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IDF Equations for Similar Rainfall Regions in Ethiopia

Fasikaw Atanaw Zimale 1, Tsegamlak Diriba Beyene2

1 Bahir Dar University, Bahir Dar Institute of Technology, Faculty of Civil and Water Resources Engineering, Bahir

Dar, P.O.Box 26, Ethiopia, Fasikaw@gmail.com

2 Bahir Dar University, Bahir Dar Institute of Technology, Faculty of Civil and Water Resources Engineering, Bahir

Dar, P.O.Box 26, Ethiopia, jipji.tse@gmail.com

ABSTRACT — Water resource engineering projects use rainfall events relating depth to a specified duration and frequency of rain fall for purposes of either planning, design or operation. One of widely known tools for such purpose is the IDF-curve. This paper uses readily available IDF curves and their associated data found in the drainage manual of the Ethiopian Roads Authority. The curves are developed classifying Ethiopia into main five hydrological regions with similar rainfall patterns. But, this curves have limitations when it comes to their practical implementation in the field of design. The main draw backs of such curves is finding intensity for any required frequency not on the curve and inducing errors while reading the curves. However these curves can be presented in an empirical equation form to ease computations of floods using computer programs and help to teach engineering hydrology with equations developed for the country. The available data were intensity of rainfall with the respective duration and frequency for all the five regions. Least square method is applied to determine the parameters of the empirical IDF equations. The results obtained showed a good match between the rainfall intensity curve and the values from the equations. The results showed that in all the hydrological regions data fitted with the IDF equations have correlations of coefficient (R2) greater

than 0.99 and NSE of 0.99. The developed equations will be useful in designing minor drainage structures using the well-known rational formula, which requires an estimate of rainfall intensity for a duration equal to the time of concentration of a catchment.

Key words- Ethiopia, IDF, ERA, Equation I. INTRODUCTION

Design of hydraulic structures for conveyance of water requires to know the magnitude of the incoming peak flood in a certain return period. This is mainly conducted to determine the discharging capacity of storm water and sewage systems, drainage systems like channels, culverts and in the design of bridges. The sole purpose is to prevent flooding and associated loss of life and property. One of the first steps in many hydrologic design projects, such as in road drainage design, is the determination of the rainfall event or events that cause peak floods. The most common approach is to use a design storm or event that involves a relationship between rainfall intensity, duration, and the frequency appropriate for the structure to be designed [1]. Design storms based on rainfall intensity-duration-frequency(IDF) curves provide a convenient alternative to continuous rainfall-runoff modeling and provide consistent standards for the evaluation of design alternatives[2]

Rainfall intensity–duration–frequency (IDF) curves are graphical representations of the amount of rainfall that falls within a given period for a specified frequency in a catchment[3]. From such graphs intensity for a known duration and frequency can be read or estimation of the return period for an observed rainfall intensity can also be known.

The establishment of IDF relationships dates back as early as 1931[4] [5]. After that, many scholars have discovered and implemented multiple forms of approaches for different watersheds [1]. However, as stated by Koutsoyiannis [6] such relationships have not been accurately constructed in many developing countries.

One of the few studies compiled on this subject matter in Ethiopia was carried out by the Ethiopian Roads Authority (ERA) in the drainage manual [7], which identified IDF-curves for all regions of the country. The curves were developed for five major hydrological regions. Similar regions of rainfall pattern are identified with analysis of monthly rainfall data employing some statistical techniques.

In many parts of the world IDF curves are developed for less than 1 hour as short duration [8-10] , 1 to 24 hours as intermediate duration[11] and 1 to 10 days as long duration [5, 12] intensities. To develop IDF curves covering the whole country rainfall depths for all ranges of durations should be available. However in case of Ethiopia short duration data is very scarce and the ERA manual has used disaggregation of the 24 hour rainfall with different methods and checked the validity with some available short duration rainfalls[7]. The manual is a highly refereed material for the design of hydraulic structures. Especially the IDF curves are the most used tools in the drainage manual for many engineering applications. Especially in the design of long

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2 distance roads where there exists many drainage structures, frequent reading of the curves is mandatory. Therefore the main purpose of this paper is to develop a single IDF empirical formula for each region based on the existing IDF curves from the ERA manual. The user is required to plug in the duration and the return period and finds the intensity easily without reading any curve. The developed equation is intended to reduce the error that could be induced in reading the curves and to facilitate automation of flood computations using programming and thereby reduce time required to design structures.The absence of an IDF equation for the Ethiopian regions reduces teaching Engineering Hydrology with practical cases of application.

