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The critical diameters for rainfall attenuation in Southern Africa.

1. INTRODUCTION

Attenuation on radio propagation paths is generally caused by various atmospheric components such as gases, water vapour, clouds and rain. Rain attenuation, caused by scattering and absorption by the water droplets is one of the most important signal impairments influencing the attenuation of microwave (3-30 GHz) and millimetre wave (30300 GHz) [1]. The nearly linear relationship existing between the rainfall rate R (mm/h) and the specific attenuation [A.sub.s] at 35 GHz has been known since 1940s and was successfully tested experimentally for estimating path-averaged rainfall rate in the early 1960s. This linearity was also found to be independent of the (rain) drop size distribution, DSD [2]. The study of drop size distribution (DSD) is however, vital for several application areas such as satellite meteorology, microwave communications, cloud physics and soil erosion [3]. The drop size distribution is an important parameter for the estimation of attenuation due to rain at microwave and millimeter-wave frequency applications because it governs all the microwave and rainfall integral relation.

It has been established that modeling of DSD in tropical and temperate regions is not the same. This is due to the presence of heavy rainfall in tropical regions compared to temperate regions. Measurements of drop size distributions in tropical regions are few when compared to temperate regions where a large database exists. The negative exponential function as proposed by Marshall and Palmer [4] or the Laws and Parsons Model [5] and the gamma distribution model often characterize modeling of raindrop size distributions in the temperate region. However, there is so much uncertainty in the preponderance and estimation of small diameter raindrops due to limitation in the sensitivity of the measuring equipment. These models grossly overestimate the concentration of the small diameter raindrops in the tropical regions hence the Ajayi and Olsen (1985) [6] model was proposed and found suitable for the modelling of tropical rain drop size distributions and equally adequate for the determination of the specific attenuation.

2. PREVIOUS WORK DONE ON RAINFALL RATE AND

DSD MODELS IN DURBAN

Several works have been done by researchers on rainfall attenuation and raindrop size distribution in the tropical and equatorial regions, as well as in Durban (29[degrees]52'S, 30[degrees]58'E), South Africa [7-17]. In 2006, Owolawi [10] estimated the M-P parameters for Durban and defined the intercept parameter (drop density per unit volume), N0. In his findings, he established a power law relation for the intercept given by [N.sub.0] = [a.sub.1][R.sup.a2] with [a.sub.1] and [a.sub.2] were estimated as 1500 and 0.26 respectively. In 2010, Odedina and Afullo [11] proposed the lognormal and the modified gamma distribution models as best fit for the measurements of rainfall in Durban. Afullo [12], while studying the rain drop size distribution model for the eastern coast of South Africa established that the optimised lognormal and gamma DSD models compete favourably well with the DSD obtained for same tropical regions using the Biweight kernel estimation technique for Durban. While using the Maximum Likelihood Estimation (MLE) technique, Owolawi [13] proposed the lognormal model as the best fit for modeling DSD in Durban. In their study, Alonge and Afullo [14] considered the rainfall drop size distributions for different seasons-summer, autumn, winter and spring; they established that the lognormal model is suitable for summer and autumn; the modified gamma for winter and Weibull distribution is best for the spring season in Durban. The values of Ra01 for the different season were also estimated. Recently, Adetan and Afullo [15, 16] in separate studies compared the two methods to evaluate the lognormal raindrop size distribution model in Durban and the three-parameter raindrop size distribution modeling for microwave propagation in South Africa. They established that the method of moment (MoM) provides a better technique to estimate the DSD parameters in Southern Africa with the lognormal model giving the best fit. In their findings, they showed that the three-parameter lognormal DSD gives a better fitting and performance when compared with the gamma distribution model. However, the gamma distribution model is also adequate as the error deviation is minimal. In 2011, Akuon and Afullo [17] derived the rain cell sizes for the southern Africa and estimated the overall [R.sub.0.01] for Durban as 60mm/h. The seasonal and overall values of [R.sub.0.01] determined by [14] and [17] are used in this work for the evaluation of the drop size distribution, N(D) parameters required for the estimation of the rainfall specific attenuation.

It is worth mentioning at this juncture that no work has been done to determine the influence of particular raindrop channels (or diameters) at which the rain attenuation is affected significantly in Durban and the Southern Africa region. This paper therefore, seeks to investigate these critical diameters that produce a major contribution to the total specific attenuation in Durban. The Mie scattering approximation at temperature of 20[degrees]C for spherical raindrop shape is adopted for the estimation of the scattering functions k and a. In this paper, the lognormal and gamma distribution models are used to characterize the measured rain drop size distribution N(D) in Durban as determined in [15].

