Grey Box Modelling of Hydrological Systems
Grey Box Modelling of Hydrological Systems
 with focus on uncertainties
PhD project by Fannar Örn Thordarson
Short introduction video with Fannar Örn Thordarson
The objective of the thesis is formulation of stochastic dynamic hydrological systems, where the focus is on modeling and interpreting the uncertainties embedded in the model structure. Two modeling approaches are applied: a grey box modeling approach, which often is referred to as a stochastic differential equation (SDE) based model, and an impulse response function modeling approach.
The SDEs that describe the hydrological system in the grey box model consist of a drift term, which describes the physical flow of the water in the system (usually obtained by ordinary differential equations), and a diffusion term that accounts for the uncertainty assigned to the SDEs. By formulating these diffusion terms, the uncertainty can be assigned to the related states and, thus, improve the prediction intervals of the model output. The impulse response function models are applied in continuous time, where the parameters are provided a physical interpretation by using known equations from hydrology as impulse response functions. Thus, applying a similar argument as the one applied to the previous method, the impulse response function model, as presented here, can in a way also be considered a grey box model. The unknown parameters in both methods are estimated by the maximum likelihood method. However, these two modeling approaches are rather different. The states in the output from the SDE based model are related to the input variables via the states in the model formulation, but for the impulse response function models the output is a direct consequence of changes in the input sequence.
The performance of the SDE based models and consequent predictions is evaluated by three measures: reliability, sharpness and resolution. For the performance comparison the skill score criterion is exploited, which aggregates the properties of the prediction with a single number. The criterion accounts for the prediction interval, but not only the point prediction of the output, i.e. the uncertainty of the prediction is included in the skill score criterion.
Case Study
Two case studies are considered in the thesis, although within fairly contrasting hydrological fields: well field modeling and sewer runoff modeling. For the well field case study, both modeling methods are applied to a water head response in operating wells in a well field, where the pumping rates from the wells were the only available input series (see the conceptual model in the figure below). Both approaches gave promising results for predicting the water head in the wells along with appropriate measures of uncertainties, despite the fact that the available data for modeling was rather limited and defected, and required substantial cleaning before it was approved for modeling purposes. The two model approaches have its advantages when it comes to groundwater management. The SDE based model approach can adequately describe the embedded uncertainty and is the optimal selection for forecasting and control of the well field. Due to the drift of the model being formulated from the main physical structure of the hydrological system, the longterm effects are also attained on the basis of details obtained from the drift term. However, the model only includes wells that penetrate the same aquifer, and the parameter estimation of the SDE based model is rather time consuming with the tools available today. On the other hand, for the impulse response function models the main advantage lie in simulations for the entire well field using a model that is robust and can generate results in a very short time. Uncertainties do not have the same constructive interpretation as is the case for the SDE based model. Considering the grey box model as a possibility for future modelling of groundwater management has several benefits, but further work is required to fully describe the properties of the well field.
For modeling the sewer runoff, only the SDE based model is considered. The suggested diffusion terms in the model includes a state dependency that is taken care of by transforming the states to obtain state independent diffusion terms. This results in a model with varying prediction intervals to describe the increasing uncertainty of the flow as rainfall is detected in the catchment. Both the model and its parameters are physically interpretable and identifiable from data, but the evaluation measures for the model (reliability, sharpness and resolution) reveal a lack of fit for larger prediction horizons when rain events occur. This is not surprising, since the rainfallrunoff process is a fairly complex hydrological phenomenon that is hardly totally obtainable with a single series of linear reservoirs (see the figure to the right).
Concluding Remarks
The simple grey box model, where the diffusion is given more attention is a reasonable approach to predict the outcome of the hydrological systems where measures of the embedded uncertainties, inform of model approximations and uncertainty in the system input, are required. The grey box model framework enables a stepwise procedure to detect the lack of fit in the hydrological model, and combined with the uncertainty assessment introduced in this thesis, the grey box model approach provides more robust models that are better suited for forecasting and control in managing hydrological systems.

Keywords:
 Grey Box Modeling
 Prediction
 Uncertainty Assessment
 Skill Score Criterion

Participants:
 Henrik Madsen
(DTU Informatics)  Henrik Madsen
(DHI)  Gianluca Dorini
 Anders Breinholt
 Peter Steen Mikkelsen
 Henrik Madsen