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

Abstract

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.

Why does EnKF suffer from analysis overconfidence? An insight into exploiting the ever‐increasing volume of observations

“…For example, since they have to be applied to the analysis perturbations after the analysis mean is updated , the gain matrix used to derive the analysis mean and the posterior covariance remains inconsistent (cf. Duc et al ., 2020 for a detailed discussion).…”

“…The inflation methods that operate on posterior perturbations such as RTPP and RTPS (see Section 5.3) are typically applied after computing the gain matrix K . As such, while the squared posterior spread tr H ′ AH ′ could be increased by these methods, they introduce inconsistency between the gain matrix K used to compute the analysis mean and the gain matrix K used to derive the posterior covariance HA ens H T = HKR (Duc et al ., 2020).…”

An Explanation for the Diagonally Predominant Property of the Positive Symmetric Ensemble Transform Matrix

“…This suggests that we can use the best approximation ̅ as a replacement for in ETKF. In fact, the diagonal ETM ̅ can be shown to be a special case of what we call Diagonal Ensemble Transform Kalman Filter (Duc et al, 2019).…”

Analysis and design of covariance inflation methods using inflation functions. Part 2: adaptive inflation

“…The prior multiplicative inflation (Pham et al, 1998;Anderson and Anderson, 1999), the relaxation-to-prior-perturbation (RTPP) method (Zhang et al, 2004), and its variant, the relaxation-to-prior-spread (RTPS) method (Whitaker and Hamill, 2012), are among the most popular CI methods in practice. In part 1 of this study (Duc et al, 2019) we have shown the existence of a unifying theory encompassing all the existing CI methods. The merit of this theory is that it can solve many open problems in CI and can reveal many new CI methods.…”