Parameter and state estimation of a groundwater model using the Ensemble Kalman Filter

Abstract

Groundwater is an essential ingredient in farming, knowledge about how this is expected to change over time can help farmers improve yields and save water. Predictions about the groundwater level can be made using a mathematical model. This model takes into account the precipitation and evaporation, the flux towards a deeper aquifer and flux to lateral open structures, with parameters for the different resistances and the storage. This model needs input data which is acquired from among others the KNMI (Koningklijk Nederlands Meteorologisch Instituut), with data about the groundwater level at the points of interest acquired from the measurements from the Water board Drenths Overijsselse Delta. The input from the model can be used to generate predictions, whilst the measurement data can be compare with the predictions to obtain more accurate predictions and calibrate the parameters in the model. Combining these factors can be done with the Kalman Filter, a mathematical tool from Data Assimilation which can be used to combine a Mathematical Model and Data into an optimal estimate. Furthermore, it is the wish to improve the parameters in the mathematical model, this causes the state to expand thereby enlarging the problem, and the parameters included in the state causes the problem to become non-linear. The Kalman Filter is only suitable for small linear models but an extension of the regular Kalman Filter can account for these problems, the Ensemble Kalman Filter. To apply all this information for groundwater modeling in a single location, first the numerical errors associated with the different model solutions are tested. It was found that a semi-analytical solution which assumed the input to stay constant over a day to produce the best result. After this the Kalman Filter is compared with the Ensemble Kalman Filter to see whether the Ensemble Kalman Filter can perform comparably in state estimation, it was found that en ensemble size of $n = 200$ this was the case. To test if the parameter calibration scheme works as desired and to test the quality of the parameters found, artificial data is created using the model with all parameters set to one. Then, the parameters are shifted to a different value and the parameter Ensemble Kalman Filter is used to try to find the original parameters back. It is found that in almost all cases the parameter calibration cause the parameter to change in the direction of one, within the first 2/3 months most change within the parameter takes place and the standard deviation of the parameters decreases. After this initial period the parameter remains more stable and the standard deviation changes little. Since the model uses multiple parameters for the resistances, the possibility of calibrating these separately was investigated. It was discovered that in the data usually one resistance parameter is lot smaller than the other, it was enough to calibrate only this parameter to account for the resistance. After this, the parameters in the model are calibrated with actual data. The model always becomes a lot closer to the measurements, and interestingly all parameter become larger than 1. Finally the quality of the predictions is tested. Here we saw that the quality improved fast, already 1/3th year was enough to find parameter far better than originally.