Journal of University of Shanghai for Science and Technology ISSN: 1007-6735

Developing Maximum Intensity Duration Frequency (IDF)Curve to

WolaitaSodo City, Ethiopia
TadewosAdema Amona1*Wondimu Elias Worajo2
1
Department of Hydraulic and Water Resources Engineering, WelaytaSodo University, Ethiopia,
2
Department of Hydraulic and Water Resources Engineering, WelaytaSodo University, Ethiopia
*Corresponding Author: TadewosAdema Amona,
tewodrosamona@gmail.com

Abstract:Modern integrated urban planning and development practice often includes time of concentration and
Rainfall-Intensity-Duration [IDF] to derive storm runoff which is important to design capacity of drainage system, culvert,
bridge and related hydraulic structure. Intensity-Duration-Frequency (IDF) curves describe the relationship between
rainfall intensity, duration and return period. In Ethiopia, it is not usual to use IDF as basic hydrologic tools, in the planning
and hydraulic design especially urban hydraulic related structure and often implemented based on assumptions. The aim of
this study was to develop IDF curve which can be used in planning, design of hydraulic structure, estimation of flash
flood risks and decision of protection measures at down part of town and updating rainfall characteristics for climate
change in WolaitaSodoCity. To develop IDF curve annual maximum observed precipitation of 32 years [1986-2017] from
ground located gauge in the town used. The IDF curves were established for return period of 2,5,10,25,50,100 and 200 to
rainfall duration in minutes from10- 240 minutes by disaggregating daily observed rainfall to short duration and
predicting extremes .To develop empirical equation formulated by using regression analysis fitting power expression and
area constants are C,m,e are 271.57,0.1305 and 0.717 respectively.The study result recommended engineers and decision
makers to use the IDF curve as decision support in need of related projects.
Keywords: IDF, Log-Normal,L-Moment,Disaggregation, Outlier-Testing

1. Introduction
Rainfall characteristics and numerical value that describes a given space is very important for climate change study, water
resources potential evaluation, urban and highway drainage design, road culverts, ditches and other drainage, flood plain
management, soil conservation studies, storm water management ponds and many other hydrological and hydraulic purposes.
Rainfall data are very important for climate study, water resources evaluation, drainage design according to Desa M.,
RakhechaP., 2004 and Wang J ,1987 ,environmental studies and many other purposes.It is very important to construct
economical and affordable structure to some level of risk to severe storm falling within short period of time by using rainfall-
intensity-frequency [IDF] curve. According to Wayal, A.S. and Menon, K. 2014, storm water management and the design
of traffic roads, as well as underground works able to withstand floods and HREs are commonly based onrainfall IDF curves.
Rainfall intensity [mm/hr.] is the amount of extreme or intense rain that falls over a period of time and rainfall frequency is the
probability that an extreme rainstorm giving intensive rainfall over a selected period of time will happen again, on average.In
many parts of the world, IDF has been practiced but such relationships have not been accurately constructed in many
developing countries according to Koutsoyiannis, D; Kozonis., and A. Manetas, 1998.Ethiopia is one of the developing
countries where some town even IDF not practiced to develop and infrastructure that require IDF are designed by simple
assumption and empirical equation insome area ofEthiopia.WolaitaSodo towntopographically characterized by upslopeand
flood plain flat at down part of town in whichexperiencing flash flooding from heavy rains falling over a short period of
timedue to heavy storm, impacted rapid changing of town increasing pavement significantly filling the ditches and over
flowing the roadway. According to Carter, T. and Jackson, C. R ., 2007, rapidurbanization processes in many municipalities,
replacing vegetation, which intercepts and storessizable amounts of rainfall, invariably alter the characteristics of surface
runoff. Consequently, low terrain regions (LTRs) will increase quickly in the runoff peaks within a short time after HREs
according to Chen, Y., Samuelson, H.W. and Tong, Z. 2016and Dakheel, A.A. ,2017,. Development of Intensity –
Duration-Frequency [IDF] curve is guide map to support risk prediction and design of engineering infrastructures of the town.
Therefore, the objective of this study is to develop rainfall IDF curves based on observed maximum daily rainfall series of 32
years by disaggregating into 0.25,0.5,1,2,3,4,6 and 12h with return periods of 2,5,10,25,50,100and 200 years.

