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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]
5.8 Comparison of Theoretical Attenuation Results with
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describes the average measured rain attenuation at 19.5 GHz is compared with theoretical attenuation model formulated by Moupfouma [1997]. The Moupfouma [1997] approach as discussed in section 3.9.4 is similar to the approach employed in this work but differs in two or three specific ways.
Firstly, while Moupfouma obtains his power law model from the scattering amplitudes calculated from oblate spheroidal raindrops of Uzunoglu et al [1977], this work computed its scattering amplitudes from spherical raindrops. Secondly, the Ray [1972] complex refractive index was used by Uzunoglu et al. [1977] for the calculation of its scattering amplitudes for oblate spheroidal drops (which Moupfouma [1977)] used), but in this work, the refractive index of the spherical rain (water) drop are calculated by using the Liebe model [Liebe et al., 1991] – a method also employed by Mätzler [2002a] and more recently by Mulangu and Afullo [2009]. Thirdly, while Moupfouma [1997] develops his rain attenuation coefficients from the imaginary part of the scattering amplitudes from the oblate spheroidal raindrop, in this work, the rain attenuation coefficients are determined from the real part of the extinction cross-sections of the spherical raindrops, which is calculated from the real part of the spherical scattering amplitudes.
For reasons stated above, the average attenuation values predicted by the theoretical model describing the average measured attenuation are compared with the attenuation values predicted by the Moupfouma theoretical model. Though some researchers like Oguchi [1973] and Morrison and Cross [1974] have also computed scattering amplitudes for oblate spheroidal drops, however their scattering amplitudes have not been really modeled out like Moupfouma [1997] did to the scattering amplitudes obtained by Uzunoglu et al. [1977]. Fig. 5-12 below shows the measured average rain attenuation at 19.5 GHz along the 6.73 km link, attenuation values from the Moupfouma [1997] theoretical model and predicted theoretical average attenuation from the lognormal model of Adimula-Ajayi [1996] (AA) for tropical thunderstorm (TT) at 19.5 GHz.
It can be observed from the figure (Fig. 5-12) that both Moupfouma [1997] theoretical attenuation and the predicted theoretical average attenuation model reasonably describe the measured average attenuation. The most conspicuous observation in this figure is that both models predicted almost the same path attenuation values, despite the differences in the formulation of each theoretical model. Knowing that the Adimula-Ajayi [1996] lognormal distribution for tropical thunderstorm is the most suitable drop-size distribution for this environment, the coefficients of this distribution
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0 5 10 15 20 25
1 10 100
Rain rate (mm/h)
Rain attenuation (dB)
Moupfouma theoretical attenuation model Measured average attenuation
Predicted theoretical average attenuation
Fig. 5- 12: Comparison between the theoretical attenuation results and the Moupfouma theoretical model along the 6.73 km radio path
was incorporated into the Moupfouma [1997] theoretical model to obtain the Moupfouma theoretical attenuation curve shown in Fig 5-12
The major thing that should be noted in the formulation of these theoretical models is that, the Moupfouma [1997] model is developed from the scattering amplitudes of an oblate spheroidal drop calculated by Uzunoglu et al. [1977], while the average theoretical model predicted in this work is developed from the scattering amplitudes calculated from a spherical drop. Then, it can easily be said that whether rain attenuation is modeled with a spherical or an oblate spheroidal drop, the same rain attenuation might be expected along a terrestrial link if the approach employed in this thesis is used, though the results may be influenced by the drop-size distribution used in the calculation of the path attenuation. Therefore, the major parameter that needs to be known if this approach is to be applied to various locations around the world to estimate the path attenuation is to know the raindrop size distribution that describes the geographical location of interest. Since the attenuation models proposed in this chapter has been formulated from different raindrop-size distributions, therefore these proposed models can be used directly to determine the attenuation caused by rain.
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