Research Article Volume 3 Issue 1
^{1}Department of civil and Environmental Engineering University of Por1t Harcourt Nigeria
^{1}Director Cooplexity Institute Spain
^{2}Professor ESADE Business School Ramon Llull University Spain
Correspondence: Ify L Nwaogazie Department of civil and Environmental Engineering University of Port Harcourt Choba Nigeria
Received: January 16, 2019  Published: February 15, 2019
Citation: Nwaogazie IL, Sam MG. Probability and nonprobability rainfall intensitydurationfrequency modeling for portharcourt metropolis, Nigeria. Int J Hydro. 2019;3(1):6675. DOI: 10.15406/ijh.2019.03.00164
This study is all about rainfall intensity – duration  frequency (IDF) modeling based on probability and nonprobability distribution function (PDF, and nPDF). A set of sixteen year rainfall amounts and durations for Port Harcourt metropolis was adopted for the modeling. The study involved the application of the following distribution functions: Gumbel Extreme Value Type1 (Gumbel EVT1), Normal, Pearson Type3 (PT3), Log Pearson Type3(LPT3), and LogNormal (LN), respectively. And the nPDF in the form of Talbot simple quotient, power, and Sherman quotientpower models. To implement the PDF modeling it was necessary to generate frequency factors for each of the five models. This was followed by nonlinear regression analysis which involved the use of Excel Solver with optimization technique in Microsoft Excel applied to estimate the parameters of the IDF models. All the PDFIDF models were calibrated using the Sherman’s equation as general models for which the intensity value is a function of return period and rainfall duration. A comparative analysis was carried out between PDF and nPDF IDF models predicted intensities that showed a good match with observed intensities. The Normal distribution IDF model ranked the best with respect to mean squared error (MSE=92.71) and goodness of fit (R^{2}=0.970) in PDF model category, while Gumbel EVT1 model was second best (MSE=109.39, R^{2}=0.975), and showed better result on each of the specified return period (2, 5, 8 and 16 years). In all, no significant difference amongst the predicted intensities of the various IDF models (PDF and nPDF models).
Keywords: probability and nonprobability distribution functions, rainfall intensity, duration, modeling, port harcourt
Flooding incidence has caused damage to properties and loss of human lives in urban centers in Nigeria and elsewhere and have since increased tremendously.^{1,2} It is very important to have measures to help estimate rainfall intensity more accurately because prolonged and intense heavy rainfall has been known to contribute more to flooding menace. Especially now that issues of climatic change seems to make worse and create uncertainties in estimating rainfall records for flood analysis needed for planning and design of storm water management facilities. The essential instrument required for deriving the characteristics of rainfall in any catchment area is a Rainfall Intensity DurationFrequency (IDF) model. The rainfall models are developed usually from a long time rainfall records for a given weather station, and these are seriously inadequate or lacking in most developing countries.
Early scholars who have made contributions in IDF modeling include Meyer,^{3} Sherman,^{4} Bernard^{5} and Bilham.^{6} The developed equations were valid for durations between 5 and 120 minutes. Bilham^{6} equations overestimated the probabilities of high intensity rainfall in the United Kingdom. Holland^{7} updated and made easier the Bilham equation and extended the duration to 25hrs. Hershfield^{8 }provided different isohyetal maps that estimated design rain depths of various specified durations and corresponding return periods for given regions in USA. Bell^{9} generalized the formula by deriving an IDF model which enabled the computation of depthduration as an index for certain locations in USA. Chen^{10} also derived applicable IDF curves for the USA. Mathematical methods have been proposed to model extreme storm probabilities taken from the scaling properties of data observed in a station rainfall records. Bara et al.^{11} worked on method of deriving IDF curves of extreme rainfall events by applying simple scaling theory to the characteristics of the IDF of rainfall with short durations in Slovakia. On the other hand, AlHassoun^{12} developed empirical formulae for calculating rainfall intensity in the region of Riyadh. He opined that there was no significant difference in the results of the curves of the IDF models obtained in Riyadh area between the LogPearson Type3 (LPT3) and Gumbel EVT1 methods. He concluded that the semiarid climate with flat topography of the region must have accounted for the results. Elsebaie,^{13} also worked in Saudi Arabia and obtained results that were very similar to that of Riyadh study using the Gumbel EVT1 and LPT3.
