Multi-criteria conditioning of integrated distributed hydrological models within GLUE: behavioural-solution discrimination, error-source separation, grid-scale choice, and calibration-data design

GLUE (Generalised Likelihood Uncertainty Estimation) conditions predictions of environmental models on observed data by retaining parameter sets that give behavioural simulations and forming uncertainty bounds from them, rather than by fitting a single optimal parameter set. Blasone, Madsen and Rosbjerg (2008) combined GLUE with Markov chain Monte Carlo sampling (GLUE-MCMC, driven by the SCEM-UA sampler) to condition the MIKE SHE integrated surface-water/groundwater model of the Karup catchment, Denmark, on discharge and groundwater head simultaneously through an aggregated two-criterion objective. They found that parameter uncertainty alone cannot explain total prediction error at all sites when systematic biases are present, and they left open four methodological issues: explicit representation and separation of non-parameter error sources within GLUE, a practical criterion for choosing model grid size, a design for incorporating spatially distributed discharge data in calibration, and — raised in their methods background as a possible future research direction — Pareto-based rather than scalar-aggregated discrimination of behavioural solutions. The task is to develop and demonstrate a methodology that addresses these four open issues.

Background

Hydrological models used for catchment-scale water assessment are often integrated and distributed: they simulate surface water and groundwater together and divide the catchment into grid cells. MIKE SHE, the model used in the source study, is a physically based example, simulating overland flow, unsaturated-zone flow, and groundwater flow on a $1\,\mathrm{km}^2$ grid for the Danish Karup catchment. Because such models have many uncertain parameters, their predictions are usually reported as uncertainty bounds rather than single values.

GLUE, the Generalised Likelihood Uncertainty Estimation framework introduced by Beven and Binley (1992), is the most widely used informal method for producing those bounds. Instead of estimating one optimal parameter set, GLUE runs the model over many candidate parameter sets, retains those whose simulations are acceptably close to observations (the behavioural sets, judged by a likelihood or performance measure above a subjectively chosen threshold), weights them, and forms prediction limits from their spread. Its informality is deliberate — it avoids assuming a statistically correct error model — but it inherits the criticism (Stedinger et al., 2008) that the bounds have no formal probabilistic interpretation. Markov chain Monte Carlo (MCMC) sampling, here the SCEM-UA algorithm (a self-adaptive metropolis-type sampler that evolves multiple chains toward the posterior/behavioural region), was combined with GLUE by Blasone and colleagues to sample behavioural sets efficiently: the GLUE-MCMC method of the source study.

The source study (Blasone, Madsen & Rosbjerg, 2008) conditioned MIKE SHE simultaneously on catchment-outlet discharge and groundwater heads in wells, combining the two criteria through the equally weighted aggregate $F_i = \mathrm{RMSE}_q(\theta_i)/\min\{\mathrm{RMSE}_q\} + \mathrm{RMSE}_h(\theta_i)/\min\{\mathrm{RMSE}_h\}$, selected after MCMC experiments whose results were inspected in the Pareto space defined by the two calibration criteria so that the sampler converges to the region of more balanced solutions (cf. multi-objective calibration, Madsen, 2003, where the Pareto trade-off surface is the set of parameter sets where no criterion can improve without another worsening). Two findings motivate the open issues. First, even after calibration, prediction bounds conditioned on parameter uncertainty failed to cover observations at all sites — systematic bias remained, notably in groundwater heads, partly because point measurements in wells were compared with $1\,\mathrm{km}^2$ grid-cell averages. Second, validation deteriorated at smaller subcatchments when only outlet discharge was used in calibration. Three unresolved issues are stated in the authors' discussion and conclusions: the proper choices of parameterization, sampling scheme and likelihood-function aggregation "can only reduce, but cannot remove, the effect of other errors" (measurement, structural, and scaling errors are not explicitly represented or separated, and distributed data at most allow "making hypothesis on the nature of the possible calibration problems, i.e. whether they might arise because of data, model or parameter errors"); it is "very important to properly define the size of the computational grid of the model", against computational time and model-stability limits, without any practical refinement criterion being given; and, for discharge, "we recommend to include spatial information from the subcatchments in the calibration process", without any design or aggregation strategy. A fourth remark appears in the methods background rather than in the discussion and conclusions: criteria may also be evaluated separately by "ranking the parameter sets according to the Pareto criterion" (Madsen, 2003), which "according to the knowledge of the authors ... has never been used in conjunction with GLUE to discriminate behavioural solutions; however, this might be a future area of research". Twenty years of GLUE reflection (Beven & Binley, 2014) still treats structural-error treatment as the framework's central open issue, and the adjacent literatures — formal Bayesian error analysis, multi-objective calibration, resolution-effect studies, and gauging-design studies — have each progressed without delivering GLUE-internal solutions to these issues.

