Variability of the propagation coefficients due to rain for microwave links in southern Africa



[1] We use the Mie scattering approach and the dielectric model of Liebe to determine the propagation coefficients and rain attenuation distribution for four locations in Botswana, southern Africa, using R0.01 = 68.9 mm/h for Gaborone, R0.01 = 137.06 mm/h for Selebi-Phikwe, R0.01 = 86.87 mm/h for Francistown, and R0.01 = 64.4 mm/h for Kasane over the frequency range of 1–1000 GHz. The results show that the extinction coefficients depend more strongly on temperature at lower frequencies than at higher frequencies for lognormal distribution. The absorption coefficient is significant but decreases exponentially with rain temperature at lower microwave frequencies. The application of the proposed model with various distributions is corroborated using practical results for Durban in South Africa.

1. Introduction

[2] Rain affects the design of any communication system that relies on the propagation of electromagnetic waves. Above a certain threshold of frequency, the attenuation due to rain becomes one of the most important limits to the performance of line-of-sight (LOS) microwave links. Rain attenuation, which is the dominant fading mechanism at these frequencies, is based on nature, which can vary from location to location and from year to year. Daily rainfall accumulations, universally recorded on hourly basis, are also fairly widely available by national weather bureaus.

[3] Rogers and Olsen [1976] employ Mie scattering theory and Laws and Parsons (LP) drop size distribution (DSD) to obtain plots of specific attenuation γ (dB/km) against frequency (from 1 to 1000 GHz) for rain rate, varying from 0.25 mm/h to 150 mm/h. For each of the nine values of rain rate, the effect of rain temperature on specific attenuation is also shown at two temperatures, namely 273°K and 293°K. Ajayi and Ofoche [1983] determined that the use of 1-min rain rates gives the best agreement with the International Telecommunication Union Radiocommunication Sector (ITU-R) [2005] stipulations for the design of microwave radio links. This was then used to estimate the signal outages at 0.01% of the time on radio communication links, since it defines the rainfall rate recommended by ITU-R [2005] to evaluate the availability of terrestrial and satellite radio links.

[4] Ajayi and Olsen [1985] have shown that the LP and MP distributions overestimate the number of drops in the small and large diameter regions; and that the lognormal model gives a better fit to the measured drop size data at Ile-Ife, Nigeria (a tropical climate). Moupfouma and Tiffon [1982] used measurement data over a 33.5-km 7 GHz terrestrial radio link in the Congo (equatorial region of Africa) to propose a new rain drop size distribution, and also confirmed that the MP distribution overestimates the number of small drops while it underestimates the larger ones for this region. Similar conclusions have been made by O. Massambani and C. A. M. Rodriguez (Una distribuicao gamma de tamanhos de gotas de nuvens, paper presented at V Congresso Brasileiro de Meteorologia, Ministério da Agricultura, Pecuária e Abastecimento, Brazil, 1988) and O. Massambani and C. A. M. Rodriguez (Specific attenuation as inferred from drop size distribution measurements in the tropics, paper presented at URSI Commission F Open Symposium, Rutherford Appleton Laboratory, Rio de Janeiro, Brazil, 3–7 December, 1990), making use of the data obtained in Brazil.

[5] To determine specific attenuation due to rain using Mie computations, Olsen et al. [1978] employed the Ray method to determine the refractive index for water at various temperatures and frequencies. However, the Ray data are regarded as inaccurate especially at frequencies above 10 GHz [Mätzler, 2002a]. On the other hand Mätzler developed MATLAB functions based on the formulation of Bohren and Huffman [1983]: he used the more accurate dielectric model of Liebe to determine the refractive index of water for computation of Mie parameters [Mätzler, 2002a]. The parameters required were the interaction cross sections by rain per unit volume of a rainy atmosphere, i.e., the propagation coefficients for rain extinction, γext, scattering, γsca, absorption, γabs, backscattering, γb, and the asymmetry parameter 〈cos θ〉 [see Mätzler, 2002a, 2002b, 2002c]. Here, 〈cos θ〉 (D) is the effective cosine of the scattering angle, and is a function of D, the rain drop size. In his approach, Mätzler computed the coefficients γj, {j = ext, abs, sca, b} from the corresponding Mie efficiencies Qj and the drop size distributions using Marshall Palmer (MP), Joss-Thunderstorm (JT), Joss-Drizzle (JD) and Laws and Parsons (LP) distributions of N(D), from the equations [see Mätzler, 2002b]:

equation image
equation image

These coefficients are used in radiative transfer theory, as in, for example, the works of Chandrasekhar [1960] and Meador and Waver [1980]. In this presentation we determine propagation coefficients, γj, due to rain in Botswana for four selected stations from the corresponding Mie efficiencies Qj and the drop size distribution based on lognormal distribution for tropical and subtropical countries, as presented by Ajayi et al. [1996]:

