299
The vector c i+1 contains the r amplitudes of the error modes that need
to be estimated at each analysis step. Finally, the scheme evaluates the
analysis error as follows
P
a
i+1 = [I K i+1 H]P
f
i+1 = S
a
i+1 S
aT
i+1
(45)
and updates the vectors of the reduced basis according to
S
a
i+1 = S
f
i+1 [I + (HS
f
i+1 )
T R
31
(HS
f
i+1 )]
31/2 .
(46)
This shows that, if the rank of P
f
i+1 is r, then the rank of P
a
i+1 is equal
to r also; therefore, recursivity in the forecast/analysis cycles is allowed.
For better numerical conditioning, it is possible to re-orthonormalize the
reduced basis by recomputing a SVD decomposition of the analysis error
covariance matrix.
There is one additional consideration that should be mentioned concerning the practical computation of the Kalman gain in the reduced
space (42). The robust estimation of small correlations associated with
remote observations is a common di!culty that can be avoided by computing EOFs with compact support, as mentioned above. Instead of
computing local EOFs, a simplification of the analysis scheme can be
designed by enforcing to zero the error covariances between distant variables which are believed to be uncorrelated in the real ocean [Houtekamer
and Mitchell, 1998].
This simplification is implemented by assuming that distant observations have negligible influence on the analysis. The global system is
split into sub-systems, and for each of these the traditional analysis is
computed. Only data points located within individual regions, centered
on a sub-domain of one or several grid points to be updated, actually
contribute to the gain in (42). This approach can be understood as a
tuning of the observation operator according to the sub-domain in question. Intuitively, this approximation makes sense because only data
points located in the neighborhood of a model grid point should objectively have an impact on the analysis for that grid point. The size of the
regions is determined in such a way that the distribution of the observations available on the model domain always provides at least a few data
points within each region of influence (if there were no data available
in the region of influence, no correction would apply). This algorithmic
simplification also improves the analysis because it enables a larger part
of the estimation space to be spanned over a particular subdomain (see,
for instance, Brusdal et al. [2003]).
Other interesting procedures exist for addressing the locality issue.
One of these is the partitioning of the large estimation problem into
OCEAN DATA ASSIMILATION
The vector c i+1 contains the r amplitudes of the error modes that need
to be estimated at each analysis step. Finally, the scheme evaluates the
analysis error as follows
P
a
i+1 = [I K i+1 H]P
f
i+1 = S
a
i+1 S
aT
i+1
(45)
and updates the vectors of the reduced basis according to
S
a
i+1 = S
f
i+1 [I + (HS
f
i+1 )
T R
31
(HS
f
i+1 )]
31/2 .
(46)
This shows that, if the rank of P
f
i+1 is r, then the rank of P
a
i+1 is equal
to r also; therefore, recursivity in the forecast/analysis cycles is allowed.
For better numerical conditioning, it is possible to re-orthonormalize the
reduced basis by recomputing a SVD decomposition of the analysis error
covariance matrix.
There is one additional consideration that should be mentioned concerning the practical computation of the Kalman gain in the reduced
space (42). The robust estimation of small correlations associated with
remote observations is a common di!culty that can be avoided by computing EOFs with compact support, as mentioned above. Instead of
computing local EOFs, a simplification of the analysis scheme can be
designed by enforcing to zero the error covariances between distant variables which are believed to be uncorrelated in the real ocean [Houtekamer
and Mitchell, 1998].
This simplification is implemented by assuming that distant observations have negligible influence on the analysis. The global system is
split into sub-systems, and for each of these the traditional analysis is
computed. Only data points located within individual regions, centered
on a sub-domain of one or several grid points to be updated, actually
contribute to the gain in (42). This approach can be understood as a
tuning of the observation operator according to the sub-domain in question. Intuitively, this approximation makes sense because only data
points located in the neighborhood of a model grid point should objectively have an impact on the analysis for that grid point. The size of the
regions is determined in such a way that the distribution of the observations available on the model domain always provides at least a few data
points within each region of influence (if there were no data available
in the region of influence, no correction would apply). This algorithmic
simplification also improves the analysis because it enables a larger part
of the estimation space to be spanned over a particular subdomain (see,
for instance, Brusdal et al. [2003]).
Other interesting procedures exist for addressing the locality issue.
One of these is the partitioning of the large estimation problem into
OCEAN DATA ASSIMILATION