Statistical estimation and nonlinear filtering in environmental pollution

Motivated by the water pollution detection, this paper studies a nonlinear filtering problem over an infinite time interval. The signal to be estimated, which indicates the concentration of undesired chemical in a river, is driven by a stochastic partial differential equation involves unknown parameters. Based on discrete observation, strongly consistent estimators of unknown parameters are derived at first. With the optimal filter given by Bayes formula, the uniqueness of invariant measure for the signal-filter pair has been verified. The paper then establishes approximation to the optimal filter with estimators, showing that the pathwise average distance, per unit time, of the computed approximating filter from the optimal filter converges to zero in probability. Simulation results are presented at last.