II. DATA AND METHODOLGY

A. Data

In this paper the data available in the ERA manual that is used to develop the IDF curves were directly used. The rainfall intensities (mm/hr) for durations of 5,10,20,30,90 and 120 minutes with the corresponding return periods of 2, 5,10,25,50 and 100 years are tabulated in the manual at page 5-42[7]. These data are available for the five major regions indicated in fig .1 such as Bair Dar - Lake Tana, D, B & C, A1 & A4 and A2 & A3. Data collected all over the country from 31 meteorological stations is used to develop the existing IDF curves and to classify the country into similar rainfall pattern sub regions.

B. IDF empirical equations

The process of developing empirical equation for IDF begins by preparing reliable rainfall intensity estimates[13]. Since we started on the existing processed data, frequency analysis for finding the extreme rainfall intensity for each duration and frequency is already performed.

Intensity is the time rate of precipitation, that is, depth per unit time (mm/h or in/h). It can be either the instantaneous intensity or the average intensity over the duration of rainfall. The average intensity is commonly used and can be expressed as;

𝑖𝑖 =𝑇𝑇𝑝𝑝

𝑑𝑑 (1) Where; i is intensity, P rainfall depth and Td is duration of rainfall

IDF empirical equations represent a relationship between maximum rainfall intensity as a dependent variable rainfall duration and frequency as independent variables. There are several used formulae relating those variables mentioned in the literature ([3, 6, 14-17]).

However in this paper we consider equations that express intensity as dependent variable, duration and return period as an independent variable[5, 11, 18] :

𝑖𝑖 =𝐶𝐶𝑇𝑇𝑟𝑟𝑚𝑚

𝑇𝑇𝑑𝑑𝑓𝑓 (2) Where; i is intensity in mm / hour, C, m, and f are coefficients, Td is duration in minutes and ,Tr is return period in years.

The other form of IDF empirical equation which has an extra one parameter added with the purpose of improving the estimation of rainfall intensity is:

𝑖𝑖 = 𝐶𝐶𝑇𝑇𝑟𝑟𝑚𝑚

(𝑇𝑇𝑑𝑑+𝑏𝑏)𝑓𝑓 ሺ͵ሻ

Where; i is intensity in mm/hr , m, b and f are constants and Td is duration in min.

C. Determining coefficients of the IDF equations

The step by step procedure of determining the coefficients in the above two equations is presented below.

Equation (2) is transformed to a linear form by using logarithmic and power-law properties.

Taking logarithm on both sides of the equation and rearranging;

Log I = Log C + m. Log Tr – f. Log Td (4) 𝑌𝑌 = 𝑏𝑏 + 𝑎𝑎𝑎𝑎 (5) 𝑏𝑏 =log𝐶𝐶 + 𝑚𝑚.log𝑇𝑇𝑟𝑟 (6) Where , y = log I, and x = log Td

Fig. 1: Rainfall regions (Source: ERA drainage manual)

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3 Calculate the logarithm of existing intensity and the corresponding logarithm of rainfall durations (Td).

Step 1

Find the slope and intercept of the logarithm of existing intensity versus the logarithm of duration for each return period. The average of the slopes of all the return periods is the value of f in eq.(4).

Step 2

Calculate the slope and intercept of the logarithm of the return periods versus the values of the intercepts calculated above in step (1)

Step 3

The slope calculated above in step (2) is the value of the coefficient m and the anti-logarithm of the intercept is equal to the value of coefficient C in eq. (6)

Equation (3) is similar to equation (2) except for the additional parameter (b) in the numerator. Equation (3) is similarly converted to a linear form as follows

Log I = Log C + m. Log Tr – f. Log (Td + b) (7) A similar procedure as in step (I) is performed but we repeat the regression analysis for different values of “b” till the coefficient of determination is maximized.

D. Performance of the IDF equations

To assess the goodness of fits, four different criteria have been considered: (i) the coefficient of determination (R2), (ii) the Nash–Sutcliffe efficiency (NSE), iii) the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE).