3. PREVIOUS INVESTIGATIONS ON CRITICAL DIAMETERS

Within the tropical region and other parts of the globe, a number of researchers have investigated the particular contribution of certain drop diameters to the rain attenuation. Lee et al. [19, 20] investigated the DSD for Singapore using the lognormal DSD model and identified the critical range of diameters as 0.771mm to 5.3 mm. They established that while removing consecutive rain diameters (channels), lower diameters raindrops (< 0.771mm) contribute insignificantly to the overall specific rain attenuation for the selected rain rates. Similarly, the influence of particular raindrop diameters on rain attenuation was also carried out by Fiser [21] in the Czech Republic. The critical range of diameter contributing to the specific rain attenuation was found to be approximately 0.7-1.5 mm. The prevailing contribution to the specific attenuation was formed by raindrops of diameters not exceeding 2 mm. In Malaysia, Lam et al. [22] while investigating the specific raindrop sizes producing major contribution to the total specific attenuation with R = 120.4 mm/h, established that small and medium-size drops contributed more to the rainfall attenuation as frequency increases. The oblate spheroid raindrop shape was however adopted. Marzuki et al. [23] observed in Equatorial Indonesia that the increasing role of small and medium-sized drops to rain attenuation is proportional to frequency of transmission. In their observation, the drop size that produced the largest contribution to the specific attenuation for rain rate of 10 mm/h did not exceed 3 mm.

In this paper, we investigate the critical raindrop diameters influencing the specific rain attenuation in Durban, South Africa using the spherical raindrop shape at temperature T = 20[degrees]C. Our method is to compute the total rainfall attenuation by integrating over all the raindrop sizes and determine the differential change in the attenuation as observed over a fixed diameter interval, d[D.sub.j] (= 0.1mm). The diameter ranges contributing 90%, 99%, 99.5%, 99.9%, 99.99% and 100% to the total specific attenuation are also discussed. This range of diameters constitutes the surface area under the curve and along the abscissa regions.

4. RAINDROP SIZE DISTRIBUTION MODELS AND EXTINCTION CROSS SECTIONS

This section considers the measurements of the raindrop size distribution models employed in the computation of the specific attenuation. There are several DSD models used for raindrop distribution modelling across the temperate and tropical regions. In this paper, the lognormal and gamma DSD models are employed for the estimation of the rainfall attenuation.

4.1. DSD Measurements and Models

Generally, the measured rain drop size distribution N([D.sub.j]) from the disdrometer data is the number of raindrops per cubic meter per millimetre diameter ([mm.sup.-3][m.sup.-1]) as adopted by [24,25] is given as (1):

N([D.sub.j]) = [n.sub.j]/v([D.sub.j] x T x A x d[D.sub.j] ([mm.sup.-3][m.sup.-1]) (1)

where [n.sub.j] is the number of drops measured in the drop size bin, v([D.sub.j]) is the terminal velocity of Gun and Kinzer's [25] water drops in m/s, T is the one-minute sampling time in 60 s, A is the measurement area of the disdrometer given as 0.005 [m.sup.2] and d[D.sub.j] is the representative change in diameters interval of the bin in mm. As earlier stated, this work considers the analysis of the data in 0.1 mm diameter interval from consecutive channels (bins). The precipitation rate, R (mm/h) can be determined from the measured DSD data as (2) [24]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

A well-known integral equation of the rain rate R as computed from the DSD model is given by [25] as (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Equation (3) must not be violated by any DSD. Disdrometer measurements of raindrop size distribution used for the estimation of the DSD parameters in this paper were obtained from the J-W RD-80 disdrometer readings mounted at the rooftop of the School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, South Africa. Sample data obtained from the disdrometer over a period of about 27 months were above 80,000 samples. It should be mentioned that rainfall samples with overall sum of drops less than 10 were removed (ignored) from the data samples to compensate for the dead- time errors. The instrument is located at an altitude of 139.7 meters above sea level. The location site is free of noise and shielded from abnormal winds.

4.1.1. The Lognormal Distribution Model

The lognormal distribution model as proposed by Ajayi-Olsen (A-O) [6] was primarily for the tropical rainfall. The tropical lognormal model was adopted because of the peculiarity of the South Africa rainfall with tropical region [26]. The model is expressed by [6, 27] in the form of (4) as:

N(D) = [N.sub.T]/[square root of 2[pi]] x [sigma] x D exp [-0.5 (ln(D) - [mu]/[sigma]] ([m.sup.-3][mm.sub.-1] (4)

where [mu] is the mean of ln(D), [sigma] is the standard deviation which determines the width of the distribution and [N.sub.T] (concentration of rainfall drops) is a function of climates, geographical location of measurements and rainfall type. One unique feature of the lognormal distribution is that N(D) approaches zero as the drop diameter tends to zero. This model was found adequate and suitable for the modeling of drop size distribution in the tropical regions characterized with heavy rain rates [6]. The three parameters in (4) above are related to the rainfall rate R by [6] in the form (5)-(7) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[mu] = [A.sub.[mu]] + [B.sub.[mu]] ln R (6)

[[sigma].sup.2] = [A.sub.[sigma]] + [B.sub.[sigma]] ln R (7)

where [a.sub.0], [b.sub.0], [A.sub.[mu]], [B.sub.[mu]], [A.sub.[sigma]] and [B.sub.[sigma]] are coefficients of moment regression determined using the least squares method of regression technique. The estimated parameters are represented in the scatter plots in Figure 1 as functions of rainfall rates using the method of moment.

