 1 1 1 ))(( ))(())((                     − −= ∑ = = ni i iactaul ipredictediactaul Y YY n PA (7) Root Mean Square Error (RMSE): The Root Mean Square Error (RMSE) is calculated as follows: ( )2 1 2 ))(())(( 1       −= ∑ = = ni i ipredictediactaul YY n RMSE (8) 88 where ))(( iactaulY is the actual rain rate given by ITU and ))(( ipredictedY is model predicted rain rate. 4. Results and Discussions Table 2a and Table 2b show the values for the empirical constants a, b, and c, the model optimization constant (Kopt), and the goodness of fit parameters, Coefficients of Determination ( 2r ), Root mean square error (RMSE) and prediction accuracy (PA) of the two models for each rain zone. For rain zone A,C,D,E,G,J,P and Q Model 1 has better prediction accuracy and hence it is used for predicting the rain rate for the rain zone A,C,D,E,G,J,P and Q (shown in Table 2a). On the other hand, Model 2 has better prediction accuracy for rain zones B,F,H,K,L,M and N and hence, Model 2 is used for predicting the rain rate for the rain zones B,F,H,K,L,M and N (shown in Table 2b). Table 2a. MODEL 1: - Values for the empirical constants a, b, and c, the model optimization constant (Kopt), and the goodness of fit parameters, namely: Coefficients of Determination ( 2r ), Root mean square error (RMSE) and prediction accuracy (PA) of the two models for each ITU rain zone a b c Kopt RMSE r2 Prediction Accuracy (%) A 1.261 -0.4148 -3.887 0.093228 0.1673 0.9996 93.0066 C 2.387 -0.4148 -3.887 0.749839 0.7819 0.9983 90.3777 D 3.976 -0.3416 -0.7553 0.161511 0.2257 0.9999 97.3823 E 2.309 -0.4945 -1.877 0.234747 0.3251 1.0000 96.8494 G 6.264 -0.3392 -0.8824 0.318893 0.4509 0.9998 97.0029 J 13.71 -0.2027 -0.6551 0.762792 1.0774 0.9977 95.9201 P 49.09 -0.2379 -2.091 3.741203 4.9247 0.9980 92.2942 Q 53.91 -0.1668 -0.8625 1.118311 1.5616 0.9995 98.2456 Table 2b. MODEL 2: - Values for the empirical constants a, b, and c, the model optimization constant (Kopt) the goodness of fit parameters, namely: Coefficients of Determination ( 2r ), Root mean square error (RMSE), and prediction accuracy (PA) of the two models for each ITU rain zone a b C Kopt RMSE r2 Prediction Accuracy (%) B 1.392 0.6512 0.01577 -0.25442 0.3640 0.9995 91.6306 F 0.6567 0.6953 0.007375 0.169725 0.8671 0.9990 93.9921 H 0.4108 0.601 0.005593 -0.48374 0.6301 0.9997 95.3553 K 0.397 0.6739 0.006244 -0.76367 0.9533 0.9996 93.0746 L 0.3157 0.7038 0.004226 -0.52586 0.6459 0.9999 93.7753 M 0.1787 0.6364 0.006129 -0.74685 0.9745 0.9997 95.1533 N 0.1083 0.6198 0.004093 -2.72616 3.4914 0.9982 92.4932 From Eq.3 and Table 2a, the empirical model for the rain rates in rain zones A,C , D, E, G, J, P and Q can be expressed as: 89 0.093228 e 1.261X -3.887X-0.4148A +=Y (9) 0.749839 e X387.2 -3.887X-0.4148C +=Y (10) 0.161511 e X976.3 -0.7553X-0.3416D +=Y (11) 0.234747 e X309.2 -1.877X-0.4945E +=Y (12) 0.318893 e X264.6 -0.8824X-0.3392G +=Y (13) 0.762792 e X71.13 -0.6551X-0.2027J +=Y (14) 3.741203 e X09.49 -2.091X-0.2379P +=Y (15) 1.118311 e X91.53 -0.8625X-0.1668Q +=Y (16) Similarly, from Eq4 and Table 2b, the empirical model for the rain rates in rain zones B, F, H, K, L,M and N can be expressed as: 25442.0 01577.0392.1 1 6512.0 − + = X YB (17) 169725.0 007375.06567.0 1 6953.0 + + = X YF (18) 48374.0 005593.04108.0 1 601.0 − + = X YH (19) 76367.0 006244.0397.0 1 6739.0 − + = X YK (20) 52586.0 004226.03157.0 1 7038.0 − + = X YL (21) 74685.0 006129.01787.0 1 6364.0 − + = X YM (22) 72616.2 004093.01083.0 