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

𝑅𝑅2= ( ∑𝑁𝑁𝑖𝑖=1(𝐼𝐼𝑝𝑝−𝐼𝐼𝑃𝑃𝑃𝑃)(𝐼𝐼𝑜𝑜−𝐼𝐼𝑜𝑜𝑃𝑃)

√∑𝑁𝑁𝑖𝑖=1(𝐼𝐼𝑝𝑝−𝐼𝐼𝑃𝑃𝑃𝑃)2√∑𝑁𝑁𝑖𝑖=1(𝐼𝐼𝑜𝑜−𝐼𝐼𝑜𝑜𝑃𝑃)2

) 2

(8) NSE (unit less) is widely used to evaluate the performance of models. It measures how well the simulated results predict the measured data. Values for NSE range from negative infinity (poor model) to 1 (perfect model). A value of 0 means, that the model predictions are just as accurate as measured data (minimally acceptable performance).

NSE = 1 −∑Tt=1(Io−Im)2

∑Tt=1(Io−I̅ )o2 (9)

MAE is an alternative measure of error that addresses the problem of RMSE which overestimates by removing the squared term

MAE =n1∑ |Ini=1 o− Ip| (10) RMSE is in the same units as the target value and tell us about what the error for predictions made by the model will be. Due to the inclusion of the squared term, the root mean squared error tends to overestimate error slightly as it overemphasizes individual large errors.

RMSE = √∑ni=1(Io−Ip)

𝑛𝑛 (11)

III. RESULTS AND DISCUSSION

The ERA IDF curves have been fitted by a function of the types in equations (2 and 3). The optimal values of the coefficients for the equations are determined by transformation and using least square method for both cases. The R2, NSE, MAE and the RMSE results for the two equations are compiled in Table 1. From these results, it can be concluded that equation 3 is an excellent choice for fitting the ERA intensity curves. Equations with their corresponding coefficients for each region are also tabulated in Table 2. The performance of the equations to capture the existing intensities of the ERA curve are very good. In fig.(2) one can see that for each region the predicted values (shown with dots) are closer to the ERA IDF curves . This can also be observed on the values of performance criterion R2 and NSE values which are more than 98% for equation (2) and 99% for equation (3) respectively. As observed from the equations only one parameter (C) has significant geographical variation, whereas the other parameters vary minimally within the sub regions. The exponent m in equation (2) varies from 0.21 to 0.26, and the exponent f is constant with a value of 0.65 in the sub regions but parameter C varies significantly from 336.98 to 482.28. Equation (3) with the addition of one extra parameter from equation (2) has different parameter sets. The exponent m has a similar range as of equation (2) but exponent f is constant within the sub regions and has a value of 0.82. Similar to equation (2) parameter C varies significantly across the sub regions with values ranging from 758.1 to 1109.18 and the extra parameter added (b) varies from 8.36 to 8.42 for rainfall intensity in mm/h and duration in minutes.

The difference between the existing ERA IDF curve and IDF equation (3) values remain less than 10% for all rainfall durations but for equation (2) it is between 11% - 15%. Because of the relative poor performance of equation (2) compared to equation (3) we hereafter talk only about equation (3) (Table 1). The maximum errors for equation (3)

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4 for the regions Bahir Dar-Lake Tana ,region D, region A2 & A3, region B & C and region A1 & A4 are 8.5%, 6.4%, 5.7%, 4.9% and 5.7% respectively. The maximum differences at Bahir Dar-Lake Tana region, region D and for region B & C occurred at the 2 year return period. A similar phenomena is also observed in the study of Yiping [9] for a similar return period. The possible reason given was that when the rainfall value is very small a small error in the rainfall intensity will translate into a significant error.

We observed from our analysis that using the 5 minutes data over predicts therefore it is

recommended that the equation developed to exclude the 5 minutes duration for all return periods. Actually the drainage manual states that the curves overestimate for periods shorter than 15 minutes. A rainfall intensity-duration equations from 10 minute to 120 minute durations for any location in Ethiopia are described. These equations will be useful in providing estimates of rainfall intensity needed to apply the rational formula in design of drainage structures where the catchment areas are less than 50 ha as stated in the manual and times of concentration less than 2 hrs.