In Nigeria, the development of IDF models is still in its growth path and is limited to the extent of available data. Oyebande^{14} derived rainfall IDF model for the western regions without adequate data. He applied the Gumbel EVT1 distribution to the maximum period of ten year records available to derive rainfall IDF models. Recent research works on IDF development are those of Nwaogazie & Duru^{15} for Port Harcourt city. Nwaogazie & Uba^{16} for Eket city; and Nwaogazie & Ologhadien^{17} for Southern Nigeria, Okonkwo & Mbajiorgu^{18} for South Eastern Nigeria. The IDF curves developed were in accord with IDF theory for shorter recurrence periods of 2 to 10 years. Akpan & Okoro^{19} developed IDF for Calabar; and Akpen et al.^{20} studied for Makurdi metropolis. However, a recent study published by Nwaogazie & Okonkwo^{21} on comparative analysis of four types of IDF models developed for Abakiliki showed that only the simple quotient 2parameter IDF model predicted the highest intensity values at short duration while the four models compared predicted approximately same intensities at higher durations. The aim of this study is to develop a Probability Distribution Function (PDF)IDF models or PDFIDF models for Port Harcourt metropolis using available rainfall intensities data with respect to the probability distribution functions (PDF) that best fit such observed data. This type of approach in IDF modeling had not been carried out for the study area.
Study area
The area of the study lies on latitude 4.46^{o} N and longitude 7.2^{o} E., (Figure 1). The study area is Port Harcourt metropolis and it stretches from Port Harcourt City through ObioAkpor to Ikwerre Local Government Areas in Rivers State. The area experiences heavy seasonal rainfall between March and October with a dry period that last from November to February producing occasional rainfall. The climate is influenced by two air masses namely; the South Westerly wind from the Atlantic Ocean and the North East Trade wind from the Sahara desert. The area has a flat topography with inadequate drainage facilities and its elevation varies about 3 to 15m above mean sea level.^{22}
Data collection
The recorded daily rainfall data of Port Harcourt was obtained from Nigeria Metrological Agency (NIMET) Oshodi Lagos. The rainfall data obtained provided the amount of precipitation and the duration for sixteen year period, this showed rainfall events from the year 1998 – 2013. The annual maximum rainfall amounts were sorted out into durations of 10, 20, 30, 40, 50, 60, 90 and 120 minutes for each year of the sixteen years rainfall data. Thereafter, the rainfall data were ranked in decreasing order of their magnitude before selecting the peak rainfall for each year.
Data analysis
These sorted observed annual maximum rainfall amounts were further divided by the corresponding durations in hours to obtain the rainfall intensities (mm/hr) which were ranked in descending order of magnitude as shown in Table 1. Similarly the logequivalent was computed including their statistical parameters (mean, standard deviation and coefficient of skewness). The California formula (see Equation 1) was used in computing the return periods for the nonprobability distribution function (nPDF) IDF models.
${T}_{r}=\frac{n}{m}$ (1)
Where T_{r} = return period (years), m = rank order and n = total number of observations
Probability distribution function (PDF)IDF models: The development of IDF relationship was achieved using probability distribution functions. The maximum rainfall intensity was computed for each of the commonly used probability distribution function (PDF) which is – Gumbel Extreme Value Type1 (Gumbel EVT1), Normal, Pearson Type3 (PT3), LogPearson Type3 (LPT3) and LogNormal (LN) distributions. The approximation of the magnitude of a random event such as rainfall intensity is given in Equation (2) by Chow (1951) as:
${X}_{T}=\overline{X}+{K}_{T}S$ (2)
Where $\overline{X}$ = mean, S= standard deviation of the sample and ${K}_{T}$ = frequency factor. The last two parameters are functions of the return period, T and the PDF type. The determination of the value of rainfall intensity requires the computation of the frequency factor for each PDF, the mean and standard deviation for the observed data are substituted into Equation (2) for evaluation. Thus for:
${K}_{T}=\frac{\sqrt{6}}{\pi}\left[0.5772+In\left[In\left[\frac{T}{T1}\right]\right]\right]$ (3)
Where T = return period (years)
${K}_{T}=Z=W\frac{2.515517+0.802853\text{w}+0.010328{\text{w}}^{2}}{1+1.432788\text{w}+\text{}0.189269{\text{w}}^{2\text{}}+0.001308{\text{w}}^{3}\text{}}$ (4)
Where $W=\text{}{\left[In\left(\frac{1}{{P}^{2}}\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}$ for (0 < p ≤ 0.5) (5)
$p=\frac{1}{T}$ = probability function, and z = standard normal variate