Recently, Adetan and Afullo [15] proposed the three-parameter lognormal DSD model using the method of moments in a representative form (4) as given by (8)-(10) as:

[mu] = -0.3104 +0.1331 ln R (8)

[[sigma].sup.2] = 0.0738 + 0.0099 ln R (9)

[N.sub.T] = [268.07R.sup.0.4068] (10)

4.1.2. The Gamma Distribution Model

The three-parameter gamma distribution model in Durban as expressed by Tokay and Short [18] in the form of (11) was similarly studied in [15] with No ([m.sup.-3][mm.sup.-1-[mu]]) indicating the scaling parameter, [mu] (unitless) is the shape parameter, and A is the slope parameter in mm-1. While the shape parameter does influence the slope of the distribution at larger diameter bound, it contributes largely on the curvature of the distribution at small diameters. The gamma distribution is particularly useful in tropical climate regions where the exponential distribution was found to be inadequate [6, 18].

N(D) = [N.sub.0][D.sup.[mu].sub.i] exp(-[LAMBDA][D.sub.i]) [[m.sup.-3][mm.sup.-1]] (11)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The parameters in (8)-(10) and (12) above are used in this paper for the computation of the drop size distribution model, N(D). The representative of the parameters discussed in Section 4.1 is shown in

Table 1.

5. EVALUATION OF THE SPECIFIC RAINFALL ATTENUATION AND THE EXTINCTION CROSS

SECTIONS

The specific attenuation [A.sub.s] (dB/km) of microwave due to rain can be computed using (13):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [Q.sub.ext] is the extinction cross sections which is dependent on the drop diameter D, the wavelength [lambda], and the complex refractive index of water drop m, which in turn is a function of the frequency and temperature. The extinction cross section [Q.sub.ext] is found by applying the classical scattering theory of Mie for a plane wave impinging upon a spherical absorbing particle. The Mie scattering theory is applied under the assumption that each spherical raindrop illuminated by a plane wave is uniformly distributed in a rain filled medium. Similarly, it is assumed that the distance between each drop is large enough to avoid any interaction between them. For more accurate modeling, the raindrops should be modelled as oblate spheroids. The cross section [Q.sub.ext] can be expressed by [28,29] as (14):

A2 o

Q(D, [lambda], m) = [[lambda].sup.2]/2[pi] [[infinity].summation over (n = 1)] (2n + 1) Re[[a.sub.n] + [b.sub.n]] (14)

where [a.sub.n] and [b.sub.n] are the Mie scattering coefficients.

The expression of the extinction cross sections, [Q.sub.ext] provided by [11] as a frequency-dependent, power law function with coefficients, k and [alpha] is expressed in (15) and adopted in this paper; where k and [alpha] are the coefficients that depend on rain rate, temperature, polarization and canting angle.

[Q.sub.ext](D) = k [(D/2).sup.[alpha] (15)

The Matzler's MATLAB [30] functions are used for the estimation of k and [alpha]. Table 2 shows the computed values of k and [alpha] of the power law relation in (15) at frequencies of 2.5 to 100 GHz. The total rainfall attenuation therefore is evaluated by integrating over all the raindrop sizes. Substituting (4) and (15) in (13) for the lognormal DSD model, the total specific rain attenuation is computed numerically over the raindrop diameters as (16):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Equation (16) is solved from the incremental values of the specific rain attenuation of (17):

d[A.sub.s] = 4.343 x [10.sup.-3] k[N.sub.T]/[2.sup.[alpha]][sigma][square root of 2[pi]] [[D.sup.[alpha] - 1.sub.j] x [e.sup.-t]] x d[D.sub.j] (17)

where

t = 1/2 [((lln[D.sub.j]) - [mu]).sup.2]/[sigma] (18)

Similarly, the total rain attenuation using the extinction cross sections (15) and integrating over the drop diameters gives the specific attenuation for the gamma DSD model in (11), as (19):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Solving (19) gives (20):

d[A.sub.s] = 4.343 x [10.sup.-3] k[N.sub.0]/[2.sup.[alpha]] [[D.sup.[alpha] + 2.sub.j] x [e.sup.-[LAMBDA][D.sub.j]] d[D.sub.j]; ([mu] = 2) (20)

6. OVERALL DETERMINATION OF CRITICAL DIAMETERS IN DURBAN

The critical diameters are the range of diameters over which the contribution to the rainfall attenuation is predominant. Figures 2(a) and (b) show the overall critical diameters versus the specific rainfall attenuation for the lognormal and gamma DSD models using the overall [R.sub.0.01] of 60 mm/h as determined by [17]. In Figure 2(a), it can be observed that the greatest shift of the attenuation towards drops of lower diameter occurs at the diameter D ~ 1.4 mm at 10 GHz; while at 100 GHz, it was observed to occur at D ~ 1.1mm. For the gamma model in Figure 2(b), the maximum shift occurs around D ~ 1.3 mm at 10 GHz and diameter D ~ 0.9 mm at 100 GHz. This compares well with the results of [21]. The specific attenuation therefore is directly proportional to the surface area under the curve and above the diameters (x-axis). Tables 3(a) and (b) show the total attenuation created by drops size in the diameter interval of 0.1 to 7 mm for the two models. The total specific rainfall attenuation increases with increasing frequency. At frequency above 40 GHz, it can be observed as illustrated in Tables 4 and 5 for the overall and seasonal determination of the critical diameters that the drop size diameters creating the prevailing contribution to the total attenuation for all the rain rates considered did not exceed 3 mm (90%). This confirms the result of [23]. Similarly, the role of small drops diameters increases with the increasing frequency for the DSD models as the prevailing contribution of raindrops diameters to the specific attenuation does not exceed 2 mm, especially at higher frequencies. This is observed in Table 6 for the DSD models.