Table 1: Performance of IDF formulas based on equation (2) and (3)

Equation (2) Equation (3)

Efficiency criteria

Regions

Bahir

Dar D A2&A3 B&C A1&A4 Bahir Dar D A2&A3 B&C A1&A4 R2 0.9871 0.9896 0.9902 0.9903 0.9898 0.9925 0.9959 0.9955 0.9965 0.9959

NSE 0.9846 0.9875 0.9880 0.9882 0.9877 0.9923 0.9959 0.9951 0.9964 0.9958

MAE 6.38 4.39 3.17 3.92 3.85 4.99 2.82 2.21 2.39 2.47

RMSE 9.8 6.7 4.85 5.96 5.87 6.7 3.68 2.99 3.17 3.28

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Fig. 2: Test of goodness of fit of IDF equation (3) to ERA IDF curves

IV. CONCLUSION

The following conclusions are drawn from this study: The ERA IDF curves are in good agreement with the empirically derived IDF equation with minimum errors. Equation (3) is more appropriate for use for design purposes than equation (2) to produce realistic estimates of rainfall intensities for various durations and different return periods for durations from 10 min to 120 minute. Using this equation is easier than reading from the curves.

REFERENCES

[1] V. T. Chow, D. R. Maidment, and L. W. Mays,

Applied hydrology, New York: McGraw-Hill, 1988.

[2] Christopher M Trefry, David W Watkins Jr, and

Dennis Johnson, “Regional rainfall frequency analysis for the state of Michigan,” Journal of

Hydrologic Engineering, vol. 10, no. 6, pp.

437-449, 2005.

[3] V.T. Chow, Handbook of Applied Hydrology: McGraw-Hill, 1988.

[4] Sherman CW, “Frequency and intensity of excessive rainfall at Boston,” Mass.Transaction

Paper ASCE, vol. 95, pp. 951-960, 1931.

[5] Merrill M Bernard, “Formulas for rainfall intensities of long duration,” Transactions of the

American Society of Civil Engineers, vol. 96, no. 1,

pp. 592-606, 1932.

[6] Demetris Koutsoyiannis, Demosthenes Kozonis,

and Alexandros Manetas, “A mathematical

framework for studying rainfall

intensity-duration-frequency relationships,” Journal of Hydrology,

vol. 206, no. 1, pp. 118-135, 1998.

[7] Ethiopian Roads Authority, "Drainage Design Manual," 2001.

[8] David C. Froehlich, “Short-duration-rainfall

intensity equations for drainage design,” Journal of

Irrigation and Drainage Engineering vol. 119, no.

5, pp. 814-828, 1993.

[9] Yiping Guo, and M.ASCE, “Updating Rainfall IDF

Relationships to Maintain Urban Drainage Design

Standards,” Journal of Hydrologic Engineering,

vol. 11, no. 5, pp. 506-509, 2006.

[10] Katarzyna Weinerowska-Bords, “Development of

Local IDF-formula Using Controlled Random

Search Method for Global Optimization,” Acta

Geophysica, 2014.

[11] David C. Froehlich, “Intermediate-duration-rainfall

intensity equations,” Journal of Hydraulic

Engineering, vol. 121, no. 10, pp. 751-756, 1995.

[12] David C. Froehlich, “Long-duration -rainfall

intensity equations,” Journal of Irrigation and

Drainage Engineering, vol. 121, no. 3, pp.

248-252, 1995.

[13] Ibrahim H. Elsebaie, “Developing rainfall

intensity–duration–frequency relationship for two

regions in Saudi Arabia,” Journal of King Saud

University – Engineering Sciences, vol. 24, pp.

131-140, 2012.

[14] Le MINH Nhat, Yasuto Tachikawa, and Kaoru

Takara, “Establishment of Intensity -Duration-Frequency curves for precipitation in the monsoon

area of Vietnam,” Annuals of Disas. Prev. Res. Inst, pp. 93-103, 2006.

[15] C.B. Burke, and T.T. Burke, Storm Drainage

Manual, LTAP, Indiana, 2008.

[16] Kee-Won Seong, “Deriving a practical form of IDF formula using transformed rainfall intensities,”

Hydrol. Process. , vol. 28, pp. 2881-2896, 2014.

[17] García-Marín A. P. , Ayuso-Muñoz J. L. ,

Jiménez-Hornero F. J. , and Estévez J. , “Selecting the best IDF model by using the multifractal approach,”

HYDROLOGICAL PROCESSES, vol. 27, pp.

433-443, 2013.

[18] Cheng-lung Chen, “Rainfall inte

nsity-duration-frequency formulas,” Journal of Hydraulic

References

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