The forward scattering amplitudes for the spherical raindrops are determined for all raindrop sizes at different frequencies by using the Mie scattering theory. The real parts of the extinction cross sections are used to generate power law models at different frequencies. These are integrated over different established raindrop-size distribution models to formulate rain attenuation models. Using the developed rain attenuation models with 5 year rain rate statistics at R0.01 determined in previous work, the specific rain attenuation is computed. The experimental results obtained from the horizontally polarized signal level measurements recorded in Durban for different rain attenuation bounds are compared with the theoretical results. Finally, the best theoretical model is used to estimate the seasonal cumulative distribution of rain attenuation for Durban, South Africa.
 Recent advances in radio communication systems have put pressure on engineers to develop microwave systems operating at higher-frequency bands. The reliability of such systems may be severely impaired due to rain induced attenuation at such frequencies. Therefore, it is necessary to establish a model capable of predicting the behavior of these systems in the presence of rain. In the calculation of rain attenuation for high-frequency radio communication systems, a high degree of accuracy is needed because overprediction of a propagation effect can result to costly overdesign of a system while in the other hand; under prediction can result into a system that is unreliable [Olsen, 1999].
 The approach of this work is similar to the work performed by Moupfouma , but differs in two ways. First, while Moupfouma obtains his scattering amplitudes from oblate spheroidal raindrop calculated by Uzunoglu et al.  and Morrison and Cross  who used the Ray complex refractive index in their simulations, in this presentation, the scattering amplitudes are calculated from spherical raindrop using the Liebe model [Liebe et al., 1991] to compute the refractive index of the rain (water) drop, a method also employed by Mätzler [2002b] and more recently by Mulangu and Afullo . Second, while Moupfouma develops his rain attenuation coefficients from the imaginary part of the scattering amplitudes from the oblate spheroidal raindrop, in this paper we determine the rain attenuation coefficients 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.
 In this work, the raindrops are assumed to be spherical so that the Mie scattering solution [Mie, 1908] is used for the calculation of the forward scattering amplitudes for the spherical raindrops at various frequencies. Based on the calculated forward scattering amplitudes, extinction cross-section coefficients are computed and these are used to generate power law models. The negative exponential, lognormal and Weibull raindrop-size distribution models are integrated over the power law model to formulate theoretical rain attenuation models. These models are used with the rain rate at R0.01 determined for 4 locations in different climatic rain zones in South Africa [Fashuyi, 2006] to compute the specific rain attenuation in these geographical locations. These locations are Brandvlei, Cape Town, Durban and Pretoria located in the M, N, P and Q climatic zones, respectively, as determined by Fashuyi et al.  and Owolawi and Afullo .
 The 1 year experimental results obtained from the horizontally polarized signal level measurements recorded in Durban for maximum, average and minimum attenuation values over a 6.73 km path at 19.5 GHz [Fashuyi and Afullo, 2007] are compared with the theoretical results obtained from the proposed rain attenuation models. The best theoretical model with a suitable raindrop-size distribution for all the seasons is used to estimate the seasonal cumulative distribution of rain attenuation for Durban.
2. Fundamentals and Theory
 In this work, the Mie theory for electromagnetic scattering by dielectric spheres is applied under the assumption that each raindrop illuminated by a plane wave are uniformly distributed in a rain-filled medium and the distance of each drop is sufficiently large to avoid any interaction between them that might result into multiple scattering. The effect of multiple scattering for propagation through a rain-filled medium has been investigated by Van de Hulst  and Uzunoglu and Evans  and it has been shown that provided the density of the scatterers (raindrops) per unit volume is small then the multiple scattering effects are negligible for all likely rain rates and for frequencies up to 30 GHz [Van de Hulst, 1957; Uzunoglu et al., 1977; Uzunoglu and Evans, 1978].
