Markov chain Monte Carlo (MCMC) methods have found widespread use in many fields of study to estimate the average properties of complex systems, and for posterior inference in a Bayesian framework. Existing theory and experiments prove convergence of well-constructed MCMC schemes to the appropriate limiting distribution under a variety of different conditions. In practice, however this convergence is often observed to be disturbingly slow. This is frequently caused by an inappropriate selection of the proposal distribution used to generate trial moves in the Markov Chain. Here we show that significant improvements to the efficiency of MCMC simulation can be made by using a self-adaptive Differential Evolution learning strategy within a population-based evolutionary framework. This scheme, entitled Differential Evolution Adaptive Metropolis or DREAM, runs multiple different chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution in randomized subspaces during the search. Ergodicity of the algorithm is proved, and various examples involving nonlinearity, highdimensionality, and multimodality show that DREAM is generally superior to other adaptive MCMC sampling approaches. The DREAM scheme significantly enhances the applicability of MCMC simulation to complex, multi-modal search problems.

[1] There is increasing consensus in the hydrologic literature that an appropriate framework for streamflow forecasting and simulation should include explicit recognition of forcing and parameter and model structural error. This paper presents a novel Markov chain Monte Carlo (MCMC) sampler, entitled differential evolution adaptive Metropolis (DREAM), that is especially designed to efficiently estimate the posterior probability density function of hydrologic model parameters in complex, high-dimensional sampling problems. This MCMC scheme adaptively updates the scale and orientation of the proposal distribution during sampling and maintains detailed balance and ergodicity. It is then demonstrated how DREAM can be used to analyze forcing data error during watershed model calibration using a five-parameter rainfall-runoff model with streamflow data from two different catchments. Explicit treatment of precipitation error during hydrologic model calibration not only results in prediction uncertainty bounds that are more appropriate but also significantly alters the posterior distribution of the watershed model parameters. This has significant implications for regionalization studies. The approach also provides important new ways to estimate areal average watershed precipitation, information that is of utmost importance for testing hydrologic theory, diagnosing structural errors in models, and appropriately benchmarking rainfall measurement devices.

In the last few decades, evolutionary algorithms have emerged as a revolutionary approach for solving search and optimization problems involving multiple conflicting objectives. Beyond their ability to search intractably large spaces for multiple solutions, these algorithms are able to maintain a diverse population of solutions and exploit similarities of solutions by recombination. However, existing theory and numerical experiments have demonstrated that it is impossible to develop a single algorithm for population evolution that is always efficient for a diverse set of optimization problems. Here we show that significant improvements in the efficiency of evolutionary search can be achieved by running multiple optimization algorithms simultaneously using new concepts of global information sharing and genetically adaptive offspring creation. We call this approach a multialgorithm, genetically adaptive multiobjective, or AMALGAM, method, to evoke the image of a procedure that merges the strengths of different optimization algorithms. Benchmark results using a set of well known multiobjective test problems show that AMALGAM approaches a factor of 10 improvement over current optimization algorithms for the more complex, higher dimensional problems. The AMALGAM method provides new opportunities for solving previously intractable optimization problems.evolutionary search ͉ multiple objectives ͉ optimization problems ͉ Pareto front E volutionary optimization is a subject of intense interest in many fields of study, including computational chemistry, biology, bioinformatics, economics, computational science, geophysics, and environmental science (1-8). The goal is to determine values for model parameters or state variables that provide the best possible solution to a predefined cost or objective function, or a set of optimal tradeoff values in the case of two or more conflicting objectives. However, locating optimal solutions often turns out to be painstakingly tedious, or even completely beyond current or projected computational capacity (9).Here, we consider a multiobjective minimization problem, with n decision variables (parameters) and m objectives:, where x denotes the decision vector, and y is the objective space. We restrict attention to optimization problems in which the parameter search space X, although perhaps quite large, is bounded: x ϭ (x 1 , . . . , x n ) ʦ X. The presence of multiple objectives in an optimization problem gives rise to a set of Pareto-optimal solutions, instead of a single optimal solution (10, 11). A Pareto-optimal solution is one in which one objective cannot be further improved without causing a simultaneous degradation in at least one other objective. As such, they represent globally optimal solutions to the tradeoff problem.Numerous approaches have been proposed to efficiently find Pareto-optimal solutions for complex multiobjective optimization problems (12-15). In particular, evolutionary algorithms have emerged as the most powerful approach for solving search and optimization pr...

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.

[1] Predictive uncertainty analysis in hydrologic modeling has become an active area of research, the goal being to generate meaningful error bounds on model predictions. State-space filtering methods, such as the ensemble Kalman filter (EnKF), have shown the most flexibility to integrate all sources of uncertainty. However, predictive uncertainty analyses are typically carried out using a single conceptual mathematical model of the hydrologic system, rejecting a priori valid alternative plausible models and possibly underestimating uncertainty in the model itself. Methods based on Bayesian model averaging (BMA) have also been proposed in the statistical and meteorological literature as a means to account explicitly for conceptual model uncertainty. The present study compares the performance and applicability of the EnKF and BMA for probabilistic ensemble streamflow forecasting, an application for which a robust comparison of the predictive skills of these approaches can be conducted. The results suggest that for the watershed under consideration, BMA cannot achieve a performance matching that of the EnKF method.

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A: CPU, central processing unit; EnKF, ensemble Kalman fi lter; GDPM, generalized dual-porosity model; KF, Kalman fi lter; LANL, Los Alamos National Laboratory; MDA, Material Disposal Area; MPI, message passing interface; MVG, Mualem-van Genuchten; PFBA, pentafl uorobenzoate; RTD, residence time distribution; SLS, simple least squares; SVE, soil vapor extraction; VOC, volatile organic compound. S S : V Z M Many of the parameters in subsurface fl ow and transport models cannot be es mated directly at the scale of interest, but can only be derived through inverse modeling. During this process, the parameters are adjusted in such a way that the behavior of the model approximates, as closely and consistently as possible, the observed response of the system under study for some historical period of me. We briefl y review the current state of the art of inverse modeling for es ma ng unsaturated fl ow and transport processes. We summariz how the inverse method works, discuss the historical background that led to the current perspec ves on inverse modeling, and review the solu on algorithms used to solve the parameter es ma on problem. We then highlight our recent work at Los Alamos related to the development and implementa on of improved op miza on and data assimila on methods for computa onally effi cient calibra on and uncertainty es ma on in complex, distributed fl ow and transport models using parallel compu ng capabili es. Finally, we illustrate these developments with three diff erent case studies, including (i) the calibra on of a fully coupled three-dimensional vapor extrac on model using measured concentra ons of vola le organic compounds in the subsurface near the Los Alamos Na onal Laboratory, (ii) the mul objec ve inverse es ma on of soil hydraulic proper es in the HYDRUS-1D model using observed tensiometric data from an experimental fi eld plot in New Zealand, and (iii) the simultaneous es ma on of parameter and states in a groundwater solute mixture model using data from a mul tracer experiment at Yucca Mountain, Nevada.