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0
0
1 2
N D e DdD
(5.37)Solving equation (5.37) by parts and applying the gamma integral function rule [Stroud and Booth, 2003], the attenuation coefficient can be written as;
0 1
1 1
2 N 1
(5.38) Substituting equation (5.38) into (5.16b) the specific attenuation with a gamma distribution model can be written as:
5 0 1
1 1
( / ) 4.343 10 1
A dB km 2 N
(5.39)
and substituting (5.38) into (5.16a), the path attenuation can be written as:
1 5
0
1 1
( ) 4.343 10 1
eff 2
A dB d N
(5.40)
Equation (5.39) and (5.40) are the theoretical attenuation model formulated from the gamma rain distribution model for specific and path attenuation estimations, respectively.
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frequencies such as 5.3 GHz, 21.3 GHz, 11.7 GHz, 19.5 GHz, and … etc without need for interpolation. This is because once the scattering amplitudes are determined for the spherical raindrops at the desired frequencies; the extinction cross section can be calculated. This can then be fitted and used to estimate the specific rain attenuation. The major parameter that will need to be known in the above proposed theoretical models, when applied to various geographical locations is the rain rate statistics of the location or the raindrop-size distribution governing the location in question.
The results from these models have been applied to the propagation measurements recorded in Durban along the 6.73 km to ascertain the best theoretical model that defines the Durban climate in section 5.7 of this chapter. But firstly, using a 5-year rain rate (mm/h) exceeded for 0.01% of the average year as determined by Fashuyi [2006], Fashuyi et al. [2006] for four different locations in South Africa, with the developed theoretical rain attenuation models, the respective specific attenuations are computed these locations. Each of this location is situated in four different climatic regions with each having different climatic rain zone. These four locations are Durban, Cape Town, Pretoria and Brandvlei with the locations having P, N, Q, and M climatic rain zones, respectively [Fashuyi et al., 2006] (see Tables 3-3 and 3-4). Durban is located in the eastern coastal area of South Africa, Cape town in the western mediterranean side, Pretoria in the temperate inland, and Brandvlei in the desert area of South Africa [from South Africa weather Services (SAWS)]. Table 5-5 below shows the annualR0.01, averageR0.01, and the geographical location of the four cities.
The theoretical rain attenuation models developed in section 5.5 above are used to calculate the specific rain attenuation. For the calculation of the negative exponential attenuation, the raindrop
Table 5- 5: Annual and averaged R0.01 statistics for the four geographical locations [Fashuyi, 2006;
Fashuyi et al., 2006; Fashuyi and Afullo, 2007]
Location Latitude south
Longitude east
2000, (mm/h)
2001, (mm/h)
2002, (mm/h)
2003, (mm/h)
2004, (mm/h)
Average (2000- 2004), mm/h Durban 29o.97' 30o.95' 108.75 88.82 138.83 139.66 121.84 119.58 Cape Town 33o.97' 18o.60' 47.70 69.96 57.03 44.50 87.04 61.25 Pretoria 25o.73' 28o.18' 159.45 114.90 100.73 106.09 113.15 118.86 Brandvlei 30o.47' 20o.48' 41.25 105.2 67.02 13.67 42.34 53.90
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size distribution coefficients proposed by Marshall Palmer [1948] for different rain types (MP) and the Joss et al. [1968] coefficients for drizzle (D) and thunderstorm (T) rain types are used (see Section 2.12.1). For the lognormal attenuation calculation, the drop-size coefficients given by Ajayi and Olsen [1985] for all tropical rains, and that of Ajayi and Adimula [1996] for tropical widespread (TW), topical shower (TS) and tropical thunderstorm (TT) are used (see Table 2-3).
For Weibull attenuation calculation, the drop-size distribution coefficients given by Sekine at al.
[1987] for drizzle, widespread and shower rains are used (see Section 2.12.4). Finally, for the gamma attenuation calculation, the coefficients given by de Wolf [2001] for the gamma distribution fit to the measurements of Laws and Parsons [1943] are used.
