Four-Dimensional Var data assimilation and POD model reduction applied to geophysical dynamics models

The first new contribution in my dissertation consists in studying a new methodology combining the dual weighted snapshots selection and trust region Proper Orthogonal Decomposition (POD) adaptivity (DWTR-POD). Another new contribution is to combine the incremental POD 4-D Var, balanced truncation techniques and method of snapshots methodology. In the linear DS, this is done by integrating the linear forward model many times using different initial conditions in order to construct an ensemble of snapshots so as to generate the forward POD modes. Then those forward POD modes will serve as the initial conditions for its corresponding adjoint system. We then integrate the adjoint system a large number of times based on different initial conditions generated by the forward POD modes to construct an ensemble of adjoint snapshots. From this ensemble of adjoint snapshots, we can generate an ensemble of so-called adjoint POD modes. Thus we can approximate the controllability Grammian of the adjoint system instead of solving the computationally expensive coupled Lyapunov equations. To sum up, in the incremental POD 4-D Var, we can approximate the controllability Grammian by integrating the TLM a number of times and approximate observability Grammian by integrating its adjoint also a number of times.;A new idea contributed in this dissertation is to extend the snapshots based POD methodology to the nonlinear system. Furthermore, we modify the classical algorithms in order to save the computations even more significantly. We proposed a novel idea to construct an ensemble of snapshots by integrating the tangent linear model (TLM) only once, based on which we can obtain its TLM POD modes. Then each TLM POD mode will be used as an initial condition to generate a small ensemble of adjoint snapshots and their adjoint POD modes. Finally, we can construct a large ensemble of adjoint POD modes by putting together each small ensemble of adjoint POD modes. To sum up, our idea in a forthcoming study is to test approximations of the controllability Grammian by integrating TLM once and observability Grammian by integrating adjoint model a reduced number of times.;We then attempt to obtain a reduced-order model (ROM) of above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D Var. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual-weighed method for snapshot selection coupled with a trust-region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual-weighing choice of snapshots is combined with POD adaptivity of the trust-region type. Results of ad-hoc adaptivity of the POD 4-D Var turn out to yield less accurate results than trust-region POD when compared with high-fidelity model.;Finally, we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets, in presence of full or incomplete observations being assimilated in a time interval (window of assimilation) presence of background error covariance terms. As an extension of this research, we attempt to obtain a reduced-order model of above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D Var for a finite volume global shallow water equations model based on the Lin-Rood [89, 90, 91, 92, 96] flux-form semi-Lagrangian semi-implicit time integration scheme. Different approaches of POD implementation for the reduced inverse problem are compared, including a dual-weighted method for snapshot selection coupled with a trust-region POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4-D Var model results combined with the trust region adaptivity exhibit similarity in terms of various error metrics to the full 4-D Var results, but are obtained using a significantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current research work, we conclude that POD 4-D Var certainly warrants further studies, with promising potential for its extension to operational 3-D numerical weather prediction models. (Abstract shortened by UMI.)...