112 Oeir Evensen
There are a few locations where the filter estimate starts diverging from the reference solution, e.g. for t = 26 and t = 34. Note however that the ensemble Kalman
filter recovers quickly and begins tracing the reference solution again. The error
estimate given in the lower plot of Fig. 6.3 shows strong error growth at these particular locations and thus indicates that the ensemble is passing through a region in
the state space which is associated with strong instability. The error estimates show
the same behavior as was found by Miller et al. (1994) with very strong error
growth when the model solution is passing through the unstable regions ofthe state
space, and otherwise rather weak error variance growth in the more stable regions.
Note for example ihe low error variance when t E [28,34] corresponding to the
oscillation of the solution around one of the attractors.
Finally, it should be pointed out that in the ensemble Kalman filter a variance
minimizing analysis is calculated at measurement times. Thus, even if the ensembIe certainly is non-Gaussian due to the forward integration of nonlinear model
equations, only the Gaussian part of the distribution is used. This is in contrast to
the work by MiHer et al. (1999) where the maximum-likelihood analysis is calculated by actually constructing the density function for the model evolution and then
calculating the conditional density in terms of analytical functions. They found that
this made a significant improvement on the analysis. However, it is stiH not clear
how this approach can be used in a practical way for high dimensional state spaces.
6.6 An ensemble Kalman fllter for an OGCM: Preliminary
results
The ensemble Kalman filter has been implemented with the Miami Isopycnic
Coordinate Ocean Model (MICOM) originally developed by R. Bleck at the University of Miami, [see e.g., Bleck and Smith (1990), and Bleck et al. (1992)]. The
currently available data assimilation applications for Ocean General Circulation
Models (OGCMs) have been based on rather simplistic assimilation schemes.
None of these take proper error statistics into account and ad hoc approaches are
used for the assimilation. Some examples are: Derber and Rosati (1989) who used
an objective analysis technique to update the model temperature in a version of the
Cox model (Rosati and Miyakoda, 1988). Mellor and Ezer, (1991) and Ezer and
Mellor, (1994), used a univariate optimal interpolation algorithm with vertical projection of surf ace information in the model by Blumberg and Mellor (1987). Cooper and Haines (1996) used a vertical projection method based on water property
conservation in the Cox model (Cox, 1987).
In the works by Mellor and Ezer, (1991), and Ezer and Mellor, (1994), sea surface height (SSH) observations were used to update the vertical density stratification to drive the geostrophic currents associated with gradients in the SSH. The
density was implicitly updated by actually updating the vertical temperature profiles using estimated correlations between temperature and SSH. Clearly, this is an
inconsistent approach, since the vertical density stratification depends on both the
There are a few locations where the filter estimate starts diverging from the reference solution, e.g. for t = 26 and t = 34. Note however that the ensemble Kalman
filter recovers quickly and begins tracing the reference solution again. The error
estimate given in the lower plot of Fig. 6.3 shows strong error growth at these particular locations and thus indicates that the ensemble is passing through a region in
the state space which is associated with strong instability. The error estimates show
the same behavior as was found by Miller et al. (1994) with very strong error
growth when the model solution is passing through the unstable regions ofthe state
space, and otherwise rather weak error variance growth in the more stable regions.
Note for example ihe low error variance when t E [28,34] corresponding to the
oscillation of the solution around one of the attractors.
Finally, it should be pointed out that in the ensemble Kalman filter a variance
minimizing analysis is calculated at measurement times. Thus, even if the ensembIe certainly is non-Gaussian due to the forward integration of nonlinear model
equations, only the Gaussian part of the distribution is used. This is in contrast to
the work by MiHer et al. (1999) where the maximum-likelihood analysis is calculated by actually constructing the density function for the model evolution and then
calculating the conditional density in terms of analytical functions. They found that
this made a significant improvement on the analysis. However, it is stiH not clear
how this approach can be used in a practical way for high dimensional state spaces.
6.6 An ensemble Kalman fllter for an OGCM: Preliminary
results
The ensemble Kalman filter has been implemented with the Miami Isopycnic
Coordinate Ocean Model (MICOM) originally developed by R. Bleck at the University of Miami, [see e.g., Bleck and Smith (1990), and Bleck et al. (1992)]. The
currently available data assimilation applications for Ocean General Circulation
Models (OGCMs) have been based on rather simplistic assimilation schemes.
None of these take proper error statistics into account and ad hoc approaches are
used for the assimilation. Some examples are: Derber and Rosati (1989) who used
an objective analysis technique to update the model temperature in a version of the
Cox model (Rosati and Miyakoda, 1988). Mellor and Ezer, (1991) and Ezer and
Mellor, (1994), used a univariate optimal interpolation algorithm with vertical projection of surf ace information in the model by Blumberg and Mellor (1987). Cooper and Haines (1996) used a vertical projection method based on water property
conservation in the Cox model (Cox, 1987).
In the works by Mellor and Ezer, (1991), and Ezer and Mellor, (1994), sea surface height (SSH) observations were used to update the vertical density stratification to drive the geostrophic currents associated with gradients in the SSH. The
density was implicitly updated by actually updating the vertical temperature profiles using estimated correlations between temperature and SSH. Clearly, this is an
inconsistent approach, since the vertical density stratification depends on both the