The conversion of rain rate to specific attenuation is a key step in analyzing the attenuation of the entire path and thus the availability of the radio link. Part of the research was also submitted for publication in the SAIEE proceedings.
S' SIt
DSD, AU gamma distribution and WB Weibull distribution 158 Figure 5-12 : Path attenuation for 19.5 GHz LOS link, using MP, JT and.
Introduction
- Specific Attenuation for Rain
- Attenuation Statistics
- Formulation of the problem
- Thesis Objectives
- Outline of the dissertation
The calculation of the specific attenuation is strongly influenced by the drop size distribution (DSD). Therefore, each DSD must be normalized before it is used in specific attenuation evaluation and subsequent calculations.
Microphysical Properties of Raindrops
Introduction
- Outline of the chapter
The numerical calculation of the scattering and attenuation of electromagnetic waves by rain requires detailed knowledge of the microphysical properties of raindrops such as raindrop size, shape, fall velocity, tilt angle and drop size distribution (DSD). Several investigations by both meteorologists and radio engineers on the microphysical properties of raindrops were examined during this study.
Terminal Fall Velocity
- Gunn and Kinzer Terminal Velocity at sea level
- Terminal Velocity for arbitrary atmospheric conditions
- Best Terminal Velocity in the Standard Atmospheres
Gunn and Kinzer (1949) made extensive measurements of the ultimate fall velocity for standard atmospheric conditions at sea level. Foot and Du Toit (1969) have shown that a polynomial of Nth degree can provide an analytical fit of some degree of accuracy to the measurements of Gunn and Kinzer (1949) by simply increasing the order of N. The polynomial has the form.
Raindrop Size and Shape
All calculations will henceforth use the equivolumetric or effective diameter and is simply referred to as the droplet diameter D. In some cases it may be more convenient to use the equivolumetric or effective radius ao' where ao=0.5D.
Drop-Size Distributions
- Historical Review
- Drop-size distribution definitions and formalism
- The Laws and Parsons DSD
- Types of Rainfall
- Exponential Drop-Size Distributions
- Gamma Drop-Size Distributions
- Lognormal Drop-Size Distributions
- Weibull Drop-Size Distributions
As mentioned, Laws and Parsons (1943) made extensive measurements of the DSD variation for various rain types in Washington DC. Stratiform rain cells cover large geographic areas typically exceeding 100 km and are associated with rain rates of less than 25 mm h-I. The rain rate is mostly uniform over the entire cell for long periods, often more than 1 hour. As will be shown, the above exponential distributions are simplifications of the common gamma DSDs.
Dielectric Properties of Raindrops
- Debye Relaxation and the Empirical Model by Ray
- The Double-Debye Model by Leibe
The relative permittivity or complex permittivity £r and the complex refractive index Sar are defined as follows. In Cole and Cole (1941, cited Ray 1972) the complex permittivity of water was represented using the Debye relaxation. The dependence of the complex permittivity on temperature is shown for 4 selected temperatures T and 30 QC in Figure 2-12.
Conclusion
This will significantly reduce the computation time and future work may include the variation of both the droplet shape and the DSD. The DSD is used to extend the scattering and extinction model of a single raindrop to the model experienced during a rainstorm. Various analytical descriptions for the DSD are then explored, including the exponential, gamma, lognormal, and Weibull functions.
Mie Scattering
Introduction
- Outline of the chapter
To estimate the specific attenuation, the scattering and extinction cross sections of the raindrops must be calculated. In section 3.5 the difference between the extinction cross section estimated for spherical raindrops and the more realistic Prupacher and Pitter (1971) is shown. Vertical polarizations will have a slightly smaller specific attenuation and horizontal polarizations a slightly larger specific attenuation than calculated using Mie theory for spheres.
Assumptions and formalisms
- Scattering Amplitudes and Cross Sections
- General properties of cross sections
According to (3.1), the extinction cross section is also called the total cross section then given by Wo is the ratio of the scattering cross section to the total cross section and is given by. It is sometimes convenient to relate the different diameters to the geometric cross section Cg of the particle.
