A Multivariate Reduced-order Optimal Interpolation Method and its Application
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the errors in the reduced space and null-space must be weakly correlated to each
other through the dynamics. The latter condition can be achieved e.g. if the dominant physical processes in both subspaces are weakly coupled over the time scale
considered. Considering these assumptions to be valid, we explicitly discard any
attempt to control the null space (we can bring this back later if needed by means of
an external algorithm). The background error covariance matrix is therefore modelled as:
(11)
The Brf matrix contains the background error covariances in the reduced space.
It is modelled as a diagonal matrix in (11), which comes back to assuming that the
columns of ST are eigenvectors of W. In other words, in the 3-D EOF case, Brf
contains the error variances. Expression (11) also implies that the null-space background errors are assumed to be zero.
Given the above assumptions, the OI problem can now be entirely expressed in
the reduced space. The Reduced-Order Optimal Interpolation (ROOI) gain in the
full-state space can be written:
KROOI= STKr
(12)
Kr= BrfHrT(HrBrfHr T + Rr)-l.
Observations are related to the reduced state space by means of the reducedorder observation operator Hr=HS~ The reduced-order observational error covariance matri x Rr includes the representativity errors in the reduced space due to the
observability ofthe null space through the Hv term in (A3), in addition to the measurement errors. At analysis time, the model restarts from its previous fields plus a
correction in reduced space which is converted back to full-state space by ST.
Whatever properties were present in the null space are preserved. The method does
not provide a built-in evo1ution of S such as in more sophisticated algorithms, but
one can impose for instance a dependency on the season.
The analysis errors now become:
(13)
with
Br a = (1 - KrHr)Brf(1 - KrHrV + KrRrKr T
(14)
The null-space errors can be added in (13) but are not accounted for by the ROOI
scheme.
As noted above, Wand its EOFs are generally unknown. One practical possibility to generate useful orthonormal vectors is to calculate EOFs of departures from a
climatology, which we will hereafter call perturbation EOFs. Either data or model
outputs can be used for this purpose. One must of course make the assumption that
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the errors in the reduced space and null-space must be weakly correlated to each
other through the dynamics. The latter condition can be achieved e.g. if the dominant physical processes in both subspaces are weakly coupled over the time scale
considered. Considering these assumptions to be valid, we explicitly discard any
attempt to control the null space (we can bring this back later if needed by means of
an external algorithm). The background error covariance matrix is therefore modelled as:
(11)
The Brf matrix contains the background error covariances in the reduced space.
It is modelled as a diagonal matrix in (11), which comes back to assuming that the
columns of ST are eigenvectors of W. In other words, in the 3-D EOF case, Brf
contains the error variances. Expression (11) also implies that the null-space background errors are assumed to be zero.
Given the above assumptions, the OI problem can now be entirely expressed in
the reduced space. The Reduced-Order Optimal Interpolation (ROOI) gain in the
full-state space can be written:
KROOI= STKr
(12)
Kr= BrfHrT(HrBrfHr T + Rr)-l.
Observations are related to the reduced state space by means of the reducedorder observation operator Hr=HS~ The reduced-order observational error covariance matri x Rr includes the representativity errors in the reduced space due to the
observability ofthe null space through the Hv term in (A3), in addition to the measurement errors. At analysis time, the model restarts from its previous fields plus a
correction in reduced space which is converted back to full-state space by ST.
Whatever properties were present in the null space are preserved. The method does
not provide a built-in evo1ution of S such as in more sophisticated algorithms, but
one can impose for instance a dependency on the season.
The analysis errors now become:
(13)
with
Br a = (1 - KrHr)Brf(1 - KrHrV + KrRrKr T
(14)
The null-space errors can be added in (13) but are not accounted for by the ROOI
scheme.
As noted above, Wand its EOFs are generally unknown. One practical possibility to generate useful orthonormal vectors is to calculate EOFs of departures from a
climatology, which we will hereafter call perturbation EOFs. Either data or model
outputs can be used for this purpose. One must of course make the assumption that