A Multivariate Reduced-order Optimal Interpolation Method and its Application
305
(1) The HS+ product must be well conditioned, i.e. the columns or modes of S+
must be chosen in the observable subspace, which comes back to the requirement
that Hr be of full rank (all eigenvalues of HrTHr must be nonzero). One example
of the exploration of a reduced-order observability problem is the work of Gavart
and De Mey (1997) on isopycnal EOFs and their signature in sea-Ievel anomaly.
(2) The SMS+ product must be well-conditioned, i.e. the columns or modes of S+
must be chosen in the control space of the full-state problem. For a nonlinear
model, however, the properties of the eigenvectors of SMS+ are not sufficient, and
one must turn to singular vector analysis (e.g. Ehrendorfer and Tribbia, 1997). In
the case of optimal interpolation, which does not have a built-in error propagation
scheme, this discussion may seem out of scope. However it is still important to
make sure that the order reduction remains efficient and meaningful even after
adjustment and time evolution by the dynamics.
(3) The observability of the null space must be marginal (the signature of null
space Hv must be a "small" contribution to By; the eigenvalues of HrTih must be
"small", with Hr = HS+) and close to normality so as to easily include the nulIspace signature in the observational noi se. Otherwise the observations will not
properly be taken into account by the reduced-order scheme. This condition and
the condition (1) together could for instance be automatically achieved by an adaptive method (such as found in Blanchet et al.; 1997), applied to the adaptive estimation of S from the sequence of innovation residuals.
(4) The dynamical coupling between reduced space and null space errors (ilIustrated e.g. in Fukumori et al., 1993) must be small and close to normality so as to
easily include the corresponding error term in the system errors (for the EKF) or
background errors (for OI). Again for a nonlinear model singular vector analysis
or an ensemble approach may be useful in exhibiting the most rapidly growing
errors and their correlations in both subspaces. There are at least two contexts in
which the dynamical coupling is weak: (a) ifthe null space only contains stable or
neutral modes or slowly growing errors; (b) if the physical processes in the null
space and reduced space are only weakly coupled for physical reasons (as in
Gavart and De Mey, 1997). Both contexts can actually be the same.
(5) Other properties can be desirable, e.g. the full-state stability of the filter. OI
methods usualIy have no useful properties in that respect, but the nature of S and in
particular the null space are expected to play an important role on full-state stability.
305
(1) The HS+ product must be well conditioned, i.e. the columns or modes of S+
must be chosen in the observable subspace, which comes back to the requirement
that Hr be of full rank (all eigenvalues of HrTHr must be nonzero). One example
of the exploration of a reduced-order observability problem is the work of Gavart
and De Mey (1997) on isopycnal EOFs and their signature in sea-Ievel anomaly.
(2) The SMS+ product must be well-conditioned, i.e. the columns or modes of S+
must be chosen in the control space of the full-state problem. For a nonlinear
model, however, the properties of the eigenvectors of SMS+ are not sufficient, and
one must turn to singular vector analysis (e.g. Ehrendorfer and Tribbia, 1997). In
the case of optimal interpolation, which does not have a built-in error propagation
scheme, this discussion may seem out of scope. However it is still important to
make sure that the order reduction remains efficient and meaningful even after
adjustment and time evolution by the dynamics.
(3) The observability of the null space must be marginal (the signature of null
space Hv must be a "small" contribution to By; the eigenvalues of HrTih must be
"small", with Hr = HS+) and close to normality so as to easily include the nulIspace signature in the observational noi se. Otherwise the observations will not
properly be taken into account by the reduced-order scheme. This condition and
the condition (1) together could for instance be automatically achieved by an adaptive method (such as found in Blanchet et al.; 1997), applied to the adaptive estimation of S from the sequence of innovation residuals.
(4) The dynamical coupling between reduced space and null space errors (ilIustrated e.g. in Fukumori et al., 1993) must be small and close to normality so as to
easily include the corresponding error term in the system errors (for the EKF) or
background errors (for OI). Again for a nonlinear model singular vector analysis
or an ensemble approach may be useful in exhibiting the most rapidly growing
errors and their correlations in both subspaces. There are at least two contexts in
which the dynamical coupling is weak: (a) ifthe null space only contains stable or
neutral modes or slowly growing errors; (b) if the physical processes in the null
space and reduced space are only weakly coupled for physical reasons (as in
Gavart and De Mey, 1997). Both contexts can actually be the same.
(5) Other properties can be desirable, e.g. the full-state stability of the filter. OI
methods usualIy have no useful properties in that respect, but the nature of S and in
particular the null space are expected to play an important role on full-state stability.