Modern search and optimization methods rely on classical measures of the quality of fit to support model-data synthesis. The limited guidance classical measures of quality of fit provide on model misspecification has led Gupta et al. (2008) to propose steps toward a more robust and powerful method of model evaluation. This so-called diagnostic approach quantifies model performance in ways that correspond to major behavioral functions of the watershed. These functions and/or patterns of watershed behavior, or hydrologic signatures, represent unique, recurring and measurable aspects of the streamflow hydrograph and may require the definition of multiple summary metrics to be meaningfully characterized. Diagnostic evaluation then proceeds with an analysis of the similarities and differences between the observed and simulated signatures. Ideally, these signatures are related to individual process descriptions and therefore help guide model improvements in a more meaningful way. The diagnostic approach to model evaluation lacks a rigorous statistical underpinning. The aim of this dissertation is to improve the statistical foundation of model diagnostics to help reduce type I errors (failure to reject “bad” models) and type II errors (falsely rejecting a “good” model).

In the first part of this dissertation, we are concerned with the characterization of the uncertainty of hydrologic signatures and the selection of robust signature formulations for diagnostic analysis. As the signatures are numeral descriptors of the discharge time series, their uncertainty stems from streamflow uncertainty. Thus, we first introduce a relatively simple data-driven method for the representation of the uncertainty in daily discharge records (Chapter 2). The proposed method relies only on hourly discharge data and takes advantage of a nonparametric difference-based estimator in the characterization of random errors in discharge time series. This procedure produces replicates of the discharge record that portray accurately the assigned streamflow uncertainty, preserve key statistical properties of the discharge record and are hydrologically realistic. Next, we address signature selection and investigate the sensitivity of hydrologic signatures to aleatory errors and to the length and period used in their computation (Chapter 3). Our results identify robustness problems in the investigated signatures and reveal that signature uncertainty stemming from aleatory errors is rather small. Chapter 4 consequently presents a more complete treatment of the uncertainty in discharge records by considering rating curve uncertainty. This allows us to provide a rigorous description of the uncertainty in hydrologic signatures, which are subsequently used in the evaluation of a previously calibrated hydrologic model. This approach, however, does not explore the full potential of a given model structure in reproducing a set of signatures. In the second part of this dissertation, we investigate the use of hydrologic signatures within a Bayesian framework for the calibration and evaluation of hydrologic models with the aim of shedding light on model structural errors and improving model fidelity. Its successful application relies on robust estimates of signature uncertainty and the use of an adequate likelihood function. Thus, we start by formulating distribution-adaptive likelihood functions and evaluating their use in uncertainty quantification of hydrologic models (Chapter 5). Next, we present a hydrologic modeling toolbox that allows many hydrologic models to be built through the combination of alternative model structures and process representations (Chapter 6). The toolbox supports our analysis of model structural errors by enabling any deficient part of a model to be easily switched. Finally, in Chapter 7, we evaluate the use of a method that couples traditional Bayesian inference with summary metrics, coined diagnostic Bayes, and illustrate the potential of this approach to improve model consistency.