Hydrometeors
- Rayleigh Scattering
- Mie Scattering
Rayleigh scattering is only applicable when the size of the scattering particle is small compared to the wavelength. Bessel functions of the fourth kind (also called spherical Hankel functions of the second kind) h~2). The cardinal sign between the brackets indicates differentiation of the Bessel function with respect to the argument as follows.
Attenuation due to raindrops
At these frequencies the specific attenuation is dominated by the number of large raindrops (radii a >2.5 mm). The influence of the MP, LP and the Wolf DSOs on the specific attenuation is shown in Figure 3-7 (d) - (f). The calculation of the specific damping is therefore strongly influenced by the droplet size distribution (DSD).
Effect of polarization
A comparison of the extinction cross section results for the vertical and horizontal polarizations is given in Figure 3-9. The deviation is more noticeable for vertical polarization and horizontal polarizations have almost the same specific attenuation as that for spherical raindrops. Also for the higher frequencies the difference in the scattering cross section for vertical and horizontal polarizations becomes smaller as indicated for 34.8 GHz in Figure 3-9.
Conclusion
In specific attenuation calculations, the number of larger raindrops (ie, > 2.25 mm) is usually small compared to the other drop sizes, especially in the gamma, lognormal, and Weibull DSDs. Thus, variations in the calculated specific attenuation due to droplet shape are not as pronounced as those observed for changes in DSD. The accuracy of the calculated specific attenuation is strongly dependent on the chosen DSD as shown in Section 3.4.
Rain-Rate statistics
Introduction
- Outline of chapter
Tattelman et al. 1994) have devised models to estimate the I minute rain rate distribution from available climatological data. These distributions are then converted to I minute rain rate distributions using the Ajayi and Ofoche (1983) relation. A comparison of all models with the I-minute rain rate measurements in Durban is given in section 4.6.
Global rain-rate climate models
- Crane's global rain-rate climate model
- ITU-R P.837 global rain-rate model
These values correspond to the rain rate intensity that is exceeded for P% of the year. The rainfall distributions for the regions applicable to Africa are given in Table 4-2 for exceedance probabilities ranging from 0.001% to 5% of the year. Bilinear interpolation of the contours in the global map is used to predict the distribution of the rain rate at any location.
Rain-rate distributions from extreme-value statistics
- Extreme-value theory
- The Gumbel and Log-Gumbel Distributions
- Depth-duration-frequency diagram for South Africa
For the Gumbel distribution XI =HI and x=h, while for the log-Gumbel distribution XI =In(HI) and x=In(h). A chi-square goodness of fit test revealed the overall acceptability of the log-Gumbel distribution for the South African data. Analysis of the log-Gumbel standard deviations (j) showed no trend as observed for the log-Gumbel means.
4.3.4 5-minute rain-rate distributions
Obtaining I-minute rain-rate distributions
From the values in Table 4-6, the 5-minute cumulative rain rate distributions can be obtained using (4.34) for each location. Using the relation in (4.39), the 5-minute cumulative rain rate distributions were converted to I-minute distributions for each location. Using the software program released by the ITU, the I-minute cumulative rain rate distributions can be obtained for each location and are shown in Figure 4-9.
Moupfouma (1987) model
There is a marked difference for Pretoria from the ITU model, which predicts a much lower probability of exceedance. The ITU forecast between Johannesburg and Pretoria is very similar as it is based on climate and due to the interpolation technique. Using Rool's ITU forecasts for eight selected locations, cumulative distributions can be calculated using Table 4-7.
Moupfouma and Martin (1995) models
This section discusses various techniques for obtaining 1-minute rainfall amount distributions. South Africa has twelve climate zones under the Koppen system and therefore the resolution of climate-based rain rate models needs to be refined and improved to improve accuracy. Apart from the scale, the shape of the rainfall distribution also varies considerably between different climates, as pointed out by Moufouma and Martin (1995) and shown in Figure 4-20 - Figure 4-23°.
