The central issue of how to obtain useful, approximate, uncertainty estimates for assimilation methods using full general circulation models is addressed. Such estimates would be used with assimilation done by either sequential methods or Pontryagin principle/adjoint techniques. The problem of computing the error covariances of realistic oceanic general circulation models is explored by finding the asymptotic solutions to the Riccati equations governing Kalman filters and related smoothers. Existence of the steady-state is established through applying the concepts of controllability and observability to a coarse-resolution primitive equation model in the presence of altimetric observations. A “doubling algorithm” is then used to solve the Riccati equation. The methodology has the added benefit of rendering sequential estimation methods much less costly. Results are presented for a “twin experiment” and for Geosat altimeter data from the North Atlantic Ocean. A realistic altimetric system improves estimates of the depth-dependent (internal mode) circulation, but the improvement of the depth-averaged (external mode) component is limited by the infrequent availability of the present data. The model is found capable of accounting for 6% of the Geosat residual variance; the data residual variance is 43 × 10−3 m2 (21 cm rms) of which the model accounts for 3 × 10−3 m2 (5 cm rms), but both model and dataset are crude.