438
4.2 Model tests
By projecting the streamfunction fields of the numerical model onto the
basis functions of the low-order-model at regular time intervals, we obtain
'experimental' time series of the low-order-model variables. We cannot
expect that these time series fulfill the equations of the low-order model
because of the neglection of relative vorticity and because of the interaction
with unresolved modes. But we can test whether the difference between
model and data can reasonably well be represented as an uncorrelated
white noise. We exemplify these test with the barotropic equations (30)
and (31).
The residuum r(t) between model and 'data' is calculated by inserting
the 'experimental' time series. We consider this residuum as a realization
of a random process with mean zero. We have assured that the mean of r
is significantly lower than the other terms in the equation. Furthermore,
the hypothesis is that the true spectrum Crr of r(t) is constant. The test
of the hypothesis follows the general procedure for a multivariate leastsquare fit as described e. g. in Hasselmann (1979). The noise level is fitted
by minimizing an error function
6 2 = L O"i/(Crr(fi) - Crr(fi)) (Crr(fj) - Crr(fj))
(48)
i,j
where Crr (fi) is the value of the spectrum at frequency fi from the hypothesis, Crr (fi) is the spectrum as estimated from the data and O"ij is
the covariance matrix of the spectral estimator for the frequencies fi, fj.
Formulas for the spectral estimator and its variance have been taken from
Jenkins and Watts (1968) and Bendat and Piersol (1986). Under the hypothesis of a true model and approximately Gaussian distributed errors in
the spectral estimation the error function 6 2 is a X 2 -distributed random
variable. We reject the hypothesis if the value of 6 2 is above the X;aquantile of the X 2 -distribution, where a = 0.05 according to a significance
level of 95% and the number of degrees of freedom n is the number of
independent spectral estimates minus one.
Figure 13 shows as an example the estimated spectrum of the residuum
of equation (30). The spectrum may be considered constant in the low
frequency range but it is surely not white in the high frequency part. The
reason is that the neglection of relative vorticity limits the applicability
4.2 Model tests
By projecting the streamfunction fields of the numerical model onto the
basis functions of the low-order-model at regular time intervals, we obtain
'experimental' time series of the low-order-model variables. We cannot
expect that these time series fulfill the equations of the low-order model
because of the neglection of relative vorticity and because of the interaction
with unresolved modes. But we can test whether the difference between
model and data can reasonably well be represented as an uncorrelated
white noise. We exemplify these test with the barotropic equations (30)
and (31).
The residuum r(t) between model and 'data' is calculated by inserting
the 'experimental' time series. We consider this residuum as a realization
of a random process with mean zero. We have assured that the mean of r
is significantly lower than the other terms in the equation. Furthermore,
the hypothesis is that the true spectrum Crr of r(t) is constant. The test
of the hypothesis follows the general procedure for a multivariate leastsquare fit as described e. g. in Hasselmann (1979). The noise level is fitted
by minimizing an error function
6 2 = L O"i/(Crr(fi) - Crr(fi)) (Crr(fj) - Crr(fj))
(48)
i,j
where Crr (fi) is the value of the spectrum at frequency fi from the hypothesis, Crr (fi) is the spectrum as estimated from the data and O"ij is
the covariance matrix of the spectral estimator for the frequencies fi, fj.
Formulas for the spectral estimator and its variance have been taken from
Jenkins and Watts (1968) and Bendat and Piersol (1986). Under the hypothesis of a true model and approximately Gaussian distributed errors in
the spectral estimation the error function 6 2 is a X 2 -distributed random
variable. We reject the hypothesis if the value of 6 2 is above the X;aquantile of the X 2 -distribution, where a = 0.05 according to a significance
level of 95% and the number of degrees of freedom n is the number of
independent spectral estimates minus one.
Figure 13 shows as an example the estimated spectrum of the residuum
of equation (30). The spectrum may be considered constant in the low
frequency range but it is surely not white in the high frequency part. The
reason is that the neglection of relative vorticity limits the applicability