Covariance Modeling by Means of Eigenfunctions of Laplace Operator

• Activity Number: 314
• Type: Contributed
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics and the Environment
• Author(s): Marie Turcicova* and Jan Mandel and Krystof Eben
• Companies: Charles University in Prague and University of Colorado Denver and Institute of Computer Science AS CR
• Keywords: data assimilation ; discrete Fourier transform ; spectral ; diagonal covariance ; maximum likelihood

Abstract

Large dimension of both state vector and data is a well-known challenge in environmental modeling (e.g., numerical weather forecast) and, in particular, in data assimilation. The Ensemble Kalman Filter addresses this problem by estimating the current system state and its uncertainty via their sample counterparts. However, sample covariance matrix based on a small ensemble is not a sufficiently good estimate of the true covariance. In this paper, we deal with techniques relying on transformation of the state to the spectral space and assuming a particular covariance structure based on the Laplace operator. Parameters, which this special structure depends on, are estimated by a least squares method and a maximum likelihood method. The behavior of both estimators is illustrated by a simulation. Both methods have a smaller error in Frobenius norm than the sample covariance, moreover, the latter method performs better than the former one, which corresponds to its stronger assumptions.