${K}_{T}=Z+({z}^{2}1)k+1/3({z}^{3}6z){k}^{2}({z}^{2}1){k}^{3}+z{k}^{4}+1/3{k}^{5}$ (6)
Where $k\text{}=\frac{{C}_{s}}{6}$ for ${C}_{s}\ne 0$ , but at ${C}_{s}=0$ , ${K}_{T}=Z$ . The ${K}_{T}$ can be seen to be dependent on the coefficient of skewness, ${C}_{s}$
$\mathrm{log}{X}_{T}=\mathrm{log}\overline{X}+{K}_{T}\mathrm{log}S$ (7)
Derivation of probability and nonprobability IDF Equations: The Sherman modified quotientpower equation given by Chow et al.^{24} and Raghumath^{25} given as Equation (8) was adopted for the derivation of the IDF model.
Other IDF equations considered in this study are those of Equations (9) and (10). We note that only Equation (8) is dimensionless (unit less).
Sherman equation, $I=\frac{c{T}^{m}}{{t}^{a}}$ (8)
Talbot equation, $I=\frac{c}{b+t}$ (9)
Power equation, $I=c{t}^{a}$ (10)
Where I=rainfall Intensity (mm/hr), t=duration (minutes) and T=return period (years); c, m, a and b are regional constants of the catchment area.
The calibrations of the IDF equations were achieved through the application of nonlinear regression analysis. This method requires the use of Excel Solver which is an optimization technique in Microsoft Excel applied to estimate the parameters of IDF models.^{26} The procedure requires feeding into the spreadsheet, PDF computed intensities as observed intensity values with their durations and return periods. Intensities corresponding to each duration and return period were calculated using the IDF equation such as Equation (8, 9 or 10) based on the assumed values of parameters of IDF equation. To obtain the values of optimum IDF parameters sum of square of deviation/error between observed intensity and predicted intensity were set to minimization using Generalized Reduced Gradient (GRG) solver as in Equation (11), thus the objective function becomes:
$Min\text{}SSE={\displaystyle \sum}_{i=1}^{n}{\left({I}_{obs}{I}_{pred}\right)}^{2}$ (11)
Where ${I}_{obs}$ = observed or computed rainfall intensity, ${I}_{pred}$ = IDF model predicted rainfall intensity
n=number of data sets and i=an index (counter)
Solving Equation (11) produces the optimum values for the constants c, m, a and b achieved through iterative process that produces the least squared error. Further, Spiegel^{27} and Zakwan^{26} provided programmable formulas to obtain the coefficient of determination (R^{2}) and the mean squared error (MSE) as in Equations (12) and (13):
${R}^{2}=\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({I}_{obs}{I}_{avg}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({I}_{obs}{I}_{pred}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({I}_{obs}{I}_{avg}\right)}^{2}}$ (12)
$MSE=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{\left({I}_{obs}{I}_{pred}\right)}^{2}$ (13)
Where ${I}_{avg}$ = average of observed or computed PDF rainfall intensity.
Similarly, the EasyFit Software was used to determine the goodness of fit statistic of the distribution models for this study on the observed rainfall intensity data. The software can be downloaded from (http://mathwave.com/easyfitdistributionfitting.html)
Results
The result of the rainfall analysis of the sorted observed rainfall records for Port Harcourt metropolis are presented in Table 1 with the log equivalent. Included in the table are the statistical parameters (mean, standard deviation and coefficient of skewness) for each of the data set. Table 2 shows the summary of the distribution factors for Gumbel EVT1 and Normal distribution factors computed. However the derived general IDF equations for each of the PDFs are shown in Table 3 alongside the corresponding goodness of fit (R^{2}) and mean squared error (MSE) values. Figures 26 are the computed rainfall intensity values for Gumbel EVT1, Normal, LogNormal, LogPearson Type3 and Pearson Type3 distributions, respectively. Figure 7 & Figure 8 are graphical plots of PDFcomputed intensities versus general PDFIDF predicted intensities for LogPearson Type3 and Gumbel EVT1 which were best and worst fit distributions of the rainfall data, respectively. Table 4 & Table 5 show the summaries of three nPDF–IDF and three selected PDF–IDF calibrated models, in addition to goodness of fit (R^{2}) and mean square errors, respectively. For sake of comparison, the results of calibrated nPDFIDF models, and PDFIDF models developed for specified return periods are shown in Figures 9  12 for 2, 5, 8 and 16 year return periods, respectively. The rainfall intensity data of Table 1 were subjected to normality test and the result confirmed not normal. Accordingly, KruskalWallis NonParametric Analysis of Variance (ANOVA) was carried out for nPDFIDF and PDFIDF models for two segments of rainfall durations, that is, 10 – 40 minutes and 50 – 120 minutes, respectively to establish any significant difference or not (Table 6).
Rank 
Duration (min) 