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2. Objectives of the Research

1. To test the outlier of annual daily maximum and disaggregate to short duration precipitation.
2. To select probability distribution methods and predict extreme precipitation to return periods of 2,5,10,50,100 and 200
3. To develop rainfall Intensity-Duration-Frequency for predicted extreme precipitation Curve of the city.
4. To develop maximum Intensity-Duration –Frequency [IDF] equation toWolaitaSodo city
3.Study Area
WolaitaSodo City is located in SNNPR,6°46’-6°53’North latitude and 37°42’-37°56’ East longitude at a distance of 329 km
and 170 km south west of Addis Ababa and Awassa respectively. The altitude of the City ranges from 1784-2346 meters above
sea level.

Figure 1.Location map of the study area :Source: WolaitaSodo Town Municipality, 2019

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4.Methodology

Figure 2.Study methodology frameworks.

4.1 Data Collection

In order to develop IDF to WolaitaSodo City, reliable daily collected precipitation by rain gauge located , longitude of
11126.634, latitude of 758089.2 National Meteorolgy Agency of Ethiopia[NMA] of Ethiopia for year 1986-2017.The data
contained less than 5% data gaps ,and applied for research study.
4.2 Screening Annual Maximum precipitation and Outlier Testing
To develop maximum intensity duration frequency curve and equation, among years of daily series data, the largest
precipitation value [peak threshold] or maximum rainfall event for each year of record [1986-2017] refer figure [3] selected .
Annual daily maximum rainfall series was subjected to test for high and low outliers. Testing was applied by using the
methodology made by US Army Corps of Engineers washing ton, DC20314-1000 Manual on Hydrologic Frequency Analysis.
The following equation is used for detecting low and high outliers:

−
X H= X ± K N S (1)
Where
X H is low/high outlier threshold in log units
−

X Mean logarithmic of the test series

S is standard deviation of the series
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K N is outlier test value for a given sample size and level of significance

4.3 Frequency Distribution and Extreme Prediction

Hydrological variables are random such as precipitation is statically described by probability distributions. In practice the true
probability distributions of the variables in question are not known and the issue is how to select reasonable and simple
parameters to describe the parameters of interest and estimate the distribution parameters. The distribution functions most often
used when estimating hydrologic events were: Normal, 2 Parameter Log Normal, Pearson Type III, Generalized Extreme-Value,
and Gumbel Type I.A number of distributions are selected to fit the data series and their parameters must be estimated by using
a variety of methods to estimate the parameters of a statistical model. Among these are the method of moments, the maximum
likelihood method, least squares, the probability weighted moments method (PWM), maximum entropy, mixed moments
(MIX), the generalized method of moments and incomplete means method. To select frequency distribution method, L-moment,
which is the recent method that gives efficient result, was applied. Given below summarizing the statistical properties of
hydrologic data as follows:

L1 = E ( X ) (2)

Let X (i/n) is the Ithlargest observation in a sample of size n (i=1 corresponding to the largest). Then, for any distribution the
second L- moment is a description of scale based on the expected difference between two randomly selected observations.

=L2 0.5 * E[ X 1/ 2 − X 2/ 2 (3)

L-moment measures of the skewness and kurtosis

=
L3 1/ 3* E[ X1/3 − 2* X 2/3 + X 3/3 ] (4)

=
L 4 1/ 4 * E[ X1/4 − 3* X 2/4 + 3* X 3/4 − X 4/4 (5)

The L-moments are independent of units of measurement, called L-moment ratios, and are definedto the quantities, according
Hosking, J.R., 1990:

τ = L 2 / L1 ( 6)

τ 3 = L3 / L 2 (7)

τ 4 = L4 / L2 (8)

Where, τ is L-coefficient of variations (L-Cv), τ3 is L-coefficient of skewness (L-Cs)

and τ 4 4 is theL-coefficient of kurtosis (L-Ck).

