Declaration 2 Publication
6.2 Sinc Function Descriptions of Rain Spikes
The proposed description of a rain spike in this thesis is essentially an entity governed by a BD process as explained in the queue approach introduction in Chapter four. Thus, standard rain spikes are expected to have an initial time with rainfall rate usually closer to zero. This time gradually peaks up until a maximum rainfall is reached and then gradually slopes down to its initial zero position. As seen in Fig. 6-1, a typical rain spike (also an event) is seen as an initially progressing phenomenon and then later regresses once it attains its peak. The shape of rainfall service time around its peak generally appears asymmetrically normal (or Gaussian distribution) with the power spectrum gradually decreasing along its left and right lobes.
Broadly speaking, the representation assumed for generic rainfall duration requires an envelope- like function to understand the shape of time-varying rainfall spike. To this end, Alonge and Afullo [2013a] initially proposed a sinc mathematical function, modified as an envelope, in early studies of rain spike. The function is an nth-powered sinc model mimicking the shape profile of a typical rain spike and is given by:
?£) = [R(`r £)]a for ≈ 1, > 0 A` ` > 0; ∀£ ∈ℤ 6.1)
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Figure 6-1: Rainfall event at University of KwaZulu-Natal campus on 25th of April, 2009 between 16:34 hrs and 17:19 hrs
Figure 6-2: Plots of different values of n for the nth-powered sinc function for β = 0.9.
where t is a random independent variable corresponding to the time progression in minutes.
Here, we assume that the random variable varies as the time progresses in the time domain. The constant, φ, is the maximum rainfall rate in the dataset but set to unity for relative rainfall rate.
The value, n, is the power of the overall sinc function.
The power, n, is a very important parameter in determining appropriately, the sinc envelope function. Fig. 6-2 shows the different plots of integer values, n, with the sinc function. The idea is to develop the simplest shape profile with a main lobe and side-lobes to approximate the
0 20 40 60 80 100 120
0 5 10 15 20 25 30 35 40 45 50
Rainfall Rate (mm/h)
Service Time (minutes)
-3 -2 -1 0 1 2 3
-0.2 0 0.2 0.4 0.6 0.8
t
F(t) = (Sinc(t))n
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10
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rainfall process. As seen in the figure, the values of n = 1 and 2 both have main lobe and side- lobes. However, the values for which n ≥ 3 produces flattened out side lobes. These values will increase the side-lobe errors in the model when applied. Therefore, the values of 1 and 2 appear to be the appropriate choices. However, the nth-powered sinc function for the value of n = 1 generates some negative values of side lobe components. On the other hand, n = 2 generates only positive values of the side-lobes. Therefore, the most ideal value of n for nth-powered sinc function from our observation is 2. It should be noted that the inner product of (6-1) is actually applied in the case of n = 2.
6.2.1 Classification of Rain Spike by Sinc Templates
A close observation of rain events in Durban show several sub-events, which are observable spikes, all which follow the typical birth and death processes. Therefore, when rainfall durations are examined, the spike’s main lobe which is of primary interest to rainfall attenuation peaks becomes significant. However, the position of the main lobe can be centrally-skewed (with or without side lobes), right-skewed or left-skewed. The main lobes when skewed have side-lobes at adjacent positions. Therefore, four possible templates are assumed for any spike according to (6.1). These templates are described in Table 6-1 by time-bounds imposed on the proposed function. The frequency domain obtained from Fast Fourier Transform (FFT) routine and Welch PSD for these templates are also depicted on Table 6-1. They are described accordingly:
(a) Template A
This template describes a rainfall spike completely specified by a main lobe with left and right side lobes. This template imitates the rainfall for a normal duration with proper rain cell formation. The bounds of the time, t, are defined as –π ≤ t ≤ π.
(b) Template B
This template describes a rainfall spike characterized by a right-skewed main lobe with left side lobes. This template represents a spike with a sudden peak towards the end of its life time. The duration bounds are specified by –π ≤ t ≤ a2π
(c) Template C
This template represents a rainfall spike for which the spike begins with a sudden peak and some few rainfall rate samples at the end. In this case, the interval for which t exists is given by – a1π ≤ t ≤ π.
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Table 6-1: Normalized nth-powered Sinc function templates for rainfall rate
TEMPLATE
TIME
BOUNDS TIME DOMAIN
FREQUENCY DOMAIN
SPECTRAL DENSITY (PSD)
1. Rainfall duration with a
Main lobe
accompanied by right and left side- lobes.
−" ≤ £ ≤ "
2. Left-Skewed rainfall duration with right side- lobes.
−" ≤ £ ≤ A#"
3. Right-Skewed rainfall duration with left side- lobes.
−A" ≤ £ ≤ "
3. Rainfall duration with main lobe and no side- lobes.
−A" ≤ £ ≤ A#"
(d) Template D
Sometimes, a rainfall spike can be characterized by a progressive increase (and eventual decrease) in the rainfall rate during its service time. This template can work as an appropriate envelope with bounds of t defined by – a1π ≤ t ≤ a2π. In most case, the assumption a1 ≈ a2 may hold for equal side lobes.
The sinc classification is useful for profiling the characteristics shape of rain spikes. The function, however, fails to explain the probability properties involved in its birth and death.
This approach therefore fails to highlight the importance of probability theory hypothesis in rainfall queueing process. The next section will discuss the subject of probability analysis of rain spikes.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(minutes)
f(t)
0 50 100 150 200 250 300 350 400 450 500
-80 -60 -40 -20 0 20 40 60 80 100
t
Frequency Spectrum
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2
Normalized Frequency (×π rad/sample)
Power/frequency (dB/rad/sample)
0 50 100 150200 250 300350 400450 500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
f (t)
0 50 100 150 200 250300 350 400 450 500
-100 -50 0 50 100 150
Frequency Spectrum
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-18 -16 -14 -12 -10 -8 -6 -4 -2
Normalized Frequency (×π rad/sample)
Power/frequency (dB/rad/sample)
0 50 100 150200 250300 350400 450500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(t)
t
0 50 100 150 200 250 300 350 400 450 500
-100 -50 0 50 100 150
Frequency Specturm
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-18 -16 -14 -12 -10 -8 -6 -4 -2
Normalized Frequency (×π rad/sample)
Power/frequency (dB/rad/sample)
0 50 100 150200 250300 350400 450500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 50 100 150 200 250 300 350400 450 500
-150 -100 -50 0 50 100 150 200 250
Frequency Spectrum
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3
Normalized Frequency (× π rad/sample)
Power/frequency (dB/rad/sample)
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