10 
20 
30 
40 
50 
60 
90 
120 
1 
285.0^{±} 
186.7 
140.5 
112.9 
96.1 
84.2 
70.1 
65.1 
2 
242.3 
162.3 
128.6 
105.7 
89 
80.4 
60.1 
1.4 
3 
184.4 
147.8 
116.8 
97.1 
85 
75.3 
54.9 
60.2 
4 
152.4 
123.6 
111.4 
84.9 
71.5 
74.4 
54.1 
52.6 
5 
121.8 
97.2 
96.6 
76.2 
68 
67.3 
52.7 
47.9 
6 
105.6 
85.3 
82.6 
75.9 
66.3 
65.8 
51.6 
44.5 
7 
84 
76.2 
73.6 
69.6 
64.9 
64.7 
48.9 
41 
8 
81 
70.5 
65 
63.9 
63.2 
62.8 
48.3 
40.5 
9 
72.6 
64.5 
60.5 
58.7 
58.3 
55.2 
46.2 
40 
10 
65.4 
58.6 
55.3 
52.8 
51.2 
50.2 
42 
39.5 
11 
53.4 
53 
50.7 
48.8 
47.4 
46.2 
40.7 
37.1 
12 
52.2 
50.4 
47.2 
44.6 
44.4 
41.7 
39.7 
36.6 
13 
51.6 
36.6 
42.6 
41.9 
35.3 
40.7 
33.9 
32.1 
14 
49.2 
35.7 
31 
36.2 
31.9 
39.7 
30.8 
30.2 
15 
36.6 
34.2 
26 
28.7 
26.4 
36.5 
29.5 
25.4 
16 
24.6 
30.3 
24.4 
27 
24.2 
32.5 
28.4 
23.1 
Mean 
103.883 
82.05 
72.0475 
64.046 
57.70125 
57.3425 
45.7417 
42.3188 
Standard Deviation 
75.8358 
48.81379 
37.0797 
26.623 
21.98262 
16.7075 
11.7196 
12.4556 
Coefficient of Skewness 
1.354518 
0.985528 
0.501281 
0.39404 
0.088316 
0.071289 
0.207761 
0.414771 
Table 1 Ranked Observed Annual Rainfall Intensities for different Durations
±Rainfall intensity value in mm/hr.
Probability distribution 
Frequency factor, K_{T} 




