Application of the Quasi-Reversibility Method in Inverse Computation of Temporal and Spatial Pollutant Concentration in Time

Document Type : Research Paper


Departement of Water Structures, Faculty of Agriculture, Tarbiat Modares University, Tehran, Iran


Pollutants are usually drained off imperceptibly and suddenly in the rivers, which can be of human or natural origin, thus finding information from contaminant source as quickly as possible is important to reduce damage. The pollutant is released by the Advection-Dispersion processes in the river. Therefore, information on contaminant release site and release time can be obtained using inverse solution of the Advection-Dispersion equation. The purpose of this study is to solve Advection-Dispersion Equation (ADE) reversely and to obtain information on the release time and time series data of pollutant concentration discharged into the studied rivers. In this research, the quasi-reversibility method is used to reverse the ADE. In this method, by adding the stability term (fourth derivative term) to ADE, the mentioned relationship can be solved reversely without the instability of the answers. A hypothetical example and a case study of Karun River have been used for model validation. The aforementioned method determines the concentration experienced at different points and intervals of the river by reversing the ADE. The highest contaminant intake in each interval, maximum and average intake time are the obtained results by this method. The results show that the quasi-reversibility method has been performed with high accuracy and the proposed method has been satisfied in stability of solving ADE.


Main Subjects

Atmadja, J. and Bagtzoglou, A.C. (2001). Pollution source identification in heterogeneous porous media. Water Resources Research, 37(8): 2113-2125.
Bavandpouri, G., Mazaheri, M. and Fotouhi Firozabadi, M. (2017). Analytical Solution of Contaminant Transport Equation in River by Arbitrary Variable Coefficients Using Generalized Integral Transform Technique. Journal of Advanced Mathematical Modeling, 7(1): 89-116. (In Farsi)
Bagtzoglou, A.C. (1992). Application of particle methods of reliable Identification of groundwater polloution sources. Water resources management. 6: 15-23.
Bagtzoglou, A.C. and Atmadja, J. (2003). Marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: Application to contaminant plume spatial distribution recovery. Water Resources Research, 39(2): 146-187.
Chapra, S.C. (1997). Surface Water Quality Modeling. New York: McGraw-Hill.
Dahmardan, A., Mazaheri, M. and Mohammad Vali Samani, J. )2018(. Identification of Location, Activity Time and Intensity of the Unknown Pollutant Source in River. Journal of Environmental Hazards Management, 5(1): 35-52. (In Farsi)
Denche, M. and Bessila, K. (2005). A modified quasi-boundary value method for ill-posed problems. Journal of Mathematical Analysis and Applications, 301(2): 419-426.
Deng ,Z. Q., Singh, V. P & ,.Bengtsson, L. (2001). Longitudinal dispersion coefficient in straight rivers. Journal of Hydraulic Engineering, 127(11), .927 -919.
Dorroh, J. R.  and Ru, Xeuping. (1998). The Application of the Method of Quasi-reversibility to the Sideways Heat Equation. Journal of Mathematical Analysis and Applications, 236(6): 503-519.
Ghane, A.,  Mazaheri, M. and Mohammad Vali Samani, J. (2016). Location and release time identification of pollution point source in river networks based on the Backward Probability Method.  Journal of Environ Manage, 180: 164-171.
Ghane, A., M‌a‌z‌a‌h‌e‌r‌i, M., M‌o‌h‌a‌m‌m‌a‌d V‌a‌l‌i S‌a‌m‌a‌n‌i, J. (2017). Location and release time T‌R‌A‌C‌I‌N‌G of pollition source in rivers based on compound model adjoint analysis and optimization method. Sharif Journal of Civil Engineering, 33.2(3.2): 95-104. (In Farsi)
Ismail-Zadeh, A.T. and Korotkil, I. A. and Tsepeler, A.I. (2006). Three-Dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibilitiy method. Water resources, 46(12): 2176-2186.
 Lattes, R., and Lions, J. (1969). The Method of Quasi-Reversibility: Applications to Partial Differential Equations. Elsevier Sci, New York.
Hossieni, P., Ildoromi, A., Hosseini, Y. (2016). The Study of Qual2kw Model Efficacy on River Self-purification (A Case Study of Karun River at Interval of Zargan to Kute Amir). Journal of Environmental Science and Technology, 18(4): 103-122. (In Farsi)
 Mazaheri, M.,  Mohammad Vali Samani, J. and Samani,   H.M.V. (2015). Mathematical Model for Pollution Source Identification in Rivers.  Environmental Forensics, 16: 310-321.
 Neupauer, R.M.,  Borchers, B. and Wilson, J.L. (2000). Comparison of inverse methods for reconstructing the release history of a groundwater contamination source. Water Resources Research, 36(9):  2469-2475.
 Qian, A. and Mao, J. (2011). Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem.  American Journal of Computational Mathematics, 01(03): 159-162.
 Skaggs, T.H. and Kabala, Z. J. (1995). Recovering the history of a groundwater contaminant plum: Method of quasi-reversibilitiy. Water resources, 31: 2669-2673.
 Tong, Y. and Deng, Z. (2015). Moment-Based Method for Identification of Pollution Source in Rivers. Journal of Environmental Engineering, 141(10): 04015026.
 Wilson, J.L. and Liu, J. 1994. Backward tracking to find the source of pollution. Water Manag, Risk Remed,1, 181-199.
 Xiong, X.-T.,  Fu, C.-L. and Qian, Z. (2006). Two numerical methods for solving a backward heat conduction problem. Applied Mathematics and Computation, 179(1), pp. 370-377.
 Yang, F.,  Fu, C. and Li, X. (2014). Identifying an unknown source in space-fractional diffusion equation. Acta Mathematica Scientia, 34(4): 1012-1024.
 Zhang, T. and Chen, Q. (2007). Identification of contaminant sources in enclosed spacey by a single sensor. Indoor Air, 17(6): 439-449.
Zhang, T. and Li, H. Wang. (2011). Identification of particulate contaminant source locations in enclosed spaces with inverse CFD modelling. 12th International Conference on Indoor Air Quality and Climate 2011, 1: 667-672.