small scale undulation in the analyzed SSH field can generate gravity waves which diminish the effect of the buoy assimilation. There exist several sources for the gravity waves. The most crucial one is the error of the mean SSH field which causes dynamical inconsistency in the observation data. The time evolution of the velocity error at 142.5｡?, 35｡? is shown in Fig. 13 and the persistent existence of the gravity waves is clearly indicated. These waves are related to the error of the observed altimetric data (mean plus time-varying part). The time evolution of the interface depth error at the same point (not shown) has the peak at the assimilation time due to the contamination of the error in the mean interface depth field. This situation is expected to be improved when the mean SSH field is corrected.
From these experiments, we found that it is important to use the accurate mean SSH to assimilate the altimetric data and the simultaneous assimilation of the drifting buoy and altimetric data is effective in constraining the SSH field. It is expected that the mean 5511 field can be corrected successively with the simultaneous assimilation of the drifting buoy and altimetric data.
5.2 Experiment 2. the model for the successive correction of mean SSH
The formulation of the basic equation for successive correction of mean SSH is similar to the optimal interpolation described earlier:
The notation is the same as in Eq. (3), where h is the mean interface depth, superscripts + and - are the corrected and a priori estimation of the mean interface depth, and h10 is the observed time-varying part of the interface depth.
With the assumption that the time-varying part has no error, and that the error of the mean SSH is uncorrelated spatially, K1 becomes follows:
The second transform is from comparing with Eq. (4) and using Eq. (3) because of R10 = 0 and R0 in Eq. (4) is equal to Ro where R10 and R0 are error covariance matrices of the altimetric observation and the estimated mean SSH field, respectively. From Eq. (13) the new estimation for the mean SSH field is obtained,
where ha is the analysis value of the interface depth obtained from Eq. (3).
The test of the successive correction model of the mean SSH field are carried out by the following two cases: only altimetric data are assimilated in Exp. 2-1 for the reference. In Exp. 2-2, observation data are the same as in Exp. 1-2 where 32 drifting buoys and the altimetric data are assimilated simultaneously. In both cases, the 1-year averaged interface depth of the simulation run (Fig. 9b) is used for the initial guess of the mean field.
The time series of the RMS errors for the interface depth are shown in Fig. 14. Exp. 2-2 using both the drifting buoy and the altimetric data shows a much smaller error than Exp. 2-1. The instantaneous error decreases less than a half of the initial field and the error in the mean interface depth field (not shown) is also steadily reduced to 38.7 m, corresponding to a 40% reduction after the 1-year experiment. This implies that the velocity data derived from the drifting buoys ensure the success of correcting the mean SSH and reducing the mean interface depth error leads to a better result of the assimilation experiment.
In contrast, the instantaneous error field in the case of assimilating only altimetric data (Exp.2-1) cannot correct the mean interface depth field and the instantaneous field does not change from Exp.1-1. The error of the mean interface depth increases at the first assimilation time. This is
