Determination of rainfall parameters for specific attenuation due to rain for different integration times for terrestrial line-of-sight links in South Africa.

Plagiarism

Publications

Power-law functions obtained over South Africa reveal that rainfall statistics at 30-second integration times provide more information compared to one-minute and 5-minute integration times. In addition, a comparison of precipitation size distributions (DSD) at 30-second and one-minute integration times over Durban was made to identify the temporal variability in the disdrometer measurements.

Introduction
Problem formulation and motivation
Objectives
Dissertation overview
Original contributions
Publications and journals

Development of rainfall rate conversion models for 30-second and one-minute data in South Africa. Correlation and comparison of DSD models over Durban for 30 second and one minute integration times.

Introduction

Clear air signal propagation

The magnitude of the atmosphere's effect on an electromagnetic wave depends on the transmission frequency, atmospheric pressure, temperature and water vapor concentration. Once the water vapor content exceeds 4%, condensation occurs and excess water vapor is released from the air [Ippolito, 2008].

Effects of hydrometeors on the signal propagation

In terms of water content, the empirical formula for estimating fog attenuation is expressed as [Altshuler, 1984]:.

Microstructural properties of rain

The stratiform rains tend to be long-lasting, widespread and associated with low rainfall rates. This type of rain is common in the tropics and is associated with high rainfall rates.

Rain rate measurement and cumulative distributions

Rainfall rate measurements
Rainfall rate cumulative distributions
Rainfall rate conversion Models

For a successful analysis of rainfall data, vital parameters including rainfall rate (R), total duration (T) of the propagation experiment, total time (t) of the experiment in which the rain rate R is exceeded, and t as a percentage (P ) of the total time must T is indicated, [Moupfouma, 1987]. From Ippolito (2008), figure 2.4 shows a general relationship between a parameter of interest, e.g. the rainfall percentage and the percentage of time a certain rainfall percentage is equal to or exceeded.

Figure 2.3 Rainfall rate measuring instruments (a) disdrometer (b) tipping bucket rain gauge (c) optical rain gauge

Rainfall Rate Distribution Models

Rice-Holmberg (R-H) rain rate model
Crane rain rate models
The Moupfouma I Model
The Moupfouma and Martin Model

This model constructs a rain rate distribution from the thunderstorm rain (state 1) and “other rain” (state 2). This is a closed-form probability distribution model that separately handles the contributions of volume cells and debris during the prediction of rain rate cumulative distribution functions (CDFs).

Rain Drop Size Distribution (DSD) Models

Lognormal Rainfall DSD Model
Modified Gamma Rainfall DSD Model

In their studies, Moupfouma and Tiffon (1982), Ajayi and Olsen (1985), Massambani and Morales (1988), Massambani and Rodriguez (1990) confirmed that the L-P and M-P models overestimate the number of raindrops in the small and large diameters. Regions. N(Di) = Nm(Di)μexp(−ΛDi) [m−3mm−1] (2.24) Where 𝑁(𝐷𝑖) is the raindrop size distribution, Nm is the scale parameter, μ is the shape parameter and Λ is the slope parameter and Di is the average raindrop diameter in the interval D to D+ΔD.

Attenuation due to rain

Pr= Pte−kd (2.26) where k is the attenuation coefficient for the rain cell, Pt and Pr are transmitted and received power respectively. Qt(r, λ, m) = Qs+ Qa [mm2] (2.28c) where 𝜌 is the droplet density, Qt is the attenuation cross section of the raindrop, Qs and Qa are scattering and absorption cross sections respectively.

Figure 2.6 Total path rain attenuation as a function of elevation angle and frequency [Ippolito, 2008]

Rain Fade Mitigation Techniques

A review on Rainfall DSD studies in South Africa

Chapter Summary

Introduction

Data Measurements and Processing

In addition to these two datasets, additional 5-minute integration time dataset, collected over a 10-year period, was obtained from the South African Weather Service (SAWD) for 10 locations in South Africa.

Table 3.1 Rainfall rate data measurements

Rainfall rate cumulative distributions and modelling over Durban

Determination of conversion factors for one-minute integration time
Determination of Conversion Factors for 30-second integration time
Error Analysis for Durban Models
Validation of Durban conversion Models

This further confirms that 30-second integration time data provides more information needed to estimate precipitation attenuation compared to one-minute data. From these results it can be seen that the R0.01 determined over Durban at one minute integration time is 59.5 mm/h. The use of the model in (3.3) to predict one-minute rainfall amounts from 5-minute gauge data over Durban is presented in Table 3.3.

