Karst water resources are valuable freshwater sources for around 10 % of the world population. Nonetheless, anthropogenic factors and global changes have been seriously deteriorating the karst water quality and dependent ecosystems. Solute transport models are powerful tools to monitor, control, and manage the water quality and dependent ecosystem functioning. By representing and predicting the spatiotemporal behavior of solute migration in karst systems, the transport models enhance our understanding about the karst transport processes, thus enabling us to explore contamination risks and potential outcomes. This paper reviews the current state of knowledge on the modelling of solute transport processes in karst aquifers, thereby unveiling the fundamental challenges underlying a successful karst transport modelling. We discuss to what extent and in what ways we can handle these challenges and derive the key challenges and directions for reliable modelling of transport processes in karst systems in the present context of global changes.

Inverse problems are ubiquitous in hydrological modelling for parameter estimation, system understanding, sustainable water resources management, and the operation of digital twins. While statistical inversion is especially popular, its sampling-based nature often inhibits the inversion of computationally costly models, which has compromised the use of the Generalized Likelihood Uncertainty Estimation (GLUE) methodology, e.g., 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 test problems demonstrate the similarity of inferred parameter posteriors and uncertainty estimates of MLGLUE and GLUE as well as increased computational efficiency. Findings are furthermore compared to inversion results from Markov-chain Monte Carlo (MCMC) and from multilevel delayed acceptance MCMC. The computation time of inversion of a groundwater flow model was decreases by ≈45% and ≈57% when using MLGLUE instead of conventional formulations of GLUE and MCMC, respectively.

Inverse problems are ubiquitous in hydrological modelling for parameter estimation, system understanding, sustainable water resources management, and the operation of digital twins. 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, e.g., 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.