Physically possible extreme precipitation for a given duration is very important parameter for hydraulic structure or water
resource projects safe life of service.Therfore to develop IDF curve of specific area extreme flow prediction by using
representative frequency distribution methods for considered return period by statically approaches as

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PTr = P +K𝝈𝝈 (9)

P =mean annual daily maximum precipitation series [1986-2917]

𝝈𝝈 = 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒐𝒐𝒐𝒐 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎

K=a frequency factor which depends upon the statically distribution of the series, number of year record and return period.

4.4 Disaggregation of Precipitation

It is convenient to divide/disaggregated daily basis rainfall to an hourly time scales rainfall data for IDF development
in the absence of hourly observed. According to Bell, F.C, 1969and Cheng-lung Chen, 1983,in the absence of
short rainfall records it is necessary to assume precipitation values for rainfall duration less than one day. More, Bell,
F.C. (1969),proposed the use of ratios between the intensity of daily rainfall and those of shorter duration.

Table1.Bell’s ratios for rainfall lasting less than one day

Duration[h] 1 2 3 4 5 6 8 12 18 24

Bell’s ratios 0.435 0.565 0.626 0.678 0.721 0.750 0.802 0.877 0.948 1.00

According to [13], the empirical reduction formula in equation (10) gives the best estimation of short-duration rainfall.

(10)
Pt = P 243 t
24
Where Pt is the required rainfall depth in mm at t-hrduration, P24 is the daily rainfall in mm
and t is the lasting duration of rainfall.

Prediction of storm rainfall according to B. Fiddes, G.A Forsgate& A.O. Grigg, 1974,used to estimate the rainfall depth to
be distributed at required duration based on a 24 hour rainfall. This method was selected to derive shorter duration extreme
precipitation.

P (b + 24 )
n
PD = * (11)
24 (b + D )n
Where: P D : Precipitation in a given duration (D) in (hr.)

P: Precipitation in 24 hours for this cases it refers daily maximum precipitation and b is constants

Based on studies of a large number of rainfall gauges in East Africa, the average values of b andn are found to be 0.3 and 0.9
respectively (range of n is 0.78 to 1.09). Assuming that the diurnal variation of these constants such as b and n is constant, the
conversion of the daily maximum rainfall into smaller hours is made.

4.5 IDF Equation Developing

To develop the IDF equation, extremerainfall for return period predicted and intensity derived for each duration by
using equation [12].The process of developing empirical equation for IDF begins by preparing reliable rainfall
intensity estimates according to Demetrius, K., Demosthenes .K., Alexandra’s .M, 1998 and this also applied to this
research study as indicated in equation[13].Intensity is depth of precipitation per duration [mm/hr.] which is either
3

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instantaneous or average over the duration of rainfall. The average intensity expressed as

I=P (12)
Td
Where; I is intensity, P rainfall depth and Td is duration of rainfall

m
I = CTr (13)
Td e
Where; I is intensity in mm / hour, C, m, and e are coefficients, Td is duration in minutes and ,Tr is
return period in years

IDF Equation coefficient Determination

The step wise producer to determine the coefficient in the above equation selected to research study presented below:

Step1: Performingsimple regression analysis intensity as independent variable or y and duration as dependent variable or x for
each Linear, Logarithmic, Power polynomial, exponential and moving average

Step2: To select the most fitting curve type, check the performance criteria orthe coefficient of determination (R2)

R2 is unit less and describes the portion of total variance in the observed as can be explained by the equation. The range is from
0[poor equation] to 1.0[perfect equation]

2
 N 

∑
(Ip − Ipm )(Io − Iom ) 
 (14)
R =  i =1
2
2 2
 N   N

  ∑ Ip − Ipm   ∑ (Io − Iom ) 

  I =1   i =1  

Step3: Based on the performance criteria, power fitting has good R2 [Perfect equation]:and written as:

Y =α*Xβ ⇒ I [independent ] = Td [ Dependent ] (15)

Step4. Perform simple power regression analysis coefficient of regression[α] of each return period determined above equation
15 independent variable and return period [Tr] of each as dependent variable.

