We present techniques for the approximate evolution of error correlation statistics used in atmospheric data assimilation. This step dominates the computational requirements of the Kalman filter, and is therefore a logical and frequent target of approximation methods. Error correlations needed by assimilation systems are influenced by the observing network, model errors and propagation due to atmospheric winds, and therefore tend to become highly localized. Wavelet functions are an efficient way to represent localized information and therefore have the potential transform the error correlations into a low dimensional system.

We apply this technique to an existing constituent assimilation system and show that the error correlations can be compressed by about a factor of 50 without loss of localized information. We also extend this work to non-linear systems using the extended Kalman Filter (EKF) applied to the one-dimensional Burgers equation and show that only a small fraction of the wavelet coefficients are required to capture the covariance dynamics.