Section 8.3: Multivariate Analysis
149
b) Theoretical Predictions
The strong constraint inherent to having to formulate apriori guesses is
not limiting for testing theoretical predictions, like whether the GCM response can be interpreted in part as a forced linear wave response, since in
this case the anticipated response is well-defined apriori. We illustrate this
approach with another small sampIe sensitivity study performed with the
GISS GCM Model II (Frankignoul and Molin, 1988b). Here a positive SST
anomaly was considered in the subtropical Pacific in order to test the applicability of stationary Rossby wave models to the atmosphere. The guesses
were constructed using a barotropic model which was linearized about both
the GCM mean zonal state and its mean wavy state, assuming for simplicity
a local relationship between SST, diabatic heating, upper level divergence,
and anticydonic vorticity anomalies. The two basic states were tested separately, so there was only one guess vector in (8.15), and the test statistic
(8.9) reduced to a (t-statistic)2. Figure 8.3 (top) shows the mean monthly
changes in 320 hPa geopotential height between 5 anomaly and 5 control
runs, after "tri angular truncation 12 (T12)". Hatched areas indicate the grid
points where the null hypothesis was rejected at the 5% level, using standard univariate t-tests (8.3) (before T12 truncation). The rejection rate in
the Northern Hemisphere was about 5%, consistent with the absence of a
true signal. Also shown is the barotropic model predition, using the 320 hPa
mean zonal (middle) and wavy (bot tom) GCM flows as basic state. At the
10% level, the prediction is significant when the basic state is wavy, but not
otherwise, and in the former case its magnitude is consistent with the GCM
signal. However, the signal-to-noise ratio is low and only a sm all fr action of
the GCM anomaly variance is explained by the barotropic model. Although
this is due in part to the differences in the numerics of the two models (it
would have been preferable to use a linearized version of the GCM to predict
the linear wave response), the limited agreement was expected, since transient forcing, barodinicity, nonlinearities and the divergent flow had been
neglected in the wave model.
From a statistical viewpoint, the use of model predictions as guess vectors
is more satisfactory than the use of empirical guesses, since the data reduction is much more effective, possibly leading to detecting very weak signals
that could not be seen otherwise. Furthermore, the results can provide interesting theoretical clues (in the example above, on the role of the mean
zonal asymmetries and the equivalent barotropic level). The multivariate
method is thus a useful tool both for hypothesis testing and for interpreting sensitivity studies. However, the implications for the GCM response are
again guess-dependent, and caution is required: another model might have
explained a larger part of the differences between anomaly and control runs,
or the use of an ill-adapted prediction might have improperly suggested that
there was no significant GCM response.
149
b) Theoretical Predictions
The strong constraint inherent to having to formulate apriori guesses is
not limiting for testing theoretical predictions, like whether the GCM response can be interpreted in part as a forced linear wave response, since in
this case the anticipated response is well-defined apriori. We illustrate this
approach with another small sampIe sensitivity study performed with the
GISS GCM Model II (Frankignoul and Molin, 1988b). Here a positive SST
anomaly was considered in the subtropical Pacific in order to test the applicability of stationary Rossby wave models to the atmosphere. The guesses
were constructed using a barotropic model which was linearized about both
the GCM mean zonal state and its mean wavy state, assuming for simplicity
a local relationship between SST, diabatic heating, upper level divergence,
and anticydonic vorticity anomalies. The two basic states were tested separately, so there was only one guess vector in (8.15), and the test statistic
(8.9) reduced to a (t-statistic)2. Figure 8.3 (top) shows the mean monthly
changes in 320 hPa geopotential height between 5 anomaly and 5 control
runs, after "tri angular truncation 12 (T12)". Hatched areas indicate the grid
points where the null hypothesis was rejected at the 5% level, using standard univariate t-tests (8.3) (before T12 truncation). The rejection rate in
the Northern Hemisphere was about 5%, consistent with the absence of a
true signal. Also shown is the barotropic model predition, using the 320 hPa
mean zonal (middle) and wavy (bot tom) GCM flows as basic state. At the
10% level, the prediction is significant when the basic state is wavy, but not
otherwise, and in the former case its magnitude is consistent with the GCM
signal. However, the signal-to-noise ratio is low and only a sm all fr action of
the GCM anomaly variance is explained by the barotropic model. Although
this is due in part to the differences in the numerics of the two models (it
would have been preferable to use a linearized version of the GCM to predict
the linear wave response), the limited agreement was expected, since transient forcing, barodinicity, nonlinearities and the divergent flow had been
neglected in the wave model.
From a statistical viewpoint, the use of model predictions as guess vectors
is more satisfactory than the use of empirical guesses, since the data reduction is much more effective, possibly leading to detecting very weak signals
that could not be seen otherwise. Furthermore, the results can provide interesting theoretical clues (in the example above, on the role of the mean
zonal asymmetries and the equivalent barotropic level). The multivariate
method is thus a useful tool both for hypothesis testing and for interpreting sensitivity studies. However, the implications for the GCM response are
again guess-dependent, and caution is required: another model might have
explained a larger part of the differences between anomaly and control runs,
or the use of an ill-adapted prediction might have improperly suggested that
there was no significant GCM response.