 When the incident electromagnetic plane wave hits the spherical raindrop (or a dielectric sphere) as shown in Figure 1, a scattered wave is generated and the corresponding scattered electric field in the far-field region for the spherical raindrop is given as (the exp(+jωt) time convention is assumed and suppressed) [Oguchi, 1973]:
3. Computation of Scattering Amplitudes of Spherical Raindrops
 In this work, scattering in the incident wave propagation direction will be considered, therefore the complex forward scattering amplitude for the spherical raindrop function can be written as [Van de Hulst, 1957; Sadiku, 2000];
where an and bn are the Mie coefficients which depend on frequency, radius of drop and the complex refractive index of water.
 The complex refractive index of the spherical water drops which is a function of temperature and frequency m(T, f) [Ray, 1972] is calculated by the dielectric function of Liebe et al.  at 20°C (293 K) for all frequencies used in this work. This is then used to calculate the scattering amplitudes for the entire spherical drop radius. The results of the scattering amplitudes are of real and imaginary parts. The scattering amplitudes are computed for frequencies up to 35 GHz. Tables 1 and 2 show examples of the scattering amplitude results computed for frequencies 2 GHz and 15 GHz for several spherical raindrop radii.
Table 1. Forward Scattering Amplitudes at f = 2 GHz, T = 293 K, m = 8.90697 + 0.490563i, and λ = 15 cm
Radius of the Sphere (cm)
Size of the Spherical Drop α = k
Scattering Amplitude of the Spherical Raindrop
Table 2. Forward Scattering Amplitudes at f = 15 GHz, T = 293 K, m = 7.3206 + 2.53811i, and λ = 2 cm
Radius of the Sphere (cm)
Size of the Spherical Drop α = k
Scattering Amplitude of the Spherical Raindrop
4. Calculation and Modeling of the Extinction Cross Section of Spherical Raindrops
 The extinction cross-section Qext which is a major parameter in the calculation of rain attenuation is determined from the forward scattering amplitude of the spherical raindrops for ϕ = 0 and is given by [Van de Hulst, 1957; Sadiku, 2000]:
Using equation (4), the extinction cross section was calculated for the entire drop radius. By applying the power law regression to the extinction cross section calculated, power law coefficients are determined for all frequencies up to about 35 GHz. Higher frequencies above 35 GHz are also fitted, but due to the negative scattering amplitudes produced by bigger raindrop size radius at the real part, the power law model may not be suitable beyond 35 GHz. Figure 2 shows the power law fitted extinction cross-section plots for various frequencies.
 From Figure 2, the extinction cross section can be described by this equation
where ReQext is the real part of the extinction cross section which is dependent on the real part of the forward scattering amplitude with ϕ = 0, radius of the sphere , wavelength λ and the water temperature T. For the computation of this work T is taken to be 20°C which equals 293K, and κ and ς are the extinction cross-section power law coefficients. Table 3 shows the extinction cross-section power law coefficients computed for some frequencies.
Table 3. Extinction Cross Section Power Law Coefficients κ and ς for ReQext = κς at T = 20°C
5. Theoretical Formulation of the Rain Attenuation Model
 The strength of an electromagnetic wave propagating through a homogeneous medium over a distance d decreases in amplitude by a factor e−γd, where γ is the attenuation factor or attenuation coefficient [Sadiku, 2000, 2007] which is given by Van de Hulst  and Sadiku  as:
The symbol n is assumed to be identical to spherical drops per unit volume.
 Using a more realistic rainfall rather than an identical spherical drops, it is necessary to know the drop-size distribution for a given rain intensity for the theoretical modeling of the rain attenuation [Sadiku, 2000]. There are several models of raindrop-size distribution, but in this paper, the negative exponential, lognormal and Weibull raindrop-size distributions are used. These models have been tested by many authors [Rogers and Olsen, 1976; Ajayi and Olsen, 1985; Sekine et al., 1987] and appear to adequately approximate most observed drop-size spectra fairly well [Rogers and Olsen, 1976]. Therefore, the specific rain attenuation can then be estimated by integrating the extinction cross-section power law over all the spherical raindrop sizes.
