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Inverse problems aim at determining model parameters that produce observed data to subsequently understand, predict or manage hydrological or other environmental systems. While statistical inversion is especially popular, its sampling‐based nature often inhibits its application to computationally costly models, which has compromised the use of the Generalized Likelihood Uncertainty Estimation (GLUE) methodology, for example, for spatially distributed (partial) differential equation based models. In this study we introduce multilevel GLUE (MLGLUE), which alleviates the computational burden of statistical inversion by utilizing a hierarchy of model resolutions. Inspired by multilevel Monte Carlo, most parameter samples are evaluated on lower levels with computationally cheap low‐resolution models and only samples associated with a likelihood above a certain threshold are subsequently passed to higher levels with costly high‐resolution models for evaluation. Inferences are made at the level of the highest‐resolution model but substantial computational savings are achieved by discarding samples with low likelihood already on levels with low resolution and low computational cost. Two example inverse problems, using a rainfall‐runoff model and groundwater flow model, demonstrate the substantially increased computational efficiency of MLGLUE compared to GLUE as well as the similarity of inversion results. Findings are furthermore compared to inversion results from Markov‐chain Monte Carlo (MCMC) and multilevel delayed acceptance MCMC, a corresponding multilevel variant, to compare the effects of the multilevel extension. All examples demonstrate the wide‐range suitability of the approach and include guidelines for practical applications. Key Points: We extend the Generalized Likelihood Uncertainty Estimation methodology to a strategy that considers multiple resolution levels (MLGLUE)We demonstrate the acceleration with MLGLUE for different spatial (groundwater flow model) and temporal (rainfall‐runoff model) resolutionsWe show that inversion results from multilevel and conventional methods align and discuss factors affecting computational efficiency [ABSTRACT FROM AUTHOR]

Abstract: Selecting a “best” model among several competing candidate models poses an often encountered problem in water resources modeling (and other disciplines which employ models). For a modeler, the best model fulfills a certain purpose best (e.g., flood prediction), which is typically assessed by comparing model simulations to data (e.g., stream flow). Model selection methods find the “best” trade‐off between good fit with data and model complexity. In this context, the interpretations of model complexity implied by different model selection methods are crucial, because they represent different underlying goals of modeling. Over the last decades, numerous model selection criteria have been proposed, but modelers who primarily want to apply a model selection criterion often face a lack of guidance for choosing the right criterion that matches their goal. We propose a classification scheme for model selection criteria that helps to find the right criterion for a specific goal, i.e., which employs the correct complexity interpretation. We identify four model selection classes which seek to achieve high predictive density, low predictive error, high model probability, or shortest compression of data. These goals can be achieved by following either nonconsistent or consistent model selection and by either incorporating a Bayesian parameter prior or not. We allocate commonly used criteria to these four classes, analyze how they represent model complexity and what this means for the model selection task. Finally, we provide guidance on choosing the right type of criteria for specific model selection tasks. (A quick guide through all key points is given at the end of the introduction.) [ABSTRACT FROM AUTHOR]