Quasi-Monte Carlo, Monte Carlo, and regularized gradient optimization methods for source characterization of atmospheric releases

An inversion technique comprising stochastic search and regularized gradient optimization was developed to solve the atmospheric source characterization problem. The inverse problem comprises retrieving the spatial coordinates, source strength, and the wind speed and wind direction at the source, given certain receptor locations and concentration values at these receptor locations. The Gaussian plume model was adopted as the forward model and derivative–based optimization was preferred to take advantage of its simple analytical nature. A new misfit functional that improves the inversion accuracy of atmospheric inverse-source problems was developed and used in the solution procedure. Stochastic search was performed over the model parameter space to identify a good initial iterate for the gradient scheme. Several Quasi-Monte Carlo point-sets were considered in the stochastic search stage and their best performance is shown to be 5 to 100 times better than the Mersenne-Twister pseudorandom generator. Newton's method with the Tikhonov stabilizer and adaptive regularization with quadratic line-search was implemented for gradient optimization. As the forward modelling and measurement errors for atmospheric inverse problems are usually unknown, issues concerning ‘model-fit’ and ‘data-fit’ were examined. In this paper, the workings and validation of the proposed approach are presented using field experiment data. Introduction The solution of inverse problems involves the retrieval of information about a physical process or phenomenon from known or observed data [48]. Inverse problems arise in various fields and hence techniques to solve such problems have been an area of extensive study. One of the contemporary applications of inversion techniques includes the source characterization problem for atmospheric contaminant dispersion. Atmospheric source characterization problems, also referred to as event reconstruction, source-inversion or inverse-source problems, comprise characterizing the source of a chemical / biological / radiological (CBR) agent released into the atmosphere. Source characterization typically involves predicting the release location and rate of the CBR agent and the meteorological conditions at the release site, based on the time-averaged concentration and wind measurements obtained from a distributed sensor network in the region of interest. In this paper, an inversion technique developed to retrieve the spatial coordinates, source strength, and the wind speed and wind Dagstuhl Seminar Proceedings 09391 Algorithms and Complexity for Continuous Problems http://drops.dagstuhl.de/opus/volltexte/2009/2299 Corresponding author: Bhagirath Addepalli, Department of Mechanical Engineering, MEB 2110, University of Utah, Salt Lake City, UT 84112, U.S.A. Page 2 direction at the source, using concentration values from known receptor locations in the domain is described. Efficient and robust event reconstruction tools can play a crucial role in the event of accidental or deliberate release of CBR agents in or close to urban centres. Under such circumstances, quick and accurate reconstruction can help government agencies evacuate people from the affected regions. Also, using the information obtained from inversion, forward models can be run to estimate the extent of the plume spread and the consequent exposure. Event reconstruction tools can also be of use to environmental monitoring agencies as they can help evaluate the contribution of the stack releases from various industries close to urban areas to the air quality within urban areas. Therefore, from the perspective of public safety and national security, a fast, robust, and accurate atmospheric event reconstruction tool is pivotal for air-quality management and to effectively deal with emergency response scenarios. Given that the subject of source characterization of atmospheric contaminant dispersion is in its infancy, researchers have examined the applicability and effectiveness of the various available inversion procedures to solve such problems. The solution methodologies used span the range of deterministic (adjoint methods), stochastic (simulated annealing (SA), genetic algorithms (GA), Bayesian inference using Markov Chain Monte Carlo sampling (MCMC)) and ‘common-sense’ methods (collector footprint methods). The inverse-source problem has been solved over local, regional and continental scales for different model parameters (m) using empirical, diagnostic and prognostic models for scalar transport as the forward operator (A). Table 1 summarizes the salient features of the inversion procedures adopted by some of the research groups across the world to solve the inverse-source problem. In this paper, an approach that has the combined benefits of stochastic search and gradient descent methods is presented. The workings of proposed approach are explained using field experiment data (The Copenhagen Tracer Experiments – TCTE) [17]. The objective of conducting stochastic search is to provide the gradient optimization scheme a good starting solution. It should be noted that the stochastic search is not a guided-search and this ensures that the misfit functional space has been adequately sampled, thereby eliminating the possibility of getting stuck in a sub-optimal region. Three strategies for solving the inverse-source problem in general and computing the ‘data-fit’ criterion for the stochastic search stage in particular are discussed. Gradient optimization is performed with the initial iterate provided by the stochastic search stage until the global minimum is reached. The ‘modelacceptancy’ criterion for the gradient scheme was based on the difference between predicted model parameters in the iteration space. The Gaussian plume dispersion model was adopted as the forward model (A) because of its theoretical and computational simplicity. Apart from the hybrid approach proposed, the present paper also investigated some of the vital aspects of the atmospheric source characterization problem when using the Gaussian plume model as the forward operator. The first feature examined was the effect of the misfit functional formulation on the accuracy and complexity of inversion. Based on this study, a new misfit functional that into account both the zero and non-zero measurements recorded by the receptors and improves the inversion accuracy of atmospheric inverse-source problems was developed and used in the solution procedure. Several Quasi-Monte Carlo point-sets were considered in the stochastic search stage and their best performance is shown to be 5 to 100 times better than the Mersenne-Twister pseudorandom generator. The choice of the descent methods (steepest descent, Newton’s, and conjugate gradient methods), stabilizing functional (Tikhonov), and the regularization parameter for gradient optimization were also examined. Gradient descent methods are an attractive choice for the current problem as analytical expressions for the Frechet and Hessian can be pre-computed for the Gaussian plume equation. For the current inverse problem, Newton's method with adaptive regularization and quadratic line-search was implemented. Since the forward modelling and measurement errors for atmospheric