Wave data assimilation with the Kalman filter

A new data assimilation method for ocean waves is presented, based on an efficient low-rank approximation to the Kalman filter. Both the extended Kalman filter and a truncated second-order filter are implemented. In order to explicitly estimate past wind corrections based on current wave measurements, the filter is extended to a fixed-lag Kalman smoother for the wind fields. The filter is tested in a number of synthetic experiments with simple geometries. Propagation experiments with errors in the boundary condition showed that the KF was able to accurately propagate forecast errors, resulting in spatially varying error correlations, which would be impossible to model with time-independent assimilation methods like OI. An explicit comparison with an OI assimilation scheme showed that the KF also is superior in estimating the sea state at some distance from the observations. In experiments with errors in the driving wind, the modeled error estimates were also in agreement with the actual forecast errors. The bias in the state estimate, which is introduced through the nonlinear dependence of the waves on the driving wind field, was largely removed by the second-order filter, even without actually assimilating data. Assimilation of wave observations resulted in an improved wave analysis and in correction of past wind fields. The accuracy of this wind correction depends strongly on the actual place and time of wave generation, which is correctly modeled by the error estimate supplied by the Kalman filter. In summary, the KF approach is shown to be a reliable assimilation scheme in these simple experiments, and has the advantage over other assimilation methods that it supplies explicit dynamical error estimates.