Building the Mathematical Foundations for Covariance Localization in Data Assimilation
Abstract
Merging computational models with observations is important in many problems of interest to theNavy. A prominent example is numerical weather prediction (NWP), where forecasts of tomorrowsweather are informed by measurements of todays atmospheric state. Merging simulations andobservations in this way can be achieved within a Bayesian framework, in which the computationalmodel and the observations jointly define a posterior distribution. The timely, and therefore navalrelevant, numerical solution of such a data assimilation (DA) problem is based on approximationsof a posterior distribution. Numerically, this is often done by ensemble methods, in particular byensemble Kalman filters (EnKF).A key requirement for the applicability of the EnKF to problems of interest to the Navy is thatthe ensemble size be moderate even if the numerical model is of a very high dimension. The reasonis that each ensemble member requires a simulation with the model, which is computationallyintensive. A small ensemble size, however, creates large sampling errors which cause spurious,long-range correlations and, ultimately, a poor forecast. The idea of covariance localization is toreduce sampling error by restricting correlations to relatively small neighborhoods, which effectivelyincreases the accuracy of ensemble covariances to exceed what one can expect from a small ensemblesize.Localization has lead to extreme gains in the computational efficacy of the EnKF and has enabled thesuccess of EnKF algorithms in NWP. There is, however, nearly no mathematical theory that justifiesthe use of localization. Within this project, I will create a mathematical theory for localization thatcan explain where the beneficial aspects of localization arise and under what conditions localizationshould be used. More generally, this project contributes to our understanding of how to effectivelysample random variables in high dimensional spaces, which is important in many problems inscience and engineering (besides NWP), e.g., in inverse problems in geophysics, model selection, orthe analysis of rare events.The proposed theoretical work is motivated by practical application. Through previous work andmany discussions with collaborators, it became apparent that a mathematical theory of localizationis particularly relevant to the Navy because it is needed to (i) effectively use localization inionospheric prediction; (ii) extend localization to nonlinear/non-Gaussian methods, such as particlefilters or Markov chain Monte Carlo (MCMC); and (iii) effectively leverage localization in multiscalesystems. The Earths ionosphere, which is the interface between the near space region and theEarths lower atmosphere, is critically important because it directly influences Over-The-Horizon-Radar (OTHR) characteristics, GPS signal propagation, and HF radio communication. Problems(ii) and (iii) are increasingly important for the Navys DA systems as the resolution of the Navysatmospheric and oceanic models increases to more tactically relevant temporal and spatial scales.This abstract is approved for public release.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 06, 2021
- Source ID
- N000142112309
Entities
People
- Matthias Morzfeld
Organizations
- Office of Naval Research
- United States Navy
- University of California, San Diego