Model reduced variational data assimilation for shallow water flow models

Abstract

Identifying uncertain parameters in large-scale numerical flow models can be done using the variational method. However, for implementing the variational method the adjoint model have to be available, which requires highly complex computer code and maintenance and thus hampers its applications. To ease this problem, this thesis has explored several methods for efficiently identifying uncertain parameters in a large-scale tidal model of the entire European continental shelf which does not require the implementation of these complex adjoint code. In this study, as a first step an estimation method based on model reduction is developed and investigated for the estimation of diffusion coefficient in a simple 2D-advection diffusion model. Two projection based model reduction methods were considered, namely proper orthogonal decomposition (POD) and Balanced proper orthogonal decomposition (BPOD). In the POD based estimation method an ensemble of forward model simulations is used to determine an approximation of the covariance matrix of the model variability and a small number of the leading eigenvectors of this matrix is used to define a model subspace. By projecting the original model onto this subspace an approximate linear reduced model is obtained. Once the reduced model is available its adjoint can be implemented easily and the minimization problem is solved completely in reduced space with very low computational cost. BPOD is also a model reduction method which considers both inputs and outputs of the system while determining the reduce subspace. The estimation method has been extended by including BPOD procedure into the estimation procedure. Numerical results from a simple pollution model demonstrate that the POD based estimation approach successfully estimate the diffusion coefficient for both advection dominated problems as for diffusion dominated problems. Another important message in this study, although lots of effort had been made in constructing a reduced order model by the BPOD method, the minimization results demonstrated that both the POD and the BPOD methods performed similarly. Preliminary results showed the validity of the POD based model reduction methods for parameter estimation. As a next step, the POD based estimation method is used to calibrate numerical tidal models. Results from (twin) numerical experiments showed that the POD based calibration method performed very efficiently to estimate depth values in the selected regions of the model domain. The computational costs of the POD based calibration method are dominated by the generation of an ensemble of forward model simulations where the simulation period of the ensemble is equivalent to the timescale of the original model. It has also been found in the study that it is not needed to use a full simulations of the original model for the generation of the ensemble. The POD based calibration method has also been implemented for the estimation of the water depth and space varying bottom friction coefficient values in a very large-scale DCSM model. The recently designed large-scale spherical grid based water level model for the northwest European continental shelf (around 1000000 computational grid points) has been used for this purpose. This has been the first application of the POD based calibration method to a very large-scale model and with real data. Results from numerical experiments showed that the calibration method performs very efficiently. An overall improvement of more than 50\% was observed after the calibration in comparison with the initial model. The results also demonstrated that the POD based calibration method offered a very efficient minimization technique compared to the classical adjoint method without the burden of implementation of the adjoint. As a concluding step, to estimate depth values in the model DCSM, a Simultaneous perturbation stochastic approximation (SPSA) method has been used. The method uses stochastic simultaneous perturbation of all model parameters to generate a search at each iteration. SPSA is based on a highly efficient and easily implemented simultaneous perturbation approximation to the gradient. This gradient approximation for the central difference method uses only two objective function evaluations independent of the number of parameters being optimized. The results from experiments showed that SPSA has a lower convergence rate than POD based calibration method, however the computational cost in each iteration of the SPSA method is usually far less then the POD based calibration method. The results also demonstrated that the SPSA algorithm proved to be a promising optimization algorithm for model calibration for cases where adjoint code is not available for computing the gradient of the objective function.