UC Berkeley

Mathematical models are often used to forecast interesting scientific phenomena. These models may provide a good approximation for the system being studied, but they are rarely perfect. Whether due to scientific misunderstandings or the necessity of numerically approximating the solution to the model, the mathematical models fail to exactly replicate the true dynamics of the system. This "model error" leads to forecast errors and degradation of forecast skill. The primary goal of this dissertation is to develop methods to correct a mathematical model with observed data, improving the accuracy of forecasts generated by the model. Correcting for model error is particularly difficult when the system is high-dimensional and when the state transition model is nonlinear. The methods developed in this dissertation are robust to both of these difficulties.

The problem of estimating model error may be phrased as a parameter estimation problem within a filtering context. Many parameter estimation methods have been developed for the particle filter (PF), a filter with strong theoretical underpinnings and well understood large sample properties. Unfortunately, the PF fails when the state dimension is large. Instead, scientists who filter high-dimensional state-spaces typically use the ensemble Kalman filter (EnKF). This algorithm is effective in practice, but its large sample properties remain to be proven and methods for parameter estimation exist only in special cases. We make a newfound connection between these two algorithms: the EnKF is a regularized PF with additional approximations that have yet to be fully justified from a theoretical standpoint. Building on this connection, we introduce a novel algorithm that combines the best features of the EnKF and the PF. The EnKF is used to filter the high-dimensional state while an auxiliary PF is used to estimate the low-dimensional parameters; we call this filter the EnKF-APF. Using the Lorenz 2005 system, we demonstrate that the true parameter values are better captured with the EnKF-APF algorithm than with the EnKF.

Having developed a robust method for parameter estimation within a filtering context, we return to the question of model error. Following previous work in the literature, we focus on a linear correction to the mathematical model. If the state has dimension d, the number of parameters in the linear correction is on the order of d^2. The EnKF-APF algorithm does not allow for the estimation of such high-dimensional parameters, and even if it did, an enormous amount of data would be required to provide good estimates. To make the problem tractable, we assume the problem of interest exhibits a large degree of spatial structure and exploit this special structure to reduce the dimensionality of the linear correction. We propose to correct the model error with a low-rank linear correction, inspired by methods used to model multivariate spatial processes in the geostatistics literature. We demonstrate our method on the Lorenz 2005 system by applying the EnKF-APF algorithm to estimate the parameters of the low-rank linear correction in both a batch and online manner. Although the model error in this test is nonlinear, we demonstrate that the proposed low-rank linear correction provides better forecasts.