A flexible additive inflation scheme for treating model error in ensemble Kalman filters

Abstract

Data assimilation algorithms require an accurate estimate of the uncertainty of the prior (background) field that cannot be adequately represented by the ensemble of numerical model simulations. Partially, this is due to the sampling error that arises from the use of a small number of ensemble members to represent the background‐error covariance. It is also partially a consequence of the fact that the geophysical model does not represent its own error. Several mechanisms have been introduced so far to alleviate the detrimental effects of misrepresented ensemble covariances, allowing for the successful implementation of ensemble data assimilation techniques for atmospheric dynamics. One of the established approaches is additive inflation, which consists of perturbing each ensemble member with a sample from a given distribution. This results in a fixed rank of the effective model‐error covariance matrix. In this article, a more flexible approach is introduced, where the model error samples are treated as additional synthetic ensemble members, which are used in the update step of data assimilation but are not forecast. This way, the rank of the model‐error covariance matrix can be chosen independently of the ensemble. The effect of this altered additive inflation method on the performance of the filter is analyzed here in an idealized experiment. It is shown that the additional synthetic ensemble members can make it feasible to achieve convergence in an otherwise divergent parameter setting of data assimilation. The use of this method also allows for a less stringent localization radius.

Representation of Model Error in Convective‐Scale Data Assimilation: Additive Noise Based on Model Truncation Error

“…The properties of model truncation error may differ in a different season, for example a training period in winter may be needed for case studies of winter storms. There are some new approaches as the Adaptive Background Error Inflation (Minamide & Zhang, ), which attempts to treat model error and non‐Gaussian sampling error adaptively, as well as a new approach that allows for using more ensemble members in the treatment of model error than forecasted with additive noise (Sommer & Janjić, ). These could be explored for convective scale data assimilation in the future together with the Adaptive Observation Error Inflation (Minamide & Zhang, ).…”

Ensemble-based kernel learning for a class of data assimilation problems with imperfect forward simulators

“…While there are already substantial efforts, e.g. [13–21], dedicated to this research topic, many of the methods have to rely on certain simplifying assumptions (e.g., Gaussianity, stationarity etc), which are avoided in our proposed integrated framework that integrates functional approximation (through a machine learning model) into data assimilation. To the best of our knowledge, such an integrated ensemble data assimilation framework is not investigated yet in the literature.…”

“…Currently, a common practice in this regard is to add some (typically) additive stochastic term into the forward simulator, as a simple way to represent simulator imperfection (see, for example, [13–21]). For practical convenience, one may presume that the stochastic term follows a Gaussian distribution, so that the effect of simulator imperfection is taken into account by including the mean and covariance matrix of the stochastic term into the assimilation algorithm.…”

Representation of Model Error in Convective‐Scale Data Assimilation: Additive Noise, Relaxation Methods, and Combinations

“…To ensure that additive noise increases the analysis ensemble covariance without changing the ensemble mean, the mean should be removed from each η ( i ) (Whitaker et al, ). Since Q is constructed by N e samples by , its rank is at most N e (see Sommer & Janjić, , for alternative formulation that increases the rank). Therefore, additive noise changes ensemble members but it cannot contribute much to the increase of rank of background covariance matrix and to reduction of sampling error.…”