Sequential Data Assimilation for Nonlinear Dynamics: The Ensemble Kalman Filter
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temperature and salinity. Updating only temperature will ultimately generate water
masses with unrealistic T -S properties.
A method for overcoming this problem has recently been presented by Oschlies
and Willebrand (1996), where both temperature and salinity were updated under a
constraint of maintaining the T -S properties of the water masses. This approach
gave satisfactory results and should now be extended to take proper error statistics
into account in the analysis scheme.
In another recent work by Forbes and Brown (1996), a nudging method was used
with the Miami Isopycnic Coordinate Ocean Model (MICOM) for assimilation of
SSH observations from altimetry. They showed that the nudging approach was
capable of pulling the model state towards the observations and that the model,
over time, also propagated information from the surf ace into the lower layers.
Essential when developing an advanced data assimilation system with an OGCM
is to estimate proper error statistics for the model prediction. That is, having a
model forecast and an observation one needs to know the inf1uence this particular
observation will have on alI the prognostic variables in the model. As an example,
assume we have one observation of the sea surface or mixed layer temperature at a
particular location. If the measured value is greater than the model predicted value
this should imply that the model mixed layer temperature is increased by some
amount in the analysis scheme. The update should of course be smooth in space. If
the model mixed layer temperature is the only variable updated this willlead to an
"unbalanced" model state. Clearly, the model density should be decreased accordingly to be consistent with the equation of state. Further ifthe mixed layer temperature is increasing we would expect a decrease of the mixed layer depth, which
requires the full vertical density structure ofthe model to change.
Important for this study is that MICOM uses density as the vertical coordinate.
Thus, in the vertical the model can be considered as a stack of layers each having a
constant density. This leads to an interesting interpretation ofhow the vertical stratification should be updated by the analysis scheme.lnstead ofupdating the thermodynamic variables themselves one rather change the locations of the model layer
interfaces, which gives the same effect. ClearlY' it becomes complicated to construct Otimal Interpolation schemes for performing such an analysis since at various locations and times different layer interfaces should be altered. As an example,
the layers with low density water masses will outcrop below the mixed layer when
going northward from the equator. However the ensemble Kalman filter holds alI
the required information about the space and time dependent error covariances
between different model variables which are used to calculate the actual inf1uence
functions for the measurements.
Here, a preliminary example is presented from an ensemble Kalman filter implementation with a coarse resolution vers ion ofMICOM for the north Atlantic. A 256
member initial ensemble is created by perturbing layer interfaces in a model state
resulting from a 10 years spin up run. Each of the ensemble members are generated
by adding smooth pseudo random fields with a specified covariance and mean
113
temperature and salinity. Updating only temperature will ultimately generate water
masses with unrealistic T -S properties.
A method for overcoming this problem has recently been presented by Oschlies
and Willebrand (1996), where both temperature and salinity were updated under a
constraint of maintaining the T -S properties of the water masses. This approach
gave satisfactory results and should now be extended to take proper error statistics
into account in the analysis scheme.
In another recent work by Forbes and Brown (1996), a nudging method was used
with the Miami Isopycnic Coordinate Ocean Model (MICOM) for assimilation of
SSH observations from altimetry. They showed that the nudging approach was
capable of pulling the model state towards the observations and that the model,
over time, also propagated information from the surf ace into the lower layers.
Essential when developing an advanced data assimilation system with an OGCM
is to estimate proper error statistics for the model prediction. That is, having a
model forecast and an observation one needs to know the inf1uence this particular
observation will have on alI the prognostic variables in the model. As an example,
assume we have one observation of the sea surface or mixed layer temperature at a
particular location. If the measured value is greater than the model predicted value
this should imply that the model mixed layer temperature is increased by some
amount in the analysis scheme. The update should of course be smooth in space. If
the model mixed layer temperature is the only variable updated this willlead to an
"unbalanced" model state. Clearly, the model density should be decreased accordingly to be consistent with the equation of state. Further ifthe mixed layer temperature is increasing we would expect a decrease of the mixed layer depth, which
requires the full vertical density structure ofthe model to change.
Important for this study is that MICOM uses density as the vertical coordinate.
Thus, in the vertical the model can be considered as a stack of layers each having a
constant density. This leads to an interesting interpretation ofhow the vertical stratification should be updated by the analysis scheme.lnstead ofupdating the thermodynamic variables themselves one rather change the locations of the model layer
interfaces, which gives the same effect. ClearlY' it becomes complicated to construct Otimal Interpolation schemes for performing such an analysis since at various locations and times different layer interfaces should be altered. As an example,
the layers with low density water masses will outcrop below the mixed layer when
going northward from the equator. However the ensemble Kalman filter holds alI
the required information about the space and time dependent error covariances
between different model variables which are used to calculate the actual inf1uence
functions for the measurements.
Here, a preliminary example is presented from an ensemble Kalman filter implementation with a coarse resolution vers ion ofMICOM for the north Atlantic. A 256
member initial ensemble is created by perturbing layer interfaces in a model state
resulting from a 10 years spin up run. Each of the ensemble members are generated
by adding smooth pseudo random fields with a specified covariance and mean