RAINFALL DROP-SIZE ESTIMATORS FOR WEIBULL PROBABILITY DISTRIBUTION USING METHOD OF MOMENTS TECHNIQUE

Transcription

1 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 83 RAINFALL DROP-SIZE ESTIMATORS FOR WEIBULL PROBABILITY DISTRIBUTION USING METHOD OF MOMENTS TECHNIQUE A. Alonge* and T. Afullo** * School of Electrical, Electronic & Computer Engineering, University of KwaZulu-Natal, Private Bag X1, Durban 41, South Africa 5@ukzn.ac.za dtanthony7@gmail.com ** School of Electrical, Electronic & Computer Engineering, University of KwaZulu-Natal, Private Bag X1, Durban 41, South Africa Afullot@ukzn.ac.za Abstract: This paper proposes a new approach for deriving the input estimators for the Weibull probability rain drop-size distribution (DSD) using the Method of Moments (MM). The Stirling s approximation is used to estimate the gamma function which is part of the raw moment function of the Weibull probability distribution. The parameters No, and are estimated using a comparative analysis with the likely measured moments of a statistical data. The new parameters are then tested with collected data at Durban, South Africa by using least squares regression fitting technique to derive their power-law relationships with rainfall rate. The results show that the proposed Weibull distribution fits well with the measured data at tested rainfall rates. In comparison with the P , the average RMSE of its specific attenuation is 0.99 for horizontal polarization and 1. for vertical polarization at selected rainfall rates for a frequency range of 0 to 0 GHz. Keywords: Method of moments, raindrop-size distribution, rainfall attenuation, Weibull probability distribution. 1. INTRODUCTION The effects of rainfall attenuation in microwave and satellite communication becomes increasingly disturbing at frequencies above GHz [1, ]. At these higher frequencies (smaller wavelengths), the perturbations from the rain droplets in rainy medium often lead to signal absorption and scattering, and therefore, signal outage and deterioration [3]. Signal outages reduce bandwidth efficiency and spectrum utilization, which comes at great cost to providers of network services [3 5]. Usually, the aim of service providers is to ensure availability of services by compensating for this inadequacies through several corrective schemes such as complex base station power control algorithm. This process can be made possible by short term (or preferably long term) studies of rainfall, which of course, depend on the availability of rainfall data. Rainfall is a complex phenomena, which is very random in nature, especially when considering the active natural variables that induce it. A method of studying its characteristics is by predicting empirically (apriori), from measurements, the behaviour of rainfall for a locality; other methods involve analytical formulations []. These methods allow for the identification of microstructural parameters, especially rainfall rate and rainfall drop-size distribution (DSD), which directly contribute to the determination of rainfall attenuation. Rainfall rate has been exhaustively used to predict attenuation in different parts of the world [1, 8]. However, the availability of rainfall DSD measurements provide a much more qualitative insight for radio engineers since it takes into reckoning the mechanics of rainfall microstructure [, ]. Efforts have been made in the past to identify appropriate statistical models to represent rainfall DSD [9 14], this is noticeable in the array and robustness of such models currently in use. Popular among these statistical models include Marshall-Palmer negative exponential model [11], modified gamma model [14] and lognormal model [9 ]. In this paper, we employ the Weibull probability DSD model to examine its suitability in modeling rainfall DSD. Sekine et al. [1 13, ] in their contribution identified the Weibull DSD model as an appropriate model and proposed a graphical method for deriving the DSD input parameters from measurement. Their empirical estimation involved direct fitting procedure for which the scale and shape parameters of the Weibull distribution model were obtained. In this paper, we present a new method of estimating the input parameters of the Weibull probability model using the Method of Moments technique. The third, fourth and sixth moments of the RD-80 disdrometer measurements is used to derive our parameters for Durban, South Africa. Our model for Durban is tested at rainfall rates of 0.91 mm/h, 5.51 mm/h, mm/h,.30 mm/h and 117. mm/h. We applied the derived model to estimate specific rainfall attenuation with frequency ranging from 0 to 0 GHz at four rainfall rates 5 mm/h, mm/h, m/h and 0 mm/h.. MATHEMATICAL DERIVATION OF PARAMETERS FOR WEIBULL RAINFALL DSD The two-parameter Weibull probability density function in [1] is given by:

2 84 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.3() June 01 In (1) above, the data series x i is the input variable of a known process either random or deterministic. is the shape parameter, while is the scale parameter. Statistically, (1) increases monotically when 1 and becomes unimodal when > 1; therefore, is largely the determinant of the shape of the Weibull function [1]. Weibull is a type III minimum value asymptotic distribution which has been extensively used for analysis of strength of materials and reliability studies [1, 17]. In related rainfall DSD research [9 ], several authors have acknowledged a minimum of two input parameters as sufficient for rainfall DSD models they are, diameter of the drop size and the number of drops per unit volume. By virtue of this, it follows that the Weibull rainfall dropsize distribution should be represented with these parameters as: and thus, our Weibull expression becomes: where N o is the number of rain drops per volumetric sweep of the point cell and D i is the diameter of the rain drops present in the same point cell. In order to estimate the parameters N o, and in (3), different estimation techniques can be utilized; among them include the method of maximum likelihood (MML), method of moments (MM), Kernel estimation and Bayesian estimation [1]. In this study, we employ the Method of Moments (MM) technique because of its exhaustive use in rainfall-related research and its unique relationship to measurable rainfall quantities [9, 18 0]. In this technique, we assume that our corresponding data sample D i exists such that its moment, m i at its point of origin is given by: Then, the equivalent Weibull probability distribution raw moments M n of the data sample at its point of origin is given in [1] as: This can be reduced to: By modifying () to suit our new parameter N o, our n th raw moment becomes: It follows from the findings of Kozu et al. [18] (see also Timothy et al. [19] and Das et al. [0]), that the typical values of n useful for radio and microwave engineering studies are the values of 3, 4 and. This is because they correspond to the liquid water content (LWC), specific attenuation and radar reflectivity of the measured data. By adopting these values, we have three sets of equation representing the third, fourth and sixth moment of the Weibull distribution as below: By basic definition of gamma functions, it follows that and thus,. In this definition, x is assumed as a positive integer. By making a further assumption that, we find a suitable expression for our gamma functions in (11) (13) so that: By restricting the results of to an integer, it follows that our representation might be difficult to express as a recurrent factorial order. Thus, in order to find an adequate representation for this problem, we employ the Stirling s asymptotic approximation [1, ]. This is defined as: where the constant, e = , is the Napierian index. By simplification, the gamma function in (11) (13) can be represented as below: where k in this case corresponds to 3, 4 and. In this paper, the maximum error bound,, attributed to Stirling s approximation is 8 % for the range 1

3 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 85

4 8 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.3() June 01

5 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 87 Fig 9: Performance of Weibull rainfall DSD model at 117. mm/h. 4. RAIN ATTENUATION PREDICTION FROM WEIBULL DSD PROBABILITY MODEL 4.1 Estimation of scattering parameters and specific attenuation Rain attenuation prediction from rainfall for signal transmision along a terrestrial path can be estimated from the expression in P [5] as: where A s is given as the specific attenuation and d eff is the effective transmission distance which is a product of the actual link distance and the path reduction factor (please see P for more information on procedures for this). By extension, the specific attenuation is given by: where the extinction cross section (ECS) is Q ext (D) in mm and dd is the change in diameter (or diameter interval) in mm. The drop size distribution is represented by N(D) and in this case will be replaced with our alternative Weibull DSD model. The ECS is computed by employing the procedure of Odedina et al. [ 7] and Mulangu et al. [8] for spherical rainfall droplets. Their work estimated the scattering parameters by using a combination of Liebe s complex refractive index for water [9], Mie scattering theory [30] and Mätzler equations [31 3]. In the Mie scattering theory, the real part of the forward scattering amplitude is used to determine the scattering parameters. By definition, the forward scattering amplitude is given as: where a n and b n correspond to the Mie scattering coefficients which are dependent on the ambient temperature during rainfall, complex refractive index of water and frequency of transmitted signals. The n th truncation of the infinite series can be determined from [33], where: for where k = / for all wavelengths and is the radius for a spherical drop assumption. The ECS of a transmitted signal can thus be estimated by multiplying the real part of s(o) by a factor as given below: Odedina et al. [ 7] concluded that the terms of Q ext for Durban at an average temperature of 0 o C can be reduced to a frequency-specific power law function in the form of : Thus, (8) can be reduced to an expression for specified 0 diametric sizes based on our disdrometer channels given by: 4. Attenuation relationships based on rainfall rate and frequencies The specific attenuation estimated from our Weibull DSD model for Durban is based on two schemes: variation of frequency, while keeping rainfall rate constant and variation of rainfall rate, while keeping frequency constant. In the case of varying the rainfall rates while keeping the assumed transmission frequency constant. This is shown in Fig. for six different frequencies: GHz, GHz, 5 GHz, GHz, GHz and 0 GHz. The graphical results indicate a progressive increase in the specific attenuation as the rainfall rate increases at all frequencies. Table 1 shows the specific attenuations from our proposed Weibull distribution at R 0.01 = 0 mm/h for Durban and at the maximum rainfall rate ( mm/h) for Durban. It is observed that the specific attenuation at R 0.01 approximately twice when R = mm/h most especially at high frequencies. Also, it is also noted that the increments in specific attenuation at frequencies above GHz are quite small, this may be due to saturation in the