7. SEASONAL DETERMINATION OF CRITICAL DIAMETERS IN DURBAN

The values of [R.sub.0.01] for summer (50.48mm/h), winter (53.37mm/h), autumn (72.15 mm/h) and spring (18.51 mm/h) seasons as determined by [14] for Durban are used to estimate the seasonal critical range of diameters in this work. Figures 3 and 4 show the specific attenuation and the range of raindrop diameters at f = 10 GHz, 40 GHz and 100GHz for the gamma and lognormal DSD models respectively. It can be observed that higher rainfall rates cause higher rainfall specific attenuation and the specific attenuation increases with increasing frequency for all seasons. Similarly as observed in Section 6, the largest contributions to the specific attenuation for DSD models considered are due to raindrop diameters not greater than 2 mm, for all seasons. Table 7 shows the contribution of the drop sizes in the range 0.1 mm [less than or equal to] D [less than or equal to] 2.0 mm, 0.5 mm [less than or equal to] D [less than or equal to] 2.5 mm, 1.0 mm [less than or equal to] D [less than or equal to] 3.0 mm, 1.5 mm [less than or equal to] D [less than or equal to] 3.5 mm and 4.0 mm [less than or equal to] D [less than or equal to] 7.0 mm for the overall and seasonal determination of critical diameters. Over 80% of the attenuation at all frequencies for the models is contributed by drop diameters in the range 1.0 mm [less than or equal to] D [less than or equal to] 3.0 mm for the overall and seasonal values of [R.sub.0.01]. The summer and winter seasons tend to have similar critical diameters at the same frequency range as shown in Table 7. At frequency of 80 GHz and above, the critical range of diameters for the gamma DSD model occurs in the range 0.5 mm [less than or equal to] D [less than or equal to] 2.5 mm. The highest contribution to the attenuation in the diameter range 4.0 mm [less than or equal to] D [less than or equal to] 7.0 is observed during the autumn season to be 6.05% (Table 7(b)). This is very low and insignificant. Therefore, larger diameters contribute little to the specific attenuation.

8. CONCLUSION

This paper considered the critical range of diameters at which the specific rainfall attenuation is most influenced. Our approach was to evaluate the total rain specific attenuation by integrating over all the raindrop sizes (diameters). The maximum (peak) value of the rain attenuation as found from the DSD models considered showed that the critical range of diameter occurs at the drops diameter in the range 0.5 mm [less than or equal to] D [less than or equal to] 2.5 mm and 1.0 mm [less than or equal to] D [less than or equal to] 3.0 mm for both the seasonal and overall values of [R.sub.0.01] in Durban. Over 80% of the attenuation at frequencies of 2.5-100 GHz for both the DSD models is created by drop diameters in the range 1.0 mm [less than or equal to] D [less than or equal to] 3.0 mm for the overall and seasonal values of [R.sub.0.01.] The summer and winter seasons tend to have similar critical diameters at the same frequency range as shown in Table 7. For instance, at a frequency of 80 GHz and above, the critical range of diameters for the gamma DSD model occurs in the range 0.5 mm [less than or equal to] D [less than or equal to] 2.5 mm. We conclude therefore that these ranges of diameter are critical to the overall determination of rain attenuation in Durban, South Africa. However, at larger diameters in the range 4.0 mm [less than or equal to] D [less than or equal to] 7.0 mm, the highest percentage contribution to rainfall attenuation was observably small (6.05%). The highest contribution of raindrops diameters to the specific rain attenuation was created by drop diameters not exceeding 2 mm, especially at higher frequencies. This confirms the results of [19-23]. The percentage contribution as created by given range of diameters at various frequencies and seasons to the total specific attenuation was also investigated. We conclude that at a frequency above 40 GHz, the drop size diameter that gives the largest contribution to the total attenuation for all the rain rates considered does not exceed 3 mm (90%). This is similar to the results obtained by [23]. A good understanding of this rainfall attenuation characteristic will be helpful to properly design adequate fade margin levels, achieve the expected quality of service in a radio communication system operating in the South Africa region and for the purpose of link budget design by the engineers and service providers in this particular area.

9. FUTURE WORK

Due to some assumptions made in the estimation of the specific attenuation may not be the same as the measured raindrop shape, our next future work will be to compare the results reported in this work with experimental attenuation measurements performed in the same region or attenuation data from satellite communications links.

ACKNOWLEDGMENT

This work was gracefully supported by Centre for Engineering Postgraduate Studies (CEPS) and the Centre of Excellence (CoE) of the University of KwaZulu-Natal, South Africa.

