Abstract. 1 Introduction

Transcription

1 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Structure of ten-days natural flows of the River Nile A.M. Negm," A.H. El-Sayed," A.A Alhamid^ o Water & Water Structures Engineering Department, College of Engineering, Zagazig University, Zagazig, Egypt * Civil Engineering Department, College of Engineering, f O Boz #, Abstract This study presents the time series analysis of the natural flow data of the River Nile at four different sites (Awasn, Hassanab, Tamaniat and Malakal) for the period of 7 years using the periodogram method. The contribution of the different components constituting the time series was computed. It was found that the natural flows consist mainly of three components (trend, periodic and stochastic). The stochastic component is dominant during both flood and dry periods. The periodic component was proved to be of complex nature. The analysis of the subseries of equal lengths revealed that the small periodicities are not dominant for all the subseries. The analysis of the annual flows at the same sites resulted in similar conclusions. For design purposes, it was suggested that any of the three components should not be ignored in any forecasting model for natural flow of the River Nile. 1 Introduction The natural flow at a certain site is the amount of water which would be observed if no reservoir is in operation or other withdrawal of water in the upper basin is existing. Investigating the structure or the pattern of the natural flows of the River Nile is essential for selecting the proper forecasting model. Many techniques are available for the purpose of analyzing and forecasting the time series data [2,,4,,1,17]. These techniques can be classified in broad sense into time domain techniques and frequency domain techniques. The frequency domain techniques is a synonym term for the spectrum analysis techniques or the periodogram methods. The following are some studies which dealt with the frequency domain techniques, Fisher [], Schuster [1], Andel

2 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software & Balek [1], Michelson [], Negm, Attia & Owais [11], Negm, El-Sayed, Attai & Hammad [] and Shaltout & Tadros [1]. Several studies concerning the analysis and the forecasting of the River Nile data were made, a few of them include the works of El-Sayed [,,7] and Negm et al [11,]. The effect of the solar activity and irradiance variation on the flooding of the River Nile was studied by Shaltout & Tadros [1]. The correlation between the sunspot series and the River Nile flood levels was made forthe period from 1 to by El-Sayed []. The power spectrum was used to investigate the pattern and the hidden periodicities of the monthly flow series at Aswan by Negm et al [11]. Later on, El-Sayed [7] found that the ARIMA models may give better forecast for the River Nile flows than the previously established models. Good information about the hydrology of the River Nile can be found in the literature, see e.g. Shahin []. In the present study, an analysis of the ten-days natural flow series of the River Nile was made by using the peridogram method or the discrete spectrum analysis. The analysis was made at main four stations on the River Nile. The location of the four stations on the River Nile is shown in figure 1. These stations are Aswan, Hassanab, Tamaniat and Malakal. Also, the analysis was carried out for subseries of equal length of the original series. The percentage contribution of the different components constituting each series to the total explained variance was computed. Also, the hidden periodicities were investigated. ASWAN RIVER ATBARA THE MAIN NILE THE WHITE NILE HASSANAB TAMANIAT 4k. KHARTOUM MOGREN -^ MALAKAL ^.,ATBARA THE BLUE NILE ROSEIRES SENNAR RIVER SOBAT BAHR EL-GEBEL BASIN Figure 1: locaton of the four investigated stations on the River Nille.

3 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Theoretical Background Hydraulic Engineering Software Generally, any time series (XJ can be represented by a system of sine and cosine functions in the form: (1) where X^ is the estimate of X<, e^ is the difference between the original and the estimated series (e^=x^-x^j, h is the total number of the harmonics which is equal to = N/2 if N is even or (N-l)/2 if N is odd, t is an index for the time ( t = 1,2,...,N ), N is the total number of observations of the series points, i/n is the frequencies of occurrence of harmonics, and ; and c^ are the Fourier coefficients. The least squares estimate of o^ and - is given by [,17]: () The unbiased estimates of the periodogram ordinates at the different frequencies P(i/N) can be estimated using the estimates c^ and ft of eqns (2) and () as given by eqn (4). N (4) The percent explained variance by each harmonic (or periodogram ordinate) can be calculated through the use of eqn (). -i)- N () To test the significance of the periodogram ordinates which corresponding to a periodicity given by the inverse of its frequency, Fisher test was used [,11,17]. It uses the following ratio:

