How Sampling Errors in Covariance Estimates Cause Bias in the Kalman Gain and Impact Ensemble Data Assimilation

Localization is the key component to the successful application of ensemble data assimilation (DA) to high-dimensional problems in the geosciences. We study the impact of sampling error and its amelioration through localization using both analytical development and numerical experiments. Specifically, we show how sampling error in covariance estimates accumulates and spreads throughout the entire domain during the computation of the Kalman gain. This results in a bias, which is the dominant issue in unlocalized ensemble DA and, surprisingly, we find that it depends directly on the number of independent observations, but only indirectly on the state dimension. Our derivations and experiments further make it clear that an important aspect of localization is a significant reduction of bias in the Kalman gain, which in turn leads to an increased accuracy of ensemble DA. We illustrate our findings on a variety of simplified linear and nonlinear test problems, including a cycling ensemble Kalman filter applied to the Lorenz-96 model.