equation image

Here μ is the mean of ln (D), σ is the standard deviation, Nt is the total number of the drops per cubic meter per mm. We use R0.01 = 68.9 mm/h for Gaborone, R0.01 = 137.06 mm/h for Selebi-Phikwe, R0.01 = 86.87 mm/h for Francistown and R0.01 = 64.4 mm/h for Kasane, for Botswana, in southern Africa, as determined b C. T. Mulangu et al. (Rainfall rate distribution for LOS radio system in Botswana, paper presented at 10th Southern Africa Telecommunication, Networks and Application Conference (SATNAC), Ericsson, Mauritius, September 2007). We also discuss the variability of the propagation coefficients due to rain with temperature for Gaborone, Selebi-Phikwe, Kasane and Francistown, comparing the same with the results of Rogers and Olsen [1976], Ajayi and Adimula [1996], and Moupfouma and Tiffon [1982]. Since our work is based on the procedures presented by the various works of Mätzler, the brief theoretical basis that follows is based purely on Mätzler [2002a, 2002b].

2. Rain Attenuation Variability

[6] The two main causes of rain attenuation are scattering and absorption. When the wavelength is fairy large relative to the size of raindrop, scattering is predominant. Conversely, when the wavelength is small compared to the raindrop size, attenuation due to absorption is dominating. An empirical procedure based on the approximate relation between γ and rain rate R is given as [Olsen et al., 1978]:

equation image

where k and α are power law parameters, which depend on frequency, raindrop size distribution, rain temperature, and polarization. The coefficients γj, j = ext, abs, sca, b, can be obtained from the rain drop size information. Mie theory also needs the complex refractive index of dielectric spheres, such as water drops. The complex refractive index m(f, T), being a function of frequency f and temperature T, is related to the complex relative dielectric permittivity ɛ(f, T) as determined by Liebe et al. [1991].

[7] The efficiencies Qi for the interaction of radiation with a sphere of radius a are cross sections σi normalized to the geometrical particle cross section, σg = πa2, where i stands for extinction (i = ext), absorption (i = abs), scattering (i = sca), and backscattering. Energy conservation requires that [see Mätzler, 2002b]:

equation image
equation image

The key parameters for Mie calculations are the Mie Coefficients an and bn to compute the amplitudes of the scattered field. The index n runs from 1 to ∞, but the infinite series occurring in Mie formulas can be truncated at a maximum nmax, given by Bohren and Huffman [1983]:

equation image

This value is used in this computation. The size parameter is given by x = ka, a is the radius of the sphere and k = 2π/λ is the wave number, λ the wavelength in the ambient medium.

3. Analytical Results From Rain Data

[8] Table 1 shows the coefficients of lognormal distributions for tropical and continental showers and thunderstorms, as per Ajayi and Ofoche [1983]. Tables 2 and 3 summarize the variability of propagation coefficients due to temperature for the four sites in Botswana.

Table 1. Coefficients for Tropical and Continental Shower and Thunderstorm for Lognormal Modelsa
Type of RainNTμgσ2g
   Showers127R−0.477−0.476 + 0.221lnR0.269 − 0.043lnR
   Thunderstorms70R−0.564−0.378 + 0.224lnR0.306 − 0.059lnR
   Showers137R−0.37−0.414 + 0.234lnR0.223 − 0.034lnR
   Thunderstorms63R−0.491−0.178 + 0.195lnR0.209 − 0.030lnR
Table 2. Summary of the Variability of Propagation Coefficients With Rain Type With Temperatures of 270°K and 318°K at 1 GHz
SitesSpecific Attenuation (dB/km)TTTSCSCT
〈cos θ〉 (D)00000000
〈cos θ〉 (D)00000000
〈cos θ〉 (D)001.51.50000
〈cos θ〉 (D)00000.0010.00100
Table 3. Summary of the Variability of Propagation Coefficients With Rain Type With Temperatures of 270°K and 318°K at 300 GHz
SitesSpecific Attenuation (dB/km)TTTSCSCT
〈cos θ〉 (D)
〈cos θ〉 (D)1.451.455.
〈cos θ〉 (D)1.51.5773.
〈cos θ〉 (D)1.41.466331.21.2

[9] A requirement of the model is that it be able to predict attenuation from rain at a frequency range of 1–1000 GHz. To this extent, we have examined predictions based on the calculation by Olsen et al. [1978], using Marshal-Palmer (MP) drop size distributions at 263°K, 273°K and 293°K; and we have applied the work of Liebe et al. [1991] to obtain the frequency dependence of the complex permittivity ɛ of rain at the same temperatures. The plots of the specific attenuation, γext (in dB/km) as a function of rain intensity R (in mm/h) for two frequencies of 5 and 1000 GHz, at temperature values of 293°K, 273°K, and 263°K, are shown in Figures 1 and 2. 1Figures 1 and 2 are compared to Figure 5 of Rogers and Olsen [1976] and Figure 3 of Olsen et al. [1978]. It is seen that a very close agreement exists between the values of γext for the plots of Olsen et al. and our model, with a variation at any rain rate not exceeding about 2 dB/km at 5 GHz, reducing to about 1 dB/km at 1000 GHz. Thus as the operating frequency rises much above 5 GHz, the specific attenuation proposed by Olsen et al. tends to be more accurate for tropical application.