4 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software ffi =, *,.,, () where C; is the maximum periodogram ordinate and S^ is the estimate of the variance of the series. The largest harmonic is statistically significant if the ratio gi exceed the ratio g given by eqn (7) for a probability level Pf for h harmonics. in which k is the greatest integer less than 1/g. Usually the first term on the right hand side of eqn (7) gives a sufficient approximation for g-critical. The values of eqn () are biased, since it produces more significant harmonics due to the subtraction of the greatest harmonics from the data. As a result of this subtraction, the variance becomes smaller and hence the g critical value becomes larger as the denominators of eqn () becomes smaller. So this method is an approximate procedure but it satisfies the needs of the analysis of the hydrologic time series which may be periodic regardless of the built in bias [17]. Analysis of Results and Discussions A total of 1 series of the River Nile ten-days natural flows was analysed using the periodogram model given by the eqn (2). The procedure consisted of: (a) Estimating the Fourier coefficients (a;& ;) using eqns (2) and (). (b) Computing the periodogram ordinates at the different frequencies (i/n) using eqn (4). (c) Calculating the percent explained variance by ecah harmonic using eqn (). (d) Identifying and testing the significance of the hidden periodicities using eqns () and (7) (e) Identifying the trend component, if any. (f) Computing the percent contribution of the periodic (P), trend (T) and stochastic (S) components. A typical time series data is shown in figure 2a. for the second ten-days of the month of July at Aswan site, for which the periodogram ordinates versus the frequencies are given by figure 2b. This figure indicates that this series has two periodicities of years and.4 years and has a trend component indicated by thefirstordinate at frequency of (1/7). The percent explained variance by thefirstperiodicity is 1.7% and by the second is only.% which shows that thefirstone is the most important. Although, the ten years periodicity is significant at p^=% probability level while the second is not but significant at Pf=% probability level. The trend component constitutes about 2% of the total explained variance. The remaining variation (%) is considered as residual which is explained by the stochastic component.

5 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software 111 1* 1 1 Year frequency, i/n Figure 2: (a) typical time series data for the second ten-days of July from Aswan station (b) corresponding periodogram ordinates versus frequencies. Table 1. presents a typical set of significant periodicities of the ten-days series for July (dry month) and Aug. (flood month) as well as the annual flow series of the four considered stations. The first periodicity (PI) is significant at Pf=% while the second (P2) and the third (P) is significant at p^=% probability level. The analysis of the subseries of equal lengths proved that most of the periodicities less than 7 years are not domninant, since it appears only either in the first or the second half of the series. This has been led to the conclusion that only the periodicities larger than 7 years should be considered in the future studies. The results of analyzing all the full length series and the subseries has been shown that the most encountered periodicities are (or 7),,, (or 11),, 17 or 1, 2 and years. Also, It can be concluded that there is no a unique periodicity which may be considered a dominant one for all the series. This means that the periodic variation of the River Nile tendays natural flows is of comlex nature. It should be mentioned that most of the periodicites which have been appeared in the series were encountered in the annual flow series. It can be seen that the presented number of periodicities is less during the flood series (Aug., Sept. & Oct.) but thier average explained variance (Table 2.) show their great importance. Table 2. presents the average percent contribution of the periodic (P%), trend (T%) and stochastic (S%) components to the total explained variance of the dry, flood and annual series of all the investigated stations. Mostly, during flood and dry periods, the trend is not dominant and the major contribution is due to the stochastic

6 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software Table 1. Typical set of significant periodicities (in years) for July (dry series), Aug. (flood series) and the annual series of the four investigated stations Month Ten *ASW St. PI P2 P HAS St. PI P2 P TAM St. PI P2 P MAL St. PI P2 P July rd Aug. Annual rd rd *ASWSt. =Aswan station, HAS =Hassanab, TAM=Tamaniat, MAL=Malakal and PI, P2, & P are thefirst,second and third significant periodicities. Table 2. Percent contribution of the periodic, trend and stochastic components of the dry, the flood and annual series to the total explained variance Period Ten *ASW St. P% T% S% HAS St. P% T% S% TAM St. P% T% S% MAL St. P% T% S% Dry rd flood rd annual rd *St. =station, P=Periodic component, T=Trend component and S =Stochastics component.

