Research Article Volume 2 Issue 2
^{1}Department of Civil & Environmental Engineering, University of Port Harcourt, Nigeria
^{2}Department of Civil Engineering, Rivers State University, Nigeria
^{3}Chattel Associates (Nigeria) Limited, Nigeria
Correspondence: Ify L Nwaogazie, Department of Civil & Environmental Engineering, University of Port Harcourt, Nigeria
Received: March 29, 2018  Published: April 27, 2018
Citation: Nwaogazie IF, Ologhadien I, Uba LO, et al. Hydrological data transposition by ratio approach for flood & rainfall frequency analyses for ungauged catchments. Int J Hydro. 2018;2(2):243251. DOI: 10.15406/ijh.2018.02.00076
This study involves the application of ratio method of hydrological data transposition for flood and rainfall frequency analysis for ungauged catchments. For the flood transposition, a gauging station, River Niger at Onitsha about 166.9km to an ungauged station, Taylor creek, both in Niger Delta were selected. Onitsha station has been established since 1914 with historic discharge records. For the Taylor creek there was evidence of bank erosion that prompted the early study on bank protection design options, either revetment or vertical face retaining wall. The Design element would require flood frequency analysis for determination of flood events such as 10, 50 and 100 year flood flows. Through bathymetric survey and velocity measurements rating curves were established for Taylor creek; at various depths discharges for Taylor creek and River Niger at Onitsha were computed with corresponding ratios. The overall average ratio is 9:100, that is, 9 percent of River Niger is equivalent to flow in Taylor creek. This ratio permitted 30years of records at Onitsha station to be transposed to Taylor creek. Analogous to ratio approach on stream flows, rainfall intensity duration frequency (IDF) models in Port Harcourt were transposed to Peremabiri by ratio 1:1.2. Both rainfall stations are in Niger Delta, Nigeria.
Keywords: hydrological data transposition, flood, rainfall IDF models, river Niger, Taylor creek
Transposition is a technique for relocating or transferring isohyetal pattern of storm precipitation within a region that is similar relative to terrain or environmental and meteorological (or climatological) features principal to the particular storm rainfall concerned. A review of pertinent literature shows that transposition has greatly increases the availability of data for evaluating rainfall potential for drainage. The factor which control limits to storm transposition are topography, storm isohyetal charts, weather maps, storm tracks and rainfalls of record for the type of storm under consideration and topographic charts.^{1} There are basically three steps in transposition; meteorological analysis, the determination of the transposition limits and the application of the appropriate adjustments for the change in storm location. The concept of storm transpostion has been applied in various hydrological studies. FoufoulaGeorgious^{2} developed a probability storm transposition method which systematically uses storm and basin data to estimate extreme precipitation frequencies. The author viewed this as the first step in evaluating the extreme flood probabilities required to apply many riskanalysis methodologies. The method was applied to two hypothetical catchments in Iowa.
The resultant depthexceedance probability curve was smooth, suggesting that extrapolating the curve to very rare events may be promising. Portela^{3} proposed the application of storm transposition to tackle the limitation of sufficient storm data for the development of a small hydropower sheme by getting hydrologic data from a gauge station with same meteorological characterisitc in Ireland. Also, in November 2004, the Alberta Transportation^{4} applied storm transposition concept in developing guidelines for extreme flood analysis which would solve the problem of apparent inconsistencies in estimates of Probable Maximum Flood and probabilitybase extreme floods employed in the design and evaluation of major hydraulic structure. England et al.^{5} worked on an integrated datamodeling of hydrologic hazard framework for physicllybased extreme ﬂood hazard estimation applying transposition technique. The study area for this work was the 12,000km^{2} Arkansas River watershed in Colorado. The study demonstrated that the size and location of extreme storms are critical factors in the analysis of basinaverage rainfall frequency and ﬂood peak distributions. The resultant runoff model was substantially improved by the availability and use of paleoﬂood nonexceedance data spanning the past 1000 years at critical watershed locations. Gan et al.^{6} investigated different forms of the regressional relationship between the concurrent monthly discharges of neigbouring catchments with the view to generalising th relationship for a region. This enables monthly streamflow data to be transposed from a gauged catchment to an ungauged catchment, provided the certain transfer coefficient can be estimated from the physical catchment and rainfall characteristics. This may be estimated in a number of ways for a pair of gauged – ungauged. However, errors in the individual transferred flows are high. The aim of this study is to apply a simple hydrological data transposition appraoch, a ratio method of finding the average ratios of peak flows between gauged and unguaged stations. For the unguage station, it is a matter of field measurements carried out at a period coincident with those of the gauged station. Both the guaged and ungauged stations are in the Niger Delta having similar terrain and meteological featurs. Apart from stream flows, is that of rainfall data for a gauged station which facilitated the development of rainfallintensitydurationfrequency (IDF) models and a nearby gauging station having monthly rainfall totals and no durations recorded. This is the case of Port Harcourt and Peremabiri cities in Niger –Delta. In this case, rainfall IDF model transposition is possible by ratio method.
Study area
The study area is within Niger Delta in Southern Nigeria bordering with Atlantic Ocean and it covers the River Niger gauging station at Onitsha, Taylor Creek at Koroama and Peremabiri town (Figure 1). Taylor Creek is a tributary of Nun River, also, a tributary of River Niger. The Onisha gauging station is located upstream of the highway bridge at Latitude 06º 10´ and Longitude 06º 45´ and on the left bank of the River Niger. The gauging station was established in 1914 and being operated by the Nigerian Inland Waterway Authority (NIWA). The highest of the gauging station is 25.41m above mean sea level. Taylor creek station is 166.9km from the Onitsha gauging station. The site lies between Longitude 06º 17´to 06º 21´E and Latitude 05º 01´ to 05º 15´N. The elevation is 8m above mean sea level (MSL).