Problem Statement

The problem is to develop and demonstrate a coherent methodology for multi-criteria conditioning of integrated distributed hydrological models within the GLUE framework, addressing the four issues that Blasone, Madsen and Rosbjerg (2008) left open in their GLUE-MCMC study of the MIKE SHE model of the Karup catchment (Denmark), for models that couple surface water and groundwater and are discretised on a spatial grid, conditioned on more than one observed variable (e.g. stream discharge and groundwater head):

1. Error-source separation within GLUE. Parameter uncertainty is insufficient to explain total predictive uncertainty when systematic biases occur. The methodology must explicitly represent and separate the additional error sources — measurement error, model structural error, and the scaling mismatch between point observations (e.g. groundwater heads measured in wells) and grid-cell averages (e.g. over $1\,\mathrm{km}^2$ cells) — instead of only widening prediction bounds by implicitly including poorly converging (pre-convergence) parameter samples, and instead of leaving those sources diagnosable only as hypotheses about "whether they might arise because of data, model or parameter errors". The separation must attain a quantitative floor on the demonstration model: the point-versus-grid scaling mismatch is to be separated quantitatively, and each of the other two sources must either be separated quantitatively on that model as well or be accompanied by a quantitative bound on its contribution to the prediction limits together with its stated assumptions.

2. Grid-scale choice. Groundwater-head bias in the source study arose partly from comparing point observations with $1\,\mathrm{km}^2$ grid-cell averages. The methodology must include a practical, stateable criterion for choosing the model grid size — balancing the more precise estimates expected from finer discretisation against computational cost and numerical stability constraints — and must cover the alternative the source explicitly allows: when the model scale cannot be reduced, the biases on the model outputs are accepted and the calibration focuses instead on reproducing the response dynamics well.

3. Calibration-data design for distributed discharge. Spatial validation at subcatchments degraded when only outlet discharge was used in calibration. The methodology must specify how spatially distributed discharge data enter calibration — the source recommends including spatial information from the subcatchments in the likelihood and notes that the locations used for calibration and validation must have similar characteristics and reproduce phenomena occurring at similar scales — together with an aggregation strategy for the spatially nested outlet and subcatchment discharge objectives.

4. Discrimination of behavioural solutions by the Pareto criterion. The source study aggregated its two criteria into the equally weighted scalar function $F_i$ described above. Its methods background notes that criteria can instead be evaluated separately by ranking parameter sets according to the Pareto criterion (multi-objective calibration in the sense of Madsen, 2003), which to the authors' knowledge had never been used in conjunction with GLUE to discriminate behavioural solutions, and which they suggest as a possible future area of research. The open question is how such Pareto-based discrimination can be used within GLUE for multi-criteria conditioning of an integrated distributed model: the behavioural set then consists of the non-dominated parameter sets — a set is non-dominated if no other set is at least as good on every criterion and strictly better on at least one, the criteria being minimised error measures such as $\mathrm{RMSE}_q$ and $\mathrm{RMSE}_h$ — which is the Pareto trade-off surface described in the Background, not a scalar ranking, so the way GLUE prediction limits are formed from such a set is part of the question.

The methodology is to be demonstrated on at least one integrated distributed hydrological model conditioned on more than one observed variable (e.g. discharge and groundwater head), as in the source study. The source study's MIKE SHE model of the Karup catchment is commercial and its setup is not publicly available, so the demonstration model may instead be a synthetic coupled surface-water/groundwater model constructed by the solver — discretised on a grid, runnable at more than one grid resolution, and providing at least groundwater-head observations and discharge observations at more than one spatial scale — or any offline-accessible integrated distributed model providing the same. All four issues are mandatory and share the same completion standard: each must be addressed by a defined method, criterion or construction — not a discussion of alternatives — with at least one worked demonstration of that method on the model used for the overall demonstration; treating an issue only conceptually, or resolving fewer than all four, does not complete the task. The source raises the four issues separately and does not require them to be resolved in a single joint workflow, but they share objects, so the levels must be kept distinct: issue 3 concerns how spatially nested discharge objectives (outlet versus subcatchments) enter the likelihood, whereas issue 4 concerns whether the resulting calibration criteria are used to discriminate behavioural sets by scalar aggregation or by Pareto non-dominance. The study condition retained from the source is GLUE's informal likelihood framing: no formally correct statistical error model is assumed, so formal Bayesian error analyses are progress evidence for issue 1 rather than answers within that framing.

