Parameter estimation in nonlinear environmental problems

Popular parameter estimation methods, including least squares, maximum likelihood, and maximum a posteriori (MAP), solve an optimization problem to obtain a central value (or best estimate) followed by an approximate evaluation of the spread (or covariance matrix). A different approach is the Monte Carlo (MC) method, and particularly Markov chain Monte Carlo (MCMC) methods, which allow sampling from the posterior distribution of the parameters. Though available for years, MC methods have only recently drawn wide attention as practical ways for solving challenging high-dimensional parameter estimation problems. They have a broader scope of applications than conventional methods and can be used to derive the full posterior pdf but can be computationally very intensive. This paper compares a number of different methods and presents improvements using as case study a nonlinear DNAPL source dissolution and solute transport model. This depth-integrated semi-analytical model approximates dissolution from the DNAPL source zone using nonlinear empirical equations with partially known parameters. It then calculates the DNAPL plume concentration in the aquifer by solving the advection-dispersion equation with a flux boundary. The comparison is among the classical MAP and some versions of computer-intensive Monte Carlo methods, including the Metropolis-Hastings (MH) method and the adaptive direction sampling (ADS) method.