Return period (years) 




2 
5 
10 
25 
50 
100 

Gumbel EVT1 
0.16425 
0.719 
1.304 
2.044 
2.592 
3.1363 
Normal 
1.00E07 
0.841457 
1.281729 
1.751077 
2.054189 
2.326785 
Table 2 Summary of Gumbel EVT1 and Normal distribution frequency factors, K_{T} values
S/N 

IDF models 
R^{2} 
MSE 
1 
Gumbel EVT1 
$I=\frac{416.54{\text{T}}_{\text{r}}^{\text{}0.2412}}{{\text{T}}_{\text{d}}^{\text{}0.5613}}$ 
0.975 
109.39 
2 
Normal 
$I=\frac{443.67{\text{T}}_{\text{r}}^{\text{}0.175}}{{\text{T}}_{\text{d}}^{\text{}0.538}}$ 
0.97 
92.71^{±} 
3 
LogNormal 
$I=\frac{407.876{T}_{r}^{0.291}}{{T}_{d}^{0.584}}$ 
0.968 
196.76 
4 
LogPearson Type3 
$I=\frac{481.679{\text{T}}_{\text{r}}^{\text{}0.30}}{{\text{T}}_{\text{d}}^{\text{}0.654}}$ 
0.961 
256.84 
5 
Pearson Type3 
$I=\frac{479.458{T}_{r}^{0.230}}{{T}_{d}^{0.600}}$ 
0.962 
170.68 
Table 3 Summary of General Rainfall IDF Models developed for each PDF
^{±}Best model with respect to R^{2} and MSE.
Figure 5 PDFIDF curves for intensities of rainfall predicted from LogPearson Type3 IDF general model.
Figure 6 PDFIDF curves for intensities of rainfall predicted from Pearson Type3 IDF general model.
Equation types 
Equation form 
Return period 
IDF models 
Coefficient of determination (R^{2}) 
Mean square error (MSE) 
1. Talbot (Quotient) 
$I=\frac{A}{B+t}$ 
16 
$I=\frac{6091.98}{11.62+t}$ 
0.985 
71.51 
8 
$I=\frac{5946.53}{15.03+t}$ 
0.984 
52.03 

5.3 
$I=\frac{6709.54}{26.455+t}$ 
0.982 
32.24 

2 
$I=\frac{10502.19}{123.24+t}$ 
0.936 
8.77 

2. Power 
$I=A{t}^{B}$ 
16 
$I=1274.52{t}^{(0.649)}$ 
0.996 
18.28 
8 
$I=971.41{t}^{(0.601)}$ 
0.996 
12.57 

5.3 
$I=590.27{t}^{(0.4899)}$ 
0.975 
43.53 

2 
$I=138.31{t}^{(0.220)}$ 
0.864 
18.53 

3. Sherman (Quotient and Power) 
$I=\frac{C{T}_{r}^{m}}{{T}_{d}^{e}}$ 
16 
$I=\frac{1.485{T}_{r}^{2.436}}{{T}_{d}^{0.649}}$ 
0.996 
18.25 
8 
$I=\frac{1.779{T}_{r}^{3.03}}{{T}_{d}^{0.601}}$ 
0.996 
12.57 

5.3 
$I=\frac{1.338{T}_{r}^{3.65}}{{T}_{d}^{0.490}}$ 
0.975 
43.53 

2 
$I=\frac{4.558{T}_{r}^{4.923}}{{T}_{d}^{0.220}}$ 
0.865 
18.53 
Table 4 Summary of three types of nPDFIDF models derived.
S/N 
Probability distribution 
Return period 
IDF models 
Coefficient of determination (R^{2}) 
Mean square error (MSE) 
1 
Gumbel EVT1 
16 
$I=\frac{6091.98}{11.62+t}$ 
0.995 
13.32 
8 
$I=\frac{5946.53}{15.03+t}$ 
0.996 
6.52 

