AUTHOR=van Leeuwen Peter Jan TITLE=Non-local Observations and Information Transfer in Data Assimilation JOURNAL=Frontiers in Applied Mathematics and Statistics VOLUME=5 YEAR=2019 URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00048 DOI=10.3389/fams.2019.00048 ISSN=2297-4687 ABSTRACT=

Non-local observations are observations that cannot be allocated one specific spatial location. Examples are observations that are spatial averages of linear or non-linear functions of system variables. In conventional data assimilation, such as (ensemble) Kalman Filters and variational methods information transfer between observations and model variables is governed by covariance matrices that are either preset or determined from the dynamical evolution of the system. In many science fields the covariance structures have limited spatial extent, and this paper discusses what happens when this spatial extent is smaller then the support of the observation operator that maps state space to observations space. It is shown that information is carried beyond the physical information in the prior covariance structures by the non-local observational constraints, building an information bridge (or information channel) that has not been studied before: the posterior covariance can have non-zero covariance structures where the prior has a covariance of zero. It is shown that in standard data-assimilation techniques that enforce a covariance structure and limit information transfer to that structure the order in which local and non-local observations are assimilated can have a large influence on the analysis. Local observations should be assimilated first. This relates directly to localization used in Ensemble Kalman Filters and Smoothers, but also to variational methods with a prescribed covariance structure where observations are assimilated in batches. This suggests that the emphasis on covariance modeling should shift away from the prior covariance and toward the modeling of the covariances between model and observation space. Furthermore, it is shown that local observations with non-locally correlated observation errors behave in the same way as uncorrelated observations that are non-local. Several theoretical results are illustrated with simple numerical examples. The significance of the information bridge provided by non-local observations is highlighted further through discussions of temporally non-local observations, and new ideas on targeted observations.