Surabaya 60111, Indonesia *Corresponding author

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1 2017 2nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2017) ISBN: Parameters Estimation of Synthetic Unit Hydrograph Model Using Multiple Linear and Non-linear Regressions I Gede TUNAS 1,*, Nadjadji ANWAR 2 and Umboro LASMINTO 2 1 Department of Civil Engineering, University of Tadulako, Palu 94118, Indonesia 2 Department of Civil Engineering, Institut Teknologi Sepuluh Nopember (ITS) Surabaya 60111, Indonesia *Corresponding author Keywords: Parameters estimation, Synthetic unit hydrograph model, Multiple linear regressions, Multiple non-linear regressions. Abstract. The use of synthetic unit hydrograph model (SUH) is remain popular used to transform rainfall into run off for water resources development. The typical feature of this model is that the main equation represents the shape of the curve expressed by the relationship between time and discharge. In addition, the SUH model is also expressed in three parameters i.e. peak time (TP), peak discharge (QP), and base time (TB), representing the hydrograph curve equation. In general, SUH model is developed based on morphometry parameters of watershed, especially watershed area (A), main river length (L) and main river slope (S). Another approach in hydrograph modelling is based on the fractal characteristics of watershed. This study aims to develop a synthetic unit hydrograph model based on a combination of morphometry and fractal characteristics of watersheds. The three model parameters (TP, QP and TB) were predicted using multiple linear regression and compared with multiple nonlinear regression. The results of the analysis show that the two methods showed excellent performance. The estimation of SUH parameters using linear regression resulted peak time equation (TP) as function of river length (L), ratio of river length (RL) and density of drainage network (D) with determination coefficient of 99.8%, a base time equation (TB) as the function of watershed area (A) and river slope (S) with determination coefficient of 98.2%. Using multiple non linear regression, estimation of SUH parameters formulated peak time equation (TP) as function of river length (L), ratio of river length (RL) and ratio of watershed area (RA) with determination coefficient of 99.9%, a base time equation (TB) as the function of watershed area (A) and ratio of watershed area (RA) with determination coefficient of 97.9%. Peak discharge equation (QP) is stated as a function of peak time and a simple single curve equation derived from Gamma Distribution Equation. Introduction In hydrological modelling, one of the rainfall-run off transformation models is hydrograph-based modelling [1,2]. Hydrograph-based modelling is generally developed to accommodate the limitations of hydrological data, especially rainfall data and discharge data in a watershed where the model is developed [3]. Oftentimes, discharge data is not available at all or available with very limited range of data [4]. This is related to the availability of hydrological and hydrometry gauge networks, especially the limited discharge gauges. Hydrometric stations that function to monitor and record river water level data, in addition to the limited number, generally only installed in certain places considered by the managers have significance function [5] in this case the potential development of a region in terms of water resources. This is due to very high installation and management costs. Besides that also due to the high level of sensitivity of this instrument, not infrequently give dubious results even not work at all. The implication is that the recording of discharge data in certain years is incomplete or even absent, so that when needed for hydrological analysis the data is not available or available in a very short period of time. On the other hand, the existing hydrometric stations are equipped with a set of automatic water level registers, so to obtain discharge data, It is needed calibration from the water 202