5.3 
$I=\frac{6709.54}{26.455+t}$ 
0.997 
3.55 

2 
$I=\frac{10502.19}{123.24+t}$ 
0.992 
1.89 

2 
LogNormal 
16 
$I=1274.52{t}^{(0.649)}$ 
0.989 
32.97 
8 
$I=971.41{t}^{(0.601)}$ 
0.991 
13.23 

5.3 
$I=590.27{t}^{(0.4899)}$ 
0.993 
5.97 

2 
$I=138.31{t}^{(0.220)}$ 
2.41 

3 
LogPearson Type3 
16 
$I=\frac{1.485{T}_{r}^{2.436}}{{T}_{d}^{0.649}}$ 
0.992 
26.45 
8 
$I=\frac{1.779{T}_{r}^{3.03}}{{T}_{d}^{0.601}}$ 
0.991 
13.89 

5.3 
$I=\frac{1.338{T}_{r}^{3.65}}{{T}_{d}^{0.490}}$ 
0.989 
9.15 

2 
$I=\frac{4.558{T}_{r}^{4.923}}{{T}_{d}^{0.220}}$ 
0.967 
4.63 
Table 5 Summary of three type of PDFIDF models derived.
Return period 
KrusKal wallis (10  40 minutes) 
KrusKal wallis (50  120 minutes) 

K (Computed) 
K (Critical) 
Pvalue 
Remark 
K (Computed) 
K (Critical) 
Pvalue 
Remark 

