Wavelet approximation of error covariance propagation in data assimilation

Estimation of the state of the atmosphere with the Kalman filter remains a distant goal in part because of high computational cost of evolving the error covariance for both linear and non-linear systems (in this case, the extended Kalman filter). Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics by implementing a wavelet approximation scheme on the assimilation of the one-dimensional Burgers’ equation. The discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6% of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.