Declaration 2 Publication
2.4 Rainfall Attenuation over Communication Links
2.4.4 Rainfall Drop Size Distribution (DSD)
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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M($) = 17.67 × (0.1$).T /R (2.10)
where D is the diameter of the rain drop in both cases of (2.9) and (2.10).
The diameter integral of the product of the rain drop volume, terminal velocity and rainfall DSD from zero to infinity describes the rainfall rate function described in Sadiku [2000] as:
?($) = U MW ($)V($)H($) $
/ℎ (2.11) Modifying (2.11) by substituting for V(D) from (2.8) yields a generic rain rate function which is related to the third moment of the rainfall DSD and is given by:
?($) = 6" × 10YZU MW ($)V($)$K$
/ℎ (2.12) Therefore, rainfall rate can be estimated given the knowledge of the terminal velocity and rainfall DSD provided the diameter range of rain drops are within satisfactory limits.
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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rain DSD mathematics has been shown to be strongly related to the Probability Density Function (PDF) of available drop sizes. Ulbrich [1983] describes the raindrop mathematical representation for rain drop size theory as:
;($) = V($)
[ V($)$\W Y for $ > 0 (2.13)
where f(D) is the PDF of the rain drop sizes, N(D) is the rain DSD function and D is the mid- value diameters of rain droplets usually ranging from 0.2 mm to about 5.5 mm.
From (2.13), it follows that denominator is the rain rate-dependent scaling constant of f(D) known as the drop concentration variable, Nt. Thus, it follows that the drop concentration per unit volume is approximately the zeroth moment of the rain drop diameter defined as:
V = U V($)$
W
\
= 3 V($)∆$ YK (2.14)
The moment of rainfall DSD is an important relation with a number of applications in rainfall statistics [Kozu and Nakamura, 1991; Timothy et al., 2002]. Mathematically, the DSD moment is defined as:
_(`) = 3 $aV($)∆$b c
d
YKa (2.15)
where n is the moment number are often used in the determination of other rainfall microstructures given the knowledge of the rain DSD. For example, setting n = 0 (zeroth moment) yields the drop concentration as earlier seen in (2.14). The third, fourth and sixth moments are useful in estimating the rainfall rate or liquid water content, rainfall attenuation and radar reflectivity respectively [Kozu and Nakamura, 1991].
The Method of Moment (MOM) parameter estimation technique is the most popular method of computing the unknown parameters of statistical distributions of rainfall DSD [Ajayi and Olsen, 1985; Kozu and Nakamura, 1991; Timothy et al., 2002]. The description of rainfall DSD patterns can be enhanced by modelling the data using statistical distribution functions. There are four popular statistical model often employed by researchers in DSD modelling: lognormal, modified gamma, Weibull and negative exponential models. The first two models have been
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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extensively demonstrated to work favourably with data from tropical areas with high rainfall rates [Awang and Din, 2004; Das et al., 2010]. The last model, on the other hand, has been proven to be the most preferable for temperate climates [Law and Parson, 1943; Marshall and Palmer, 1974; Bhattacharyya et al., 2000]. The discussion will focus on lognormal and modified gamma models based on their importance in the evaluation of rainfall drop sizes in tropical areas.
2.4.4.1 Lognormal Rainfall DSD
The lognormal model rainfall DSD model is a three-parameter function defined from Ajayi and Olsen [1985] and Maitra [2004] as:
V($) = Ve
f$√2"exp h−0.5 ln($) − j
f '#k YKY (2.16)
where NT is the total number of rain drops per unit volume, µ is the mean of the drop size data and σ is the standard deviation of the drop sizes. These parameters are dependent on the prevailing local rainfall rate, R, given as:
Ve = A\?Fl (2.17a) j = n+ n In (?) (2.17p) f#= q+ q In (?) (2.17r)
where the coefficients, ao, bo, Aµ, Bµ, Aσ and Bσ can be obtained through regression analysis.
It is usual that the MOM technique as given in (2.15) is equated to the lognormal moment generator given by [Kozu and Nakamura, 1991]:
_a= Ve exp s`j + 1
2 (`f)#t (2.19)
where n is the moment index and other parameters are components of the lognormal distribution. The third, fourth and sixth moments of (2.19) are solved to obtain the input parameters as required. The lognormal DSD model is most preferred model in the descriptive statistics of rainfall DSD in tropical areas around the world because it works well at regions with high rainfall rate occurrence [Ong and Shan, 1997; Awang and Din, 2004; Das et al., 2010]. Figure 2-4 gives the variation of rainfall DSDs at different tropical locations around the world at 75 mm/h. The locations compared are Singapore [Ong and Shan, 1997], Nigeria [Ajayi
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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Figure 2-4: Comparison of rainfall DSDs at different part of the world
and Olsen, 1985], India [Maitra, 2004], Malaysia [Tharek and Din, 1992] and South Africa [Alonge, 2011]. At this particular rain rate, the probability of having maximum rain drops beyond 3 mm in Durban is almost zero, compared to results from other locations.
2.4.4.2 Modified Gamma DSD
The modified gamma model is a modification of the classical exponential DSD function proposed by Marshall and Palmer [1974], where $n represents the exponential modifier. This distribution is given by Atlas and Ulbrich [1974]:
V($) = V\$nexp(−Λ$) YKY (2.20)
where No represents the constant related to number of rainfall drops, µ is the shape parameter and Λ is the slope parameter of the distribution. These parameters are also related to the rainfall rate given by:
V,= A,?Fv (2.21A) Λ = Aw?Fx (2.21p)
The coefficients in (2.21a) and (2.21b) can be obtained from regression analysis from results of the MOM technique. The modified gamma DSD moment generator is equated to raw moments of the data as given by Kozu and Nakamura [1991]:
1E-05 0.0001 0.001 0.01 0.1 1 10 100 1000 10000
0 1 2 3 4 5 6
Rainfall DSD (mm-1m-3)
Rain Drop Diameter (mm)
Singapore Nigeria India Malaysia South Africa
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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Λnzaz (2.22)
The moment generator (2.20) is solved for the third, fourth and sixth moments as (2.19) and the estimators are obtained simultaneously. This model is suitable for rainfall DSDS at both temperate and tropical areas [Ulbrich, 1983; Bhattacharya et al., 2000; Awang and Din, 2004].