REFERENCES

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[16.] Adetan, O. and T. J. Afullo, "Comparison of two methods to evaluate the lognormal raindrop size distribution model in Durban," The SouthernAfrica Telecommunication Networks and Applications Conference (SATNAC), Fancourt, Western Cape, South Africa, Sep. 2-5, 2012.

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Oluwumi Adetan * and Thomas J. O. Afullo

School of Electrical, Electronic & Computer Engineering, University of KwaZulu-Natal, Durban 4041, South Africa

Received 6 November 2012, Accepted 1 December 2012, Scheduled 5 December 2012

* Corresponding author: Oluwumi Adetan (211559250@stu.ukzn.ac.za).

Table 1. DSD parameters from disdrometer measurements in Durban.

                                    Gamma DSD Model

[mu] = 2               [N.sub.o] =            [LAMBDA] =
                       78259[R.sup.-0.156]    6.3209[R.sup.-0.168]

                       Lognormal DSD Model

[N.sub.T] =            [mu] = -0.3104 +       [sigma] = 0.0738 +
268.07[R.sup.0.4068]   0.1331 ln R            0.0099 ln R

Table 2. k and [alpha] values at f = 2.5-100 GHz at T = 20[degrees]C.

Frequencies (GHz)     k      [alpha]

2.5                 0.0048   3.3911
10                  0.3857   4.5272
19.5                1.6169   4.2104
25                  2.4567   4.0186
40                  4.3106   3.5077
60                  6.0493   3.0094
80                  7.0623   2.6621
100                 7.6874   2.4156

Table 3. Total rainfall specific attenuation created by drops in the
diameter range 0.1 [greater than or equal to] D [greater than or
equal to] 7mm at f = 10-100GHz. (a) Overall ([R.sub.0.01] [17]). (b)
Seasonal ([R.sub.0.01] [14]) for the lognormal (L-M) and gamma (G-M)
DSD models in Durban.

            L-M         G-M

f (GHz)   A.sub.s]   [A.sub.s]
          (dB/km)     (dB/km)

10        0.961007    0.985026
19.5      3.977033    4.027874
40        10.73367    10.72919
60        15.72329    15.80689
80        19.23337    19.6010
100       21.82271    22.58165

(a)

Season                                     L-M         G-M
                              f (GHz)   [A.sub.s]   [A.sub.s]
                                         (dB/km)     (dB/km)

Winter                          10       0.843837    0.865123
([R.sub.0.01] = 50.48 mm/h)    19.5      3.514999    3.559706
                                40       9.621152    9.614014
                                60       14.23015    14.30342
                                80       17.52132    17.85711
                                100      19.97118    20.67386
Summer                          10       0.793249    0.813361
([R.sub.0.01] = 53.37 mm/h)    19.5      3.314549    3.356609
                                40       9.133414    9.125235
                                60       13.57084    13.63964
                                80       16.76166    17.08379
                                100      19.14682    19.82421
Autumn                          10       1.179336    1.208381
([R.sub.0.01] = 72.15 mm/h)    19.5      4.830721    4.893022
                                40       12.75207    12.75333
                                60       18.39855    14.30342
                                80       22.27487    22.69526
                                100      25.09265    25.94920
Autumn                          10       0.260292    0.267478
([R.sub.0.01] = 18.51 mm/h)    19.5      1.150270    1.164359
                                40       3.356328    3.563353
                                60       5.771739    5.792821
                                80       7.540564    7.692947
                                100      8.957303    9.305668

(b)

Table 4. Percentage fraction (%) of the overall specific attenuation
created by particular diameter intervals to the total specific
attenuation at f = 10/100 GHz for [R.sub.0.01] = 60 mm/h [17].

           G-M                                      L-M

f        Diameter    [summation]    Diameter    [summation]     %
(GHz)   Range (mm)    [A.sub.s]    Range (mm)    [A.sub.s]
                       (dB/km)                    (dB/km)

10       0.7-3.5       0.8885       1.0-3.3       0.8681       90
         0.8-5.3       0.9759       0.6-4.6       0.9514       99
         0.7-5.4       0.9797       0.9-5.6       0.9562      99.5
         0.6-6.3       0.9837       0.5-5.9       0.9600      99.9
         0.3-6.8       0.9849       0.4-6.7       0.9609      99.99
         0.2-6.9       0.9850       0.3-6.9       0.9601       100

40       0.6-3.1       9.6613       0.8-2.9       9.6996       90
         0.7-5.7       10.616       0.4-4.0       10.605       99
         0.6-5.8       10.678       0.6-4.5       10.683      99.5
         0.4-5.6       10.718       0.5-5.3       10.723      99.9
         0.3-6.7       10.728       0.5-6.4       10.732      99.99
         0.2-6.7       10.729       0.3-6.8       10.733       100

80       0.5-2.8       17.744       0.6-2.6       17.329       90
         0.5-4.2       19.380       0.6-3.7       19.029       99
         0.5-5.9       19.485       0.7-4.3       19.140      99.5
         0.3-5.3       19.585       0.4-4.7       19.208      99.9
         0.2-6.5       19.599       0.5-6.6       19.232      99.99
         0.2-6.9       19.600       0.3-6.7       19.233       100

100      0.5-2.7       20.387       0.5-2.5       19.551       90
         0.4-4.0       22.374       0.6-3.6       21.590       99
         0.3-4.2       22.475       0.4-3.9       21.706      99.5
         0.3-5.4       22.564       0.4-4.7       21.801      99.9
         0.2-5.9       22.579       0.5-6.1       21.820      99.99
         0.1-6.9       22.581       0.4-6.7       21.822       100

Table 5. Percentage fraction (%) of the seasonal specific
attenuation created by particular diameter intervals to the total
specific attenuation at (a) f = 10 GHz, (b) f = 100 GHz for
[R.sub.0.01] [14].