Y = α 1 X β 1 ⇐ α [ Independent ] = Tr[ Dependent (16)

Step5.for equation (13) , Equation (15) and equation (16)

m
β1 β
I = CTr e (10) & Y = α 1 X = (13) C= α1 , m = β1 , Y = αX = (12) e= β ( 17)
Td
5. Results and Discussions
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5.1. Annual Maximum of Rainfall Data Series and Outlier Testing

Daily rainfall data series of WolaitaSodo City meteorological rain gauge station used in this study was obtained from the
national meteorological agency for a period of 32 years [1987-2017] which is adequate number of data size and less than 5%
missed data observation.
Data series was organized andmaximum rainfall was selected foreach year in excel. The average, maximum and minimum
daily historical rainfall is613mm, 334mm and 972mm respectively and value of eachyear shown infigure 3.

Figure 3.Annual MaximumDaily Rainfall [WolaitaSodo Station].

1
Table 2.Annual Maximum Daily Rainfalls.
Year Max daily PCP[mm] Year Max daily PCP[mm]
1986 500 2003 640
1987 583 2004 638
1988 426 2005 763
1989 710 2006 573
1990 733 2007 538
1991 428 2008 818
1992 438 2009 557
1993 785 2010 483
1994 523 2011 647
1995 550 2012 830
1996 630 2013 972
1997 473 2014 813
1998 632 2015 515
1999 406 2016 627
2000 792 2017 334
2001 838 Mean 580.7
2002 425 Stand Dev 146.3
Annual daily maximum rainfallseries was subjected to test for high and low outliers.
For the Log-formed series
−

X =2.77, S= 0.11; and KN=2.59 for N=32 and 10% level of significance
=
X H 2.77 ± 2.59 * 0.11
XH [Low] = 2.485 and, XH (High) =3.0549
Upper Limit of high outlier=10^3.0549=1134.749mm
Lower Limit of low outlier=10^2.485=305.5625mm,
Hence, the upper limit for high outliers is 1135mm mm and the lower limit for low outliers becomes 306 mm. The result
show that data series has no outliers and all the data series will be used for the further analysis.

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5.2Rainfall Distribution Methods and Prediction

Distribution type to observe rainfall tested by commonly applied: Gamble’s, Log Pearson, Log Normal, GEV [Generalized
extreme value distribution, exponential and uniform distribution.

Figure 4.Moment ratio diagram.

Based L-moment analysis and moment ratio diagram, Log-Normal distribution method fit and adopted to predict extreme
precipitation to return period of 2,5,10,25,50,100 and 200 years.
To predict the precipitation for required return period the variants is first transformed in to logarithmic form (base 10) and the
transformed data is then analyzed

Z = log X (18)

ZT= Z + K Z σ Z (19)

P = anti log[ZT ] (20)

Where
K z = a frequency factor which is a function of recurrence interval T and the coefficient of skew C s ,
𝜎𝜎𝜎𝜎[Standard deviation of Z variates sample]=�∑(𝑧𝑧 − 𝑧𝑧)^2/�(𝑁𝑁 − 1)
𝑁𝑁 ∑(𝑧𝑧−𝑧𝑧)3
Cs=coefficient of skew of variants𝐶𝐶𝐶𝐶 = (𝑁𝑁−1)(𝑁𝑁−2)𝜎𝜎𝜎𝜎3
𝑍𝑍=mean value of z values
N=sample size=number of years of recorded
The variation kz=f (cs, T) is given in the table according to V.T. Chow, 1998.

Table 3.Predicted Maximum Precipitation to return period.

Return Period KZ ZT PT[mm]
2 0.0247 2.773 592.627
5 0.848 2.865 733.643
10 1.264 2.912 817.196
25 1.7 2.961 914.998
50 1.975 2.992 982.619
100 2.215 3.019 1045.705
200 2.44 3.045 1108.521

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5.3. Rainfall Disaggregation

Conversion of the daily maximum rainfall into smaller hours is made based on studies of a large number of rainfall gauges in
East Africa, the average values of b andn are found to be 0.3 and 0.9 respectively .

Table 4.Short duration rainfall.

