Reduced Order Models and Data Assimilation for Hydrological Applications

The present thesis work concerns the study of Monte Carlo (MC)-based data assimilation methods applied to the numerical simulation of complex hydrological models with stochastic parameters. The ensemble Kalman filter (EnKF) and the sequential importance resampling (SIR) are implemented in the CATHY model, a solver that couples the subsurface water flow in porous media with the surface water dynamics. A detailed comparison of the results given by the two filters in a synthetic test case highlights the main benefits and drawbacks associated to these techniques. A modification of the SIR update is suggested to improve the performance of the filter in case of small ensemble sizes and small variances of the measurement errors. With this modification, both filters are able to assimilate pressure head and streamflow measurements and correct model errors, such as biased initial and boundary conditions. SIR technique seems to be better suited for the simulations at hand as they do not make use of the Gaussian approximation inherent the EnKF method. Further research is needed, however, to assess the robustness of the particle filters methods in particular to ensure accuracy of the results even when relatively small ensemble sizes are employed. In the second part of the thesis the focus is shifted to reducing the computational burden associated with the construction of the MC realizations (which constitutes the core of the EnKF and SIR). With this goal, we analyze the computational saving associated to the use of reduced order models (RM) for the generation of the ensemble of solutions. The proper orthogonal decomposition (POD) is applied to the linear equations of the groundwater flow in saturated porous media with a randomly distributed recharge and random heterogeneous hydraulic conductivity. Several test cases are used to assess the errors on the ensemble statistics caused by the RM approximation. Particular attention is given to the efficient computation of the principal components that are needed to project the model equations in the reduced space. The greedy algorithm selects the snapshots in the set of the MC realizations in such a way that the final principal components are parameter independent. An innovative residual-based estimation of the error associated to the RM solution is used to assess the precision of the RM and to stop the iterations of the greedy algorithm. By way of numerical applications in synthetic and real scenarios, we demonstrate that this modified greedy algorithm determines the minimum number of principal components to use in the reduction and, thus, leads to important computational savings

Multivariate data assimilation in snow modelling at Alpine sites

The knowledge of snowpack dynamics is of critical importance to several real-time applications such as agricultural production, water resource management, flood prevention, hydropower generation, especially in mountain basins. Snowpack state can be estimated by models or from observations, even though both these sources of information are affected by several errors

Turbulent velocity field reconstruction using four-dimensional variational data assimilation

Important progress in computational fluid dynamics has been made recently by applying the data assimilation (DA) techniques. In this thesis, we apply the four-dimensional variational approach to rebuild the small scales of the velocity fields of fully developed three-dimensional turbulence, given a time sequence of measurement data on a coarse mesh of grid points. In this problem, we deal with new challenges since the flow is governed by the processes of nonlinear vortex stretching and forward energy cascade, which are absent in two-dimensional flows that have been investigated so far. Two different models are presented to examine their effects on the reconstruction quality: the Navier-Stokes equations as the model and when the large eddy simulations are applied as the model (Smagorinsky model). The investigations examine different statistics of the reconstructed fields. The results show that the agreement improves over time within the optimization horizon, where the rebuilt fields tend to the DNS target. Reasonable agreements are accomplished between the optimal initial fields and the target data. To assess the quality of the reconstruction of non-local structures, minimum volume enclosing ellipsoids are introduced, which enables us to perform quantitative comparisons for the geometry of non-local structures. The rebuilding of non-local structures with strong vorticity, strain rate and subgrid-scale energy dissipation gives satisfactory results. A small misalignment between the MVEE's axes can be obtained; structures in the rebuilt fields are reproduced with sizes smaller by a small percentage to what exists in the target field; the locations of the MVEE are different on average by around 20% of the axes lengths. Both Navier-Stokes equations and filtered Navier-Stokes equation (as models for the data) show satisfactory reconstruction. The imperfect model (Smagorinsky model) demonstrate the capability for recovering the target fields quicker in most of the presented examinations