Implementation of a reduced rank square-root smoother for high resolution ocean data assimilation

Abstract Optimal smoothers enable the use of future observations to estimate the state of a dynamical system. In this paper, a square-root smoother algorithm is presented, extended from the Singular Evolutive Extended Kalman (SEEK) filter, a square-root Kalman filter routinely used for ocean data assimilation. With this filter algorithm, the smoother extension appears almost cost-free. A modified algorithm implementing a particular parameterization of model error is also described. The smoother is applied with an ocean circulation model in a double-gyre, 1/4° configuration, able to represent mid-latitude mesoscale dynamics. Twin experiments are performed: the true fields are drawn from a simulation at a 1/6° resolution, and noised. Then, altimetric satellite tracks and sparse vertical profiles of temperature are extracted to form the observations. The smoother is efficient in reducing errors, particularly in the regions poorly covered by the observations at the filter analysis time. It results in a significant reduction of the global error: the Root Mean Square Error in Sea Surface Height from the filter is further reduced by 20% by the smoother. The actual smoothing of the global error through time is also verified. Three essential issues are then investigated: (i) the time distance within which observations may be favourably used to correct the state estimates is found to be 8 days with our system. (ii) The impact of the model error parameterization is stressed. When this parameterization is spuriously neglected, the smoother can deteriorate the state estimates. (iii) Iterations of the smoother over a fixed time interval are tested. Although this procedure improves the state estimates over the assimilation window, it also makes the subsequent forecast worse than the filter in our experiment.