New Approaches for Adaptive Covariance Relaxation (Inflation) and Adaptive Covariance Localization

For ensemble filters, covariance inflation and localization are often necessary to help improve filter performance by accounting for the unrepresented sampling and model errors. However, tuning inflation and localization parameters can be very costly and adaptive methods are desirable.

In this talk, we first present a new adaptive covariance relaxation (ACR) method for covariance inflation through online estimation of the relaxation parameter according to the innovation statistics. We demonstrate with low-order model experiments that the ACR method is able to ensure filter performance for a range of sampling/model error severity. The newly proposed ACR method borrows ideas from existing techniques that adaptively or non-adaptively inflate prior covariance. Its reliability and ease of implementation suggests its potential for future applications with atmospheric models.

We then propose a variational approach to adaptively determine the optimum radius of influence for ensemble covariance localization when uncorrelated observations are assimilated sequentially. The probabilistic approach is based on the premise of finding an optimum localization radius that minimizes the distance between the Kalman update using the localized sampling covariance versus using the true covariance, when the sequential Ensemble Kalman square-root Filter method is used. We first examine the effectiveness of the proposed method for the cases when the true covariance is known or can be approximated by a sufficiently large ensemble size. Not surprisingly, it is found that the smaller the true covariance distance or the smaller the ensemble, the smaller the localization radius is needed. We further generalize the method to the more usual scenario that the true covariance is unknown but can be represented or estimated probabilistically based on the ensemble sampling covariance. The mathematical formula for this probabilistic and adaptive approach with the use of the Jeffery's prior is derived.

Promising results and limitations of both newly developed adaptive methods are discussed through experiments using the Lorenz-96 system. Testing in complex real-data systems are underway.