Figure 1.

Specific attenuation as function of rain intensity R (mm/h), using the MP drop size distribution at 263°K, 273°K, and 293°K at 5 GHz.

Figure 2.

Specific attenuation as function of rain intensity R (mm/h), using the MP drop size distribution at 263°K, 273°K, and 293°K at 1000 GHz.

[10] In Figure 3, we show the variability of the propagation coefficients with frequency and drop size distribution for four Botswana stations at rain rates of R0.01 = 86.87 mm/h, R0.01 = 68.9 mm/h, R0.01 = 64.4 mm/h, and R0.01 = 137.06 mm/h, respectively. The drop size distributions assumed in calculating the Mie coefficients are four exponential DSDs (MP, JD, JT and LP) and four lognormal DSDs (CS, TS, CT and TT). In almost all the four stations, for frequencies below 15 GHz, the upper bound of γ is due to the Tropical Showers (TS) DSD, while the lower bound is due to the JD (Joss-Drizzle) distribution. For the frequency range of 15 to 50 GHz, the upper bound of γ is due to the Weibull (WB) DSD, while the lower bound is due to the Continental Thunderstorm (CT) distribution. Finally, at frequencies between 50–1000 GHz, the upper bound of specific rain attenuation is due to JD (Joss-Drizzle) DSD, while the lower bound is still due to the CT DSD.

Figure 3.

Specific attenuation due to rain, γ, for (a) Francistown (R0.01 = 86.87 mm/h), (b) Gaborone (R0.01 = 68.9 mm/h), (c) Kasane (R0.01 = 64.4 mm/h), and (d) Selebi-Phikwe (R0.01 = 137.06 mm/h) in Botswana using the MP, JD, JT, LP, WB, CS, TS, CT and TT lognormal DSDs.

[11] Compared to Figure 5 of Olsen et al. [1978], the value of the specific attenuation due to rain for each of the four stations should reasonably fall (between the curves for 50 mm/h and 150 mm/h) in the range of about 15–30 dB/km. This range is also confirmed for another tropical region (Nigeria) in Figure 4 of Ajayi and Adimula [1996]. For frequencies above 50 GHz, one observes that the JD (Joss-Drizzle) DSD does overestimate the value of γ, as it rises above 50 dB/km while the CT (Continental Thunderstorm) distribution underestimates γ with values of 2–3 dB/km. For the frequency range of 15 to 50 GHz, Figure 5 of Olsen et al. [1978] shows that the range of γ should be about 10–25 dB/km. Again, this is overestimated by the upper bound of WB (Weibull) DSD of 20–30 dB, while it is again underestimated by the lower bound CT distribution of 2–3 dB/km.

[12] We also compare Figure 3b for Gaborone (R0.01 = 68.9 mm/h) and Figure 3c for Kasane (R0.01 = 64.4 mm/h) with the measurements of γ at 13 GHz in Figure 2 of Moupfouma and Tiffon [1982]. From Figure 3, the specific attenuation range at 13 GHz extends between 2 and 13 dB/km, at both values of R0.01. Through extrapolation of the measurements of Moupfouma and Tiffon [1982] for vertical and horizontal polarization, as well as using their proposed model, they find γ falls within the range 3–4 dB/km for the two rain rates. The closest DSD model corresponding to this attenuation range in Figure 3 is the TT (Tropical Thunderstorm) model. Again, this result is consistent with the findings in Table 4, for rain rate range 21 < R < 72 mm/h, as discussed below.

Table 4. RMS and χ2 Statistic for Various Rain DSD Modelsa
Rain Rate Range: 1 ≤ R < 21 mm/hBest Model
DSD ModelRMS Testχ2 Statistic Threshold Equal to 33.4
  • a

    At 1% significance level.

