Declaration 2 Publication
4.5 Measurement and Data Processing
The measurements for this study were undertaken at the city of Durban, KwaZulu-Natal province, South Africa from electronic logs generated by the Joss Waldvogel (JW) distrometer RD-80 series, between January 2009 and December 2010. Details of this installed equipment in Durban have been discussed in subsection 3.3 of chapter 3.
The instrument has a rain rate sampling time of one minute (or 60 seconds) with a sampling error of ±5%. During the period of measurement, a few outages occurred but this is assumed to have little significance on the overall collected data for this work. As a precaution, only rainfall
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
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events with a maximum rainfall rate greater than 3 mm/h were considered in this work. This is because the overall contributions of rain droplets to rain attenuation at this 3 mm/hr threshold are often minimal below 300 GHz operating frequency. Also, the effects of dead-time errors were assumed to be minimal in the data processing, as other rainfall microstructures such as rainfall drop and radar reflectivity, were of little concern.
For processing the measurements, a regime demarcation based on maximum rainfall rates observed per rain event is proposed for this study. This is based on the four rainfall regime classification already undertaken in a number of studies on Durban [Afullo, 2011; Alonge and Afullo, 2012c], and also other areas [Adimula and Ajayi, 1996; Mandeep and Allnut, 2007]. The bounds considered for each designated rainfall regime are drizzle (? < 5 mm/h), widespread (5 mm/h ≤ ? < 10 mm/h), shower (10 mm/h ≤ ? < 40 mm/h) and thunderstorm (? > 40 mm/h) [Alonge and Afullo, 2012c]. Firstly, it is important to examine the variation of the service times and inter-arrival times, based on the maximum rainfall rate observed in an event. This also provides statistics on the typical Markov metrics, as well as the frequencies of single event occurrences, for the examined events. We expect that these properties differ with different regimes, as are their rainfall microstructural properties.
Figures 4-5(a) – (d) depict the procedural steps taken to process the collected data. A three- stage method is applied on the data namely: isolation, segmentation and identification. Firstly, rain data series in a distinct and singular rain event A is isolated from its composite data of two independent rain events A and B as seen in Fig 4-5(a) and 4-5(b). Thereafter, event A is segmented into service time bounds distinguishable by rain spike peaks as seen in Fig 4-5(c).
As observed, event A consists of a finite number of rain spikes with a total of five distinct segments from 1-5. Thus, at segmentation, the theory of the BD Markovian theory is applied to locate the possible points of a dying spike, which also coincides with re-emergence of another spike. In most cases, these points occur far above 1 mm/h as seen for segments 4 and 5 in Fig.
4-5d, which obviously indicates the existence of an overlap between the two spikes. Since two spikes are overlapping (see Fig 4-5d), there is need to determine the ‘ground-zero’ threshold for the commencement and end tails. An extrapolation technique is undertaken to determine these thresholds (corresponding to 0.003 mm/h), for which the Newton’s Divided Difference (NDD) interpolation function is adopted. The NDD function with rain rate, r in mm/h, and time, t, in minutes, is given from [Abramowitz and Stegun, 1972] as:
∅a2) = 2 − 2+)
a +d
, 4.16A)
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Afric
Figure 4-5: The three-stage procedure for processing rainfall time series data from RD Distrometer
(a) Composite Rain Events (b) Isolation of Single Events
(c) Segmentation of Segment A into Fractal Spikes (d) Identifying some points of Event Overlaps (in red dott
0 5 10 15 20 25
0
Rainfall Rate (mm/h)
0 5 10 15 20 25
0
Rainfall Rate (mm/h)
0 5 10 15 20 25
0
Rainfall Rate (mm/h)
0 1 2 3 4 5 6 7 8
0
Rainfall Rate (mm/h)
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Afric
65 (a)
(b)
(c)
(d)
stage procedure for processing rainfall time series data from RD ) Composite Rain Events
) Isolation of Single Events
) Segmentation of Segment A into Fractal Spikes
) Identifying some points of Event Overlaps (in red dotted line)
100 200 300 400 500
Duration (minutes)
10 20 30 40 50 60
Duration (minutes)
10 20 30 40 50 60
Duration (minutes)
5 10 15 20 25 30
Duration (minutes) A
5 4
2
1 3 4 5
A B
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
stage procedure for processing rainfall time series data from RD-80
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
66
£2) = £\+ 3 ∅+Y2)[2\, 2, … , 2+] + Êa 4.16p)
a +d
for which the remainder Hn of (7b) is given as,
Êa2) = ∅a2)£az)Ë)
` + 1)! 4.17)
where 2+ is the rainfall rate index from data and £ refers to the indices of time along the time series axis. A MATLAB© code was written to separately calculate these thresholds from our data samples with an assumption of 0.003 mm/h as near-zero rain rate threshold. A depiction of the identified predicted terminal points for each spike as shown in Fig 4.5(d), with the trails for each spike, in red dotted line.
Table 4-1: Summary of BD parameters obtained from measurement for different rainfall regimes in Durban
REGIME/CLASS TIME BOUNDS (minutes)
OCCURRENCE NUMBER SERVICE
TIME
INTER- ARRIVAL
TIME
OVERLAPPING TIME
DRIZZLE
0 < t ≤ 10 61 108 124
10 < t ≤ 20 66 17 3
20 < t ≤ 30 15 2 0
30 < t ≤ 40 5 0 0
TOTAL 147 127 127
WIDESPREAD
0 < t ≤ 10 56 125 145
10 < t ≤ 20 91 31 11
20 < t ≤ 30 26 2 1
30 < t ≤ 40 2 0 0
40 < t ≤ 50 1 0 0
TOTAL 176 158 157
SHOWER
0 < t ≤ 10 35 101 108
10 < t ≤ 20 61 20 13
20 < t ≤ 30 22 2 3
30 < t ≤ 40 4 0 1
40 < t ≤ 50 5 2 0
TOTAL 127 125 125
THUNDERSTORM
0 < t ≤ 10 21 58 79
10 < t ≤ 20 40 19 8
20 < t ≤ 30 20 7 3
30 < t ≤ 40 11 2 0
40 < t ≤ 50 10 4 0
TOTAL 102 90 90
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The summary of the data processed from the distrometer measurements are provided in Table 4- 1. For the service time, over 90% of the available data for all regimes have spike service times less than 30 minutes. However, thunderstorm spikes have less than 22% of its service times longer than 30 minutes. For the inter-arrival time and overlap time, we observe that over 90% of the data is within the 30 minutes domain. Although, we can see that thunderstorm spikes have some samples with inter-arrival times greater than 30 minutes. Generally, it is seen that all the queueing parameters appear to have uniform periods of 30 minutes, with about 90% data conformity. Also, it is observed that the queueing parameters for all regimes tend to decrease as the time bounds increases. For the service time data, we observe that the data peaks at time bounds between 10 minutes ≤ t ≤ 20 minutes for each regime.