1 6198.0 − + = X YN (23) Fig. 2 and Table 3 show the graph of the actual and Model 1 predicted rain rate versus percentage of time exceeded for rain zone E in which Model 1 has prediction accuracy of 96.8494% (as shown in Table 2a). Similarly, Fig. 3 and Table 4 show the graph of the actual and Model 2 predicted rain rate versus percentage of time exceeded for rain zone F in which Model 2 has prediction accuracy of 93.9921% (as shown in Table 2b). Table 3. Actual and Model 1 Predicted Rain Rate for Rain Zone E. Percentage of Time Exceeded (%) Actual Rain Rate (mm/h) for Rain Zone E Model 1 Predicted Rain Rate (mm/h) for Rain Zone E 1 0.6 0.588136 0.3 2.4 2.619461 0.1 8 6.210696 0.03 12 12.59514 0.01 22 22.32864 0.003 41 40.83624 0.001 70 70.39785 90 Figure 2. The Actual and Model 2 Predicted Rain Rate for Rain Zone E Table 4. The Actual and Model 2 Predicted Rain Rate for Rain Zone F. Fig. 3 The Actual and Model 2 Predicted Rain Rate For Rain Zone F 5 Conclusion In this paper, two different empirical models referred to as Model 1 and Model 2 are developed and evaluated for their suitability for predicting the rain rate for the 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 R a in R a te ( m m / h ) Percentage Of Time Exceeded (%) Actual Rain Rate (mm/h) for Rain Zone E Model 1 Predicted Rain Rate (mm/h) for Rain Zone E 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 R a n i R a te ( m m / h ) Percentage of Time Exceeded (%) Actual Rain Rate (mm/h) for Rain Zone F Model 2 Predicted Rain Rate (mm/h) for Rain Zone F Percentage of Time Exceeded (%) Actual Rain Rate (mm/h) for Rain Zone F Model 1 Predicted Rain Rate (mm/h) for Rain Zone F 1 1.7 1.505854 0.3 4.5 3.428223 0.1 8 7.151562 0.03 15 15.45074 0.01 28 29.33349 0.003 54 52.79358 0.001 78 78.34848 91 fifteen ITU rain zones across the world. The goodness of fit of the models are indicated in terms of coefficient of determination (otherwise called ��), root mean square error and prediction accuracy. For rain zone A,C,D,E,G,J,P and Q Model 1 has better prediction accuracy and hence it is used for predicting the rain rate for the rain zone A,C,D,E,G,J,P and Q. On the other hand, Model 2 has better prediction accuracy for rain zones B,F,H,K,L,M and N and hence, Model 2 is used for predicting the rain rate for the rain zones B,F,H,K,L,M and N. The models are useful for the estimation of rain rate and hence rain attenuation for any given link percentage availability in all the fifteen ITU rain zones. References [1] Panagopoulos, A.D., Arapoglou, P.D., & Cottis, P.G. (2004). Satellite communications at KU, KA, and V bands: Propagation impairments and mitigation techniques. IEEE Communications Surveys & Tutorials, 6(3), 2-14. [2] Tewari, R. K., Swarup, S., & Roy, M. N. (1990). Radio wave propagation through rain forests of India. Antennas and Propagation, IEEE Transactions on, 38(4), 433-449. [3] Crane, R. K. (1971). Propagation phenomena affecting satellite communication systems operating in the centimeter and millimeter wavelength bands. Proceedings of the IEEE, 59(2), 173-188. [4] Mandeep, J. S., Hassan, S. I. S., Ain, M. F., & Tanaka, K. (2008). Rainfall propagation impairments for medium elevation angle satellite‐to‐earth 12 GHz in the tropics. International Journal of Satellite Communications and Networking, 26(4), 317-327. [5] Yeo, J. X., Lee, Y. H., & Ong, J. T. (2014). Rain Attenuation Prediction Model for Satellite Communications in Tropical