Figures 5-6 to 5-9 below show the specific rain attenuation models calculated from the various developed theoretical models, and the ITU-R model, for Durban, Cape Town, Pretoria and Brandvlei. In Fig. 5-6 and 5-8 where the rain rate exceeded for 0.01% of the average year R0.01 is 119.58 mm/h and 118.86 mm/h for Durban and Pretoria, respectively, at about 4 GHz, the ITU-R model gives the highest attenuation values for the frequencies up to 15 GHz in Durban and 16 GHz in Pretoria. At 16 and 17 GHz for Durban, the ITU-R attenuation results overlaps with the Weibull attenuation model, after which the Weibull attenuation model takes the lead up to 35 GHz. For Pretoria, the overlap between the ITU-R and Weibull model occurred at 17 GHz after which the Weibull model also takes the lead up to 35 GHz.
The Weibull model is seen to give lower attenuation values at lower frequencies, but increases rapidly at frequencies above 5 GHz. This behaviour tends to be pronounced for lower rain rate environments, like Brandvlei withR0.0153.90mm h/ , and Cape Town with R0.0161.25mm h/ . With this, in the high rain rates environment, of figures 5-6 and 5-8, the Weibull model gives the lowest attenuation values at 1 GHz and the highest attenuation values at 35 GHz when compared to other models. The lognormal model of AA-tropical widespread (TW) also gives lower attenuation results in these figures (Fig. 5-6 and 5-8), but the attenuation results do not increase rapidly like that of the Gamma or Weibull model. Thus, the lognormal model of AA-tropical widespread (TW) gives the lowest attenuation results at 35 GHz for the two locations (Durban and Pretoria).
For figures 5-7 and 5-9 below, at frequency above 5 GHz, the gamma model is seen to give to lowest attenuation values up to 25 GHz, after which it also rises rapidly up to 35 GHz. Also in the same figures, the lognormal model of AA-tropical shower (TS) gives a higher attenuation from 7
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0.0001 0.001 0.01 0.1 1 10 100
0 5 10 15 20 25 30 35
Frequency (GHz)
Specific attenuation (dB/km)
Expo. model (MP) Expo. model (Joss et al.)-T Expo model (Joss et al.)-D Gamma model (de Wolf) Weibull model (Sekine et al.) Logn. model (AO) Logn. model (AA)-TT Logn. model (AA)-TS
Logn. model (AA)-TW ITU-R model
Fig. 5- 6: Specific rain attenuation from theoretical models for Durban
0.0001 0.001 0.01 0.1 1 10 100
0 5 10 15 20 25 30 35
Frequency (GHz)
Specific attenuation (dB/km)
Expo. model (MP) Expo. model (Joss et al.)-T Expo. model (Joss et al.)-D Gamma model (de Wolf) Weibull model (Sekin et al.) Logn. model (AO) Logn. model (AA)-TT Logn. model (AA)-TS
Logn. model (AA)-TW ITU-R model
Fig. 5- 7: Specific rain attenuation from theoretical models for Cape Town
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0.0001 0.001 0.01 0.1 1 10 100
0 5 10 15 20 25 30 35
Frequency (GHz)
Specific attenuation (dB/km)
Expo. model (MP) Expo. model (Joss et al.)-T Expo. model (Joss et al.)-D Gamma model (de Wolf) Weibull model (Sekine et al.) Logn. model (AO) Logn. model (AA)-TT Logn. model (AA)-TS
Logn. model (AA)-TW ITU-R model
Fig. 5- 8: Specific rain attenuation from theoretical models for Pretoria
0.0001 0.001 0.01 0.1 1 10 100
0 5 10 15 20 25 30 35
Frequency (GHz)
Specific attenuation (dB/km)
Expo. model (MP) Expo. model (Joss et al.)-T Expo. model (Joss et al.)-D Gamma model (de Wolf) Weibull model (Sekine et al.) Logn. model (AO) Logn. model (AA)-TT Logn. model (AA)-TS Logn. model (AA)-TW ITU-R model
Fig. 5- 9: Specific rain attenuation from theoretical models for Brandvlei
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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.