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GHz up to 35 GHz. At 6 GHz, the attenuation results of the ITU-R and the lognormal model overlap with a specific attenuation of about 0.45 dB/km in Cape Town. Beyond this point, the lognormal model of AA-tropical shower (TS) takes the lead. For Brandvlei, the attenuation results of the ITU-R and the lognormal model of AA-tropical shower (TS) also overlap, but in this location it is at 4 – 6 GHz. Above this, the lognormal attenuation values lead.
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inlands areas such as Pretoria and Johannesburg have a temperate climate, and coastal areas such as Durban, Pietermaritzburg, and Richards Bay exhibit sub-tropical climate. Rains in the latter part of this climatic region, especially in the summer exhibit some tropical characteristics and in the winter, the rains may show some temperate characteristics. This is because the rains in such environment are usually heavy and come with larger raindrops (but sometimes alternate with light rains). In the spring/winter rains are usually light with smaller raindrops that can occur for longer periods of time. Examples of such rains are drizzle and widespread rains.
Therefore, Fig. 5-10a which shows that most of the theoretical attenuation results fall on the minimum measured attenuation in Durban might have reflected the effect of light rains with smaller raindrop sizes. Since the raindrop-size distribution models employed in formulating these theoretical attenuation models are developed from research works in the temperate zones, [Green, 2004], the minimum measured attenuation which is as a result of lower rain rates (smaller raindrop-sizes) tend to be reasonably described by these theoretical models in this figure.
It is also observed from this figure (Fig. 5-10a) that the ITU-R model [2007] gives attenuation values that fall within the measured minimum, average and increases rapidly towards the measured maximum attenuation. Though it has been confirmed by some researchers in the tropics and equatorial climates that there are large disparities between measured attenuation results and the ITU-R predictions, but for the locality of study in this work (South Africa), the attenuation disparities from the ITU-R model may not be as large as those for the tropical or equatorial climates. This is because of the location and the climatic characteristics of South Africa which is quite different from the tropical or equatorial climate.
From Fig 5-10b, it is observed that theoretical attenuation results falls within the average and the maximum attenuation bounds which are produced by rain-rates with bigger drop-sizes. Knowing the limitation of the former raindrop-size distributions developed in the northern temperate regions, which are mostly suitable for lower rain rates, authors like Ajayi and Olsen, [1985];
Feingold et al., [1986]; Zainal et al., [1993]; Ajayi and Adimula, [1996]; etc, developed alternative raindrop-size distribution models that can best describe drop-sizes in the tropics and lower latitudes regions with higher rain rates. That is why the higher rain-rates produced by bigger raindrops seems to be reasonably described with the lognormal attenuation model developed in this work.
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0.1 1 10 100
0 10 20 30 40 50 60 70 80
Rain rate (mm/h)
Rain Attenuation (dB)
Min.Measured Attenuation Max. Measured Attenuation Ave. Measured Attenuation ITU- R model Expo. Model (Joss et al.)-T Expo. Model (Joss et al.)-D Expo. Model (MP) Weibull Model (Sekine et al.) Gamma Model (de Wolf)
Fig. 5- 10a: Measured and theoretical rain attenuation along the 6.73 km path length at 19.5 GHz in Durban
0.1 1 10 100
0 10 20 30 40 50 60 70 80
Rain rate (mm/h)
Rain attenuation (dB)
Min. Measured Attenuation Max. Measured Attenuation Ave. Measured Attenuation Logn. Model (AO) Logn. Model (AA)-TT Logn. Model (AA)-TS
Logn. Model (AA)-TW ITU-R Model
Fig. 5-10b: Measured and theoretical rain attenuation along the 6.73 km path length at 19.5 GHz in Durban
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5.7.1 Statistical Analysis of the Rain Attenuation Results
From Fig. 5-10a and 5-10b, the theoretical attenuation models that best fit into the measured attenuation values for the minimum, average and the maximum bounds were determined by using the chi-square (2) statistics. The 2 statistics which has been given in equation (4.3) as [Freedman et al., 1978];
, ,
22
, 1
N mea i pre i
pre i i
X X
X
(5.41)in this context, Xmea is the measured rain attenuation at the 3 bounds (minimum, average and maximum attenuation values), Xpreis the predicted theoretical rain attenuation values at 19. 5 GHz and Nis the number of measured or predicted attenuation points ranging fromi1, 2...N.
Table 5-6 shows the 2 results of all the theoretical attenuation models as compared to the
Table 5- 6: The 2 statistic (1% significance level) for the theoretical attenuation models as compared to the maximum, average and minimum measured attenuation at 19.5 GHz on 6.73 km
path length [Odedina and Afullo, 2010]
*For the purpose of comparison
**Lowest 2 values of the theoretical attenuation model that best fit the measured rain attenuation Theoretical attenuation models Degrees of freedom is 20; 2 statistic threshold = 37.57
Minimum measured attenuation
Average measured attenuation
Maximum measured attenuation
Negative exponential model (MP) 17.496 112.741 596.298
Negative exponential model (Joss et al.)-T **16.656 156.69 753.963 Negative exponential model (Joss et al.)-D 23.308 243.898 1051.059
Gamma model 20.416 61.761 403.978
Weibull model 44.530 238.740 963.557
Lognormal model (AO) 88.811 16.347 60.432
Lognormal model (AA)-TT 55.534 **3.819 103.606
Lognormal model (AA)-TS 185.181 73.55 **15.935
Lognormal model (AA)-TW 105.245 41.455 103.558
*ITU-R model 44.559 38.774 256.386
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maximum, average and minimum measured attenuation. From this Table, the theoretical attenuation models developed from the negative exponential model of Joss et al. for thunderstorm (T) rain type, lognormal model of Adimula-Ajayi (AA) for tropical thunderstorm (TT), and the lognormal model of Adimula-Ajayi (AA) for tropical shower (TS) rains, give the lowest chi- square values for the minimum, average and maximum attenuation measurements, respectively.
Hence these three theoretical models are accepted to give the best fit that describes the minimum, average and maximum rain attenuation values for the path. Fig. 5-11 below shows the measured and the best fit theoretical rain attenuation for Durban at 19.5 GHz on the 6.73 km link.
0.1 1 10 100
0 10 20 30 40 50 60 70 80
Rain rate (mm/h)
Rain attenuation (dB)
Min. Measured Attenuation Max. Measured Attenuation Ave. Measured Attenuation
Min. Theoretical Attenuation Max. Theoretical Attenuation Ave. Theoretical Attenuation
Fig. 5- 11: Best fit theoretical rain attenuation in Durban at 19.5 GHz along the 6.73 km link for the maximum, average and minimum measured attenuation [Odedina and Afullo, 2010]