Regions. Antennas and Propagation, IEEE Transactions on, 62(11), 5775-5781. [6] Sudarshana, K. P. S., & Samarasinghe, A. T. L. K. (2011). Rain rate and rain attenuation estimation for Ku band satellite communications over Sri Lanka. In Industrial and Information Systems (ICIIS), 2011 6th IEEE International Conference on (pp. 1-6). IEEE. [7] Dissanayake, A. (2002). Ka-band propagation modeling for fixed satellite applications. Online Journal of Space Communication, 2, 1-5. [8] Dissanayake, A., Allnutt, J., & Haidara, F. (1997). A prediction model that combines rain attenuation and other propagation impairments along earth-satellite paths. Antennas and Propagation, IEEE Transactions on, 45(10), 1546-1558. [9] Karasawa, Y., Yasukawa, K., & Yamada, M. (1988). Tropospheric scintillation in the 14/11-GHz bands on Earth-space paths with low elevation angles. Antennas and Propagation, IEEE Transactions on, 36(4), 563-569. [10] Hossain, S. (2014). Rain Attenuation Prediction for Terrestrial Microwave Link in Bangladesh. Journal of Electrical and Electronic Engineering, Vol. 7, No. 01, May 2014, 63-68. [11] Kestwal, M. C., Joshi, S., & Garia, L. S. (2014). Prediction of Rain Attenuation 92 and Impact of Rain in Wave Propagation at Microwave Frequency for Tropical Region (Uttarakhand, India). International Journal of Microwave Science and Technology, 2014. Article ID 958498, 6 pages. [12] Cakaj, S. (2009). Rain attenuation impact on performance of satellite ground stations for low earth orbiting (LEO) satellites in Europe. Int'l J. of Communications, Network and System Sciences, 2(06), 480-485. [13] Fuentes, M., Reich, B., & Lee, G. (2008). Spatial-temporal mesoscalemodeling of rainfall intensity using gage and radar data.The Annals of Applied Statistics, 1148-1169. [14] Kwarteng, A. Y., Dorvlo, A. S., &Vijaya Kumar, G. T. (2009). Analysis of a 27‐ year rainfall data (1977–2003) in the Sultanate of Oman. International Journal of Climatology, 29(4), 605-617. [15] Lepore, C., Veneziano, D., &Molini, A. (2015). Temperature and CAPE dependence of rainfall extremes in the eastern United States. Geophysical Research Letters, 42(1), 74-83. [16] Mulangu, C. T., Owolawi, P. A., &Afullo, T. J. (2007). Rainfall rate distribution for LOS radio systems in Botswana. In Southern Africa Telecommunication Networks and Applications Conference (SATNAC) Mauritius. [17] Recommendation ITU-R P.837-1,2,3,4 “Characteristics of Precipitation for Propagation Modelling,” International Telecommunication Union, Geneva, Switzerland, 2001. [18] Recommendation ITU-R837-1,2,3,4, “Characteristics of Precipitation for Propagation Modelling,” International Telecommunication Union, Geneva, Switzerland. Vol. 1992-2003, Pseries. [19] Sulochana, Y., Chandrika, P., &Rao, S. V. B. (2014). Rainrate and rain attenuation statistics for different homogeneous regions of India. Indian Journal of Radio & Space Physics, 43, 303-314. [20] Liu, W., & Michelson, D. G. (2009). Fade slope analysis of Ka-band Earth-LEO satellite links using a synthetic rain field model. Vehicular Technology, IEEE Transactions on, 58(8), 4013-4022. [21] Michelson, D. G., & Liu, W. (2009, May). Simulation of rain fading and scintillation on Ka-band Earth-LEO satellite links. In Electrical and Computer Engineering, 2009. CCECE'09. Canadian Conference on (pp. 635-640). IEEE. Copyright © 2016 Orogun Avuvwakoghene Jonathan, Kufre M. Udofia, Constance Kalu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.