5.1. Negative Exponential Raindrop-Size Distribution Model
 The negative exponential raindrop-size distribution models used in this work are the ones by Marshall and Palmer  (MP) given for all rain types and that of Joss et al.  (J) for drizzle and thunderstorm type of rains. Integrating the extinction cross-section power law coefficients over exponential raindrop-size distribution models, we have
 The general tropical lognormal raindrop-size distribution models given by Ajayi and Olsen  (AO) for all rain types, and that of Adimula and Ajayi  (AA) 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]. Integrating the extinction cross-section power law coefficients over lognormal raindrop-size distribution models, we have:
The specific attenuation with a lognormal distribution model can be written as:
and path attenuation can be written as:
5.3. Weibull Raindrop-Size Distribution
 The Weibull distribution models for drizzle, widespread and shower rains given by Sekine et al.  are used in this work. Integrating the extinction cross-section power law coefficients over the Weibull raindrop-size distribution model, we have:
Therefore the specific attenuation from a Weibull distribution model can be written as:
and the corresponding path attenuation is given as:
 Until now, the kRα expression has been widely used for the calculation of specific attenuation due to rain [Olsen et al., 1978; ITU-R, 2005]. Oftentimes, for intervening frequencies, the corresponding values of k and α are determined by interpolation [Ajayi et al., 1996; Moupfouma, 1997] which may sometimes bias the attenuation results [Moupfouma, 1997]. The method presented in this work enables us to calculate the specific rain attenuation even at some intermediate 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 such frequencies, the extinction cross section can be calculated, which can then be fitted and used to estimate the specific rain attenuation. Therefore, 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.
6. Computation of Specific Rain Attenuation From the Theoretical Models
 A fundamental quantity in the calculation of rain attenuation statistics for terrestrial and earth-space links is the specific rain attenuation A(dB/km) [Olsen et al., 1978]. Using the rain rate exceeded for 0.01% of the time (R0.01) from a 5 year rain rate statistics determined by Fashuyi , Fashuyi and Afullo , and Owolawi and Afullo  for four locations situated in different rain climatic zones in South Africa with the developed theoretical rain attenuation models, the specific attenuation are computed. These four locations are Durban, Cape Town, Pretoria and Brandvlei situated in the eastern, western, inland and northern parts of South Africa, respectively (see http://places.co.za/maps/south Africa map.html). Table 4 shows the geographical climatic locations and the respective values of R0.01.
Table 4. Description of the Geographical Climatic Locations and Their R0.01
 The theoretical rain attenuation models developed in section 5 are used to estimate rain attenuation. Figures 3–6 show the specific rain attenuation models calculated from the theoretical models and the ITU-R model [ITU-R, 2005] for Durban, Cape Town, Pretoria and Brandvlei. In Figures 3 and 5, where R0.01 = 119.58 mm/h and R0.01 = 118.86 mm/h, respectively, the ITU-R model gives the highest attenuation values for the frequencies up to 30 GHz; above this frequency, the Joss drizzle negative exponential model takes the lead. The AA-topical widespread (TW) gives the lowest attenuation values for Figures 3–6, but it is observed that at lower frequencies, the Weibull model gives lower attenuation values than the AA-(TW), but increases rapidly at frequencies above 14 GHz. This situation tends to be pronounced for lower rain rate environments, like Brandvlei with R0.01 = 53.90 mm/h, and Cape Town with R0.01 = 61.25 mm/h.
 In Figures 4 and 6, the ITU-R model gives high attenuation values, but at 20 GHz, the lognormal model of AA-tropical shower (TS) produces a higher attenuation value of 6.61 dB/km as compared to that of ITU-R of 6.19 dB/km in Brandvlei; while the two models overlap for Cape Town at this frequency. Above this frequency, the ITU-R model gives higher attenuation values up to 30 GHz, where the negative exponential model of (MP) and Joss drizzle gives higher attenuation.
