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

4.6. ESTIMATION OF FRICTION PARAMETERS AND LAWS IN 1.5D SHALLOW-WATER GRAVI
Ocean Dynamics
tel-00545911, version 1 - 13 Dec 2010
information can travel to the next assimilation point
1 km away, but it cannot reach over the distance of two
assimilation grid points. In between assimilation points,
h is interpolated linearly.
In general, the observed value of the vertical extension
of the gravity current h obs (x, t) includes measurement
errors η(x, t) and is related to the true value
by h obs (x, t) = h true (x, t) + η(x, t). For consistency, the
measurement error η(x, t) has the same first- (zero
mean) and second-order moments (˜σ 2 ) as the noise
vectors ǫ i (x, t), but it does not depend on the actual
realisation, that is, i. When assimilating data, σ has to be
provided prior to the experiment, whereas ˜σ is usually
not known and can only be estimated.
In our parameter estimation experiments, the ensemble
size is m = 100; this is much larger than the
number of parameters to estimate, that is, two, equal
to the number of observations at each assimilation
time, but smaller than the dimension of the augmented
state vector x = (h, u, v, τ, c D ), that is, 302. Using an
ensemble size an order of magnitude larger did not
improve the convergence significantly, reducing the ensemble
size an order of magnitude leads to a frequent
divergence of the assimilation.
Another important point is that the parameter values
are set to be constant in time. This allows us to iterate
our estimation experiment, that is, once we performed
an entire estimation run, we could take the analysed
values of the parameters after the last assimilation and
use them for the first guess of a new experiment, iterating,
thus, the same gravity current dynamics several
times, always keeping the same data.
The values of the friction parameters are clearly
non-negative, so every time the assimilation scheme
provides a negative value of one of these parameters,
which is possible due to the linearity of the analysis step
and the statistical nature of the EnKF, the value is put
to zero.
4 Twin experiments
The goal of our data assimilation experiment is to
determine the friction law acting on the gravity extension
(h) of the gravity current. The vertical extension,
that is, the density structure of a gravity current, is
the variable that is easiest to measure and observe in
the ocean and in laboratory experiments. In our twin
experiments, the data from observations or laboratory
experiments are replaced by data from a control run
(dynamics not subject to data assimilation) using the
same numerical model as in the data assimilation experiments
(see Section 2). In some experiments, the
initial conditions in the control runs differ from those
of the assimilation runs. In Fig. 2, the vertical extension
is shown for the control runs of three different sets of
the friction parameters. The differences in the shapes of
these curves are a prerequisite for the possible success
of our experiments. It is a time series of these shapes,
and only these, which is provided to the assimilation
run and which has to determine the friction parameters
on their basis. The observed variables must be sensitive
to the parameters to be controlled. By performing
twin experiments, we thus evaluate the feasibility of the
data assimilation strategy by assimilating data that were
produced by the very same model that is used for the assimilation
experiment with the actual parameter values
fixed. We thus explore the prerequisites under which
the parameter estimation scheme is able to come up
with the right set of parameters under these favourable
conditions.
To explore the feasibility of the parameter estimation,
we performed a series of experiments, which are
summarised in Table 1, where σ gives the standard
deviation of the noise added to the “observation” of
the vertical extension of the layer. When the perturbed
layer thickness is smaller than 10 −6 m, it is put to
10 −6 m. The initial distribution of the ensemble of
parameters (τ i , c i D ) i=1,...,100 has a normal distribution,
with a mean of (8. · 10 −3 , 7. · 10 −4 ) and a standard variation
of (5. · 10 −3 , 3.5 · 10 −4 ). Please note that the initial
ensemble has a mean that is significantly different from
the true values (τ 0 , c D0 ) (values used in the control
run). The evaluation of the assimilation procedure is
based on the convergence of the ensemble mean to the
true values together with the decrease of the ensemble
dispersion.
All pseudo-random numbers were generated by a
“Mersenne Twister” (Matsumoto and Nishimura 1998).
Other experiments, not shown here, with different
mean values and standard deviations of the initial dis-
Fig. 2 Vertical extension of the gravity current (control runs) for
experiments G0001 (blue), G0011 (red) and G0021 (green). At
t = 0 (all curves superpose on black line), at t = 96 h, shapes in
the three experiments are different