On the mathematics of data assimilation

Abstract

The problem of convergence of a “forward-backward” assimilation is considered for the most general dynamical system. Using elementary techniques of stability theory, it is shown that the variation, over one assimilation cycle, of the difference between the assimilating model and the state to be reconstructed is, to the first order, determined by a perfectly defined amplification matrix. This leads to a straightforward criterion for convergence, depending on the eigenvalues of that matrix. This convergence criterion has in effect already been shown in a previous article to be verified by the linearized meteorological equations. It is shown here to be verified by the non-linear shallow-water equations, in the case of successive observations of the geopotential field, at least when these observations are sufficiently close in time. Numerical experiments support the theoretical results, and they together lead to the following description of the effects of an assimilation of geopotential observations. The divergent part of the wind field is reconstructed more rapidly, because it is directly influenced by introductions of observations. The rotational part is reconstructed indirectly and more slowly, mostly through the effect of Coriolis acceleration. These results are independent of whether or not the flow to be reconstructed is geostrophic, and do not require the presence of any dissipative process in the assimilating model. The effect of observing and/or modelling errors is considered, and small errors are shown not to modify the convergence properties of an assimilation. Finally, a theoretical method is defined through which the amplification matrix over one assimilation cycle can be made equal to zero, thereby optimizing the convergence of the process. DOI: 10.1111/j.2153-3490.1981.tb01755.x