7. Theoretical and Experimental Path Attenuation
 The time intervals for the measured rain rates are 1 min and their corresponding signal level measurements are recorded at the same time along a 6.73 km propagation path for 1 year in Durban. But it was observed from the measurements that the signal level measurements varied with the same rain rate intensity along the propagation path. For this reason, the minimum, average and maximum attenuation values (i.e., the attenuation bounds) for each value of rain rate along the path were defined [see Fashuyi and Afullo, 2007]. These attenuation bounds determined by Fashuyi and Afullo  were compared with the results obtained from the theoretical rain attenuation models. The theoretical rain attenuation models are used in conjunction with the 1 min rain rates recorded at two points under the 6.73 km link, with one rain gauge at the transmitter end and the other at the receiver end. The chi-square (χ2) statistics [Freedman et al., 1978] is then employed to determine the theoretical attenuation models that best fit into these measured attenuation bounds. Figures 7a and 7b show the measured rain attenuation values as well as the theoretical rain attenuation values for Durban at 19.5 GHz along the 6.73 km line-of-sight path.
Figure 7a shows the measured rain attenuation values at the 3 bounds (maximum, minimum and average attenuation values per rain rate), rain attenuation calculated from the ITU-R model, and theoretical attenuation models developed from the negative exponential and Weibull raindrop-size distribution model. From Figure 7a it is observed that the attenuation results from these models fall largely on the minimum measured attenuation. This is because the raindrop-size distribution models employed in formulating these theoretical attenuation models are developed from research works in the northern hemisphere temperate zones which based their distribution models on propagation data obtained in the temperate climate zones [Green, 2004]. And in recent times, evidences have shown the limitation of these distribution models when applied to equatorial, tropical or lower-latitudes regions [Ajayi, 1990; Maciel and Assis, 1990; Yeo et al., 1993; Zainal et al., 1993; Li et al., 1995; Zhou et al., 2000], etc. For these reasons, attenuation values in the minimum attenuation bounds which are produced by lower rain rates or small raindrop sizes (which seems to be common in the temperate zones) tend to be reasonably described by these theoretical models in Figure 7a.
 It is also observed from Figure 7a that the ITU-R  model gives attenuation values that fall within the measured minimum, average and increases rapidly toward the measured maximum attenuation. Works in the tropics and equatorial climates from Nigeria [Ajayi, 1990], Brazil [Maciel and Assis, 1990], Malaysia [Zainal et al., 1993], Singapore [Yeo et al., 1993; Li et al., 1995; Zhou et al., 2000], and India [Maitra, 2004] have confirmed large disparities between measured attenuation results and the ITU-R predictions. This is because factors that make important contributions to propagation impairments in these climates (tropical or equatorial) are different from the northern temperate climates [Green, 2004]. 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 as shown by the χ2 statistic results in Table 5. This is because South Africa, located on the latitude 29°S and longitude 24°E in the most southern tip of the Africa continent, is classified as a subtropical region. Thus, attenuation results from the tropics or equatorial climates cannot be directly mapped as South Africa results due to their climatic differences.
Table 5. The χ2 Statistic for the Theoretical Attenuation Models as Compared to the Maximum, Average, and Minimum Measured Attenuation at 19.5 GHz on 6.73 km Path Lengtha
Theoretical Attenuation Models
Degrees of Freedom is 20; χ2 Statistic Threshold is 37.57
Minimum Measured Attenuation
Average Measured Attenuation
Maximum Measured Attenuation
The χ2 statistic is at the 1% significance level.
Lowest χ2 values of the theoretical attenuation model that best fit the measured rain attenuation.
