On Asymptotic Distributions of Analysis Characteristics for the Linear Data Assimilation Problem*

Abstract

A commonly investigated linear data assimilation problem as a correction of the numerical model output is defined. This problem means that a numerical model state vector is corrected by observations through a system of linear equations. This paper shows that the asymptotic behavior of the characteristics of objective analyses produced by data assimilation under various conditions exists. In particular, the existence of a stationary regime for this problem is demonstrated, and a special case is discussed when the norm of the Kalman gain matrix approaches zero. For this case the limit theorem for the characteristics of the analysis state vector is proved under certain conditions. Another limit theorem asserts that the model variables after assimilation approach a diffusion stochastic process and the parameters of this process are determined. As a corollary, a new method to determine the gain matrix and the confidence intervals for the analysis state is derived. This led to a new approach on how to realize the data assimilation problem. A few numerical experiments are performed to illustrate the usefulness and feasibility of those theorems.