Figure 1 Map showing the study area, locations of Onitsha gauging station, peremabiri and Taylor creek basin.
Data collection
Data collection consists of two aspects:
In general, no one vertical section included more than 10 percent of the total flow, thus, a minimum of ten verticals per crosssection were employed for mean discharge estimation. Velocity measurements were carried out using current meter located at 0.2, 0.6 and 0.8m depths below the water surface.
Data analyses
Crosssectional velocity &discharge computations
The mean velocities in each vertical segment were calculated according to the formula:
$\underset{}{\overset{\_\_}{V}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\scriptscriptstyle \frac{1}{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({V}_{0.2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{0.6}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{0.}{}_{8}\right)$ (1)
Where: V_{0.2}, V_{0.6} and V_{0.8} are point velocities calculated at 20%, 60% and 80% of the depth.
Data transposition for discharge or rainfall gauged to ungauged stations was facilitated by use of discharge or rainfall ratios between both stations. Temporary discharge or rainfall measurements are necessary for the ungauged stations to match the same period of time (rainfall period) for the gauged station in other to validate the computation of rainfall or discharge ratio. Two methods were adopted for discharge computation namely the mean section method and the midsection method according to Equations 2 & 3, respectively.
$Q={\displaystyle \sum {q}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \sum \stackrel{\_\_}{V}}}A\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}^{m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\frac{{\overline{v}}_{i\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{\_\_}{V}{}_{i}}{2}\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}*\text{\hspace{0.17em}}\left(\frac{{d}_{i1\text{\hspace{0.17em}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{i}}{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({b}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}\right)$ (2)
and
$Q={\displaystyle \sum {q}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \sum \stackrel{\_\_}{V}}}A\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}^{m}\text{\hspace{0.17em}}{\stackrel{\_\_}{V}}_{idi}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\frac{{b}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}}{2}\right)$ (3)
where: Q is the calculated total discharge; b_{i} is the width of the i^{th} Section; d_{i} is the depth of the i^{th} Section$\stackrel{\_\_}{V}$ ; is the mean velocity in the i^{th} vertical; m is the total number of sections; q_{i} is the calculated segment discharge; and b_{i–1}, v_{i–1}, d_{i–1} refers to the previous section. Although, both methods were employed in discharge computations for comparison purposes. The Q values obtained using the midsection was used for flood frequency analysis. The midsection values were generally higher.
Stagedischarge equation
The rating curves were developed according to BS 3680:3c^{8} & BS ISO^{9}. The stagedischarge relation was expressed by an Equation of the form:
$Q\text{}=\text{}C{h}^{\beta}$ (4)
Where: Q is the discharge, h is the gauge height and C and β are coefficients, over the whole range of discharges.
If the zero of the gauge does not coincide with zero discharge, as the case of Taylor Creek, a correction factor, “a” must be applied to h, as depicted in Equation (5).
$Q\text{}=\text{}C\text{}{\left(h\text{}+\text{}a\right)}^{\beta}$ (5)For the determination of factor, “a” three values of discharge Q_{1}, Q_{2} and Q_{3} are selected in geometric progression. i.e.
${Q}_{2}^{2}={Q}_{1}{Q}_{3}$ (6)
If the corresponding values of the gauge or depth readings from the curve are h_{1}, h_{2} and h_{3}, it is possible to verify that:
$\text{a}=\frac{{{\displaystyle h}}_{2\text{\hspace{0.17em}}\text{\hspace{0.17em}}}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{1}{h}_{3}}{{h}_{1\text{\hspace{0.17em}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{3}^{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2{h}_{2}}$ (7)
Flood frequency analysis
LogPearson Type III and Gumbel Extreme Value Type I were the basis of flood frequency analysis, to obtain the design flood for return period of 10, 50 and 100yrs.^{10}
LogPerson type III
For Log Pearson Type III parameter estimate, we have Equations (8 – 11):
$Mean=Log\overline{Q}=\frac{{\displaystyle \sum LogQ}}{n}$ (8)
Standard deviation, ${\sigma}_{LogQ}=\sqrt{\frac{{{\displaystyle \sum}}^{\text{}}{\left(LogQLog\overline{Q}\right)}^{2}}{n1}}$ (9)
Skew Coefficient, $Gs\text{}=\frac{n\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \sum {(LogQ\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{\_\_}{\mathrm{log}Q})}^{3}}}{(n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1)(n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2){({\sigma}_{LogQ})}^{3}}$ (10)
where: Q=value at any probability level is obtained as:
$Log{Q}_{T}=\overline{Q}+{K}_{T}({\sigma}_{LogQ},S)$ (11)where K_{T}=coefficient for the LogPearson Type III; $\overline{Q}$ = Mean of Logs of annual floods;
${\sigma}_{LogQ}$ =standard deviation of Logs of annual floods; and G_{s} = coefficient of skewness of logs of annual floods.
Gumbel extreme value type I (EV1)
The probability density function of Gumbel extreme value type I distribution is given by
$f\left(x\right)=exp\left\{\left(x\alpha \right)/\beta exp\left(\left(x\alpha \right)/\beta \right)\right\}/\beta $ (12)
The cumulative Density function
$F\left(x\right)exp\left\{exp\left\{\left(x\propto \right)/\beta \right\}/\beta \right\}$ (13)
Velocity measurement & discharge computation
Crosssection number eleven (CS11) was used for the discharge computation and development of stagedischarge relation for Taylor creek. The velocities and depths for the crosssection are shown in Table 1. Equations (2) and (3) were applied to Table 1 for discharge computations. Tables 2, Table 3 show the calculation procedures and the discharge values obtained. Water depths above LowLow Waters (LLWs) were varied at intervals of 0.5m, until the depth/discharge table was obtained (Table 4). Data in Table 4 were used to generate the Depthdischarge curves for Taylor Creek.
Distance from right bank(m) 
0 
5 
15 
25 
35 
45 
55 
65 
75 
85 
90 

