Significance testing for variational assimilation

The hypothesis test associated with a variational data-assimilation algorithm is examined in detail. It can be shown that the test statistic, (J) over cap, should have a chi(M)(2) distribution, where M is the number of scalar data assimilated, if the hypothesis regarding the model and data residuals is consistent with the true physical system. The skill of the Kolmogorov-Smimov (KS) test in identifying whether the hypothesis used in the assimilation is `true' (that is, the hypothesis correctly reflects the residuals in the model and the data) or not (that is, the hypothesis has either incorrect parameters in the covariance or an incorrect bias) is evaluated. It is shown that the KS test does indeed have some skill in identifying a false hypothesis. However, as one might expect, the skill is limited to cases where the particular component of the hypothesis that is incorrect has an impact on the test statistic and therefore on its distribution.