6 88 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.3() June 01 Fig. : Specific attenuation due to Weibull probability DSD model for rainfall rates up to mm/h at different frequencies at 0 o C. Fig. 11: Specific attenuation due to Weibull probability DSD model for frequencies from 0 to 0 GHz at different rainfall rates at 0 o C. scattering coefficients. In order to obtain a specific attenuation law for our Weibull distribution, we follow the [8] report which represents specific attenuation as a powerlaw function of rainfall rate in which the power law coefficients k and are given at different frequencies. By using this approach, we can also find the power-law equivalent of our Weibull Rainfall DSD model which is given as: where k weibull and weibull are the power-law coefficients for our Weibull probability distribution. Table gives values of k weibull and weibull for different frequencies of our Weibull rainfall DSD and their respective coefficient of determination (R ). The specific attenuation coefficients in the Table show a progressive increase in the value of k weibull at all frequencies, while there is increase in value of weibull until a decline starts between GHz and 0 GHz. From our observation, it appears that the contribution of the scale parameter, Table 1: Specific attenuation due to Weibull probability model at R 0.01 and maximum rainfall rate in Durban (GHz) SPECIFIC ATTENUATION (db/km) R 0.01 = 0 mm/h R= mm/h Table : Power-law coefficients for specific attenuation due to Weibull probability model in Durban for 0 mm/h > R > mm/h (GHz) k weibull weibull R k weibull, to the A s is much more prominent at frequencies above GHz. The shape parameter, weibull, however appears to contribute less to A s above GHz. Thus, there is a likelihood that the shapes of the specific attenuation function for our Weibull distribution are similar above this frequency. Fig. 11 also shows the variation in the frequency while the rainfall rate is kept constant. The graph confirms the increment in specific attenuation with an increase in rainfall rate. 4.3 Comparison of proposed Weibull model with existing models By considering the various different rain drop-size distribution models used globally for rainfall attenuation estimation, we compared them with results from the proposed Weibull model. The four rainfall DSD models used are: lognormal model by Ajayi and Olsen [9], negative exponential model by Marshall and Palmer [11], modified gamma models by Atlas and Ulbrich [14] and Weibull model by Sekine and Lind [1]. The model for horizontal polarization and vertical polarization are used to compare the various specific attenuations at different frequencies. The comparison is done at four rainfall rates: 5 mm/h, mm/h, mm/h and 0 m/h, while the maximum frequency is 0 GHz.

7 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 89 The input parameters for the other models are summarized in Table 3. In Fig. 1, it was noticed that at lower extreme boundary of rainfall rate (1 mm/h), the modified gamma model (AU) least fitted both the proposed Weibull model and estimation; Table 4 gives an indication of the compared models at 5 mm/h to verify this. However, compared models in Fig. 13 show that at an upper extreme boundary of rain rate (0 mm/h), the negative exponential model least fitted our proposed model and estimation; Table 7 shows the values of all compared models at this rainfall rate. Tables 5 and also shows the performances of our compared models at mm/h and mm/h respectively. Table 3: Parameters of existing models for model comparison MODELS model (Ajayi and Olsen) exponential model by (Marshall and Palmer) Modified gamma model (Atlas and Ulbrich) Weibull model by (Sekine and Lind) INPUT PARAMETERS Fig 1. Comparison of specific attenuation for different models with varying frequencies at 1 mm/h. Fig 13. Comparison of specific attenuation for different models with varying frequencies at 0 mm/h. In all the presented results, it is shown that our proposed model performs reasonably well especially when compared with specifications for both vertical and horizontal polarization. However, an error test can be used to verify the performance of our model. Table 4: Comparison of specific attenuation of different models with our proposed model in Durban at 0 o C with R = 5 mm/h (GHz) SPECIFIC ATTENUATION (db/km) Modified Proposed gamma Weibull Weibull (AU) (Durban) (SL) (H) (V)

8 90 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.3() June 01 Table 5: Comparison of specific attenuation of different models with our proposed model in Durban at 0 o C with r = mm/h (GHz) SPECIFIC ATTENUATION (db/km) Modified Proposed gamma Weibull Weibull (AU) (Durban) (SL) (H) (V) Table : Comparison of specific attenuation of different models with our proposed model in Durban at 0 o C with R = mm/h (GHz) SPECIFIC ATTENUATION (db/km) Modified Proposed gamma Weibull Weibull (AU) (Durban) (SL) (H) (V) Table 7: Comparison of specific attenuation of different models with our proposed model in Durban at 0 o C with R = 0 mm/h (GHz) SPECIFIC ATTENUATION (db/km) Modified Proposed gamma Weibull Weibull (AU) (Durban) (SL) (H) (V)