Depth of flow(m) 
0 
3.0 
6.3 
7.8 
8.7 
10.5 
11.3 
9.0 
3.9 
2.0 
0 

Velocity, 
At 0.2 depth 
0 
0.47 
0.2 
0.2 
0.2 
0.5 
0.5 
0.5 
0.19 
0.135 
0 
At 0.6 depth 
0 
0.03 
0.43 
0.81 
0.188 
2.08 
2.08 
2.08 
0.17 
0.165 
0 

At 0.8 depth 
0 
0.011 
0.133 
0.153 
0.57 
0.57 
1.32 
1.32 
0.154 
0.153 
0 
Table 1 Velocity measurement at crosssection No.11
Distance from bank b_{i}(m) 
Depth d_{i} 
Mean velocity, 
$\frac{{b}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}}{2}$ 
Segment discharge, q_{i} 
0 
0 
0 
0 
0 
5 
3 
0.017 
7.5 
0.383 
15 
6.3 
0.254 
10 
16.002 
25 
7.8 
0.39 
10 
30.42 
35 
8.7 
0.53 
10 
46.11 
45 
10.5 
1.05 
10 
110.25 
55 
11.3 
1.3 
10 
146.9 
65 
9.0 
1.3 
10 
117 
75 
3.9 
0.17 
10 
6.63 
85 
2.0 
0.151 
10 
2.265 
90 
0 
0 
0 
0 
å 
476m^{3}/sec 
Table 2 Discharge computation by midsection method
Distance 
Depth d_{i}(m) 
Mean velocity, m/s 
$\frac{{V}_{i}{{}_{+\text{\hspace{0.17em}}1}}_{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{\_\_}{V}}_{i\text{}\text{\hspace{0.17em}}}}{2}$ 
$\frac{{d}_{i}{{}_{+\text{\hspace{0.17em}}1}}_{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{i\text{}\text{\hspace{0.17em}}}}{2}$ 
$\frac{{b}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}}{2}$ 
q_{i} 
0 
0 
0 
0.0 
0.0 
0 
0 
5 
3 
0.017 
0.0085 
1.5 
5 
0.0638 
15 
6.3 
0.254 
0.1355 
4.65 
10 
6.30075 
25 
7.8 
0.39 
0.322 
7.05 
10 
22.701 
35 
8.7 
0.53 
0.46 
8.25 
10 
37.95 
45 
10.5 
1.05 
0.79 
9.6 
10 
75.84 
55 
11.3 
1.3 
1.175 
10.9 
10 
128.075 
65 
9.0 
1.3 
1.3 
10.15 
10 
131.95 
75 
3.9 
0.17 
0.735 
6.45 
10 
47.4075 
85 
2.0 
0.151 
0.1605 
2.95 
10 
4.73475 
90 
0 
0 
0.0755 
1.0 
5 
0.3775 
å 
å 
455.4005m^{3}/s 
Table 3 Discharge computation by meansection method
Development of depthdischarge relation
Given the discharge and stage (depth) values from a typical rating curve for Taylor creek, Equation (5) is calibrated for values of C and β. However, the constant “a” is evaluated via Equation (7). In other to estimate the value of “a” of Equation (7), we select from Table 3, h_{1}=6.94 m, h_{2}=8.94m and h_{3}=11.44m. Thus
$\text{a}=\frac{{8.94}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}(6.94)(11.44)}{(6.94\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}11.44)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2(8.94)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{0.53}{0.5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1.1$The values of C and β in Equation (5), are obtained by linear regression modeling, after logarithmic linearization. Equations (14 & 15) are the resulting logarithmic normal equations,^{11‒13} & shows in Table 5.
$\sum log\text{}Q\text{}=\text{}N\text{}logC\text{}+\beta \sum \left(log\text{}h\text{}\u2013a\right)$ (14)
$\sum log\text{}Q\text{}log\left(ha\right)\text{}=\sum log\text{}\left(ha\right)\text{}log\text{}C\text{}+\beta \sum {\left[log\text{}\left(h\text{}\u2013a\right)\right]}^{2}$ (15)
Using Table 4, Equations (14) and (15) become:
$27.6865\text{}=\text{}Loc\text{}C\text{}+\text{}9.01314\beta $ (16)
$24.9966\text{}=\text{}9.01314\text{}Log\text{}C\text{}+\text{}8.18505\beta $ (17)
Solving Equations (16) and (17) simultaneously gives:
Log C = 2.126 or C = 133.66; and β = 0.70368.
Thus,
$Q\text{}=\text{}133.66\text{}{\left(h\text{}\u2013\text{}1.1\right)}^{0.70368}$ (18)
S/No. 
Mean depth, 
Midsection method, 
Meansection method, 
Q_{avg} (m^{3}/sec) 
1 
6.94 
476 
455.4 
465.7 
2 
7.44 
501.56 
480.895 
491.23 
3 
7.94 
527.16 
506.39 
516.78 
4 
8.44 
552.76 
531.885 
542.32 
5 
8.94 
578.36 
557.38 
567.87 
6 
9.44 
603.96 
582.875 
593.42 
7 
9.94 
629.56 
608.37 
618.97 
8 
10.44 
655.16 
633.865 
644.51 
9 
10.94 
680.76 
659.36 
670.06 
10 
11.44 
706.36 
684.855 
695.61 
Table 4 Summary of depthdischarge computations
S/No. 
A 
B 
C 
D 
E 
F 
H 
Q 
$Log\left(h1.1\right)$ 
Log Q 
$\begin{array}{l}logQ*log\left(h1.1\right)\hfill \end{array}$ 
${\left[Log\left(h1.1\right)\right]}^{2}$ 