2 
1.48 
11.07 
0.915 
NSD^{+} 
3.61 
11.0705 
0.609 
NSD 
5 
3.23 
11.07 
0.665 
NSD 
2.03 
11.07 
0.845 
NSD 
8 
1.16 
11.07 
0.949 
NSD 
1.04 
11.07 
0.959 
NSD 
16 
1.27 
11.07 
0.938 
NSD 
1.52 
11.07 
0.911 
NSD 
Table 6 Kruskal Wallis NonParametric test result on distribution of nPDF and PDF IDF models
+NSD = No Significant Difference
Figure 9 Distribution of nPDF and PDFIDF curves for Port Harcourt metropolis for 2 year return period.
Figure 10 Distribution of nPDF and PDF IDF curves for Port Harcourt metropolis for 5 year return period.
Figure 11 Distribution of nPDF and PDF IDF curves for Port Harcourt metropolis for 8 year return period.
Discussion
Developing PDFIDF models: The purpose of this study was to derive IDF equations and develop IDF curves using both probability distribution function techniques and conventional empirical methods (nPDF models) to estimate the rainfall intensity for Port Harcourt metropolis that can serve as design tool for drainage sizing, storm management, economic and safe flood control measures. Five PDF models namely Gumbel EVT1, Normal, LogNormal, LPT3 and PT3 distributions were used for frequecncy analysis of rainfall data collected. The analysis applied Chow^{23} formula for estimation of magnitude of rainfall intensity IDF curves for which rainfall estimates decreased with increase in duration and increased with increase in return period. The results obtained from the five PDFs methods showed small differences and had good consisitency as was reported by Elsebaie.^{13} Goodness of fit test carried out to ascertain the best option (model) among the PDFs showed that all the five distributions statistic were less than the critcal value at 5% significance level. Thus, the data fitted the distributions and can be considered reliant.
Figure 12 Distribution of nPDF and PDF IDF curves for Port Harcourt metropolis for 16 year return period.
The general rainfall PDFIDF models and curves were further developed for the various PDFs. The general PDFIDF models as in Figure 7 & Figure 8 predicted lower rainfall intensities for the five PDFs at lower return periods of 2, 5 and 10years, but predicted higher intensities at higher return periods of 25, 50 and 100 years. The predicted rainfall intensities obtained from the general PDFIDF models relatively showed a good match with the computed rainfall intensity from the five PDFs. The predicted rainfall intensities plotted in arithmetic graph paper showed good closeness with some degree of parallelness for specified return periods. The PDF IDF curve results show that by specifying rainfall return period or duration, intensity value can be obtained in agreement with publication in literature.^{17,21} The goodness of fit, (R^{2}) values obtained are all within the range of 0.961 to 0.975 indicating very high fit in each of the model. The mean squared error (MSE) also varied betweeen 92.71 and 256.84. Thus, the PDFIDF model of choice should be the Normal distribution with MSE 92.71 and seconded by the Gumbel EVT1 distribution with MSE of 109.39.
Comparative analysis of PDF and nPDF IDF models: The three PDFIDF and nPDFIDF models considered for comparative analysis were calibrated by nonlinear regression analysis method for specified return periods of 2, 5, 8 and 16 years. The three nPDF–IDF model types selected were: Type 1 – Talbot simple quotient, Type 2 – power, and Type 3 – Sherman quotientpower models. While the PDFIDF models selected were: Gumbel EVT1, LogNormal and LPT3 models. The results of goodness of fit (R^{2}) for nPDF–IDF models ranged between 0.864 and 0.996, while that of PDFIDF models ranged between 0.967 and 0.997 which are clear indication that the variables have good correlationship. This relationship however are better exhibited in the PDFIDF models which showed a stronger correlationship. The R^{2} for the PDFIDF at 2 and 5 year return periods gave: Gumbel EVT1 0.992  0.997; Normal are 0.995  0.997; LogNormal are 0.9850.993; LPT3 are 0.9670.989; and PT3 are 0.9710.995, respectively. The mean squared error (MSE) results for the PDFIDF showed consistency from lower values at 2 year return period to higher values at 16 year return period. The MSE for the PDFIDF at 2 and 5 year return periods gave: Gumbel EVT1 1.893.55; Normal are 1.664.33; LogNormal are 2.415.97; LPT3 are 4.639.15; and PT3 are 5.695.85, respectively. In nPDFIDF model, the TalbotType 1 equation gave MSE value varying from 8.77 at 2 year to 32.24 at 5 year return period. The Type 2 and 3 equations gave 18.5345.53 at 2 and 5 years, each. The MSE and R^{2} values of the PDFIDF models showed that the models are better than those of nPDFIDF models.
Furthermore, the comparative study of the IDF models sought to determine whether significant difference exist in the predicted intensity distribution between the three PDF and three nPDFIDF models selected. The Kruskal Wallis nonparameter test was performed on two segments of the IDF model distribution, that is, for 10 – 40 minute and 50–120 minute durations. The results of the test shown in Table 6 indicated that the calculated Kvalues are less than the K critical at 0.05 level of significance. The Pvalues are also greater than alpha value of 0.05, therefore we accept the null hypothesis (i.e no significant difference exist). The result indicates that there are no significant differences found amongst the predicted intensities of the different IDF models (PDF & nPDF models).^{28}
Model derivation of PDF and nPDFIDF types were accomplished using nonlinear regression analysis with respect to model calibration and verification. Results of the derived IDF models are consistent with IDF theory of higher intensity occuring at lower duration and lower intensity occuring at higher duration. The intensity also increased with higher return period. The PDFIDF models predicted rainfall intensities which showed a good match with the observed rainfall intensity from the five PDFs. The Normal distribtution model ranked as the best with respect to MSE=92.71 and R^{2}=0.970 in the general PDFIDF model category, while the Gumbel EVT1 model was second best (MSE=109.39 and R^{2}=0.975) and showed better result for each of the specified return periods (2, 5, 8 and 16 years). The nPDFIDF models showed higher prediction at lower durations of 10 – 40 minutes, while the PDF–IDF models showed higher prediction at higher durations of 50 – 120 minutes. The nPDFIDF models return period are limited by the number of years of data collection, while the PDFIDF models predicted higher rainfall intensities for longer return period of 25, 50, and 100 years. There are no significant differences amongst the predicted intensities of the various IDF models compared (i.e PDF and nPDF models).
None.
The author declare that there is no conflicts of interest.
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