Anti-diffusion and source identification with the ‘RAW’ scheme: A particle-based censored random walk

Abstract

This paper presents a novel methodology for time reversal in advective-diffusive pollutant transport in groundwater systems and other environmental flow systems (specifically: time reversal of diffusive terms). The method developed in this paper extends previous particle-based approaches like the Reversed Time Particle Tracking Method of Bagtzoglou [6]. The reversal of the 'diffusive' and/or 'macrodispersive' component of pollutant migration is especially under focus here. The basis of the proposed scheme for anti-diffusion is a continuous time, censored, non-local random walk capable of tracking groundwater solute concentration profiles over time while conserving the (reverse) Fickian properties of the anti-diffusing particle cloud in terms of moments. This scheme is an alternative to the direct solution of the eulerian concentration-based diffusion PDE, which is notoriously unstable in reverse time. Our analysis leads to the conclusion that an adaptive time stepping scheme-with decreasing time step-is necessary in order to maintain a constant amount of anti-diffusion (the reverse form of Fick's law). Specifically, we study the relations between the following parameters: time step evolution vs. time (or vs. number of steps); variance evolution (decrease rate); total time (or number of steps) required to reach a fully anti-diffused solution. The proposed approach is shown to be quite efficient; typically, for every ten time steps, one to two orders of magnitude reduction of the dispersion width of the plume can be attained. Furthermore, the method is shown to be asymptotically exact for reverse Fickian diffusion. The method is applied with success to several situations involving the diffusive transport of a conservative solute in the following cases: (i) Single source recovery in onedimensional space with constant diffusion parameters (this example serves as a validation test for the theory); (ii) Single source recovery in two-dimensional space with constant isotropic diffusion (this example also serves as a test for the theory); (iii) Multiple source recovery R. Ababou (B) · A. Mallet 123 42 Environ Fluid Mech (2010) 10:41-76in two-dimensional space, assuming isotropic diffusion. It is expected that the methodology tested in this paper is applicable more generally to complex environmental pollution problems involving multiple sources, anisotropic hydrodynamic dispersion, and space-time variable advection-diffusion flow systems; the modeling of reverse diffusion/dispersion in such systems is currently under investigation.

Optimal Unknown Pollution Source Characterization in a Contaminated Groundwater Aquifer—Evaluation of a Developed Dedicated Software Tool

“…Some of the initial contributions in identification of unknown groundwater pollution sources proposed the use of linear optimization model based on linear response matrix approach (Gorelick et al, 1983) and statistical pattern recognition (Datta et al, 1989). Some of the important contributions to solve the unknown groundwater pollution sources identification problem include: non-linear maximum likelihood estimation based inverse models to determine optimal estimates of the unknown model parameters and source characteristics (Wagner, 1992); minimum relative entropy, a gradient based optimization for solving source identification problems (Woodbury et al, 1998); embedded nonlinear optimization technique for source identification (Mahar and Datta, 1997; inverse procedures based on correlation coefficient optimization (Sidauruk et al, 1997); Genetic Algorithm (GA) based approach (Aral et al, 2001;Singh & Datta, 2006); Artificial Neural Network (ANN) approach , 2007; constrained robust least square approach (Sun et al, 2006); classical optimization based approach (Datta et al, 2009a; inverse particle tracking approach (Bagtzoglou, 2003;Ababou et al, 2010); heuristic harmony search for source identification (Ayvaz, 2010); Simulated Annealing (SA) as optimization for source identification (Jha & Datta, 2011;Prakash & Datta, 2012, 2013, 2014a. A review of different optimization techniques for solving source identification problem is presented in Chadalavada et al (2011) and Amirabdollahian and Datta (2013).…”

Estimation of river pollution source using the space-time radial basis collocation method

“…[9]. Commonly employed techniques include the Tikhonov regularization method [38], the nonlinear optimization model [2], particlebased methods [1,7], the backward beam equation method [3,4,8] and geostatistical based methods [6,32]. Some other newly emerged techniques have also been under intensive investigation, such as artificial neural networks [37], a global search algorithm with a constrained least squares estimator [40], geostatistical kriging [36] and iterative regularization [23].…”

Optimal characterization of pollutant sources in contaminated aquifers by integrating sequential-monitoring-network design and source identification: methodology and an application in Australia

“…Some of the important contributions include: reverse-time random particle method for finding the most probable source (Bagtzoglou et al 1992); inverse models based on nonlinear maximum-likelihood estimation to determine optimal estimates of the unknown model parameters and source characteristics (Wagner 1992); minimum relative entropy, a gradient-based optimization for solving source identification problems (Woodbury and Ulrych 1996;Woodbury et al 1998); embedded non-linear optimization technique for source identification (Mahar and Datta 1997; inverse procedures based on correlation coefficient optimization (Sidauruk et al 1997); Tikhonov regularization (Skaggs and Kabala 1994;Liu and Ball 1999); quasi-reversibility (Skaggs and Kabala 1995;Bagtzoglou and Atmadja 2003); marching-jury backward beam equation (Atmadja and Bagtzoglou 2001a;Bagtzoglou and Atmadja 2003;Bagtzoglou and Baun 2005); genetic algorithm (GA) based approach (Aral et al 2001;Mahinthakumar and Sayeed 2005;Singh and Datta 2006); artificial neural network (ANN) approach Datta 2004, 2007;); constrained robust least square approach (Sun et al 2006a, b); classical optimization based approach (Datta et al 2009a; probabilistic support vector machines (PSVMs) and probabilistic neural networks (PNNs) based probabilistic models ; inverse particle tracking approach (Bagtzoglou 2003;Ababou et al 2010); heuristic harmony search for source identification (Ayvaz 2010); simulated annealing (SA) as optimization for source identification (Jha and Datta 2011, 2012bPrakash and Datta 2014a). A review of different optimization techniques for solving source identification problem is presented in Chadalavada et al (2011) and Amirabdollahian and Datta (2013).…”