                     G-M

Season             Diameter    [summation]
                  Range (mm)    [A.sub.s]
                                 (dB/km)
Winter             0.8-3.5       0.7858
([R.sub.0.01] =    0.4-4.8       0.8533
50.48 mm/h)        0.6-5.1       0.8606
                   0.5-6.1       0.8645
                   0.4-6.7       0.8650
                   0.3-6.8       0.8651

Summer             0.4-3.4       0.7359
([R.sub.0.01] =    0.8-5.7       0.8070
53.37 mm/h)        0.7-5.6       0.8099
                   0.7-5.9       0.8126
                   0.4-6.7       0.8132
                   0.3-6.9       0.8133

Autumn             0.5-3.6       1.0908
([R.sub.0.01] =    0.4-4.8       1.1932
72.15 mm/h)        0.6-5.4       1.2027
                   0.5-6.1       1.2071
                   0.3-6.7       1.2082
                   0.2-6.8       1.2084

Spring             0.7-2.9       0.2416
([R.sub.0.01] =    0.4-3.8       0.2639
18.51 mm/h)        0.6-4.8       0.2663
                   0.5-5.7       0.2672
                   0.3-6.1       0.2674
                   0.2-6.8       0.2675

                     L-M

Season             Diameter    [summation]     %
                  Range (mm)    [A.sub.s]
                                 (dB/km)
Winter             0.6-3.2       0.7665        90
([R.sub.0.01] =    0.3-4.5       0.8356        99
50.48 mm/h)        0.9-5.5       0.8392       99.5
                   0.7-6.1       0.8432       99.9
                   0.3-6.6       0.8437      99.99
                   0.2-6.9       0.8438       100

Summer             0.6-3.2       0.7245        90
([R.sub.0.01] =    0.7-4.5       0.7861        99
53.37 mm/h)        0.9-5.6       0.7890       99.5
                   0.5-5.5       0.7921       99.9
                   0.3-6.6       0.7931      99.99
                   0.3-6.9       0.7932       100

Autumn             1.0-3.4       1.0634        90
([R.sub.0.01] =    1.0-5.2       1.1703        99
72.15 mm/h)        0.8-5.3       1.1735       99.5
                   0.6-6.3       1.1786       99.9
                   0.6-6.8       1.1791      99.99
                   0.2-6.9       1.1793       100

Spring             0.8-2.6       0.2359        90
([R.sub.0.01] =    0.3-3.7       0.2582        99
18.51 mm/h)        0.7-4.0       0.2590       99.5
                   0.7-5.3       0.2599       99.9
                   0.5-5.6       0.2602      99.99
                   0.2-6.9       0.2603       100

(a) 10 GHz

                     G-M

Season             Diameter    [summation]
                  Range (mm)    [A.sub.s]
                                 (dB/km)

Winter             0.4-2.6       18.613
([R.sub.0.01] =    0.5-4.0       20.374
50.48 mm/h)        0.4-4.6       20.584
                   0.3-5.9       20.660
                   0.2-6.1       20.672
                   0.2-6.8       20.673

Summer             0.6-2.7       17.874
([R.sub.0.01] =    0.5-4.6       19.613
53.37 mm/h)        0.4-4.5       19.732
                   0.3-6.4       19.811
                   0.2-6.2       19.823
                   0.2-6.8       19.824

Autumn             0.5-2.8       23.507
([R.sub.0.01] =    0.4-4.0       25.668
72.15 mm/h)        0.4-4.6       25.828
                   0.3-5.7       25.932
                   0.2-6.2       25.947
                   0.2-6.8       25.948

Spring             0.6-2.4        8.384
([R.sub.0.01] =    0.4-4.0        9.237
18.51 mm/h)        0.3-3.5        9.260
                   0.5-5.6        9.293
                   0.2-6.9        9.304
                   1.0-6.9        9.306

                     L-M

Season             Diameter    [summation]        %
                  Range (mm)    [A.sub.s]    Contribution
                                 (dB/km)

Winter             0.6-2.5       18.095           90
([R.sub.0.01] =    0.3-3.6       19.801           99
50.48 mm/h)        0.4-3.8       19.861          99.5
                   0.6-5.5       19.958          99.9
                   0.4-6.1       19.969         99.99
                   0.3-6.7        19.97          100

