Description

The original development of the Generalised likelihood uncertainty Estimation (GLUE) methodology (Beven and Binley, 1992) arose out of a dissatisfaction with an optimisation approach to model calibration and with the assumptions of statistical models of “measurement error” in representing model uncertainties. Arguments in favour of the GLUE approach have been rehearsed elsewhere (Beven, 2002; Beven, 2006; Beven and Young, 2003).

As noted above, GLUE allows that multiple models may provide acceptable simulations of the response of the system of interest. A search for the set of behavioural or acceptable models is generally made by uniform random sampling of the parameter space, assuming that upper and low limits for each parameter can be specified. For models with significant run times, this is only generally possible for a small number of parameter dimensions.

In GLUE, modelling errors associated with each acceptable model are treated implicitly, under the assumption that error series associated with a particular model (such as over- or under-prediction of flood peaks) will be similar in prediction to those found in evaluation or calibration. Each model can be given a likelihood weight to express relative belief in that particular model, based on the evaluation of the model performance for a calibration data set. Informal likelihood measures can be used in deciding on model acceptability, and in determining likelihood weights. Models that are rejected during calibration will have a likelihood of zero and need not be used in prediction. Different model structures, as well as different parameter sets in a particular model structure, are easily combined within this framework.

Prediction bounds are formed by calculating the weighted cumulative distribution (cdf) of a predicted variable over all models retained as behavioural. Once the cdf has been formed, any prediction quantiles can be extracteted (typically the centroid, 5 and 95% quantiles). These are quantiles of the the behavioural model predictions; they are not directly equivalent to probabilistic prediction limits (unless specific assumptions have been made about the nature of the residual errors and a formal statistical likelihood used in the model evaluation. In this case GLUE can be equivalent to a formal Bayesian analysis; it is generalised in that sense). Thus the prediction bounds may not bracket 90% of the observations. In this way it can be more obvious when the model cannot predict the observations, because of either input data error or model structural error, and requires further investigation.

As more data become available, a variety of methods can be used to combine likelihood measures, including Bayes equation, fuzzy operations, averaging and other methods. Recently (Beven, 2006) the method has been extended to be more rejectionist in approach by allowing that, to be retained for use in prediction, a model must provide predictions that lie within the range of some “effective observational error” defined prior to making any model runs (rather than by a measure based on a residual series).

Approaches based on formal statistical error model structures can be considered to be special cases of GLUE in which the modeller is prepared to make strong assumptions about the nature of the errors and for which additional parameter dimensions for the error structure are added (e.g. Romanowicz et al., 1994). In this case, however, if there is a well-defined likelihood surface, uniform Monte Carlo sampling may not be an efficient way of sampling for behavioural models and other approaches (such as Bayesian Monte Carlo Markov Chain methods) should be considered.

The application of GLUE requires a number of choices as follows:

• Choice of one or more model structures to be considered

• Choice of ranges for each parameter values (including error model parameters if included)

• Choice of sampling strategy for searching the parameter space (e.g. discrete interval sampling, uniform random sampling, latin hypercube sampling, importance sampling)

• Choice of likelihood measure or measures for model evaluation, including criteria for model rejection

Software

http://www.es.lancs.ac.uk/hfdg/glue.html

Advantages

• Allows for equifinality in model structures and parameter sets, which may mitigate against an optimised model being overparameterised when used in prediction.

• Many different types of performance measure

• Subjective assumptions in applying the method must be made explicit and can be assessed.

• May result in all models being rejected as nonbehavioural, leading to review model structures or data sets.

Disadvantages

• The method has been criticised for its lack of formal assumptions in assessing the likelihood of different models, leading to subjectivity in its approach.

• May require high performance or parallel computing resources to obtain sufficient samples of acceptable models from the full range of potential models

• Whether a model produces acceptable simulations or not may depend on (unknown) error in input and boundary conditions.

• May result in all models being rejected as nonbehavioural (unless an explicit error model is added to compensate for other sources of error).

Case studies

• Flood forecasting cascade from ensemble rainfall forecasts to inundation maps (case study)

• Vulnerability weighted flood inundation risk estimation (case study)

References and Further reading in flood related applications

Aronica, G, Hankin, B.G., Beven, K.J., 1998, Uncertainty and equifinality in calibrating distributed roughness coefficients in a flood propagation model with limited data, Advances in Water Resources, 22(4), 349-365.