2 level to the discharge. This calibration process actually also provides a large deviation because the river cross section used to establish the calibration curve always changes at any time due to the interaction between the river wall with the flow in the morphodynamic process of the river. The consequence of this limitation is that the analysis of rainfall transformation into discharge is the only commonly used method in various purposes. To solve these problems, a synthetic hydrograph model was developed based on the characteristics of watershed morphometry such as SUH of Snyder, Nakayasu, GAMA I and some other SUH models and based on the fractal characteristics of watershed such as HSS ITS-2 developed at the Institut Teknologi Surabaya (ITS) in 2017 by Tunas et al. [6] The most important part in the hydrograph modelling are the three main hydrograph parameters consisting of peak time (TP), peak discharge (QP) and base time (TB) as shown in Figure 1b. In general these three parameters play a major role in controlling the hydrograph curve (Figure 1a). Basically, these three parameters can be estimated based on the characteristics of the watershed, using regression analysis of either multiple linear regression or multiple non-linear regression, depending on the tendency of the relationship between parameters. Therefore, this study tries to compare the estimation of these parameters using both methods so that the performance can be known. Discharge (m 3 /s) T r (2) (1) (1) Total flow (2) Surface run-off) (3)Iinterflow) (4) Baseflow) Discharge (m 3 /s) T r (1) (2) (1) Rising limb (2) Peak discharge,qp (3) Recession limb (4) Time to peak, TP (5) Base Time (3) (4) (3) (4) (5) Time (hour) Time (hour) (a) (b) Figure 1. Hydrograph component (a) and hydrograph parameters (b). Materials and Methods This study was conducted at eight measured watersheds in Indonesia as shown in Figure 2, covering the watersheds of Bahomoleo, Pinamula, Toaya, Bangga, Singkoyo, Tambun, Malino and Bunta. Administratively the eight watersheds are located in Central Sulawesi Province at coordinates '32"E '47"E and 02 28'34"S 01 09'48"N. The data used in this research are rainfall and discharge data which are found in each observed watershed, and topographic map data. Rainfall and discharge data are used to derive unit hydrographs, while topographic maps are used to analyze the characteristics of watersheds. Estimation of SUH model parameters are done using multiple linear regression analysis and compared with non-linear regression. The use of these regression models is based on the output variable that is influenced by more than one input variable and predicted to have a linear or non-linear relationship (then this prediction will be verified) [8.9]. For the practicality of the analysis, regression modelling will be done using the Mini Tab software and all the testing required to get the best model. The performance of model equations can be measured by determination coefficient (r 2 ). This method is one technique to determine how far the mathematical model of regression can describe the existing 203

3 data. The coefficient of determination (r 2 ) shows how far the error in estimating the magnitude of the dependent variable (y) which can be reduced using the information of the independent variable (x). Figure 2. Location of research (red colour border) [7]. Regression model will be better when the coefficient of determination close to 1 and otherwise worse if the coefficient of determination close to 0. Coefficient of determination is expressed by equation [10]: (1) Results and Discussions Peak Time Equation In order to construct the peak time equation, the following in Table 1 are presented the values of parameters which become dependent and independent variables of eight watersheds used as modelling. Table 1. Variables to construct peak time equations. Watershed Dependent Variable Independent Variable TP L FB D RB RL RA Bahomoleo Pinamula Toaya Bangga Singkoyo Tambun Malino Bunta Based on Table 1, the peak time is assumed to be a function of river length (L), form factor (FB), river network density (D), ratio of river length (RB), ratio of river branching (RB) and ratio of watershed area (RA) or can be written as: 204

4 ,,,,, (2) Using multiple linear regression analysis, the peak time equation (TP) was obtained as: (3) The test results show that the regression model is optimal (good fit) and has the coefficient of determination (r 2 ) of 99.9%, close to 100% means the resulting model has a very good performance where r 2 explains the diversity of independent variables which can be explained by the dependent variable. Using multiple non-linear analysis, the peak time equation (TP) is obtained as: (4) The residual result shows that the residual data follows the normal distribution, residual data are not autocorrelated (no residual autocorrelation line exceeds the curve boundary) and the residual variant is homogeneous. Based on the results of the analysis the resulting equation has a value of r 2 of 99.9%. Base Time Equation The base time equations as identified by the preceding parameters and also based on the base time equations in various SUH models are considered to be functions of watershed area (A), river length (L), main river slope (S), form factor (FB ) and ratio of watershed area (RA) and can be expressed as:,,,,,, (5) The following values of parameters are used as the compilers of the base time equation as dependent and independent variables (Table 2). Table 2. Variables to construct base time equations. Dependent Watershed Variable Independent Variable TB A L S FB D RB RL RA Bahomoleo Pinamula Toaya Bangga Singkoyo Tambun Malino Bunta The results of multiple linear regression analysis showed that the regression model is optimal (good fit), this model has the coefficient of determination (r 2 ) = 98.7% with the following equation: (6) Using multiple non-linear regression, the resulting equation is: (7) The residual test shows that the residual data follows the normal distribution, the residual data is not autocorrelated (no residual autocorrelation line exceeds the curve limit) but the residual variant is not homogeneous. Based on the results of the analysis the resulting equation has a value of r 2 of 97.9%, Peak Discharge Equation The peak discharge equation is constructed using several watershed parameters that affect the hydrograph and peak time parameters, following the equilibrium volume of rain and hydrograph volume under the unit hydrograph curve. The hydrograph curve equation is based on the hydrograph 205

