Analysis and Design of Covariance Inflation Methods Using Spectral Transformations

Covariance inflation (CI) in the Ensemble Kalman Filter (EnKF) is approached from a functional viewpoint. That is each CI method is identified with an inflation function that alters the structure of 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 singular values of ensemble transform matrices. These singular values are indeed the reduction factors of background singular values in data assimilation, which results in analysis singular values.

To be an inflation function, a function has to satisfy three conditions: (1) the functional condition: all reduction factors have to increase; (2) the no-observation condition: when impact of observations becomes negligible, analysis perturbations should be identical to background perturbations; and (3) the order-preserving condition: the resulting analysis singular values have to keep the same increasing order as the uninflated background singular values.

It turns out that the relaxation-to-prior-spread method (RTPS) and the relaxation-to-prior-perturbation method (RTPP) belongs to the class of linear inflation functions. Whereas RTPS is a realization of multiplicative inflation functions, RTPP is a fixed-parameter version of linear inflation functions. In this class, we also have constant inflation functions, other realizations of multiplicative inflation functions and parameter-varying linear inflation functions. More surprisingly, the Deterministic EnKF is found to belong to the class of quadratic inflation functions with the linear terms omitted. This quadratic class introduces an elegant form for computing analysis perturbations through the Kalman gain without a need for ensemble transform matrices. Higher-order polynomial and non-polynomial forms of inflation functions can be handled by certain EnKF methods.