Comparison with I-minute rain-rate measurements
The ITU model, and the Moupfouma (1987) and Moupfouma (1995) models for providing 1-minute rain rate distributions, are also shown for comparison. For South Africa, the Seeber (1985) model together with the Ajayi and Ofoche (1983) relationship can also be used to obtain a I-minute rain rate distribution. The Moupfouma and Martin (1995) models for tropical/subtropical regions showed good comparative results with the rain rate measurements.
Obtaining path attenuation from surface-point rain-rate
In the absence of reliable, long-term rainfall data, the ITU-R P.837 global rain rate models can be used to provide a valuable estimate of the I-minute rain rate distribution. The only required inputs are the power law parameters for the specific attenuation and the surface point rain rate. To account for the variation of the rain rate along the propagation path, L can be replaced by an effective path lengthLelf•.
Conclusion
These climate-based rain rate models provide a useful estimate of rain rate distribution when there is a lack of meteorological data and are discussed in Section 4.2. Section 4.3 describes the procedure for obtaining 5-minute rainfall amount distributions for each location in South Africa using the depth-duration-frequency curve. The rainfall amount exceedance for other percentages of the year is also useful. Therefore, a model for evaluating these percentages using 0.01% of annual rainfall is also provided.
Path Attenuation
Introduction
- Outline of chapter
The total specific attenuation has several peaks arising from the absorption spectra of oxygen and water vapor. Although the total specific attenuation for the bands around 75, 128 and 215 GHz is relatively less, the attenuation at these bands still exceeds that of the peak absorption due to water vapor at 22 GHz. Knowledge of the specific attenuation will therefore be invaluable in determining the path lengths, operating frequency and fading margins required for communication systems.
The Line-of-sight Link
- Path Profile
- Link Calculations
It contains an adjustment to account for vegetation, buildings and the curvature of the earth. This is a scenario where these rain cells only cover part of the path length. This was the noise limit of the receiver and caused dropouts at these intervals.
Specific Attenuation at 19.5 GHz
As discussed in Chapter 3, the specific attenuation at 19.5 GHz is affected by the medium-sized raindrops. The specific attenuation using the four lognormal distributions, CS, TS, CT and TT is shown in Figure 5-10. Specific attenuation for all the lognormal DSDs lies between that of the MP and JT DSDs.
Path Attenuation for LOS link
The power law parameters K and a were obtained for each of the above OSOs and are shown in Table 5-2. Of the three exponential OSOs shown in Figure 5.12, IT has the largest path weakening, followed by MP and then lO. At other frequencies, the variations due to the OSO are quite significant and can result in changes of up to 50% in the calculated path attenuation.
Path Attenuation exceedance probabilities
This means that for approximately 53 minutes per year, one can expect the road attenuation to exceed 28.3 dB - 35.6 dB. The results for the Moupfouma (1987) model and the Moupfouma and Martin (1995) temperate and tropical/subtropical rain rate models are given in Figure 5-17 - Figure 5-19. The Moupfouma and Martin (1995) temperate and tropical/subtropical rain rate models may be better suited for the lower probabilities.
Conclusion
These cumulative rain rate distributions along with path smoothing can be used to analyze when a particular path smoothing is exceeded. With such models, the system designer can then determine the appropriate fade margin and resulting blackout period for the link. For example, with the proposed link operating at 19.5 GHz, if the de WolfDSD model is applied, along with the proposed extreme value rain rate model, the hourly excess attenuation) will be 14 dB.
Conclusion
Summary of Dissertation
First, the fall velocity of the raindrops is investigated using Gunn and Kinzer (1949) measurements of the terminal velocity at sea level, and several analytical descriptions of this are discussed in section 2.2. It is shown that a polynomial model of order 3 can provide a good approximation of the terminal velocity at sea level. The size and shape of a raindrop are required for calculations of the scattering and extinction cross sections of a single raindrop.
