Towards a data assimilation system for morphodynamic modeling

Abstract

Currently, available models are not able to accurately predict the temporal evolution of coastal morphodynamic processes. The main reason for this, is that essential components governing this evolution are neither fully identified nor understood. Additionally, there is high uncertainty in some of the parameterizations and their input parameters. A proper management of coastal systems depends on a thorough understanding of their present and future state. Regarding the present state of the system, detailed analyses that take into account the inference of human activities and interactions with other natural process may provide the required information to the decision makers. The evaluation of future states, on the other hand, is not possible without the use of data assimilation, i.e. integration of measurements and predictions in the context of formal uncertainty analysis. Despite the need to implement data assimilation, only recently the necessary coastal observations became available. Since then, the integration of data assimilation methods and coastal morphodynamic models have become the focus of ongoing research. Unfortunately, the implementation of these methods is not straightforward. Sequential data assimilation methods, such as ensemble Kalman filter, are practical to implement but their update usually breaks conservation laws and may easily result in model instabilities. Variational (adjoint based) schemes, on the other hand, preserve the physical integrity but require the implementation of an adjoint model which commonly matches in complexity the forward model. This thesis is concerned with the implementation of a variational data assimilation method to improve the predictive skills of a commercial model by estimating its input parameters. Model reduced 4DVar was used to address the problem. This method is an adjoint-free variational method that uses a truncated first order Taylor approximation of the morphodynamic model to implement the data assimilation process. The construction of the Taylor approximation is considerably simpler than the development of a full adjoint model. Additionally, the method provides the sensitivity of the morphodynamic model with respect to the state and/or input parameter vector. Regarding the commercial morphodynamic model, on the other hand, the Delft3D suite was selected. Delft3D is a module based simulator that merges a wave module and a flow and morphodynamic module to produce a fully coupled wave, flow and morphodynamic model. To gain a better understanding of the method and the potential challenges of its implementation on a real application, the technique is first applied to a small study case with synthetic observations (twin experiment). In this \emph{proof of concept}, three wave properties were estimated by assimilating observations of bathymetry. Different reduction strategies were implemented to characterize the method's performance and to assess the effects of uncertain observations on the results. The data assimilation method is able to improve the performance of the morphodynamic model and shows to be robust in the presence of noisy observations. Nevertheless, some difficulties are identified regarding the size and direction (sign) of the perturbations required for the finite difference estimations. In light of this finding, the implementation shows that it is necessary to increase the number of model executions to make the method more robust. Also, the test show problems in connection with the restarting capability of Delft3D. These problems have a significant impact on the accuracy of the finite difference estimations. To address the issues found in the proof of concept, an alternative novel model reduction method is proposed: \emph{ensemble model order reduction} (enMOR). The method computes the low rank linear approximation of the full model based on an ensemble similar to those used in Monte Carlo approaches (e.g. ensemble Kalman filter). This offers a number of advantages over the more common finite difference approach. For instance, the model executions necessary for this type of ensembles are not prone to numerical instabilities; making the implementation significantly more robust. The implementation time is considerably lower and the availability of such ensemble allows computing the model error covariance matrix, which was not available when the finite differences are used. Finally, the method does not requires the model to be restarted, solving altogether the problems identified during the implementation of this first experiment. The enMOR method was partially implemented in a more realistic study case based on a set of laboratory measurements. The measurements were taken as part of an experiment aimed at studying rip current circulations in a morphodynamic system. A set of 12 bathymetric observations were assimilated into the morphodynamic model to estimate a set of 19 flow, wave and morphodynamic parameters. One data assimilation problem for each state transition (LBT, RBB, TBR, and LTT) is implemented in order to characterize the parameter changes in each case independently. The enMOR technique is only applied to the dynamic components of the linear approximation, the model sensitivities to input parameters are estimated with finite differences. This serves two purposes, (1) it allows to assess possible shortcomings in the implementation of the method, and (2) reliable model sensitivities are estimated to analyze the influence of each parameter in the dynamic evolution of the system. The results show that the evolution of the morphodynamic system cannot be easily captured by a constant set of parameters; it is necessary to consider some of the parameters as time dependent. The results also show that the 2D model used is not able to reconstruct shallow and deep morphodynamic processes simultaneously. Nevertheless, the data assimilation is able to improve the overall performance of the model. The implementation of the enMOR shows no significant practical difficulty other than potential long estimation times for reduced order models of considerable size. A full implementation of the enMOR method with a 3D model of the Egmond aan Zee system in The Netherlands is finally presented. A set of time exposure images are used to estimate roller energy dissipation maps that are assimilated into the model to estimate a set of 13 input parameters that include the time-dependent wave height and peak period. The results show that the data assimilation method is able to find a parameter set that considerably improves the model performance. The results also suggest that the predicted morphodynamic evolution of the model has been improved, enhancing the forecasting capacity of the system. The implementation of the method proved significantly more practical than the common finite difference approach. The complete data assimilation implementation only required a set of 40 forward model runs (commonly known as ensemble). The model is not restarted at any point to produce the piece wise linear approximation necessary in model reduced 4DVar and no finite difference was implemented in the process. Consequently, the implementation difficulties observed in the proof of concept are not an issue in this implementation. In conclusion, enMOR is a practical model order reduction method that can be used for data assimilation without any loss of performance. Further developments are necessary to take advantage of the available information about the adjoint of the model. Its use in applications for observation network optimization and process characterization should be considered. Also, state estimation is probably the best manner for bathymetry estimation. On this regard, further research is necessary to assess the best manner to implement state estimation using model reduced 4DVar. Finally, an estimation of the expected errors of the reduced order models can be achieved by means of Monte Carlo methods. This information could be valuable to identify weaknesses in the reduced order model and therefore in the assimilation process.