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123
0.0000001 0.00001 0.001 0.1 10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Raindrop radius (cm)
Extinction cross-section (2 GHz) Extinction cross-section (5 GHz) Extinction cross-section (10 GHz) Extinction cross-section (19.5 GHz) Extinction cross-section (23 GHz)
Power fit law (2 GHz; 5 GHz; 10 GHz; 19.5 GHz; 23 GHz)
extReQ(=0)
Fig. 5- 5: Modeling the extinction cross-section real part at different frequencies
where
Re
Qext is the real part of the extinction cross-section calculated which is dependent on the real part of the forward scattering amplitude with
0
, radius of the spherical raindrop a, wavelength ,and the water (raindrop) temperature T.
and are the extinction cross-sections power law coefficients. Table 5-4 below shows the extinction cross-section power-law coefficients determined for frequencies 1-35 GHz.A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
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Table 5- 4: The extinction cross-section power-law coefficients κ and ςfor ReQext=κa at ς T=20 Co
Frequencies (GHz)
ext
ext1 0.0058 3.0408
2 0.0315 3.1478
3 0.107 3.2934
4 0.3086 3.4613
5 0.8542 3.6594
6 2.3114 3.8787
7 5.9654 4.0972
8 11.71 4.228
9 19.431 4.2975
10 30.043 4.3555
12 51.378 4.3217
14 87.188 4.4109
15 95.823 4.3751
16 100.89 4.3265
17 99.89 4.263
18 97.091 4.2223
19.5 83.956 4.1142
20 81.36 4.0887
23 51.931 3.7151
25 36.769 3.489
28 15.566 3.0001
30 10.2283 2.6873
35 8.7185 2.5773
The symbol n is assumed to be identical to spherical drops per unit volume. Therefore, the wave is attenuated by [Crane, 1996; Sadiku, 2000];
10
( ) 10log 1d (ratio of power to reference power [ , 1996])
A dB Crane
e
(5.15a)
A dB( )d10log10e (5.15b)
where e2.718, and thus;
(A dB)4.343d (5.16a) or
(A dB km/ )4.343 (5.16b)
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125
Equations (5.16a) and (5.16b) are path attenuation and specific rain attenuation respectively.
Thus, using equation (5.14) and (5.16b), the specific attenuation can be given as
4.343 ext
A n Q (5.17)
or
2
4.343 nRe (0)
A S
(5.18)
for nidentical drops.
For the formulation of rain attenuation models, a more realistic rainfall must be used rather than identical spherical drops, because rainfall comprises of drops of various sizes. It is therefore necessary to know the drop-size distribution of a given rain intensity for the theoretical modeling of the rain attenuation [Hogg, 1968; Sadiku, 2000]. Several raindrop-size distribution models have been proposed by different authors, but for the formulation of rain attenuation models in this study, negative exponential, lognormal, Weibull, and gamma raindrop-size distributions are used.
These models have been tested by many researchers [Rogers and Olsen, 1976; Ajayi and Olsen, 1985; Sekine et al., 1987; de Wolf, 2001] and they appear to adequately approximate most observed drop-size spectra fairly well [Rogers and Olsen, 1976]. Thus, the specific rain attenuation can then be estimated by integrating the extinction cross-section power law over all the spherical raindrop sizes [Hogg, 1968].
5.5.1 Negative Exponential Attenuation Model
The negative exponential attenuation model is formulated based on the negative exponential raindrop-size distribution model. The negative exponential drop-size distributions used in this work are the ones given by Marshall and Palmer [1948] (MP) for all rain types and that of Joss et al (J) [1968] for drizzle and thunderstorm type of rains (see Table 2-1). The negative exponential raindrop-size distribution as shown in equation (2.19) as a function of drop diameter can be represented by [Marshall and Palmer, 1948; Joss et al., 1968]:
( ) 0 a
n a N e (5.19)
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126
as a function of drop radius, where ais the radius of the drop, N0is the number of concentration of drops n a( )for radius a 0 on the exponential approximation and is its slope per unit volume of space (defined in equation 5.19) [Marshall and Palmer, 1948].
From equation (5.13) the extinction cross-section power law model can be written as:
Qext
a (5.20)To determine the attenuation coefficient, the extinction cross-section power law coefficients was integrated over exponential raindrop-size distribution models, to have
0 a n a da
( )
= 0 0a N e ada
(5.21)where n a da
( )
is the number of drops of radius between aand adain a unit volume of space.Applying integral gamma function in solving the integral in equation (5.21) [Stroud and Booth, 2003]:
0 0
0
0
( 1)
a a
N a e da N e da
0
(
11)
N
(5.22) with as the integral gamma function.
Substituting equation (5.22) into (5.16b), the specific rain attenuation can be written as:
5 0 ( 11)
( / ) 4.343 10
A dB km N
(5.23)
Substituting equation (5.22) into (5.16a), the path attenuation becomes:
5 0 ( 11) ( ) 4.343 10
A dB N d
(5.24)
A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
127 where dis the propagation path length.