5 of dimensionless units according to the principle of mass conservation as done in SUH of ITB-1 and ITB-2 [11], which stated as:. Where R0=unit rainfall and A SUH=Area under unit hydrograph curve (dimensionless). (8) Curve Hydrograph Equation The hydrograph curve equations are developed based on the Gamma Distribution equation, and expressed as SUH of ITS-2, as follows [6]: exp 1 (9) Conclusion and Recommendations Based on the analysis result from the study, the two methods which used to estimate synthetic unit hydrograph parameters showed an excellent performance. The estimation of SUH parameters using linear regression resulted peak time equation (TP) as function of river length (L), ratio of river length (RL) and density of drainage network (D) with determination coefficient of 99.8%, a base time equation (TB) as the function of watershed area (A) and river slope (S) with determination coefficient of 98.2%. Using multiple non-linear regression, estimation of SUH parameters formulated peak time equation (TP) as function of river length (L), ratio of river length (RL) and ratio of watershed area (RA) with determination coefficient of 99.9%, a base time equation (TB) as the function of watershed area (A) and ratio of watershed area (RA) with determination coefficient of 97.9%. Peak discharge equation (QP) is stated as a function of peak time and a simple single curve equation derived from Gamma Distribution Equation. Given that this analysis uses only a limited number of watersheds, verification is necessary to see the performance of both methods used on a wider scale. Acknowledgement This article is part of doctoral research in Civil Engineering Department, Institut Teknologi Sepuluh Nopember (ITS) Surabaya. Author thanks to Ministry of Research, Technology and Higher Education which support with BPPDN Scholarship and Doctoral Research Grant. References [1] K. Subramanya, Engineering Hydrology, McGraw Hill, New Delhi, [2] I.G. Tunas, N. Anwar, and U. Lasminto, Fractal characteristic analysis of watershed as variable of synthetic unit hydrograph model, The Open Civil Engineering Journal. 10 (2016) [3] I.G. Tunas, N. Anwar, and U. Lasminto, The improvement of synthetic unit hydrograph performance by adjusting model parameters for flood prediction, International Journal of Engineering and Technology (IJET). 9 (2017) [4] A.W. Salami, S.O. Bilewu, A.M. Ayanshola, and S.F. Oritola, Evaluation of synthetic unit hydrograph methods for the development of design storm hydrographs for rivers in South-West, Nigeria, Journal of American Science. 5 (2009) [5] Sri Harto, GAMA I Synthetic Unit Hydrograph, Publisher of Indonesian Public Work Ministry, Jakarta, 1985 [in Indonesian). 206

6 [6] I.G. Tunas, N. Anwar, and U. Lasminto, Synthetic unit hydrograph model based on fractal characteristics of watersheds: submitted to International Journal of River Basin Management (2017). [7] Indonesian Geospatial Information Agency, Indonesian Topography Map, Jakarta, [in Indonesian). [8] C.T. Haan, Statistical Methods in Hydrology, The Iowa State University Press, Iowa, (1995) [9] N.T. Kottegoda, and R. Rosso, Applied Statistics for Civil and Environmental Engineers, Blackwell Publishing, Oxford, (2008). [10] R.E. Walpole., R.H. Myers, S.L. Myers, and K. Ye, Probability and Statistics for Engineers and Scientist, Prentice Hall, Boston, (2012). [11] D.K. Natakusimah, W. Hatmoko, and D. Harlan, General procedure calculation of synthetic unit hydrograph using ITB method and its application examples, Journal of Teknik Sipil, 18(2011) [in Indonesian]. 207