The results of Table 3.3 show a comparison of the regression coefficients μ and λ obtained for Durban at one minute integration time with those obtained by Flavin [1982] in Australia, USA, Europe and Canada and by Ajayi and Ofoche (1984) for Ile-Ife in Nigeria. Rainfall conversion models from 5-minute and 1-minute data to 30-second data were determined using the regression additions in Figure 32 | P a g e From Table 3.4, he observed that the predicted 30-second rainfall rates are higher than the measured one-minute and 5-minute rainfall rates, especially for the 99.99% system availability requirements.

Figure 3.3 Conversion factors for one-minute integration time over Durban

Development of Rainfall conversion factors for other locations in South Africa

Cumulative Distributions for 5-minute Data
Rain rate conversion factors for other locations in South Africa

Rain rate conversion factors for one-minute conversion factors
Rain rate conversion factors for one-minute conversion factors

Rain rate distributions for predicted rainfall rates

At the lower end are Mossel Bay and Cape Town with low precipitation figures and the same chance of exceedance. Precipitation rate exceeded at 5 minute integration time for 10 locations Location Precipitation rate [mm/h] Precipitation statistics. 39 | P a g e Accordingly, Durban, classified with a Cfa climate, recorded higher rainfall rates of 55.2 mm/h compared to other locations.

Cumulative distributions at different locations in South Africa of one-minute and 30-second predicted data are presented in the figure. The observations from this table show that the precipitation rates at the 30-second integration time are higher than those at the one-minute integration time for the same exceedance probability. Overall, there is an average margin of 0.95 mm/h at 99% system availability between the one-minute and 30-second rainfall rates, considering all 10 locations.

Figure 3.6 CSIR Koppen-Geiger map based on 1985-2005 SAWS data [Conradie, 2012]

Error Analysis for Proposed Power-law conversion Models over Durban

Specific Attenuation Prediction at Ku, Ka and V bands

48 | P a g e The observations from Table 3-14 show that, for example, there is a need to allocate more slack margins for communication links over Durban than Mossel Bay for the same link length. Otherwise, the communication links over Durban should be shorter than those over Mossel Bay on the same frequency. Another observation drawn from this table shows that the predicted specific attenuation values using horizontal polarization are higher than those obtained when vertical polarization is used.

This suggests that specific attenuation at 30-second integration time requires a slightly higher margin to achieve rain fading at all these locations. It is also confirmed that specific attenuation due to rainfall increases as the frequency of operation increases, as expected. This implies that a consequent increase or decrease in rainfall also affects the availability of the system.

Figure 3-9 Specific attenuation estimates from predicted one-minute data at 30 GHz Band for (a) Horizontal polarization (b) vertical polarization

Chapter Summary

As a result, the occurrence of high rainfall in Durban will affect the performance of radio links.

Introduction

Rainfall Drop Size Distributions Over Durban

Method of Moments (MM) Parameter Estimation Technique

Lognormal DSD Input Parameters from MM Parameter Estimation
Modified Gamma DSD Method of Moments Estimation

Lognormal rainfall DSD Method of Maximum Likelihood (ML) Estimation

50 | Sheet rainfall DSD models are used in this study with method of moment (MM) and maximum likelihood estimation (ML) parameter estimation techniques. Research has indicated that the third, fourth and sixth moments are very useful in estimating distribution parameters NT, μ and  [Timothy et al., 2002; Das et al, 2010; Kozu and Nakamura , 1991 ]. The three parameters of the lognormal model in (4.13) were estimated using the method of moment parameter estimation technique which estimates input parameters of a statistical distribution by equating theoretical moments of a known distribution to actual moments of the sample data.

N(Di) = Nm(Di)μexp(−ΛDi) [m−3mm−1] (4.21) where 𝑁(𝐷𝑖) is the raindrop size distribution, Nm is the scaling parameter, μ is the shape parameter and Λ is the slope parameter and Di is the mean raindrop diameter in the interval D to D+ΔD. In finding a point estimator, the maximum likelihood estimation method selects the best parameter value that maximizes the likelihood of the observed data. For the observed data Xn = (X1, X2,…Xn), the maximum likelihood method selects the parameter that maximizes the probability of occurrence of Xn.

Measurement and Data Processing

Seasonal variations of rainfall DSD

59 | P a g e second DSDs, although lower in magnitude, vary evenly compared to the pattern at 60-second integration time. From this figure, it is observed that droplet distributions are higher at 30 sec for droplet sizes in the range 2 mm < D < 4 mm than at 60 sec integration time. The model that agrees well with the measured data at 60-sec integration time is the gamma model, especially at higher rainfall rates around 24 mm/h.