1 
6.94 
465.7 
0.966413 
2.66811 
2.04487 
0.58739 
2 
7.44 
491.23 
0.8021 
2.6913 
2.15868 
0.64335 
3 
7.94 
516.78 
0.83506 
2.713306 
2.265773 
0.69732 
4 
8.44 
542.32 
0.8657 
2.73426 
2.367045 
0.74943 
5 
8.94 
567.87 
0.89432 
2.75425 
2.4632 
0.7998 
6 
9.44 
593.42 
0.92117 
2.773362 
2.55474 
0.84855 
7 
9.94 
618.97 
0.94645 
2.79167 
2.6412 
0.89577 
8 
10.44 
644.51 
0.97035 
2.80923 
2.72594 
0.94157 
9 
10.94 
670.06 
0.993 
2.826114 
2.80633 
0.98604 
10 
11.44 
695.61 
1.01452 
2.8424 
2.88367 
1.02925 
9.009083 
å27.604002 
å24.91142 
å8.17847m^{3}/s 
Table 5 Evaluation of parameters for normal equations
Discharge transposition for Taylor creek
Bank erosion for Taylor creek at Koroama was of concern that prompted a study on possible bank protection works (revetment or vertical face retaining wall). For instance, the revetment design elements would require flood frequency analysis for determination of flood events such as Q_{10}, Q_{50} and Q_{100}. Given nonexistent historic data on Taylor Creek, the need to transpose River Niger discharges with over 30 years of records was necessitated. Rating curves for both Taylor Creek and River Niger were plotted for sake of comparison and establishment of flow ratios or weighting (Figure 2). Taylor Creek discharges were obtained for 30 years by analyzing the mean ratio of its discharges with those of River Niger at corresponding depths (Table 6). On the average, 9% of River Niger discharge is equivalent to that of Taylor Creek. The flow transposition approach exemplified in this study is possible given that the gauged station (Onitsha) and ungauged station (Taylor Creek) are within Niger Delta, having similar meteorological features. The approach is simple as compared with methods proposed by FoufoulaGeorgious,^{2} Portela,^{3} & Gan et al.,^{5} It is also interesting to note that FoufoulaGeorgious^{2} proposed the probability storm transformation approach, While Gan et al.^{5} worked on regressional relationships for monthly streamflow data, for which high errors were observed.
S/No. 
Stage 
Discharge Q(m^{3}/sec) 
$\frac{{Q}_{Talor}}{{Q}_{Niger}}\times 100\%$ 

