Shay Gilpin, Department of Applied Mathematics, University of Colorado Boulder

Continuum Covariance Propagation for Understanding and Ameliorating Variance Loss in Advective Systems

Historically, data assimilation has been formulated primarily in discrete terms. Subsequently, when problems arise during data assimilation, such as spurious loss of variance, they are typically analyzed and addressed in discrete space. In this presentation, I will interpret data assimilation as a continuum problem, and in particular study continuum covariance propagation in an effort to understand spurious loss of variance. Motivated by atmospheric and other high-dimensional data assimilation schemes, I consider the problem for states governed by the continuity and other related hyperbolic partial differential equations. By analyzing the continuum covariance propagation, my research is the first to identify a peculiar behavior in the continuum covariance dynamics and directly connect the spurious loss of variance observed in discrete space to this continuum phenomenon. Through a series of numerical experiments, I also illustrate the inaccuracies of full-rank discrete covariance propagation, suggesting that this problem may also be occurring in associated low-rank approximations commonly practiced in data assimilation. Based on these results, I propose two methods to mitigate spurious loss of variance which are motivated by and derived from the continuum dynamics. This also results in the introduction of a new correlation function to the data assimilation community. By studying the continuum problem, this work contributes interesting and valuable discoveries that will strengthen our understanding of covariance propagation and the theoretical basis of data assimilation.

Passcode for this talk is covariance