As described in [Wahba and Wold, 1975, Craven and Wahba, 1979, Golub et al., 1979, Wahba, 1990, Brankart and Brasseur, 1994], the Ordinary Cross-Validation (OCV) and furthermore the Generalised Cross-Validation (GCV) provide a much more robust tools to evaluate and optimise the analysis parameters, i.e. the signal/noise ratio and the characteristic length. The first method is not that interesting because it needs as many analysis as there are data. The GCV a contrario, only needs two analysis for the estimation of the noise variance: one applied to the real data and the other one applied to a random vector. The minimum estimator found minimises the statistical error. The GCV is only valid for sufficient data ( ). For small data bases, one could generate several random vector and take the mean estimator. This method however would increase dramatically the cost of the generalised cross validation method. Despite the approximate results, we used this approach in a first time. Future work will investigate other modified GCV calibration methods. A third method, a kind of 'Sampling Cross Validation' is shown to be equivalent to the GCV [Brankart, 1996], but it is relatively expensive too.
We have seen that the displacement of stations leads to undercovered and overcovered regions. We can expect the later to gain accuracy. Also the statistical features of the field might be modified.
Figure: Covariance of temperature field at the surface.
Figure: Covariance of salinity field at the surface.