9 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 91 Table 8: RMS errors of specific attenuation for all frequencies due to existing models at 0 o C for horizontal polarization ROOT-MEAN-SQUARE ERROR Rainfall Rate (mm/h) Modified gamma (AU) Proposed Weibull (Durban) Weibull (SL) Table 9: RMS errors of specific attenuation for all frequencies due to existing models at 0 o C for vertical polarization ROOT-MEAN-SQUARE ERROR Rainfall Rate (mm/h) Modified gamma (AU) Proposed Weibull (Durban) Weibull (SL) Error estimates for our proposed model were obtained using RMS error test with the estimates considered as the actual model. The RMS error in Timothy et al. [19] is given by: Where x model, i are the samples of our proposed Weibull model, x actual, i are the samples of the estimates and N is the total number of samples. Two RMS tests were undertaken: one for horizontal polarization and the other, for vertical polarization. This is considered because the spherical assumption for our scattering parameters is almost independent of polarization sequence. Results from the RMS tests are given in Table 8 and 9. The average RMSE for all the models are arranged in the following order as they appear in the table. For Table 8 (horizontal), we have:.04,.33, 1.38, 0.99 and 1.83 respectively and for Table 9 (vertical), we have:.79, 1.55, 1.87, 1. and. respectively. Our proposed Weibull model has an average RMSE of 0.99 for horizontal polarization and 1. for vertical polarization. From all indications, our proposed Weibull model has the lowest RMSE for both horizontal and vertical polarization in Durban and therefore, is clearly the best and closest among other existing models with respect to recommendation. 5. CONCLUSION In this paper, we proposed a method of estimating the parameters for the Weibull rainfall drop-size distribution using the third, fourth and sixth moments of the method of moments. With our new estimators, we fitted the Weibull parametric relationships for our locality Durban, South Africa. It was also shown that power-law estimates can be obtained for Weibull distribution at different frequencies. Using horizontal polarization and vertical polarization, the specific attenuation due to the proposed model for Durban was compared with existing models. Our proposed Weibull model compares well with the estimation for specific attenuation with an average RMS error of 0.99 (horizontal polarization) and 1. (vertical polarization) for all selected rainfall rates and frequencies. In general, the results from our modeling have shown that the Weibull probability distribution is an appropriate rainfall distribution for Durban.. FUTURE WORK The assumption of drop-size sphericity used in this study, usually becomes less accurate as the rainfall drop diameter and frequency increases, with consequent effects on depolarization and asymmetrical drop axial influences [34 35] from rainfall dynamics. The resulting specific attenuation prediction from our Mie scattering will therefore be lower than the prediction and will have a multiplier effect on the path attenuation. In order to correct this, the Pruppacher and Pitter drop-size shape model [3] can be applied to cater for polarization effects particularly at frequencies beyond GHz.

