Theory for Reconstruction of an Unknown Number of Contaminant Sources using Probabilistic Inference

Abstract

We address the inverse problem of source reconstruction for the difficult case of multiple sources when the number of sources is unknown a priori. The problem is solved using a Bayesian probabilistic inferential framework in which Bayesian probability theory is used to derive the posterior probability density function for the number of sources and for the parameters (e.g., location, emission rate, release time and duration) that characterize each source. A mapping (source-receptor relationship) that relates a multiple source distribution to the concentration measurements made by an array of detectors is formulated based on a forward-time Lagrangian stochastic model. A computationally efficient methodology for determination of the likelihood function for the problem, based on an adjoint representation of the source-receptor relationship and realized in terms of a backward-time Lagrangian stochastic model, is described. An efficient computational algorithm based on a parallel tempered Metropolis-coupled reversible-jump Markov chain Monte Carlo (MCMC) method is formulated and implemented to draw samples from the posterior probability density function of the source parameters. This methodology allows the MCMC method to initiate jumps between the hypothesis spaces corresponding to different numbers of sources in the source distribution and, thereby, allows a sample from the full joint posterior distribution of the number of sources and the parameters for each source to be obtained. The proposed methodology for source reconstruction is tested using synthetic concentration data generated for cases involving two and three unknown sources.

Evaluation of a new inference method for estimating ammonia volatilisation from multiple agronomic plots

“…Mukherjee et al (2015) highlighted the dependency of the inferred source on background concentration and plot disposition by means of an inverse footprint approach. Yee et al (2008) have shown how to retrieve the number, location and intensity of multiple sources with dispersion models coupled with Bayesian inference methods. Yee and Flesch (2010) have evaluated the inversion and inference methods for determining four point sources using several laser transects.…”

Locating and quantifying greenhouse gas emissions at a geological CO2storage site using atmospheric modeling and measurements

“…In our current formulation, we assume implicitly that the number of localized sources (Ns) is known a priori (viz., Ns is a fixed quantity that does not need to be estimated). The significantly more difficult problem (not considered herein) of the reconstruction of an a priori unknown number of localized sources was studied by Yee [2008Yee [ , 2010 as a generalized parameter estimation problem. Note that in this case, the source parameter vector q consists of the characteristics (e.g., location and emission rate) for each source as well as Ns, and the dimensionality of the source parameter space depends on Ns.…”

“…Note that in this case, the source parameter vector q consists of the characteristics (e.g., location and emission rate) for each source as well as Ns, and the dimensionality of the source parameter space depends on Ns. To address this problem, Yee [2008Yee [ , 2010 used a reversible jump Markov chain Monte Carlo algorithm to sample from the generalized parameter space, allowing changes in the dimensionality of the model (viz., changes in Ns). An alternative (and arguably simpler) approach to address the problem of an unknown number of sources is to treat the problem as a model selection problem rather than as a generalized parameter estimation problem.…”

“…The equilibrium distribution of the generated samples corresponds to the posterior PDF, which can then be used to determine summary statistics related to each source parameter. Past applications of MCMC include: Keats et al [2007] and Chow et al [2008] in urban environments, Keats et al [2009] for source apportionment for particulate matter, and Yee [2008Yee [ , 2010Yee [ , 2012 for source reconstruction for an a priori unknown number of sources.…”

Control of emission rates

“…The method allows identifying the main sources of excessive pollution in a selected zone (residential area, park, forest, etc.). In particular, it can be used for the evaluation of pollution levels due to oil spills (Skiba, 1996(Skiba, , 1999Skiba and Parra-Guevara, 1999;Dang et al, 2012), or to vehicular emissions along the main roads (Skiba and Davydova-Belitskaya, 2003;Chiou and Chen, 2010;Li et al, 2012b;Shafiq and Iqbal, 2012); for estimating parameters which describe the source location and strength (Keats et al, 2007a, b); for the detection of industrial plants which violate the emission rates, prescribed by some control strategy (Skiba, 2003); for the reconstruction of an unknown number of contaminant sources (Yee, 2008); or for the optimal location of a new industrial enterprise, so that its operation will not violate health standards in ecologically most important zones (Marchuk, 1982(Marchuk, , 1986Skiba et al, 2005). The method can also be used to install safety devices in high-risk areas to prevent accidents or unauthorized discharges of contaminants and design emission control strategies for already existing industries (Penenko and Raputa, 1983;Jhih-Shyang, 1998;Parra-Guevara and Skiba, 2000a, b;Zundel and Rentz, 1995;Zhou, 2008, 2009).…”