Mapping tropical Pacific sea level: Data assimilation via a reduced state space Kalman filter

The well‐known fact that tropical sea level can be usefully simulated by linear wind driven models recommends it as a realistic test problem for data assimilation schemes. Here we report on an assimilation of monthly data for the period 1975–1992 from 34 tropical Pacific tide gauges into such a model using a Kalman filter. We present an approach to the Kalman filter that uses a reduced state space representation for the required error covariance matrices. This reduction makes the calculation highly feasible. We argue that a more complete representation will be of no value in typical oceanographic practice, that in principle it is unlikely to be helpful, and that it may even be harmful if the data coverage is sparse, the usual case in oceanography. This is in part a consequence of ignorance of the correct error statistics for the data and model, but only in part. The reduced state space is obtained from a truncated set of multivariate empirical orthogonal functions (EOFs) derived from a long model run without assimilation. The reduced state space filter is compared with a full grid point Kalman filter using the same dynamical model for the period 1979–1985, assimilating eight tide gauge stations and using an additional seven for verification Miller et al. , 1995. Results are not inferior to the full grid point filter, even when the reduced filter retains only nine EOFs. Five sets of reduced space filter assimilations are run with all tide gauge data for the period 1975–1992. In each set a different number of EOFs is retained: 5, 9, 17, 32, and 93, accounting for 60, 70, 80, 90, and 99% of the model variance, respectively. Each set consists of 34 runs, in each of which one station is withheld for verification. Comparing each set to the nonassimilation run, the average rms error at the withheld stations decreases by more than 1 cm. The improvement is generally larger for the stations at lowest latitudes. Increasing the number of EOFs increases agreement with data at locations where data are assimilated; the added structures allow better fits locally. In contrast, results at withheld stations are almost insensitive to the number of EOFs retained. We also compare the Kalman filter theoretical error estimates with the actual errors of the assimilations. Features agree on average, but not in detail, a reminder of the fact that the quality of theoretical estimates is limited by the quality of error models they assume. We briefly discuss the implications of our work for future studies, including the application of the method to full ocean general circulation models and coupled models.