10 9 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.3() June REFERENCES [1] G.O. Ajayi, S. Feng, S.M. Radicella, B.M. Reddy: Handbook on Radiopropagation Related to Satellite Communications in Tropical and Subtropical Countries, ICTP, Trieste, pp. 7 14, 199. [] R.K. Crane: Electromagnetic Wave Propagation Through Rain, John Wiley and Sons Inc., New York, pp. 1, 199. [3] R.K. Crane, Prediction of attenuation by rain, IEEE Trans. Antennas, Vol. 8, no. 9, pp , Sept [3] L. Li, T. Yeo, P. Kooi and M. Leong, An efficient calcululation approach to evaluation of microwave specific attenuation, IEEE Trans. Antennas, Vol. 48, No 8, pp. 19, Aug [4] L. Li, P. Kooi, M. Leong and T. Yeo, Microwave attenuation by realistically distorted raindrops: part I theory, IEEE Trans. Antennas, Vol. 43, no. 8, pp , Aug [5] C.T. Mulangu and T.J Afullo, Variability of the propagation coefficients due to rain for microwave links in Southern Africa, Radio Science, Vol. 44, RS3000, 009. [] R.K. Crane: Propagation Handbook for Wireless Communication System Design, CRC Press, Florida, pp. 74 7, 003. [7] Rec. P.837-5, Characteristics of Precipitation for Propagation Modelling,, Geneva, 007. [8] Rec., Specific Attenuation Model for Rain for use in Prediction Methods,, Geneva, 005. [9] G.O. Ajayi and R.L. Olsen, Modeling of a tropical raindrop size distribution for microwave and millimeter wave applications, Radio Science, Vol. 0, number, pp , Apr [] I.A Adimula and G.O. Ajayi, Variation in raindrop size distribution and specific attenuation due to rain in Nigeria, Ann. Telecom, Vol. 51, No. 1-, pp , 199. [11] J. S. Marshall and W. Palmer, The distributions of raindrop with size, Journal of Meteorology, 5, pp. 1, [1] M. Sekine and G. Lind, Rain attenuation of centimeter, millimeter and submillimeter radio waves, Proc. of 1 th European Microwave Conference, pp , 198. [13] H. Jiang, M. Sano and M. Sekine, Weibull raindropsize distribution and its application to rain attenuation, IEE Proc Microw. Antennas propag., Vol. 144, no. 3, June [14] D. Atlas and C.W. Ulbrich, The physical basis for attenuation-rainfall relationships and the measurement of rainfall parameters by combined attenuation and radar methods, J. Rech. Atmos., 8, pp , [] M. Sekine, C. Chen and T. Musha, Rain attenuation from log-normal and Weibull raindrop-size distribution, IEEE Trans. Antennas propagat., Vol. 35, no. 3, Mar [1] D.N. Murthy, M. Xie and R. Jiang: Weibull models, John Wiley and Sons Inc., New York, pp. 58, 8 74, 004, [17] W. Weibull, A statistical distribution function of wide applicability, J. of App. Mechanics, pp , Sept [18]T. Kozu and K. Nakamura, Rainfall parameter estimation from dual-radar measurements combining reflectivity profile and path-integrated attenuation, J. of Atmos. and Oceanic tech., pp. 59, [19] 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., pp , October 00. [0] S. Das, A. Maitra and A.K. Shukla, Rain attenuation modeling in the -0 GHz frequency using drop size distributions for different climatic zones in tropical India, Progress in Electromagnetics Research, Vol. 5, pp. 11 4, 0. [1] A.E. Taylor and W.R. Mann: Advanced Calculus, John Wiley and Sons Inc., pp. 99 3, [] H.J. Weber and G.B. Arfken: Essential Mathematical Methods for Physicists, Academic Press, San Diego, pp , 003. [3] M.O. Odedina and T.J. Afullo, Characteristics of seasonal attenuation and fading for line-of-sight links in South Africa, Proc. of SATNAC, pp , Sept [4] P.A. Owolawi and T.J. Afullo, Rainfall rate modelling and worst month statistics for millimetric line-of sight radio links in South Africa, Radio Sci., vol. 4, 007. [5] Rec. P , Propagation data and prediction methods for the design of terrestrial lineof-sight systems,, Geneva, 009. [] M.O. Odedina and T.J. Afullo, Determination of rain attenuation from electromagnetic scattering by spherical raindrops: Theory and experiment, Radio Sci., Vol. 45, 0. [7] M.O. Odedina and T.J. Afullo, Analytical modeling of rain attenuation and its application to terrestrial LOS links, Proc. of SATNAC, 009. [8] C.T. Mulangu and T.J. Afullo, Variability of the propagation coefficients for microwave links in Southern Africa, Radio Sci., vol. 44, 009. [9] H.J. Liebe, G.A. Hufford and T. Manabe, A model for the complex permittivity of water at frequencies below 1 THz, Inter. J. of Infrared and Millimeter Waves, Vol. 1, no. 7, pp , [30] G. Mie, Beiträge zur optik trüber medien, speziell kolldaler metallösungen, Ann. Phys., 5, pp , doi:np [31] C. Mätzler, Drop-size distributions and Mie computation, IAP Res. Rep. 00-1, Univ. Of Bern, Bern, Nov. 00. [3] C. Mätzler, MATLAB functions for Mie scattering and absorption, IAP Res. Rep , Univ. Of Bern, Bern, June 00.

11 Vol.3() June 01 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 93 [33] C.F. Bohren, D.R. Huffman, Absorption and scattering of light particles, Wienheim: John Wiley, 004. [34] C. Mätzler, Advanced model of extinction by rain and measurements at 38 GHz and 94 GHz and in the visible range, IAP Res. Rep , Univ. of Bern, Bern, February 003. [35] H.R. Pruppacher and J.D. Klett, Microphysics of clouds and precipitation. Dordrecht:Riedel, [3] H.R. Pruppacher and R.L. Pitter, A semi-empirical determintaion of the shape of cloud and raindrops, J. Atmos. Sci., 8, pp. 8 94, 1971.