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Abstract
Kurau River Basin (KRB), which covers an area of 322 km2 and is the main drainage artery pouring into Bukit Merah Reservoir (BMR), is located in Perak State of Malaysia. The study of rainfall-runoff processes in KRB is important because BMR plays a vital role in rice production, flood control, ecosystems, and tourism in the region. This study proposes a new approach to rainfall-runoff modeling based on the fractional calculus. A dataset of daily rainfall and streamflow has been acquired. Then, the standard linear autoregressive with exogenous input (ARX) model is identified from the dataset in the sense of least square error. We consider the ARX model as a discretized differential equation with fractional orders. Such a model with fractional derivatives is versatile to represent hysteresis, which is intrinsically linked to the real runoff processes in tropical catchment basins like KRB.
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References
- Akaike, H. (1974). A new look at statistical-model identification. IEEE T. Automat. Contr., AC19, 716-723. https://ieeexplore.ieee.org/document/1100705
- Bushnaq, S., Ali, S., Shah, K., & Arif, M. (2018a). Exact solution to non-linear biological population model with fractional order. Therm. Sci., 22, S317-S327.https://pdfs.semanticscholar.org/a360/787b5e0f1abde25a787b8f3c28c2132b0c03.pdf
- Bushnaq, S., Khan, S.A., Shah, K., & Zaman, G. (2018b). Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative. Cogent Math., 5, 1432521. https://www.cogentoa.com/article/10.1080/23311835.2018.1432521
- Fadhil, R. M., Rowshon, M. K., Ahmad, D., Fikri, A., & Aimrun, W. (2017). A stochastic rainfall generator model for simulation of daily rainfall events in Kurau catchment: model testing. Acta Horticulturae, 1152, 1-10. https://www.ishs.org/ishs-article/1152_1
- Fernández-Pato, J., Gracia, J.L. & Garcia-Navarro, P. (2018). A fractional-order infiltration model to improve the simulation of rainfall/runoff in combination with a 2D shallow water model. J. Hydroinform., 20, 898-916. http://scholar.google.es/citations?user=dD4cnfYAAAAJ&hl=es
- Ghani, A. A., Azamathulla, H. M., Chang, C. K., Zakaria , N. A., & Abu Hasan, Z. (2011). Prediction of total bed material load for rivers in Malaysia: A case study of Langat, Muda and Kurau Rivers. Environmental Fluid Mechanics, 11, 307-318.
- Guinot, V., Savean, M., Jourde, H., & Neppel, L. (2015). Conceptual rainfall-runoff model with a two-parameter, infinite characteristic time transfer function. Hydrological Processes, 29, 4756-4778. https://doi.org/10.1002/hyp.10523
- Hamidon, N., Harun, S., Malek, M. A., Ismail, T., & Alias, N. (2015). Prediction of paddy irrigation requirements by using statistical downscaling and CROPWAT models: a case study from the Kerian Irrigation Scheme in Malaysia. Journal Teknologi, 76, 281-288. https://doi.org/10.11113/JT.V76.4038
- Hassan, Z., & Harun, S. (2011). Hydrological response of a catchment to climate change in the Kurau River Basin, Perak, Malaysia, 3rd International Conference on Managing Rivers in the 21th Century: Sustainable Solutions for Global Crisis of Flooding, Pollution and Water Scarcity, Penang, Malaysia, 216-225.
- Hassan, Z., Harun, S., & Malek, M.A. (2012). Application of ANNs model with the SDSM for the hydrological trend prediction in the sub-catchment of Kurau River, Malaysia. Journal of Environmental Science and Engineering, B: 1(5B), 577-585.
- Ismail, W. R., & Najib, S. A. M. (2011). Sediment and nutrient balance of Bukit Merah Reservoir, Perak (Malaysia). Lakes and reservoirs: Research and Management, 16, 179-184. https://doi.org/10.1111/j.1440-1770.2011.00453.x
- Jarad, F. & Abdeljawad, T. (2018). A modified Laplace transform for certain generalized fractional operators. Results in Nonlinear Analysis, 1, 88-98. https://dergipark.org.tr/en/download/article-file/527716
- Lohani, A.K., Kumar, R., & Singh, R. D. (2012). Hydrological time series modeling: A comparison between adaptive neuro-fuzzy, neural network and autoregressive techniques. Journal of Hydrology, 442, 23-35. https://doi.org/10.1016/j.jhydrol.2012.03.031
- Malaysia DID (2011). Review of the national water resources study (2000-2050) and formulation of national water resources policy: Final report. Kementerian Sumber Asli Dan Alam Sekitar Malaysia. https://www.water.gov.my/jps/resources/PDF/Hydrology%20Publication/Vol2WaterGovernance.pdf
- Oldham, K. B., & Spanier, J. (1974). The Fractional Calculus. Academic Press, Inc.: 234pp. https://link.springer.com/chapter/10.1007/978-3-7091-2664-6_5