Summer             0.6-2.5       17.439           90
([R.sub.0.01] =    0.4-3.5       18.959           99
53.37 mm/h)        0.6-3.9       19.059          99.5
                   0.6-4.9       19.127          99.9
                   0.5-6.3       19.145         99.99
                   0.3-6.5       19.147          100

Autumn             0.3-2.6       22.562           90
([R.sub.0.01] =    0.2-3.7       24.820           99
72.15 mm/h)        0.5-4.1       24.972          99.5
                   0.4-4.7       25.057          99.9
                   0.5-6.2       25.090         99.99
                   0.2-6.7       25.092          100

Spring             0.7-2.1        8.119           90
([R.sub.0.01] =    0.6-3.1        8.895           99
18.51 mm/h)        0.4-3.1        8.910          99.5
                   0.5-3.8        8.943          99.9
                   0.4-4.8        8.956         99.99
                   0.2-6.6        8.957          100

(b) 100GHz

Table 6. Contributions (dB/km) of raindrop diameters to specific
attenuation at f = 10/100 GHz. (a) Gamma. (b) Lognormal DSD models
([R.sub.0.01] = 60mm/h [17]).

Diameter     10 GHz       19.5 GHz       40 GHz
(mm)

0.5        0.000664581   0.004322293   0.030523641
1          0.012517079   0.065358612   0.283590164
1.5        0.036054359   0.16556628    0.540282505
2          0.04814263    0.201819597   0.538044131
2.5        0.042184486   0.164772675   0.375526933
3          0.028317604   0.104400918   0.209324254
3.5        0.015816249   0.055531905   0.099911229
4          0.007721478   0.02598769    0.042568511
4.5        0.00340138    0.011028519   0.016630033
5          0.001381641   0.004332728   0.006067134
5.5        0.000525588   0.001599184   0.002094276
6          0.000189396   0.000560599   0.000690612
6.5        6.52142E-05   0.000188196   0.000219162
7          2.16018E-05   6.08923E-05   6.73133E-05

Diameter     60 GHz        80 GHz        100 GHz
(mm)

0.5        0.085469317    0.1614904    0.247393232
1          0.562162419   0.834930961   1.078171325
1.5        0.875074998   1.128957357   1.319192745
2          0.755066663   0.881508157   0.95953242
2.5        0.471539663   0.509451002   0.524864766
3          0.240016267   0.243402605   0.239746372
3.5        0.106090339   0.101978768   0.09670172
4          0.042291453   0.038810205   0.035610277
4.5        0.015580019   0.013724484   0.012232532
5          0.005393336    0.0045803    0.003977728
5.5        0.001775341   0.001458622   0.001237316
6          0.000560598   0.000446878   0.000371032
6.5        0.000170947   0.000132533   0.000107889
7          5.0601E-05    3.82336E-05   3.05609E-05

(a) 10 GHz

Diameter     10 GHz       19.5 GHz       40 GHz
(mm)

0.5        2.44494E-05   0.00015901    0.00112294
1          0.009552573   0.04987928    0.21642554
1.5        0.044693481   0.20523825    0.66974166
2          0.055845165   0.23410953    0.62412799
2.5        0.040321344   0.15749524    0.35894122
3          0.022364563   0.08245333    0.16531926
3.5        0.010830001   0.03802486    0.06841311
4          0.004884991    0.0164411    0.02693095
4.5        0.002126469   0.00689479    0.01039674
5          0.000911798   0.00285933    0.00400393
5.5        0.000389847   0.00118617     0.0015534
6          0.000167452   0.00049565    0.00061059
6.5        7.2591E-05    0.00020948    0.00024395
7          3.18489E-05   8.9777E-05    9.9244E-05

Diameter     60 GHz        80 GHz        100 GHz
(mm)

0.5        0.003144345   0.005941097   0.009101391
1           0.4290216    0.63718848    0.822820544
1.5        1.084755054   1.399471134   1.635289547
2          0.875872843   1.022544224   1.113051905
2.5        0.450713399   0.486950326   0.501683318
3          0.189559075   0.192233523   0.189345918
3.5        0.072644183   0.06982883    0.06621543
4          0.026755673   0.024553263   0.02252883
4.5        0.009740292   0.008580251   0.007647515
5          0.003559268   0.003022715   0.002625054
5.5        0.001316832   0.00108191    0.000917759
6          0.000495645    0.0003951    0.000328043
6.5        0.000190284   0.000147525   0.000120094
7          7.46042E-05   5.63702E-05   4.50578E-05

(b) Lognormal DSD model

Table 7. Percentage (%) fraction of the attenuation created by range
of raindrop diameters (mm) to the total attenuation within the given
diameter range for (a) [R.sub.0.01] = 60 mm/h [17] and (b) different
seasons, [R.sub.0.01] [14].

