On shadowing methods for data assimilation

Combining orbits from a model of a (chaotic) dynamical system with measured data to arrive at an improved estimate of the state of a physical system is known as data assimilation. This thesis deals with various algorithms for data assimilation. These algorithms are based on shadowing. Shadowing is a concept from the theory of dynamical systems. When a dynamical system has the property that an exact orbit of the dynamical system is located in a neighborhood of each pseudo-orbit, then this exact orbit shadows the pseudo-orbit. Shadowing can be used to show that a numerical solution of a dynamical system is located in a neighborhood of an exact solution. Shadowing refinement is a numerical technique in which an improved approximation to an exact solution is found from a pseudo-orbit. It is possible to use a shadowing refinement technique for data assimilation. Starting from observations, Newton's method is applied to approximate a zero of a cost operator, where the cost operator assigns costs to deviations from model solutions.

The algorithms of Chapter 2 are based on a numerical time-dependent split between stable and unstable directions. The algorithm uses time-dependent projections onto the unstable subspace determined by using Lyapunov exponents and Lyapunov vectors. A shadowing algorithm is used in the unstable subspace, while synchronization is used in the stable subspace. The method is further extended to include parameter estimation and to some cases where only partial observations are available.

Chapter 3 discusses data assimilation for imperfect models. Through regularization according to the Levenberg-Marquardt method, imperfections in the model are considered. It also describes how the shadowing method compares, both analytically and numerically, with the weak constraint 4DVar method and shows that the shadowing method is consistent with the measurement error distribution, which is not the case for the weak constraint 4DVar method. This effect is particularly evident when there are fewer observations. Moreover, when there are few observations, they have a smaller impact on unobserved variables in the shadowing method than in the weak constraint 4DVar method.

Chapter 4 extends the method of Chapter 2 to other cases of partial observations, in a similar way to Chapter 3. Local convergence to a solution manifold is proved and a lower bound on an algorithmic time step is provided. Numerical experiments with the Lorenz-'63 and Lorenz-'96 models show convergence of the algorithm and further show that the method compares favorably with the weak constraint 4DVar method and another shadowing method called pseudo-orbit data assimilation.

Chapter 5 further develops the method of the previous chapters. The algorithm is extended to an ensemble of states for estimating uncertainties of the algorithm, based on the concept of indistinguishable states. The chapter also includes some proofs on uniqueness, accuracy and consistency of the algorithm. The algorithm is applied to an imperfect model to show how the unmodeled components of the model can be estimated using the data assimilation algorithm.