7 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software 11 component while the minor contribution is due to the periodic one. For example at Aswan station, the average contribution of the trend ranged between and 11%, and that of the periodic is between 1% and 2% while the stochastic component represented the major contribution which ranged from 1% to 4%. These results have been shown the relative importance of the stochastic, periodic and trend components respectively in designing forecasting models for the River Nile ten-days naturl flows. In the series of Malakal station, the contribution of the trend component is larger than in the other stations while the that of the stochastic is smaller due to the special nature of the Nile Basin upstream of Malakal station []. 4 Conclusions A frequency domain technique (periodogram method) was applied to the River Nile ten-days natural flows series at main four stations (Aswan, Hassanab, Tamaniat and Malakal) to investigate the structure of the series. The analysis revealed that the majority of the series either during flood or dry periods consisted of three components, namely, the periodic, the trend and the stochastic.. The most important encountered periodicities of the periodic component has been shown to be (7,,,, 11,,, 1, 2, & ) which indicated a complex nature of the periodic variation. The importance of the trend component is more for the dry periods than for the flooding ones. The stochastics component has been proved to constitute the major contribution to the total explained variance of the investigated series. The Malakal station has a structure in which the trend and the periodic components represented more than about 4 % of the total explained variance. The study recommended that all the three components should be considered in any future forecasting model for the ten-days natural flows of the River Nile to ensure better forecast. References 1. Andel, J. & Balek, J. Analysis of periodicity in hydrological sequences, /. tfydroz.,171,, Bloomfield, P. Fourier analysis of time series: an introduction, John Willey & Sons, New York, 17.. Brillinger, D.R. Time series: data and analysis, Holt, Rinehart & Winston, New York, Box, G.E.P. & Jenkins, G.M. Time series analysis: forecasting and control, San Francisco, California, Holden Day, 17.. El-Sayed, A.M. Hydrology ofrivers,m.sc. Thesis Presented to Faculty ofeng., Alex. University, Alex., Egypt, 17.. El-Sayed, A.H. Hydrological parameters and their effect on River Nile, Ph. D. Thesis Presented to Faculty of Eng., Zagazig University, Zagazig, Egypt, El-Sayed, A.H. Forecasting River Nile discharges using ARIMA

8 Transactions on Ecology and the Environment vol 7, 14 WIT Press, ISSN Hydraulic Engineering Software modelling technique, Int. conf. on Efficient Utilization and Management of Water Resources in Africa, Khartoum, sudan, Feb. 1-4, 14.. Fisher, R. A. Tests of significance in harmonic analysis, Proceedings of R. Soc. London, Ser. A.,.. Kottegoda, N.T. Stochastic water resources technology, Machillan Press LTD, London and Basingstoke, 1.. Michelson, A.A. Determination of periodicities by the harmonic analyzer with an application to the sunspot cycle, /. Astrophys., Vol., Negm, A.M.; Attia, B.B. & Owais, T.M. Power spectrum for the River Nile natural flows times series, Proceedings of Int. Seminar on Climatic Fluctuations and Water Management, 11- Dec., Paper 1., Cairo, Egypt, 1.. Negm, A.M., El-Sayed, A.H., Attia, B.B. and Hammad, M.N. Forecasting the River Nile natural flows via spectrum analysis, Proceedings of Al-Azhar rd Int. Conf. (AE1C-), Dec. 1-21, Faculty of Eng., Al-Azhar University, Cairo, Egypt, 1, 4, Schuster, A. On Investigation of hidden periodicities, Terr. Nagn., 14,, Shahin, M. Hydrology of the Nile Basin, Elservier Science Publishing Co., New York, Shaltout, M. & Tadros, M.T.Y. Variation of the solar activity and irradiance and their influence on the flooding of the River Nile. Research report, WRC, Ministry of Public Works and Water Resources, Cairo, Egypt, Wei, W.W.S. Time series analysis (Univariate and multivariate methods), Addison-Wesley Publishing Co., Inc., New York, Yevjevich, V. Stochastic process in hydrology, Water Resources Publications, Fort Collins, Colorado, USA., 172. Nomenclature C; g h k i/n N t X. the estimates of the Fourier coefficients; the amplitude of the harmonic i; the ratio of Fisher test; the critical value used in Fisher test, eqns (,7); the total number of harmonics; an integer in eqn (7); the frequencies of occurrence of harmonics; the total number of the series points; the estimate of the variance of the sequencex,.. an index for the time; the estimated time series; the times series and the Fourier coefficients;