The ever increasing pace of computational power, along with continued advances in measurement technologies and improvements in process understanding has stimulated the development of increasingly complex hydrologic models that simulate soil moisture flow, groundwater recharge, surface runoff, root water uptake, and river discharge at different spatial and temporal scales. Reconciling these high‐order system models with perpetually larger volumes of field data is becoming more and more difficult, particularly because classical likelihood‐based fitting methods lack the power to detect and pinpoint deficiencies in the model structure. Gupta et al. (2008) has recently proposed steps (amongst others) toward the development of a more robust and powerful method of model evaluation. Their diagnostic approach uses signature behaviors and patterns observed in the input‐output data to illuminate to what degree a representation of the real world has been adequately achieved and how the model should be improved for the purpose of learning and scientific discovery. In this paper, we introduce approximate Bayesian computation (ABC) as a vehicle for diagnostic model evaluation. This statistical methodology relaxes the need for an explicit likelihood function in favor of one or multiple different summary statistics rooted in hydrologic theory that together have a clearer and more compelling diagnostic power than some average measure of the size of the error residuals. Two illustrative case studies are used to demonstrate that ABC is relatively easy to implement, and readily employs signature based indices to analyze and pinpoint which part of the model is malfunctioning and in need of further improvement.

Multi-model averaging is currently receiving a surge of attention in the atmospheric, hydrologic, and statistical literature to explicitly handle conceptual model uncertainty in the analysis of environmental systems and derive predictive distributions of model output. Such density forecasts are necessary to help analyze which parts of the model are well resolved, and which parts are subject to considerable uncertainty. Yet, accurate point predictors are still desired in many practical applications. In this paper, we compare a suite of different model averaging techniques by their ability to improve forecast accuracy of environmental systems. We compare equal weights averaging (EWA), Bates-Granger model averaging (BGA), averaging using Akaike’s information criterion (AICA), and Bayes’ Information Criterion (BICA), Bayesian model averaging (BMA), Mallows model averaging (MMA), and Granger-Ramanathan averaging (GRA) for two different hydrologic systems involving water flow through a 1950 km² watershed and 5 m deep vadose zone. Averaging methods with weights restricted to the multi-dimensional simplex (positive weights summing up to one) are shown to have considerably larger forecast errors than approaches with unconstrained weights. Whereas various sophisticated model averaging approaches have recently emerged in the literature, our results convincingly demonstrate the advantages of GRA for hydrologic applications. This method achieves similar performance as MMA and BMA, but is much simpler to implement and use, and computationally much less demanding.

Although plant photosynthetic capacity as determined by the maximum carboxylation rate (i.e., V_c, max25) and the maximum electron transport rate (i.e., J_max25) at a reference temperature (generally 25 °C) is known to vary considerably in space and time in response to environmental conditions, it is typically parameterized in Earth system models (ESMs) with tabulated values associated with plant functional types. In this study, we have developed a mechanistic model of leaf utilization of nitrogen for assimilation (LUNA) to predict photosynthetic capacity at the global scale under different environmental conditions. We adopt an optimality hypothesis to nitrogen allocation among light capture, electron transport, carboxylation and respiration. The LUNA model is able to reasonably capture the measured spatial and temporal patterns of photosynthetic capacity as it explains ∼ 55 % of the global variation in observed values of V_c, max25 and ∼ 65 % of the variation in the observed values of J_max25. Model simulations with LUNA under current and future climate conditions demonstrate that modeled values of V_c, max25 are most affected in high-latitude regions under future climates. ESMs that relate the values of V_c, max25 or J_max25 to plant functional types only are likely to substantially overestimate future global photosynthesis.

Formal and informal Bayesian approaches have found widespread implementation and use in environmental modeling to summarize parameter and predictive uncertainty. Successful implementation of these methods relies heavily on the availability of efficient sampling methods that approximate, as closely and consistently as possible the (evolving) posterior target distribution. Much of this work has focused on continuous variables that can take on any value within their prior defined ranges. Here, we introduce theory and concepts of a discrete sampling method that resolves the parameter space at fixed points. This new code, entitled DREAM_(D) uses the recently developed DREAM algorithm (Vrugt et al., 2008, 2009a, b) as its main building block but implements two novel proposal distributions to help solve discrete and combinatorial optimization problems. This novel MCMC sampler maintains detailed balance and ergodicity, and is especially designed to resolve the emerging class of optimal experimental design problems. Three different case studies involving a Sudoku puzzle, soil water retention curve, and rainfall – runoff model calibration problem are used to benchmark the performance of DREAM_(D). The theory and concepts developed herein can be easily integrated into other (adaptive) MCMC algorithms.

The quest for a more powerful method for model evaluation has inspired Vrugt and Sadegh (2013) to introduce “likelihood‐free” inference as vehicle for diagnostic model evaluation. This class of methods is also referred to as Approximate Bayesian Computation (ABC) and relaxes the need for a residual‐based likelihood function in favor of one or multiple different summary statistics that exhibit superior diagnostic power. Here we propose several methodological improvements over commonly used ABC sampling methods to permit inference of complex system models. Our methodology entitled DREAM_(ABC)uses the DiffeRential Evolution Adaptive Metropolis algorithm as its main building block and takes advantage of a continuous fitness function to efficiently explore the behavioral model space. Three case studies demonstrate that DREAM_(ABC)is at least an order of magnitude more efficient than commonly used ABC sampling methods for more complex models. DREAM_(ABC)is also more amenable to distributed, multi‐processor, implementation, a prerequisite to diagnostic inference of CPU‐intensive system models.

Nonstationary extreme value analysis (NEVA) can improve the statistical representation of observed flood peak distributions compared to stationary (ST) analysis, but management of flood risk relies on predictions of out‐of‐sample distributions for which NEVA has not been comprehensively evaluated. In this study, we apply split‐sample testing to 1250 annual maximum discharge records in the United States and compare the predictive capabilities of NEVA relative to ST extreme value analysis using a log‐Pearson Type III (LPIII) distribution. The parameters of the LPIII distribution in the ST and nonstationary (NS) models are estimated from the first half of each record using Bayesian inference. The second half of each record is reserved to evaluate the predictions under the ST and NS models. The NS model is applied for prediction by (1) extrapolating the trend of the NS model parameters throughout the evaluation period and (2) using the NS model parameter values at the end of the fitting period to predict with an updated ST model (uST). Our analysis shows that the ST predictions are preferred, overall. NS model parameter extrapolation is rarely preferred. However, if fitting period discharges are influenced by physical changes in the watershed, for example from anthropogenic activity, the uST model is strongly preferred relative to ST and NS predictions. The uST model is therefore recommended for evaluation of current flood risk in watersheds that have undergone physical changes. Supporting information includes a MATLAB^® program that estimates the (ST/NS/uST) LPIII parameters from annual peak discharge data through Bayesian inference.