Etudes et évaluation de processus océaniques par des hiérarchies ...

124 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
Ocean Dynamics
Table 1 Parameters varied in
the assimilation experiments:
linear friction parameter τ 0 ,
quadratic drag c D0 , noise σ
and geostrophic adjustment γ
(γ = 0 no initial velocity;
γ = 1 current initially in
geostrophic equilibrium)
Exp. τ 0 (5.10 −4 m s −1 ) c D0 (5.10 −3 ) σ (m) ˜σ (m) γ
A00001 1.0 0.0 10 0.0 1.0
A00011 0.0 1.0 10 0.0 1.0
G00001 1.0 0.0 10 0.0 1.0
G00011 0.0 1.0 10 0.0 1.0
G00021 1.0 1.0 10 0.0 1.0
G00121 1.0 1.0 5 0.0 1.0
G00221 1.0 1.0 20 0.0 1.0
G00321 1.0 1.0 2.5 0.0 1.0
G00421 1.0 1.0 40 0.0 1.0
G01021 1.0 1.0 10 0.0 0.0
G02021 1.0 1.0 10 0.0 0.8
G02221 1.0 1.0 20 0.0 0.8
G10021 1.0 1.0 10 10 1.0
tel-00545911, version 1 - 13 Dec 2010
tribution of the parameter values show qualitatively the
same behaviour. One has to take care, however, that
the true value (the one used in the control run) is a
likely member of the initial ensemble to avoid slow
convergence. The initial v component of the velocity in
the control run is given by v = γv g , only for γ = 1, the
control run starts from geostrophic equilibrium. The
assimilation runs always start from a geostrophically
adjusted state, as an unperturbed gravity current is
close to such a state.
5 Results
5.1 Estimating one parameter
We started, in Exp. A0001 and A0011, to estimate the
parameters τ and c D , respectively, keeping the other
parameters equal to zero. In Fig. 3, a convergence to
the true value is seen in both cases. As stated in the
previous section, the gravity dynamics spans 7 days
(168 h) and the ensemble of the (τ i , c i D ) i=1...m at the
end of a 7-day assimilation experiment is then reused
as the first guess of a new data assimilation run, with
the same control run. By this, we are able to iterate
the assimilation experiment many times to improve the
values for (τ, c D ). This explains why we present results
for times larger than 7 days.
Reusing the data might seem, at a first glance, contradictory
to the concept of the Kalman filter, which uses
all data in an optimal way at the first passage. This is,
however, only true if the problem is linear (parabolic
cost-function), the initial ensemble of the parameter
values are perfectly chosen (mean and variance), the
ensemble size is infinite and the covariance matrix is
complete. Iterating the procedure provides for a better
first guess of the initial ensemble of the parameter
values and increases the ensemble size. A possible
consequence of the iteration is that the variance of the
ensemble is underestimated due to the fact that the
procedure supposes the data to be independent, which
it is not. This can be seen in Fig. 3, where the mean
value converges rapidly to the correct value and the
subsequent iterations mostly reduce the variance. The
situation is different in the other experiments where
the non-linearity is increased due to the interplay of two
different friction laws and we see a continued convergence
to the exact values in subsequent iterations. The
ensemble of parameter values could be resampled after
each iteration to avoid the artificial decrease in variance,
but it is a result of our research that this was not
necessary to do so in the here-presented experiments
Fig. 3 Mean value (red) and standard deviation (black) normalised
by the true value for the estimated parameter in experiment
A0001 (upper) and A0011 (lower)