Taylor creek, at 
Niger river, at onisha, 

1 
6.94 
465.7 
2800 
16.63 
2 
7.44 
491.23 
3500 
14.04 
3 
7.94 
516.78 
4200 
12.3 
4 
8.44 
542.32 
5000 
10.85 
5 
8.94 
567.87 
5800 
9.791 
6 
9.44 
593.42 
6400 
9.272 
7 
9.94 
618.97 
6600 
9.38 
8 
10.44 
644.51 
7700 
8.37 
9 
10.94 
670.06 
7800 
7.702 
10 
11.44 
695.61 
10000 
6.96 
11 
12.2 
728.763 
11600 
6.28 
12 
12.81 
756.62 
12700 
5.96 
13 
13.42 
784.1 
14000 
5.6 
14 
14.03 
811.1 
15700 
5.2 
Mean Percentage 
8.6% 9% 
Table 6 Summary of discharge ratio computations river Niger and Taylor creek
Log Pearson type III & gumbel extreme value type I flood frequency predictions
The Taylor Creek discharges taken as 9% of River Niger discharges for 30 years period are as presented in Table 7. Equations (810) are evaluated using Table 6 for Taylor Creek, as:
$Mean\text{}=\frac{{\displaystyle \sum Log\text{Q}}}{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{}3.156859$Standard deviation ${s}_{logQ}{}^{=}{\left[\frac{{\displaystyle \sum {(Log\text{Q}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Q})}^{2}}}{n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left[\frac{0.34492}{29}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}=0.11$
Skew coefficient $\left(G\right)\text{}=\frac{n{\displaystyle \sum {(Log\text{Q}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Log\text{Q})}^{3}}}{(n1)(n2)\text{}\text{\hspace{0.17em}}({\sigma}_{\text{\hspace{0.17em}}Log}){}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{30(0.009156903\text{\hspace{0.17em}})}{29x28x{(0.11)}^{3}}=\text{}0.254$
Q 
Gumbel 
Log. pearson III 
% Difference/Error 
Q_{10}: 
1947.2 m^{3}/sec 
1971 m^{3}/sec 
1.22 
Q_{50}: 
2415.4 m^{3}/sec 
2382 m^{3}/sec 
1.38 
Q_{100}: 
2608.1 m^{3}/sec 
2468.1 m^{3}/sec 
5.4 
Table 7 Gumbel versus log Pearson type III computed values
Adopting Equation (11) for flood frequency modeling, we have skew coefficient (Gs) of 0.254, return periods (T) of 10, 50 & 100 years, the corresponding K_{T} values are for 10 year flood, K_{T} = 1.252; 50 year flood, K_{T} = 1.918 and 100 year flood; K_{T} = 2.141; Thus,
${Q}_{10}=\text{}LogQ\text{}=\text{}Log+{K}_{T}{s}_{logQ}{}_{}=\text{}3.156859\text{}+\text{}1.252\text{}\left(0.11\right)\text{}=\text{}3.294579$Thus,
Q_{10}=Antilog (3.2504155)=1971m^{3}/sec
Q_{50}=3.156859+1.918 (0.11)=2332 m^{3}/sec
Q_{100}=3.156859+2.141 (0.11)=2468.1m^{3}/sec
Gumbel extreme value type 1 parameters and predictions
For the Extreme Value Type 1 distribution, the K_{T} values were obtained as follows:
${K}_{T}=\frac{\sqrt{6}}{\Pi}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[0.5772\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ell n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[\ell n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\frac{T}{T\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}\right)\right]\right]$ (19)
Where: K_{10}=1.305; K_{50}=2.608; and K_{100}=3.1445 and
${Q}_{T}=\stackrel{\_\_}{{\displaystyle \text{Q}}}+{K}_{T}S$ (20)
where S=standard deviation, and S is calculated from data (Table 7), thus:
Q_{10} =1478.3+1.305 (359.3)=1947.2
Q_{50} =1478.31+2.608 (359.3)=2415.4
Q_{100} =1478.31+3.1445 (359.3)=2608.1
Results comparison of Gumbel and log–Pearson III
% Difference at 10 and 100 years:
