In the past decades, Bayesian methods have found widespread application and use in environmental systems modeling. Bayes theorem states that the posterior probability of a hypothesis is proportional to the product of the prior probability of this hypothesis and the likelihood of the hypothesis given the new/incoming observations. In science and engineering, this hypothesis often constitutes some numerical simulation model, which summarizes using algebraic, empirical, and differential equations, state variables and fluxes, all our theoretical and/or practical knowledge of the system of interest, and unknown model parameters which are subject to inference using some data of the observed system response. The Bayesian approach is intimately related to the scientific method and uses an iterative cycle of hypothesis formulation (model), experimentation and data collection, and theory/hypothesis refinement to elucidate the rules that govern the natural world. Unfortunately, model refinement has proven to be very difficult in large part because of the poor diagnostic power of residual based likelihood functions.

In this thesis, I present seven different chapters (publications) that introduce the theory, concepts and implementation of a diagnostic approach to model identification and evaluation. This approach, coined approximate Bayesian computation (ABC), relaxes the need for an explicit likelihood function in favor of one or more summary statistics, which when rooted in the relevant environmental theory have a much stronger and compelling diagnostic power than some average measure of the size of the error residuals. The proposed methodology is statistically coherent, and provides new insights into and guidance on model structural (epistemic) errors, model (hypothesis) refinement, system nonstationarity, and the information content of experimental data. What is more, the proposed diagnostic approach provides a much-needed statistical underpinning of the popular generalized likelihood uncertainty framework (GLUE) of Beven and co-workers, and presents a powerful alternative to subjective regularized inversion methods used in geophysical inversion. Finally, the DREAM(ABC) algorithm, developed to solve the diagnostic model evaluation problem, is orders of magnitude more efficient than commonly used ABC sampling methods, thereby permitting inference of parameter-rich system models.