References

1. Blasone, R.-S., Madsen, H., & Rosbjerg, D. (2008). Uncertainty assessment of integrated distributed hydrological models using GLUE with Markov chain Monte Carlo sampling. Journal of Hydrology, 353(1-2), 18-32. https://doi.org/10.1016/j.jhydrol.2007.12.026
2. Blasone, R.-S., Vrugt, J. A., Madsen, H., Rosbjerg, D., Robinson, B. A., & Zyvoloski, G. A. (2008). Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlo sampling. Advances in Water Resources, 31(4), 630-648. https://doi.org/10.1016/j.advwatres.2007.12.003
3. Beven, K., & Binley, A. (1992). The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6(3), 279-298. https://doi.org/10.1002/hyp.3360060305
4. Beven, K., & Binley, A. (2014). GLUE: 20 years on. Hydrological Processes, 28(24), 5897-5918. https://doi.org/10.1002/hyp.10082
5. Madsen, H. (2003). Parameter estimation in distributed hydrological catchment modelling using automatic calibration with multiple objectives. Advances in Water Resources, 26(2), 205-216. https://doi.org/10.1016/S0309-1708(02)00092-1
6. Stedinger, J. R., Vogel, R. M., Lee, S. U., & Batchelder, R. (2008). Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research, 44(12). https://doi.org/10.1029/2008WR006822
7. Efstratiadis, A., & Koutsoyiannis, D. (2010). One decade of multi-objective calibration approaches in hydrological modelling: a review. Hydrological Sciences Journal, 55(1), 58-78. https://doi.org/10.1080/02626660903526292
8. Renard, B., Kavetski, D., Kuczera, G., Thyer, M., & Franks, S. W. (2010). Understanding predictive uncertainty in hydrologic modeling: The challenge of identifying input and structural errors. Water Resources Research, 46(5). https://doi.org/10.1029/2009WR008328
9. Wu, B., Zheng, Y., Tian, Y., Wu, X., Yao, Y., Han, F., Liu, J., & Zheng, C. (2014). Systematic assessment of the uncertainty in integrated surface water-groundwater modeling based on the probabilistic collocation method. Water Resources Research, 50(7), 5848-5865. https://doi.org/10.1002/2014WR015366
10. Leta, O. T., Nossent, J., Velez, C., Shrestha, N. K., van Griensven, A., & Bauwens, W. (2015). Assessment of the different sources of uncertainty in a SWAT model of the River Senne (Belgium). Environmental Modelling & Software, 68, 129-146. https://doi.org/10.1016/j.envsoft.2015.02.010
11. Shafii, M., Tolson, B. A., & Matott, L. S. (2015). Addressing subjective decision-making inherent in GLUE-based multi-criteria rainfall-runoff model calibration. Journal of Hydrology, 523, 693-705. https://doi.org/10.1016/j.jhydrol.2015.01.051
12. von Gunten, D., Wöhling, T., Haslauer, C., Merchán, D., Causapé, J., & Cirpka, O. A. (2014). Efficient calibration of a distributed PDE-based hydrological model using grid coarsening. Journal of Hydrology, 519, 3290-3304. https://doi.org/10.1016/j.jhydrol.2014.10.025
13. Zhou, R., Li, Y., Lu, D., Liu, H., & Zhou, H. (2016). An optimization based sampling approach for multiple metrics uncertainty analysis using generalized likelihood uncertainty estimation. Journal of Hydrology, 540, 274-286. https://doi.org/10.1016/j.jhydrol.2016.06.030
14. Seibert, J., & Beven, K. J. (2009). Gauging the ungauged basin: how many discharge measurements are needed? Hydrology and Earth System Sciences, 13(6), 883-892. https://doi.org/10.5194/hess-13-883-2009
15. Maneta, M. P., & Wallender, W. W. (2013). Pilot-point based multi-objective calibration in a surface-subsurface distributed hydrological model. Hydrological Sciences Journal, 58(2), 390-407. https://doi.org/10.1080/02626667.2012.754987
16. Akay, H., Baduna Koçyiğit, M., & Yanmaz, A. M. (2018). Effect of using multiple stream gauging stations on calibration of hydrologic parameters and estimation of hydrograph of ungauged neighboring basin. Arabian Journal of Geosciences, 11, 282. https://doi.org/10.1007/s12517-018-3642-z
17. Usman, M. N., Leandro, J., Broich, K., & Disse, M. (2024). Multi-objective calibration and uncertainty analysis for the event-based modelling of flash floods. Hydrological Sciences Journal, 69(4), 456-473. https://doi.org/10.1080/02626667.2024.2322599