$\left[\frac{Q{}_{10}\text{\hspace{0.17em}}(Gumbel)\text{\hspace{0.17em}}\text{\hspace{0.17em}}Q{}_{10}\text{\hspace{0.17em}}(PIII)}{Q{}_{10}\text{\hspace{0.17em}}(Gumbel)}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}100\%\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1947.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1971}{\text{\hspace{0.17em}}1947.2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1.22\%$ $\frac{Q{}_{100}\text{\hspace{0.17em}}(Gumbel)\text{\hspace{0.17em}}\text{\hspace{0.17em}}Q{}_{{}_{100}}\text{\hspace{0.17em}}(\mathrm{log}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Pearson)\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}100}{Q{}_{100}\text{\hspace{0.17em}}(Gumbel)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{2608.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2468.1}{2608.1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}5.4\%$Errors between the two distributions are within 10% limit, thus they are both acceptable. However, we recommend the LogPearson III values for design purposes Table 8.
Year 
Q+ 
$Q{Q}_{m}$ 
${\left(Q{Q}_{m}\right)}^{2}$ 
Log Q 
$Log\text{}QLog\text{}{Q}_{m}$ 
${\left(Log\text{}QLog\text{}{Q}_{m}\right)}^{2}$ 
${\left(Log\text{}QLog\text{}{Q}_{m}\right)}^{3}$ 
1 
1134.54 
343.77 
118178 
3.05482 
0.10204 
0.010412 
0.001062438 
2 
1444.95 
33.36 
1112.89 
3.159853 
0.002994 
8.96E06 
2.68276E08 
3 
2275.2 
796.89 
635034 
3.35702 
0.20016 
0.040064 
0.008019259 
4 
1939.59 
461.28 
212779 
3.28771 
0.130851 
0.017122 
0.002240414 
5 
2007.81 
529.5 
280370 
3.302723 
0.145863 
0.021276 
0.003103409 
6 
1621.08 
142.77 
20383.3 
3.209804 
0.052945 
0.002803 
0.000148416 
7 
1943.82 
465.51 
216700 
3.288656 
0.131797 
0.01737 
0.002289364 
8 
1827.27 
348.96 
121773 
3.261803 
0.104944 
0.011013 
0.001155757 
9 
1785.6 
307.29 
94427.1 
3.251784 
0.094925 
0.009011 
0.000855345 
10 
1640.07 
161.76 
26166.3 
3.214862 
0.058003 
0.003364 
0.000195144 
11 
1600.74 
122.43 
14989.1 
3.204321 
0.047462 
0.002253 
0.000106912 
12 
1338.12 
140.19 
19653.2 
3.126495 
0.03036 
0.000922 
2.79952E05 
13 
1108.71 
369.6 
136604 
3.044818 
0.11204 
0.012553 
0.001406481 
14 
1798.74 
320.43 
102675 
3.254968 
0.098109 
0.009625 
0.000944341 
15 
1809.99 
331.68 
110012 
3.257676 
0.100817 
0.010164 
0.00102471 
16 
1079.37 
398.94 
159153 
3.03317 
0.12369 
0.015299 
0.001892308 
17 
1418.22 
60.09 
3610.81 
3.151744 
0.00512 
2.62E05 
1.33873E07 
18 
1515.87 
37.56 
1410.75 
3.180662 
0.023803 
0.000567 
1.34859E05 
19 
1624.05 
145.74 
21240.1 
3.210599 
0.05374 
0.002888 
0.000155202 
20 
1508.22 
29.91 
894.608 
3.178465 
0.021605 
0.000467 
1.00854E05 
21 
1614.51 
136.2 
18550.4 
3.208041 
0.051182 
0.00262 
0.000134072 
22 
1342.98 
135.33 
18314.2 
3.12807 
0.02879 
0.000829 
2.38622E05 
23 
1059.39 
418.92 
175494 
3.025056 
0.1318 
0.017372 
0.002289704 
24 
910.26 
568.05 
322681 
2.959165 
0.19769 
0.039083 
0.00772643 
25 
945.45 
532.86 
283940 
2.975639 
0.18122 
0.032841 
0.005951454 
26 
1331.37 
146.94 
21591.4 
3.124299 
0.03256 
0.00106 
3.452E05 
27 
1065.69 
412.62 
170255 
3.027631 
0.12923 
0.0167 
0.002158108 
28 
1096.11 
382.2 
146077 
3.039854 
0.11701 
0.01369 
0.001601822 
29 
1605.69 
127.38 
16225.7 
3.205662 
0.048802 
0.002382 
0.000116232 
30 
955.89 
522.42 
272923 
2.980408 
0.17645 
0.031135 
0.005493822 
Sum 
44349.3 
1E11 
3743217 
94.70578 
0.34492 
0.009156903 

