The multivariate statistical analysis of sensitivity experiments with atmospheric GCMs is difficult because the sample size is always much smaller than the dimensionality of the GCM fields. Thus, Hasselmann has suggested using a hypothesis testing method, where the anticipated GCM response is represented by an a priori sequence of guessed patterns characterized by only a few parameters. Here we extend it to the more realistic case where the sample size is limited. When only a few CYCM runs are available, it is shown that the statistical significance of the guessed patterns is best established in the full GCM space, using a Hotelling T2test and no optimization procedure. Only in the case of large sample size might it be advantageous to work in the subspace defined by the empirical orthogonal functions of the sample GCM noise field, and to consider rotated guess vectors leading to an optimal signal-to-noise ratio. However, the distribution of the test statistic is then only known asymptotically, and the method is sensitive to the correctness of the guesses and to sampling errors in the noise field.
The method is used to evaluate the sensitivity of the Goddard Institute for Space Studies (GISS) GCM Model I to a North Pacific sea surface temperature anomaly. After discussing standard univariate tests of significance, the multivariate procedure is applied, using a sequence of large scale spherical harmonics as a priori guesses. The analysis is done both in the full GCM space and in the subspace of the sample noise field. It is found that the SST anomaly has a significant scale influence on the wintertime circulation of the model. A two mode linear wave model is then used to provide dynamical guesses for the GCM response, but only the barotropic response is consistent with the GCM data This is due to uncertainties in the heating data, and to the oversimplicity of the linear model.