Non-local Observations and Information Transfer in Data Assimilation - van Leeuwen, Peter Jan (Frontiers in Applied Mathematics and Statistics, 2019)
Nonlocal observations are observations that cannot be allocated one specific spatial location. Examples are observations that are spatial averages of linear or nonlinear functions of system variables. In conventional data assimilation such as (ensemble) Kalman Filters and variational methods information transfer \pj{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 \pjvl{prior} covariance structures by the nonlocal observational constraints, building an information bridge \pj{(or information channel)} that has not been studied before: \pjvl{the posterior covariance can have nonzero 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 nonlocal observations are assimilated can have a large influence on the analysis. Local observations should be assimilated first. This relates directly to localisation used in Ensemble kalman Filters and Smoothers, but also to variational methods with a prescribed covariance structure where observations are assimilated in batches.} \pjvl{This suggests that the emphasis on covariance modelling should shift away from the prior covariance and towards the modelling of the covariances between model and observation space.} \pjvl{Furthermore, it is shown that local observations with non-locally correlated observation errors behave in the same way as uncorrelated observations that are nonlocal. Several theoretical results are illustrated with simple numerical examples.} The significance of the information bridge provided by nonlocal observations is highlighted \pj{further} through discussions of \pj{temporally nonlocal} observations, and new ideas on targeted observations.