Cramer-Rao Lower Bound for a Gas Source Localization Based on Poisson's Equation

This paper focuses on the performance bounds of methods for Gas Source Localization (GSL) that utilize partial differential equations (PDE) for gas dispersion modeling. Specif-ically, Poisson's equation in 2D is used as a simplified gas dispersion model, with the right-hand side modeling a source as a Dirac measure with unknown support. By utilizing the Green's function method for solving the PDE and embedding the solution into the Sparse Bayesian Learning framework, a likelihood function of the source location parameters is formulated. Using the latter the corresponding Cramer-Rao lower bound for the variance of the source location estimate is derived and analyzed. The computed bound not only provides a theoretical quantification of the gas source localization precision but also allows deriving information-seeking criteria for sampling schemes that identify sampling locations with increased information content. The dependency of the bound as well as of the information-seeking criteria on some model parameters are studied with synthetic examples.