Aronica, G., Bates, P.D. and Horritt, M.S., 2002. Assessing the uncertainty in distributed model predictions using observed binary pattern information within GLUE. Hydrological Processes, 16(10): 2001-2016.

Beven, K.J., 2001. Rainfall-Runoff Modelling: The Primer. Wiley, New York.

Beven, K.J., 2002. Towards a coherent philosophy for modelling the environment. Proceedings of the Royal Society of London Series A- Mathematical Physical and Engineering Sciences, 458(2026): 2465-2484.

Beven, K.J., 2005. A Manifesto for the equifinality thesis. Journal of Hydrology, 320: 18-36.

Beven, K.J. and Binley, A., 1992. The Future of Distributed Models: Model Calibration and Uncertainty Prediction. Hydrological Processes, 6: 279-298.

Beven, K.J. and Young, P., 2003. Comment on "Bayesian recursive parameter estimation for hydrologic models" by M. Thiemann, M. Trosset, H. Gupta, and S. Sorooshian. Water Resources Research, 39(5): art. no.-1116.

Blazkova, S. and Beven, K., 2004. Flood frequency estimation by continuous simulation of subcatchment rainfalls and discharges with the aim of improving dam safety assessment in a large basin in the Czech Republic. Journal of Hydrology, 292(1-4): 153-172.

Blazkova, S. and Beven, K.J., 2002. Flood frequency estimation by continuous simulation for a catchment treated as ungauged (with uncertainty). Water Resources Research, 38(8): art. no.-1139.

Cameron, D., Beven, K.J., Tawn, J. and Naden, P., 2000. Flood frequency estimation by continuous simulation (with likelihood based uncertainty estimation). Hydrology and Earth System Sciences, 4(1): 23-34.

Cameron, D., Beven, K. and Naden, P., 2000, Flood frequency estimation under climate change (with uncertainty). Hydrology and Earth System Sciences, 4(3), 393-405 Freer, J. et al., 1997. Topographic controls on subsurface storm flow at the hillslope scale for two hydrologically distinct small catchments. Hydrological Processes, 11(9): 1347-1352.

Lamb, R., K.J. Beven and S. Myrabø, S., 1998, Use of spatially distributed water table observations to constrain uncertainty in a rainfall-runoff model., Advances in Water Resources, 22(4), 305-317.

Pappenberger, F., Beven, K., Horritt, M., Blazkova, S., 2005, Uncertainty in the calibration of effective roughness parameters in HEC-RAS using inundation and downstream level observations, Journal of Hydrology, 302, 46-69.

Pappenberger, F., Beven, K.J., Hunter N., Gouweleeuw, B., Bates, P., de Roo, A., Thielen, J., 2005, Cascading model uncertainty from medium range weather forecasts (10 days) through a rainfall-runoff model to flood inundation predictions within the European Flood Forecasting System (EFFS). Hydrology and Earth System Science, 9(4),381-393.

Pappenberger, F., Frodsham, K., Beven, K J, Romanovicz, R. and Matgen, P., 2006. Fuzzy set approach to calibrating distributed flood inundation models using remote sensing observations. Hydrology and Earth System Sciences, 10,1-14.

Pappenberger, F., Beven, K.J., Frodsham, K., Romanovicz, R. and Matgen, P., 2006. Grasping the unavoidable subjectivity in calibration of flood inundation models: a vulnerability weighted approach. Journal of Hydrology, 333, 275-287.

Romanowicz, R. and Beven, K.J., 2003. Estimation of flood inundation probabilities as conditioned on event inundation maps. Water Resources Research, 39(3): art. no.-1073.

Romanowicz, R, and Beven, K J, 2006, Comments on Generalised Likelihood Uncertainty Estimation, Reliability Engineering and System Safety, 91, 1315–1321

Romanowicz, R., Beven, K.J. and Tawn, J.A., 1994. Evaluation of predictive uncertainty in nonlinear hydraulic models using a Bayesian Approach. In: V. Barnett and K.F. Turkman (Editors), Statistics for the Environment 2, Water Related Issues. Wiley & Sons, New York, pp. 297-317.

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