             L-M             G-M             L-M             G-M

f         0.1 [less       0.1 [less       0.5 [less       0.5 [less
(GHz)   than or equal   than or equal   than or equal   than or equal
         to] D [less     to] D [less     to] D [less     to] D [less
        than or equal   than or equal   than or equal   than or equal
            to] 2           to] 2          to] 2.5         to] 2.5

10          45.97           39.58           70.71           62.75
19.5        50.24           44.32           74.28           67.06
40          59.63           55.31           81.31           75.94
60          65.99           63.16           85.50           81.37
80          70.19           68.46           88.00           84.59
100         73.02           72.08           89.59           86.54

             L-M            G-M            L-M             G-M

f       1 [less than   1 [less than     1.5 [less       1.5 [less
(GHz)   or equal to]   or equal to]   than or equal   than or equal
        D [less than   D [less than    to] D [less     to] D [less
        or equal to]   or equal to]   than or equal   than or equal
              3              3           to] 3.5         to] 3.5
10
19.5        85.02          78.04          80.64           77.32
40          86.99          80.29          74.46           76.12
60          90.13          83.33          75.19           71.03
80          91.30          83.50          70.92           65.42
100         91.60          82.49          67.42           60.63
            91.54          81.15          64.71           56.85

             L-M             G-M

f       4 [less than    4 [less than
(GHz)   or equal to]    or equal to]
        D [less than    D [less than
        or equal to]    or equal to]
              7               7
10
19.5        3.28            4.94
40          2.57            3.93
60          1.45            2.25
80          0.93            1.44
100         0.68            1.04
            0.53            0.81

(a)

Season                        L-M             G-M             L-M

            f (GHz)     0.1 [less       0.1 [less       0.5 [less
                      than or equal   than or equal   than or equal
                       to] D [less     to] D [less     to] D [less
                      than or equal   than or equal   than or equal
                          to] 2           to] 2          to] 2.5

Winter        10          48.42           41.58           72.89
             19.5         52.67           46.35           76.30
              40          61.92           57.29           82.96
              60          68.13           65.01           86.87
              80          72.18           70.18           89.19
              100         74.89           73.69           90.65

Summer        10          49.59           42.54           73.91
             19.5         53.83           47.32           77.24
              40          63.01           58.23           83.71
              60          69.13           65.88           87.49
              80          73.11           70.99           89.73
              100         75.77           74.45           91.13

Autumn        10          42.21           36.52           67.16
             19.5         46.47           41.20           70.96
              40          56.01           52.20           78.57
              60          62.59           60.21           83.18
              80          66.98           65.69           85.99
              100         69.97           69.47           87.77

Spring        10          70.77           60.57           89.02
             19.5         74.16           65.01           90.81
              40          80.89           74.32           93.99
              60          84.93           80.20           95.67
              80          87.39           83.86           96.59
              100         88.95           86.21           97.14

Season        G-M             L-M             G-M             L-M

           0.5 [less       1.0 [less       1.0 [less       1.5 [less
         than or equal   than or equal   than or equal   than or equal
          to] D [less     to] D [less     to] D [less     to] D [less
         than or equal   than or equal   than or equal   than or equal
            to] 2.5          to] 3           to] 3          to] 3.5

Winter       64.82           86.35           79.43           80.23
             69.02           88.11           81.45           78.79
             77.57           90.79           83.94           74.00
             82.72           91.65           83.69           69.43
             85.73           91.74           82.39           65.76
             87.53           91.55           80.86           62.96

Summer       65.79           86.94           80.06           79.98
             69.94           88.61           81.97           78.42
             78.32           91.07           84.19           73.39
             83.34           91.79           83.75           68.68
             86.25           91.78           82.31           64.94
             87.97           91.52           80.69           62.10

Autumn       59.45           82.71           75.67           80.94
             63.92           85.02           78.27           80.19
             73.27           88.88           82.17           76.79
             79.13           90.55           83.00           73.02
             82.67           91.17           82.45           69.81
             84.87           91.35           81.43           67.28

Spring       81.47           93.16           87.54           69.18
             84.27           93.12           87.38           65.86
             89.36           92.01           84.89           57.75
             91.89           90.36           81.16           51.61
             93.06           88.78           77.52           47.25
             93.55           87.44           74.44           44.16

Season        G-M             L-M             G-M

           1.5 [less     4 [less than    4 [less than
         than or equal   or equal to]    or equal to]
          to] D [less    D [less than    D [less than
         than or equal   or equal to]    or equal to]
            to] 3.5            7               7

Winter       77.32           2.79            4.32
             75.84           2.18            3.42
             70.17           1.21            1.93
             64.24           0.77            1.23
             59.28           0.56            0.88
             55.41           0.44            0.68

Summer       77.27           2.58            4.04
             75.66           2.01            3.19
             69.73           1.11            1.79
             63.65           0.71            1.14
             58.61           0.51            0.81
             54.71           0.39            0.63

Autumn       77.05           4.17            6.05
             76.31           3.29            4.86
             72.15           1.89            2.83
             67.09           1.23            1.84
             62.61           0.90            1.33
             59.00           0.72            1.04

Spring       71.45           0.47            1.02
             67.82           0.35            0.77
             58.23           0.17            0.39
             50.44           0.10            0.23
             44.74           0.07            0.15
             40.63           0.05            0.12

(b)
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Article Details
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Author:Adetan, Oluwumi; Afullo, Thomas J.O.
Publication:Progress In Electromagnetics Research B
Article Type:Abstract
Geographic Code:6SOUT
Date:Jan 1, 2013
Words:7766
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