LP3.01247941.766236WB or ITU-R model for Horizontal Polarization
ITU-R model for Horizontal Polarization2.42258623.85383
ITU-R model for Vertical Polarization3.18175136.40273666
Rain Rate Range: 21 ≤ R ≤ 79 mm/hBest Model
DSD ModelRMS Testχ2 Statistic Threshold Equal to 13.3
ITU-R model for Horizontal Polarization10.0511312.03571
ITU-R model for Vertical Polarization5.5740766.768724

[13] Figure 4 shows the variations of γj, j = ext, abs, sca, b with temperature, for four types of lognormal drop size distribution, for frequencies of 1 GHz and 300 GHz. It is seen from Figure 4 that at lower frequencies, the absorption coefficient, γabs, contributes more strongly to the extinction coefficient, γext, than the scattering coefficient, γsca, for all drop size distributions. At higher frequencies, γabs and γsca are almost constant; thus making γext depend just slightly on temperature.

Figure 4.

Attenuation coefficient due to rain versus temperature for Gaborone at (a) 300 GHz with CT lognormal distribution, (b) 300 GHz with TT lognormal distribution, (c) 1 GHz with TT lognormal distribution, and (d) 1 GHz with CT lognormal distribution.

[14] Finally, the analytical model is applied in Durban where a 6.73-km terrestrial line-of-sight millimetric radio link operating at 19.5 GHz has been set up between the Howard College campus and the Westville campus [see Naicker and Mneney, 2006; Fashuyi and Afullo, 2007]. The comparison of the attenuation using raindrop size distribution models by Mie scattering with average measured rain attenuation is presented in Figure 5 according to ITU-R [2007]. ITU-R [2007] gives a simple technique that may be used for estimating the long-term statistics of rain attenuation. To determine the effective path length, deff, of the link of path length d, use is made of a reduction factor r, given by ITU-R [2007]:

equation image

The above expression is valid for R0.01 ≤ 100 mm/h. For R0.01 > 100 mm/h, use is made of the value 100 mm/h in place of R0.01. In the case of the actual path in Durban, r is 0.61 and deff is 4.1. Thus the estimated path attenuation exceeded 0.01% of the time is:

equation image

Table 4 shows the values for the χ2 statistic and the root-mean-square (RMS) error for LP, JT, JD, WB, ITU-R-Horizontal Polarization and ITU-R-Vertical Polarization models. For 17 degrees of freedom, the threshold value, tα for α = 1% significance level, is 33.4. The values of χ2 statistic for the rain rate 1 ≤ R < 21 mm/h, with the above distributions, when compared with the threshold value, show that the results for WB and ITU-R-Horizontal Polarization are the lowest and below the threshold value; and therefore, along with the RMS error test, the ITU-R-Horizontal Polarization and the WB models give the best fit at 1 ≤ R < 21 mm/h. Similarly, for the range 21 ≤ R ≤ 79 mm/h, with 4 degrees of freedom, the threshold value, tα for α = 1% significance level, is 13.3. With the same distributions, the lognormal model for tropical thunderstorm (TT) gives the best fit at 21 ≤ R ≤ 79 mm/h, with a χ2 statistic of 4.16, and an RMS error of 4.21. The curves are displayed in Figure 5, where those for the ITU-R-Horizontal Polarization model and the TT DSD model are seen to fall within the minimum and maximum bounds of the measured rain attenuation for the Howard College-Westville campus radio link, for their respective rain rates.

Figure 5.

Rain attenuation for Durban along the 6.73 km link at 19.5 GHz for the year 2004: Maximum, minimum, and average measured values versus the analytical representations using WB, TT, and ITU-R models.

4. Conclusion

[15] In this presentation, the extinction coefficient γext, is seen to depend more strongly on temperature at lower frequencies than at higher frequencies for lognormal rain drop size distribution as confirmed by Olsen et al. [1978] and Mätzler [2002b, 2002c] using exponential distributions. The variation of the specific rain attenuation, γ, for Botswana (a tropical area) at 5 GHz and 1000 GHz shows a very close agreement with the results obtained by Rogers and Olsen [1976], at three temperatures of 263°K, 273°K, and 293°K, even though the latter used the less accurate Ray model for Mie scattering parameter evaluation, while we have employed the Liebe model. Representations of γ for four different values of R0.01 in Botswana for different rain drop size distributions and varying frequency shows that while they are comparable to results obtained by Olsen et al. [1978] and Ajayi and Adimula [1996], at frequencies above 50 GHz, the Joss-Drizzle distribution overestimates the attenuation, while the Continental Thunderstorm model underestimates it. In the frequency range 15–50 GHz, the upper bound of γ is provided by the Weibull DSD, while the lower bound is due again to the CT model. In comparing these results with the measurements over a 13 GHz microwave link by Moupfouma and Tiffon [1982], it is seen that the appropriate model for Gaborone and Kasane in Botswana is the Tropical Thunderstorm. Finally, applying the model to measurements over the 6.73-km 19.5 GHz line-of-sight link in Durban, it is concluded that the TT model would again be the most suitable for Durban, for rain rates greater than 21 mm/h.