8. Conclusion remarks

The same pictures are obtained for the estimates <sup>b</sup>ck, k <sup>¼</sup> <sup>2</sup>; <sup>3</sup>; 4. Mention that in the CHF,

In Table 3, the performances of the three filters are displayed. The errors are the averaged (spatially and temporally) rms of PE for the SSH and for the two velocity components u and v. The results in Table 3 show that two filters PEF (SCH) and PEF (SSP) are practically of the same performance, and their estimates are much better compared to those of the CHF, with a slightly better performance for the PEF (SSP). We note that as the PEF (SCH) is constructed on the basis of an ensemble of samples tending to the first dominant SchV, its performance must be theoretically better than that of the PEF (SSP). The slightly better performance of PEF (SSP) (compared to that of PEF (SCH)) may be explained by the fact that the best theoretical performance of PEF (SCH) can be obtained only if the model is linear, stationary, and the number of PE samples in En SCH ð Þ at each iteration must be large enough. The ensemble size of En SCH ð Þ in the present experiment is too small compared with the dimension of the

rms CHF (cm) PEF (SCH) (cm=s) PEF (RAN) (cm=s)

Figure 3. Variance of PE resulting from three filters PEF (SCH), PEF (SSP), and AF. It is seen that the AF yields much

better performance compared to the two other nonadaptive filters PEF (SCH) and PEF (SSP).

ssh(fcst) 7.39 5.09 4.95 u(fcst) 7.59 5.36 5.29 v(fcst) 7.72 5.73 5.64

c<sup>2</sup> ¼ c<sup>3</sup> ¼ 0.

MICOM model.

7.2. Performance of different filters

78 Perturbation Methods with Applications in Science and Engineering

Table 3. rms of PE for ssh, and u, v velocity components.

We have presented in this chapter the different types of OPs—deterministic, stochastic, or optimal, the invariant subspaces of the system dynamics. The ODPs and OSPs play an important role in the study on the predictability of the system dynamics as well as in construction of optimal OFS for environmental geophysical systems.

One other class of perturbation known as SSP is found to be a very efficient tool for solving optimization and estimation problems, especially with Hd matrices and in computing the optimal perturbations.

The numerical experiments presented in this chapter confirm the important role of the different types of OPs in the numerical study of Hd assimilation systems.
