A New Perspective on Covariance Propagation for Data Assimilation Applications

Gilpin, Shay

Graduate Thesis Or Dissertation

A New Perspective on Covariance Propagation for Data Assimilation Applications Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/cj82k894f

Abstract

The propagation of the error covariance is an important, but not well-understood, aspect of the statistical estimation of dynamical systems, such as data assimilation. One problem often encountered during covariance propagation in data assimilation is spurious loss of variance, in which the estimated variance under-approximates the exact variance. Current approaches to understanding and combatting this issue are formulated primarily in discrete terms and often fall short of addressing fundamental causes. In an effort to understand the root cause of this spurious loss of variance, this thesis closely examines the underlying continuum covariance dynamics. Motivated by atmospheric data assimilation, the problem for states governed by the continuity and other related hyperbolic partial differential equations is considered. By analyzing the continuum covariance propagation, this thesis is the first to identify a discontinuous change in the continuum covariance dynamics along the hyperplane 𝔁₁=𝔁₂. Through a series of numerical experiments and error analysis of full-rank covariance propagation, this work then demonstrates that standard methods of variance propagation are inherently inaccurate because of this discontinuous change in the continuum covariance dynamics. The result in discrete space is inaccurate variance propagation, in which full-rank covariance propagation can produce both spurious loss and gain of variance by approximating the incorrect continuum dynamics. Based on the insights gained from the continuum analysis and numerical experiments, an alternative method to mitigate this inaccurate variance propagation is proposed. As part of this attempt, this thesis develops a new correlation function required to implement the proposed alternative method of covariance propagation, and with immediate uses in other areas of data assimilation.

By studying the continuum problem, this research provides a new perspective on covariance propagation that can strengthen our understanding of its practice in data assimilation applications.

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