Efficient Parameterization of the Observation Error Covariance Matrix for Square Root or Ensemble Kalman Filters: Application to Ocean Altimetry

Abstract

In the Kalman filter standard algorithm, the computational complexity of the observational update is proportional to the cube of the number y of observations (leading behavior for large y). In realistic atmospheric or oceanic applications, involving an increasing quantity of available observations, this often leads to a prohibitive cost and to the necessity of simplifying the problem by aggregating or dropping observations. If the filter error covariance matrices are in square root form, as in square root or ensemble Kalman filters, the standard algorithm can be transformed to be linear in y, providing that the observation error covariance matrix is diagonal. This is a significant drawback of this transformed algorithm and often leads to an assumption of uncorrelated observation errors for the sake of numerical efficiency. In this paper, it is shown that the linearity of the transformed algorithm in y can be preserved for other forms of the observation error covariance matrix. In particular, quite general correlation structures (with analytic asymptotic expressions) can be simulated simply by augmenting the observation vector with differences of the original observations, such as their discrete gradients. Errors in ocean altimetric observations are spatially correlated, as for instance orbit or atmospheric errors along the satellite track. Adequately parameterizing these correlations can directly improve the quality of observational updates and the accuracy of the associated error estimates. In this paper, the example of the North Brazil Current circulation is used to demonstrate the importance of this effect, which is especially significant in that region of moderate ratio between signal amplitude and observation noise, and to show that the efficient parameterization that is proposed for the observation error correlations is appropriate to take it into account. Adding explicit gradient observations also receives a physical justification. This parameterization is thus proved to be useful to ocean data assimilation systems that are based on square root or ensemble Kalman filters, as soon as the number of observations becomes penalizing, and if a sophisticated parameterization of the observation error correlations is required.