Analysis and design of covariance inflation methods using inflation functions. Part 1: Theoretical framework

We propose a unifying theory for covariance inflation (CI) in the Ensemble Kalman Filter (EnKF) that encompasses all existing CI methods and can explain many open problems in CI. Each CI method is identified with an inflation function that alters analysis perturbations through their singular values. Inflation functions are usually considered as functions of singular values of background or analysis perturbations. However, we have shown that it is more fruitful if inflation functions are viewed as functions of reduction factors of background singular values after assimilation. These factors indeed comprise the spectra of linear transformations between background and analysis perturbations. To be an inflation function, a function has to satisfy three conditions: (a) the functional condition: all reduction factors must increase, (b) the no‐observation condition: when no observations are assimilated, analysis perturbations are identical to background perturbations, and (c) the order‐preserving condition: inflated analysis singular values must have the same order as background singular values. If the upper‐bound condition, that is, inflated analysis error variances must be less than observation error variances, is imposed, the resulting inflation functions are shown to be equivalent to prior inflation functions which are functions of singular values of background perturbations. This condition is necessary if we want to inflate analysis increments in posterior CI. It turns out that the relaxation‐to‐prior‐spread method and the relaxation‐to‐prior‐perturbation method belong to the class of linear inflation functions. In this class, we also have constant inflation functions, multiplicative inflation functions and parameter‐varying linear inflation functions. More interesting, the Deterministic EnKF is found to belong to the class of quadratic inflation functions. This quadratic class introduces an elegant form for computing analysis perturbations through the Kalman gain. Higher‐order polynomial and non‐polynomial forms of inflation functions are less appealing in practice due to high computation cost and difficulty in determining free parameters. A unifying theory for covariance inflation (CI) is proposed in which each CI method is identified with an inflation function that alters linear transformations between background and analysis perturbations through their spectra. The theory encompasses all existing CI methods including multiplicative inflation, RTPS, RTPP