Identify the source of pollution with an Inverse-time analytical solution to the pollution transport equation

Document Type : Original Article


Department of Water Structures Engineering Faculty of Agriculture،Tarbiat Modares University


Despite the existence of the regulations that penalize wastewater discharging into water sources, repeated violations in terms of discharging has been occurred for many years. Therefore, identifying the pollutant source is of high significance. Pollution introduced into the river is transmitted to the downstream via the advection-dispersion process. Direct advection-dispersion equation (Direct ADE) or positive time step can be solved using traditional numerical and analytical methods. In the true solution, the upstream Pollutant concentration time pattern is already specified, and the aim is to find the temporal and spatial distributions of pollutant concentration at downstream.In contrast, in inverse solution, a totally different approach is adopted. Inverse solution of ADE in the form of negative reverse time step falls into the category of the partial differential equations (III-posed), and the answers to this equation do not achieve convergence. Therefore, the quasi-reversibility method is used. The quasi-reversibility method is basically built upon the addition of the stability term, which includes the fourth-order term and the stabilization coefficient and plays a major role in the convergence of the inverse answers to ADE. In the next step, ADE and stability term are solved by Fourier transform (FT). This solution leads to a highly oscillatory integral using FT. By solving this integral, Pollutant sources time pattern (time series) can be identified. To resolve the integral, two hypothetical examples (simple and intricate) were used, and the validation of the numerical results indicates the high capability of the analytical model.


Main Subjects

[1] Kamble SM. Water pollution and public health issues in Kolhapur city in Maharashtra. International journal of scientific and research publications. 2014 Jan;4(1):1-6.
[2] Tong Y, Deng Z. Moment-based method for identification of pollution source in rivers. Journal of Environmental Engineering. 2015 Oct 1;141(10):04015026.
[3] مهدی مظاهری. مدل ریاضی تشخیص منابع آلاینده در رودخانه: بازیابی مکان و شدت زمان منابع آلاینده [رسالۀ دکتری]. تهران: دانشگاه تربیت مدرس؛1390.
[4] Hosseinkhani MR, Omidvar P. Smoothed Particle Hydrodynamics for the rising pattern of oil droplets. Journal of Fluids Engineering. 2018 Aug 1;140(8).
[5] Lattès R, Lions JL. The method of quasi-reversibility: applications to partial differential equations. 1969.
 [6] Dorroh JR, Ru X. The application of the method of quasi-reversibility to the sideways heat equation. Journal of mathematical analysis and applications. 1999 Aug 15;236(2):503-19.
[7] Ismail-Zadeh A, Korotkii A, Schubert G, Tsepelev I. Numerical techniques for solving the inverse retrospective problem of thermal evolution of the Earth interior. Computers & structures. 2009 Jun 1;87(11-12):802-11.
[8] Xiong XT, Fu CL, Qian Z. Two numerical methods for solving a backward heat conduction problem. Applied mathematics and Computation. 2006 Aug 1;179(1):370-7.
[9] Qian A, Mao J. Quasi-reversibility regularization method for solving a backward heat conduction problem. American Journal of Computational Mathematics. 2011 Sep 1;1(3):159.
[10] Skaggs TH, Kabala ZJ. Recovering the history of a groundwater contaminant plume: Method of quasi-reversibility. Water Resources Research. 1995 Nov;31(11):2669-73.
[11] Liu C, Ball WP. Application of inverse methods to contaminant source identification from aquitard diffusion profiles at Dover AFB, Delaware. Water Resources Research. 1999 Jul;35(7):1975-85.
[12] Atmadja J, Bagtzoglou AC. Pollution source identification in heterogeneous porous media. Water Resources Research. 2001 Aug;37(8):2113-25.
[13] Wang Z, Liu J. Identification of the pollution source from one-dimensional parabolic equation models. Applied Mathematics and Computation. 2012 Dec 15;219(8):3403-13.
[14] Fan YA, Chuli FU, Xiaoxiao LI. Identifying an unknown source in space-fractional diffusion equation. Acta Mathematica Scientia. 2014 Jul 1;34(4):1012-24.
[15] Zhang T, Chen Q. Identification of contaminant sources in enclosed spaces by a single sensor. Indoor air. 2007 Dec 1;17(6):439-49.
[16] de Barros FP, Colbrook MJ, Fokas AS. A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line. International Journal of Heat and Mass Transfer. 2019 Aug 1;139:482-91.
[17] Ghane A, Mazaheri M, Samani JM. Location and release time identification of pollution point source in river networks based on the backward probability method. Journal of environmental management. 2016 Sep 15;180:164-71.
[18] Mazaheri M, Mohammad Vali Samani J, Samani HM. Mathematical model for pollution source identification in rivers. Environmental Forensics. 2015 Oct 2;16(4):310-21.
[19] Rump SM. Reliability in computing. Academic Press; 1988 Jan 1. Algorithms for verified inclusions: Theory and practice; p. 109-26.
[20] Guerrero JP, Pimentel LC, Skaggs TH, Van Genuchten MT. Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique. International Journal of Heat and Mass Transfer. 2009 Jun 1;52(13-14):3297-304.
[21] Chapra SC. Surface water-quality modeling. Waveland press; 2008 Dec 17.
[22] Hadamard J. Thesis on the problem of analysis relating to the equilibrium of embedded elastic plates. National printing press; 1908.
[23] لوشابی محمد.کاربرد روش شبه‌معکوس‌پذیری در شناسایی منبع آلاینده در رودخانه [پایان‌نامه کارشناسی ارشد]. تهران: دانشگاه تربیت مدرس؛ 1396.
[24] Haberman R. Applied partial differential equations. Prentice Hall; 2003.
[25] Gray RM, Goodman JW. Fourier transforms: an introduction for engineers. Springer Science & Business Media; 2012 Dec 6.
[26] Polyanin AD, Nazaikinskii VE. Handbook of linear partial differential equations for engineers and scientists. CRC press; 2015 Dec 23.
[27] Santos A. Finite-size estimates of Kirkwood-Buff and similar integrals. Physical Review E. 2018 Dec 3;98(6):063302.
[28] جورک خیراله. حل عددی انتگرال‌های نوسانی [پایان‌نامه کارشناسی ارشد]. سیستان و بلوچستان:دانشگاه سیستان و بلوچستان؛ 1371.
[29] مشهدگرمه ندا، مظاهری مهدی، محمدولی سامانی جمال. حل تحلیلی معادله دوبعدی و غیرماندگار انتقال آلودگی برای شرایط اولیه و مرزی دلخواه.  هیدروفیزیک. 1398؛ 5(1):111-123.