But because of the non-homogeneity of rain along propagation path length [Crane, 1980; 1996;
Moupfouma, 1984; 2009; Hall et al., 1996; ITU-R 530-12, 2007; Forknall et al., 2008], as discussed in the previous chapters, the concept of effective path length deff was introduced into the path attenuation model formulated in equation (5.24). The ITU-R effective path length model is used in this work. This is because it seems to be the only model that cut across the three measured attenuation bounds (minimum, average, and maximum measured attenuation)and the model that has the minimum root mean square percentage error across the attenuation bounds predicted in Chapter four of this work. The effective path length is achieved by multiplying the actual path length of the propagation path by a distance factorrknown as the reduction factor [Moupfouma, 1984; ITU-R 530-12, 2007]. The guiding equations for the effective path length are shown in Section 3.2.1.
Therefore equation (5. 24) for the path attenuation can be written as:
5 0 ( 11) ( ) 4.343 10 eff
A dB d N
(5.25)
where deff is the effective path length of the terrestrial radio link, and is the extinction cross- section power-law coefficient, is the integral gamma function, N0 is the number of concentration of drops n a
( )
for radius a 0, and is the slope (the rain dependent parameter), and is given as =R- where Ris the rain rate in mmh-1, and and are the power law coefficients for the slope [ Marshall and Palmer, 1948; Konwar, 2006]Equation (5.23) and (5.25) are the theoretical attenuation model formulated from the negative exponential rain distribution model for specific and path attenuation estimations, respectively.
5.5.2 Lognormal Attenuation Model
The lognormal attenuation model is formulated from the lognormal raindrop-size distribution. In this attenuation formulation, the general tropical lognormal raindrop-size distribution models given by Ajayi-Olsen (AO) [Ajayi and Olsen, 1985] for all rain types and that of Adimula-Ajayi
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(AA) [Adimula and Ajayi, 1996] for drizzle, widespread, shower, thunderstorm and tropical thunderstorm rain are used in this work. The tropical lognormal models were chosen as against the continental lognormal models because of the similarities that the rain type in the South Africa has with the tropical environment [Fashuyi et al., 2006]. The lognormal raindrop-size distribution model as given in equation (2.23) can be written as:
1 ln( ) 2
( ) exp
2 2
Nt D
n D D
(5.26)
where Dis the raindrop diameter, n D
( )
is the number of raindrops in a unit volume of air,
is the standard deviation, is the mean of lnDand Ntis the total number of drops of all sizes.With
,
and Nt depending on the geographical characteristics of the location concerned [Ajayi and Olsen, 1985; Adimula and Ajayi, 1996].Integrating the extinction cross-section power law coefficients over lognormal raindrop-size distribution models, we have:
0 ( )
2
D n D dD
= 0 21 ln( )
2 2 exp 2
Nt
D D
D dD
2 1
0 2
1 1 (ln )
2 2 exp 2
t
N D D dD
(5.27)Integrating by parts and applying the integration bounds, the attenuation coefficient for lognormal models can be given as [Moupfouma, 1997]:
0 ( )
2
D n D dD
=2Ntexp22
(5.28)
Substituting equation (5.28) into (5.16b) the specific attenuation with a lognormal distribution model can be written as:
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2
( / ) 4.343 105 exp
2 2
A dB km Nt
(5.29) and path attenuation can be written as:
2
( ) 4.343 105 exp
2 2
eff t
A dB d N
(5.30)
whereNt,, and are the rain dependent parameters
Equations (5.29) and (5.30) are the theoretical rain attenuation model developed from a lognormal rain distribution model for specific and path attenuation, respectively.
5.5.3 Weibull Attenuation Model
The Weibull attenuation model is formulated from the Weibull drop-size distribution models.
The Weibull distribution model for drizzle, widespread and shower rains given by Sekine et al.
[1987] are used in this work. The Weibull raindrop-size distribution model given in equation (2.25) can be represented by:
1
( ) 0 e c
c
c D D b
n D N
b b
(5.31)
where bandcare parameters depending on the geographical characteristics of the location concerned.
Integrating the extinction cross-section power law over equation (5.31), we have:
0 ( )
2
D n D dD
= 0 0 1e 2c
c
D c D D b
N dD
b b
(5.32)
1
0 0 e
2
c c
c D D D b
N dD
b
b
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Integrating equation (5.32) by parts and applying the integral gamma functions [Stroud and Booth, 2003], the attenuation coefficient can be given as:
0 ( )
2
D n D dD
N0 2b c c (5.33)with as the integral gamma function. Therefore the specific attenuation from a Weibull distribution model can be written as:
A dB km( / )4.343 10 5N0 2 b
c c
(5.34)
and the corresponding path attenuation is given as:
A dB( )4.343 10 5deff N0 2 b
c c
(5.35)
Equation (5.34) and (5.35) are the theoretical attenuation model formulated from the Weibull rain distribution model for specific and path attenuation, respectively.
5.5.4 Gamma Attenuation Model
The gamma attenuation model is formulated based on the gamma drop-size distribution model discussed in section 2.12.2 of this thesis. The gamma drop-size distribution given in equation (2.20) can also be expressed as:
n D( )N D0 exp(D)
(5.36)
Integrating the extinction cross-section power law model over the gamma drop-size distribution in equation (5.36), to obtain the attenuation coefficient
0 ( )
2
D n D dD
= 0 2D
N D0 exp(D dD)A Semi-Empirical Formulation for Determination of Rain Attenuation on Terrestrial Radio Links
131
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.