On the other hand they all underestimate the measured DSDs for spots larger than 3 mm in diameter. 62 | P a g e In figure 4.4, the DSD of precipitation for winter is presented, where the Lognormal (MM) models estimate the rain fall below 2 mm in diameter compared to other models at the integration time of 30 seconds. But in general, the gamma model best estimates the measured data at low rainfall rates, especially at the 60-second integration time.

Table 4.4 Estimation of Modified Gamma parameters using Method of Moments technique with parameter μ = 3

Annual variations of rainfall DSD

Determination of DSD Conversion between two integration times

Channel-by-channel DSD correlations at 30-second and 60-second integration times
Error analysis for the DSD conversion model

Medium to large raindrops dominate higher rainfall events and thus the channel-to-channel DSD conversions performed between these two integration times will result in small deviations at higher rainfall rates. Ci,30 = aci(Ci,60)bci (4.42) where Ci,30 and Ci,60 are the average number of drops in the ith channel at 30 second and 60 second integration times, respectively, and aci and bci are regression coefficients. A × T × Vt × dDi (4.44) where Ci is the number of rainfall in the ith channel, A is the surface area of the disdrometer, Vt is the fall velocity and dDi is the droplet diameter interval.

From these two figures, it is observed that the estimated 30-second DSD well overestimates the raindrops especially in the middle range of the average drop diameters of the disdrometer. At R = 34.5 mm/h, it is observed that the proposed model slightly overestimates channel dips above 2 mm.

Figure 4.7 Fitting of coefficient a against drop mean diameter, D

Error Analysis for Seasonal and Annual DSD models

Chapter Summary

Introduction
Extinction Cross Section

Complex Refractive Index of water
Computation of the Extinction of Cross Section

Estimation of specific Attenuation over Durban

Estimation of specific attenuation from seasonal data
Estimation of specific attenuation from annual data over Durban

Error Analysis on Annual Models
Chapter Summary

For the summer season, it is observed that the specific attenuation estimates from the measured DSD precipitation are higher for the 30-s integration time compared to their 60-s counterparts for the same frequencies, except at 10 GHz. The estimates of specific attenuation in this subsection are realized using the expression given in (5.7) and DSD precipitation data measured and modeled at 30-s integration time and 60-s integration time. Furthermore, the influence of the integration time on the specific attenuation estimate is also supported here.

For example, it is observed that the specific attenuation estimates from the measured DSDs are higher at 30 second integration time compared to their counterparts at 60 second integration time and especially for frequencies above 100 GHz, as shown in Figure 5.6. Also, it is observed that both ITU-R models maintain an advantage in the overestimation of specific attenuation at both integration times, with higher margins at frequencies below 150 GHz. 90 | P a g e Table 5.5 Analysis of errors in specific attenuation due to seasonal variations. sec) Seasons Lognormal Gamma model.

Figure 5.1 Specific attenuation estimation for Summer Season at (a) 30-second and (b) 60-second integration times

Conclusion

Suggestions for Future Work

Afullo, 2014: "Microstructural Analysis of Precipitation for Microwave Link Networks: Comparison in Equatorial and Subtropical Africa", Progress in Electromagnetic Research B, Vol. A., 2011: “Correlation of rain size distribution with rain rate derived from disdrometers and rain gauge networks in Southern Africa, Msc. Hufford (1989), "Millimeter Wave Attenuation and Rate of Delay Due to Fog/Cloud Conditions", IEEE Transactions on Antennas and Propagation, Vol.37, No. -Communication system planning in a tropical country: Nigeria, IEEE Antennas and Propagation Magazine, Vol.51, No.

A., 2011: "Derivation of one-minute rain rate from five-minute equivalent for calculating rain attenuation in South Africa," PIERS Online, Vol.7, No.6, pp. 524-535. Kottek, 2010: "Observed and projected climate change 1901-2100 depicted by world maps of the Köppen-Geiger classification," Meteorologische Zeitschrift, Vol. 1986), "Influence of the Raingauge Integration Time on measured Rainfall-Intensity Distribution Functions", Journal of Atmospheric and Oceanic Technology, Vol. Choo, 2002: “Raindrop size distribution using moment methods for terrestrial and satellite communication applications in Singapore,” IEEE Transactions on Antennas and Propagation, Vol. 1983): 'Natural variation in the analytical form of the raindrop size distribution', J. of Climate and Applied Meteor., vol.