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

Part 1 of this study shows the existence of a unifying theory that encompasses all exiting covariance inflation (CI) methods under a framework of inflation functions operating on the eigenvalues of the ensemble transform matrix. Given a vast space of potential inflation functions, a natural question is how to choose the optimal one. This part 2 shows how adaptive inflation can be implemented in the context of inflation functions to estimate inflation functions on-the-fly from observations. Estimation of prior inflation functions (PIF) is equivalent to estimation of background error covariances G in observation space. However, the conventional CI methods confront a difficult problem of how to convert G back to model space to calculate analysis increments and perturbations, which is possible in very limited cases such as multiplicative inflation. The solution here is to exploit the equivalence between PIFs and inflation functions to transplant the estimated PIF into its equivalent inflation function. Analysis increments and perturbations can be calculated using this intermediate inflation function. Maximum likelihood (ML) estimations of three families of PIFs with two data assimilation systems show that this approach works well and that the adaptive methods can even beat a well-tuned CI method. Estimation of the inflation functions that do not have equivalent PIFs is more difficult since we lack a theoretical guidance, such as Bayesian inference as in the case of PIFs. Those inflation functions are proposed to be estimated from posterior innovation statistics. There are two supporting facts for this estimation: (a) ML estimation of nonparametric PIFs leads back to the prior innovation statistics, and (b) posterior innovation statistics is an effective diagnostic tool for assessing performance of inflation functions. It is interesting to find that ML estimates of inflation functions yield analyses and forecasts better than ML estimates of PIFs.