A stochastic approach is presented in view that a
time series modelling is achieved through an Autoregressive Moving Average
(ARMA) model. The applicability of the ARMA model is then further presented
using the Great Letaba River as a case study. River flow discharge for 25 years
(1989-2014) for the Great Letaba River was obtained from the Department of
Water and Sanitation, South Africa and analysed by Autoregressive (AR),
Autoregressive Moving Average (ARMA) and Autoregressive Integrated Moving
Average (ARIMA) models. Monte Carlo simulation approach was used to generate
forecasts of the ARIMA error model for the next 25 years. Initial model
identification was done using the Autocorrelation function (ACF) and Partial
Autocorrelation function (PACF). The model analysis and evaluations provided
proper predictions of the river system. The models revealed some degree of
correlation and seasonality behaviour with decreasing river flow. Hence, in
conclusion, the Great Letaba River flow has shown a decreasing trend and
therefore, should be effectively used for sustainable future development.
References
[1]
Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis Forecasting and Control. San Francisco, CA: Holden Day.
[2]
Box, G. E. P., & Jenkins, G. M. (1977). Time Series Analysis. San Francisco, CA: Holden Day.
[3]
Dessalegn, T. A., Moges, M. A., Dagnew, D. C., & Gashaw, A. (2017). Applicability of Galway River Flow Forecasting and Modeling System (GFFMS) for Lake Tana Basin, Ethiopia. Journal of Water Resource and Protection, 9, 1319-1334. https://doi.org/10.4236/jwarp.2017.912084
[4]
DWA (2004). River System Annual Operating Analysis. Pretoria, South Africa: Department of Water Affairs (DWA).
[5]
DWAF (2006). National Water Resources Strategy. Annual Report 2006, Pretoria, South Africa: Department of Water Affairs and Forestry (DWAF).
[6]
Garrote, L., & Bras, R. L. (1995). A Distributed Model for Real-Time Flood Forecasting using Digital Elevation Models. Journal of Hydrology, 167, 279-306. https://doi.org/10.1016/0022-1694(94)02592-Y
[7]
Granger, C. W. J., & Newbold, P. (1976). Forecasting Transformed Series. Journal of Royal Statistical Society, B38, 189-203. https://doi.org/10.1111/j.2517-6161.1976.tb01585.x
[8]
Makridakis, S., & Hibon, M. (1995). ARMA Models and the Box-Jenkins Methodology. 95/45/TM Revised Version of 95/33/TM.
[9]
Mcleod, A. I., Hipel, K. W., & Lennox, W. C. (1977). Advances in Box-Jenkins Modelling Applications. Journal of Water Resources Research, 13, 577-586. https://doi.org/10.1029/WR013i003p00577
[10]
Mujumdar, P. P., & Kumar, N. D. (1990). Stochastic Models of Stream Flow: Some Case Studies. Hydrological Sciences, 35, 395-410. https://doi.org/10.1080/02626669009492442
[11]
Mukherjee, D., & Mansour, N. (1996). Estimation of Flood Forecasting Errors and Flow Duration Joint Probabilities of Exceedance. Journal of Hydrological Engineering, 122, 130-140. https://doi.org/10.1061/(ASCE)0733-9429(1996)122:3(130)
[12]
Musa, J. J. (2013). Stochastic Modelling of Shiroro River Stream Flow Process. American Journal of Engineering Research, 2, 49-54.
[13]
Nyabeze, W. R., Mallory, S., Hallowes, J., Mwaka, B., & Sinha, P. (2007). Determining Operating Rules for the Letaba River System in South Africa Using Three Models. Journal of Physics and Chemistry of the Earth, 32, 1040-1049. https://doi.org/10.1016/j.pce.2007.07.003
[14]
Otache, Y. M., Isiguzo, E. A., & Sadeeq, A. M. (2011a). Parametric Linear Stochastic Modelling of Benue River flow Process. Open Journal of Marine Sciences, 3, 73-81.
[15]
Otache, Y. M., Sadeeq, A. M., & Isiguzo, E. A. (2011b). ARMA Modelling River Flow Dynamics: Comparative Study of PAR Model. Open Journal of Modern Hydrology, 1, 1-9. https://doi.org/10.4236/ojmh.2011.11001
[16]
Peng, G., Lu, F., Song, Z., & Zhang, Z. (2018). Key Technologies for an Urban Overland Flow Simulation System to Support What-If Analysis. Journal of Water Resource and Protection, 10, 699-724. https://doi.org/10.4236/jwarp.2018.107040
[17]
Slutsky, E. (1937). The Summation of Random Causes as the Source of Cyclic Processes. Econometrica, 5, 105-146. https://doi.org/10.2307/1907241
[18]
Vandaele, W. (1983). Applied Time Series and Box-Jenkins Models. New York: Academic Press.
[19]
Wold, H. (1938). A Study in the Analysis of Stationary Time Series. Stockholm: Almgrist and Wiksell.
[20]
WRC (2004). Monthly Multi-Site Stochastic Stream Flow Model. Report K5 909/1/04, Pretoria, South Africa: Water Research Commission (WRC).
[21]
Yule, G. U. (1926). Why Do We Sometimes Get Nonsense-Correlations between Time Series? A Study in Sampling and the Nature of Time Series. Journal of Royal Statistical Society, 89, 1-64. https://doi.org/10.2307/2341482