Figure 7b also shows the measured rain attenuation at the 3 bounds, the theoretical attenuation models developed from the lognormal raindrop-size distribution model of Ajayi-Olsen and Adimula-Ajayi. 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. Having observed the limitation of the former raindrop-size distributions developed in the northern temperate regions, authors like Ajayi and Olsen , Feingold and Zev , Ajayi , Zainal et al. , and Adimula and Ajayi , etc. developed alternative raindrop-size distribution models that can best describe drop-size distribution in the tropics and lower-latitudes regions. 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. From Figures 7a and 7b, the theoretical attenuation models that best fit into the measured attenuation values for each bounds were determined by using the chi-square statistics (see Table 5).
 From Table 5, 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. Figure 8 shows the measured and the theoretical path attenuation for Durban at 19.5 GHz on the 6.73 km link.
8. Cumulative Distribution of Rain Attenuation
 The cumulative distributions of rain attenuation give the different percentages of time for which certain rain attenuation levels are exceeded on a particular radio link [Schnell et al., 2002]. Using a 1 min rain rate recorded in Durban for a period of 5 years (2000–2004) with the theoretical attenuation model developed from the lognormal raindrop-size distribution which gives the best fit for the measured average attenuation in the Durban, the seasonal cumulative distribution of rain attenuation is determined for a 6.73 km link at 19.5 GHz. This gives the average attenuation expected on the link at different percentages of time for each seasons in Durban, South Africa, which is climatically controlled by four major seasons; summer, autumn spring and winter (see http://www.sa-venues.com/no/weather.htm). Tyson  records most of its high rain rates in the summer and autumn seasons [Odedina and Afullo, 2008] with the exception of Cape Town which records its highest in spring and winter, because of the Mediterranean climate [Odedina and Afullo, 2008].
Figure 9 gives the seasonal cumulative distribution of rain attenuation in Durban determined from the theoretical attenuation model developed from the tropical thunderstorm lognormal raindrop-size distribution. From Figure 9, the attenuation values exceeded for various percentages of time ranging from 99.99% to 90% can be determined for different seasons in Durban. The autumn and summer seasons gives the higher rain attenuation due to their higher rain rates statistics than the winter and spring seasons which has lower rain rates. The attenuation values at these percentages of time are very useful in the design of radio communication links, for it helps to estimate the necessary allowance that may be required on a radio link that can guarantee a reliable service and reduce the effect of loss of signal due to rain to some infrequent heavy rains. The theoretical models employed here can be extended to other locations around the world to estimate the rain attenuation that may be expected on any radio links.
 This paper gives a new theoretical approach for the calculation of rain attenuation on a communication radio link based on the assumption that the raindrops are of a spherical shape. The theoretical rain attenuation model was formulated by integrating the extinction cross-section power law models developed from the scattering amplitudes of the spherical raindrops over different established raindrop-size distribution models. This is then used to compute the specific rain attenuation for four locations situated in different rain climatic zones in South Africa. To validate these theoretical models, it was compared with the experimental signal level measurement recorded in Durban. And for the purpose of cross referencing, it was also compared with the ITU-R rain attenuation model.
 The results show that the theoretical attenuation developed from the negative exponential model of Joss et al. thunderstorm rain type, lognormal model of Adimula-Ajayi tropical thunderstorm rains and tropical shower type of rains give the best fits for the minimum, average and maximum measured attenuation, respectively. Hence the average theoretical attenuation model is then used to calculate the cumulative distribution of rain attenuation in Durban using a 1 min integration time rain rate statistics recorded in Durban for a period of 5 years. This allows for the determination of the average attenuation values for each season at different percentages of time.
 These proposed theoretical attenuation models can be applied to various geographical locations around the world to estimate the behavior of a radio link in the presence of rain, especially when either the rain rate statistics or the raindrop-size distribution governing the locality is known. However, a lot of work needs to be done to expand the power law coefficients of the extinction cross section beyond 35 GHz. This is the intention for further studies.
 The authors are grateful to Tracey Gill of South African Weather Service for facilitating access to the 5 year rainfall data.