Mean 
1478.31 
4E13 
3.156859 

S. Dev. 
359.272 
Table 8 Flood frequency analysis for Taylor creek basin
+ Source of data, NIWA.^{7}
Developing IDF models by rainfall transposition
Rainfall Transposition between Peremabiri and Port Harcourt is possible (Figure 3). The rainfall IntensityDurationFrequency (IDF) models for Port Harcourt were transposed by multiplying with a conversion factor of 1.2 to obtain the equivalent for Peremabiri in Taylor Creek basin. The rainfall distribution of Port Harcourt and Peremabiri were assumed to be similar in this study by reason of nearness and both are in the Niger Delta with same meteorological conditions. The average of the ratios of annual rainfall amounts in Peremabiri and that of Port Harcourt yielded 1: 1.2. That is 1mm of rainfall in Port Harcourt as equivalent to 1.2mm in Peremabiri. In effect, rainfall models for Port Harcourt are multiplied by a factor of 1.2 to obtain that of Peremabiri (Table 9).
Return period 
IDF models 

Port Harcourt± 
Peremabiri/Taylor creek basin 

5.0 
$\frac{4595.1}{{\left(t+50\right)}^{1.004}}$ 
$\frac{\left(1.2\right)\left(4595\right)}{{\left(t+50\right)}^{1.004}}$

7.5 
$\frac{6696.95}{{\left(t+50\right)}^{1.048}}$ 

10 
$\frac{8273.47}{{\left(t+50\right)}^{1.072}}$ 

20.0 
$\frac{12196.28}{{\left(t+50\right)}^{1.112}}$ 
Table 9 Rainfall models for Port Harcourt
^{±}Source, Ologhadien & Nwaogazie.^{14}
Based on this study the following conclusions can be drawn:
None.
The authors of this article have declared that no conflict interests exist in the course of preparing this document.
©2018 Nwaogazie, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.