Short duration rainfall(mm) depth to return period
Duration[D]
2 5 10 25 50 100 200
1/4h 143.55 177.71 197.91 221.65 238.03 253.31 268.52
1/2h 225.60 279.28 311.03 348.34 374.07 398.10 422.00
1h 317.85 393.48 438.21 490.77 527.03 560.88 594.56
2h 403.74 499.81 556.63 623.40 669.45 712.44 755.23
4h 473.65 586.35 653.01 731.33 785.36 835.80 885.99
6h 506.94 627.55 698.90 782.73 840.56 894.54 948.26
8h 527.73 653.29 727.56 814.83 875.03 931.22 987.15
12h 553.87 685.66 763.61 855.20 918.38 977.36 1036.06
16h 570.68 706.47 786.78 881.16 946.26 1007.02 1067.50
20h 582.96 721.67 803.71 900.12 966.62 1028.69 1090.47
24h 592.60 733.60 817.00 915.00 982.60 1045.70 1108.50

5.4. Establishment ofIDF Curve

To derive IDF curve for the study area ,maximum rainfall intensity I T (mm/hr) for return period T to short duration maximum
rainfall obtained above table 3 in the form of:

IT = P / D (21)

Where D is duration in hours calculated below in table for respective return period, intensity tabulated in the table 4 below.

Table 5.Intensity to corresponding return period.

Intensity(mm/hr.) to return period
Duration[D] 2Yr 5Yr 10Yr 25Yr 50Yr 100Yr 200Yr
0.25 574.21 710.83 791.64 886.60 952.10 1013.24 1074.09
0.5 451.20 558.56 622.06 696.68 748.15 796.19 844.01
1 317.85 393.48 438.21 490.77 527.03 560.88 594.56
2 201.87 249.90 278.31 311.70 334.73 356.22 377.61
4 118.41 146.59 163.25 182.83 196.34 208.95 221.50
6 84.49 104.59 116.48 130.45 140.09 149.09 158.04
8 65.97 81.66 90.95 101.85 109.38 116.40 123.39
12 46.16 57.14 63.63 71.27 76.53 81.45 86.34
16 35.67 44.15 49.17 55.07 59.14 62.94 66.72
20 29.15 36.08 40.19 45.01 48.33 51.43 54.52
24 24.69 30.57 34.04 38.13 40.94 43.57 46.19

Regression applied to most fitting power function by plotting dependent variable or duration in table 4 and independent
variable intensity in table 4 to each return period shown figure 5-8.

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Figure 5.Regression of Intensity versus duration curve for 2 &5 years return period.

Figure 6.Regression of Intensity versus duration curve for 10 & 25 years return period.

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Figure 7.Regression of Intensity versus duration curve for 50 & 100 years return period.

Figure 8.Regression of Intensity versus duration curve for 200 years return period.

Table 6.Summary of Regression.

Return period Coefficient of Regression Exponent of Regression
2 277.63 -0.717
5 343.68 -0.717
10 382.76 -0.717
25 428.67 -0.717
50 460.34 -0.717
100 489.9 -0.717
200 519.32 -0.717
Average 414.6142857 -0.717

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Figure 9.Regression of coefficient regression versus return period.

The individual coefficient value C value increases when the return period increases and it is both dependent of return period
and Intensity of rainfall. The power coefficient of duration is constant to all return period and Intensity.
Fitting power expression of the data, equation (15) for performance criteria of to R2=0.963 which is excellent observed data
simulation and independent of return period of observed data. Therefore, Equation of the intensity Duration Frequency curve
(IDF)becomes determined constants

e = 0.717, m = 0.1305, C = 271.57

Tm T 0.1305
=I c= 271.57 (22)
De D 0.717
Where
I=Maximum Intensity of precipitation [mm/hr]
T=Return period in years
t=Duration of precipitation in minute

The performance criteria R2while compared to research finding according to Fasikaw A, TsegamlakDiriba ,2017to similar
equation or equation [1]with this research study as tabulated below table 6

Region Performace R2 Equation

Bahir Dar 0.9871

D 0.9896

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A2&A3 0.9902

B&C 0.9903

A1&A4 0.9898

Great geographical variation of equation constant parameter C of Bahir Dar 482.28 to 271.57 to WolaitaSodo City and less
performance criteria of WolaitaSodo City R2 96.3%.The other constant parameter to the same equation has little geographical
variation.To develop IDF curve of city by using derived equation [22] to time duration considered tabulated below table 7 and
figure [10] of IDF curve city.