- Osman, Y., Al-Ansari, N., & Abdellatif, M. (2019). Climate change model as a decision support tool for water resources management in northern Iraq: a case study of Greater Zab River. Journal of Water and Climate Change, 10, 197-209. https://doi.org/10.2166/wcc.2017.083
- Unami, K., & Kawachi, T. (2005). Systematic assessment of flood mitigation in a tank irrigated paddy fields area. Paddy and Water Environment, 3, 191-199.https://doi.org/10.1007/s10333-005-0022-6
- Yeh, W. W. G. (1985). Reservoir management and operations models - a state-of-the-art review. Water Resources Research, 21, 1797-1818. https://doi.org/10.1029/WR021i012p01797
References
Akaike, H. (1974). A new look at statistical-model identification. IEEE T. Automat. Contr., AC19, 716-723. https://ieeexplore.ieee.org/document/1100705
Bushnaq, S., Ali, S., Shah, K., & Arif, M. (2018a). Exact solution to non-linear biological population model with fractional order. Therm. Sci., 22, S317-S327.https://pdfs.semanticscholar.org/a360/787b5e0f1abde25a787b8f3c28c2132b0c03.pdf
Bushnaq, S., Khan, S.A., Shah, K., & Zaman, G. (2018b). Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative. Cogent Math., 5, 1432521. https://www.cogentoa.com/article/10.1080/23311835.2018.1432521
Fadhil, R. M., Rowshon, M. K., Ahmad, D., Fikri, A., & Aimrun, W. (2017). A stochastic rainfall generator model for simulation of daily rainfall events in Kurau catchment: model testing. Acta Horticulturae, 1152, 1-10. https://www.ishs.org/ishs-article/1152_1
Fernández-Pato, J., Gracia, J.L. & Garcia-Navarro, P. (2018). A fractional-order infiltration model to improve the simulation of rainfall/runoff in combination with a 2D shallow water model. J. Hydroinform., 20, 898-916. http://scholar.google.es/citations?user=dD4cnfYAAAAJ&hl=es
Ghani, A. A., Azamathulla, H. M., Chang, C. K., Zakaria , N. A., & Abu Hasan, Z. (2011). Prediction of total bed material load for rivers in Malaysia: A case study of Langat, Muda and Kurau Rivers. Environmental Fluid Mechanics, 11, 307-318.
Guinot, V., Savean, M., Jourde, H., & Neppel, L. (2015). Conceptual rainfall-runoff model with a two-parameter, infinite characteristic time transfer function. Hydrological Processes, 29, 4756-4778. https://doi.org/10.1002/hyp.10523
Hamidon, N., Harun, S., Malek, M. A., Ismail, T., & Alias, N. (2015). Prediction of paddy irrigation requirements by using statistical downscaling and CROPWAT models: a case study from the Kerian Irrigation Scheme in Malaysia. Journal Teknologi, 76, 281-288. https://doi.org/10.11113/JT.V76.4038
Hassan, Z., & Harun, S. (2011). Hydrological response of a catchment to climate change in the Kurau River Basin, Perak, Malaysia, 3rd International Conference on Managing Rivers in the 21th Century: Sustainable Solutions for Global Crisis of Flooding, Pollution and Water Scarcity, Penang, Malaysia, 216-225.
Hassan, Z., Harun, S., & Malek, M.A. (2012). Application of ANNs model with the SDSM for the hydrological trend prediction in the sub-catchment of Kurau River, Malaysia. Journal of Environmental Science and Engineering, B: 1(5B), 577-585.
Ismail, W. R., & Najib, S. A. M. (2011). Sediment and nutrient balance of Bukit Merah Reservoir, Perak (Malaysia). Lakes and reservoirs: Research and Management, 16, 179-184. https://doi.org/10.1111/j.1440-1770.2011.00453.x
Jarad, F. & Abdeljawad, T. (2018). A modified Laplace transform for certain generalized fractional operators. Results in Nonlinear Analysis, 1, 88-98. https://dergipark.org.tr/en/download/article-file/527716
Lohani, A.K., Kumar, R., & Singh, R. D. (2012). Hydrological time series modeling: A comparison between adaptive neuro-fuzzy, neural network and autoregressive techniques. Journal of Hydrology, 442, 23-35. https://doi.org/10.1016/j.jhydrol.2012.03.031
Malaysia DID (2011). Review of the national water resources study (2000-2050) and formulation of national water resources policy: Final report. Kementerian Sumber Asli Dan Alam Sekitar Malaysia. https://www.water.gov.my/jps/resources/PDF/Hydrology%20Publication/Vol2WaterGovernance.pdf
Oldham, K. B., & Spanier, J. (1974). The Fractional Calculus. Academic Press, Inc.: 234pp. https://link.springer.com/chapter/10.1007/978-3-7091-2664-6_5
Osman, Y., Al-Ansari, N., & Abdellatif, M. (2019). Climate change model as a decision support tool for water resources management in northern Iraq: a case study of Greater Zab River. Journal of Water and Climate Change, 10, 197-209. https://doi.org/10.2166/wcc.2017.083
Unami, K., & Kawachi, T. (2005). Systematic assessment of flood mitigation in a tank irrigated paddy fields area. Paddy and Water Environment, 3, 191-199.https://doi.org/10.1007/s10333-005-0022-6
Yeh, W. W. G. (1985). Reservoir management and operations models - a state-of-the-art review. Water Resources Research, 21, 1797-1818. https://doi.org/10.1029/WR021i012p01797