Table 7Intensity (mm/hr.) calculated by using equation developed for given duration and return period.
Duration Intensity I(mm/hr) to Return periods
(minute) 2 5 10 25 50 75 100 200 500 1000
10 57.04 64.28 70.37 79.31 86.82 91.53 95.03 104.03 117.25 128.35
20 34.70 39.11 42.81 48.25 52.82 55.69 57.82 63.29 71.33 78.08
30 25.95 29.24 32.01 36.08 39.49 41.64 43.23 47.32 53.33 58.38
40 21.11 23.79 26.04 29.35 32.13 33.88 35.17 38.50 43.39 47.50
50 17.99 20.27 22.19 25.01 27.38 28.87 29.97 32.81 36.98 40.48
60 15.78 17.79 19.47 21.95 24.02 25.33 26.30 28.79 32.45 35.52
70 14.13 15.93 17.44 19.65 21.51 22.68 23.55 25.78 29.05 31.80
80 12.84 14.47 15.84 17.86 19.55 20.61 21.40 23.42 26.40 28.90
90 11.80 13.30 14.56 16.41 17.96 18.94 19.66 21.53 24.26 26.56
100 10.94 12.33 13.50 15.22 16.66 17.56 18.23 19.96 22.50 24.63
110 10.22 11.52 12.61 14.21 15.56 16.40 17.03 18.64 21.01 23.00
120 9.60 10.82 11.85 13.35 14.62 15.41 16.00 17.51 19.74 21.61
130 9.07 10.22 11.19 12.61 13.80 14.55 15.11 16.54 18.64 20.40
140 8.60 9.69 10.61 11.95 13.09 13.80 14.33 15.68 17.67 19.35
150 8.18 9.22 10.10 11.38 12.45 13.13 13.63 14.92 16.82 18.41
160 7.81 8.81 9.64 10.86 11.89 12.54 13.02 14.25 16.06 17.58
170 7.48 8.43 9.23 10.40 11.39 12.00 12.46 13.64 15.38 16.83
180 7.18 8.09 8.86 9.98 10.93 11.52 11.96 13.10 14.76 16.16
190 6.91 7.78 8.52 9.60 10.51 11.08 11.51 12.60 14.20 15.54
200 6.66 7.50 8.21 9.26 10.13 10.68 11.09 12.14 13.69 14.98
210 6.43 7.25 7.93 8.94 9.79 10.32 10.71 11.73 13.21 14.47
220 6.22 7.01 7.67 8.65 9.46 9.98 10.36 11.34 12.78 13.99
230 6.02 6.79 7.43 8.37 9.17 9.67 10.04 10.99 12.38 13.55
240 5.84 6.58 7.21 8.12 8.89 9.37 9.73 10.66 12.01 13.15

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Figure 10.IDF Curve developed to WolaitaSodo City

6. Conclusion
Due to little attention of precipitation data collection and knowledge of IDF curve consideration carrying out engineering
design water resource projects, different project works damaged in Ethiopia.It is professional duty of hydrologists, hydraulic
and environmental engineers need IDF curve for different water resource related engineering projects.Adequate historical
rainfall record is needed to design estimates of water resource projects. This study tried to obtain 32 year daily data of the city
and fitted to log-normal statically distribution by L-moment approach an empirical equation for maximum rainfall intensity-
duration-frequency was developed for generation of IDF curves, rainfall characteristics of WolaitaSodo city.The constants for
the empirical equation developed are c, m, e is 271.57, 0.1305 and 0.717 respectively and performance criteria adopted to
develop the equation R2 is 96.3% to capture the observed information.Therefore, established IDFcurves and equation can be
adopted for drainage works, culverts and other related structure to the city.

7. Data Availability Statement

The data used for this research study was obtained from the Ethiopian National meteorology Agency.Datasets are
available in a funder-mandated or public and it is not easily accesses by through center distribution. For this
research study case the data obtained through legal written letter from the government representative head from
WolaitaSodouniversity to National meteoroidal Agency of Ethiopia

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Climatic Regions of Argentina. Journal of Hydrologic Engineering, 9, 311-327.
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Rainfall Ratio Method (Transport and Road Research Laboratory, Department of Environment, TRRL
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• .Carter, T. and Jackson, C. R. (2007), “Vegetated roofs for storm water management at multiple spatial
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