Assimilation of Satellite Altimetry in Ocean Models
119
consistent with these observations. We have a better chance that the 3D assimilation scheme will converge if we are confident that at each assimilation time the
post assimilation state could have been achieved by advection from the state prior
to assimilation, which would guarantee that the analysed states are consistent in the
important conserved properties of the ocean. Fig. 7.1 illustrates the nature of the
convergence of the 2 assimilation schemes. Although the sequential scheme does
not provide a good analysis of the ocean state before the end-time of the assimilation, it is possible to work backwards to obtain the solution at earlier times using
the 'Kalman Smoother' operation.
In Cooper and Haines (1996) a practical water redistribution assimilation scheme
was developed for use with altimeter data. It is well known that in mid-Iatitudes a
high sea level is usually correlated with a deeper thermocline, of ten with a warmer
water layer at the surface, as exemplified in a warm core ocean ring. Similarly a
low sea level usually indicates a raised thermocline, with cold water at the surface,
as in a cold core ring. The Cooper and Haines (1996) assimilation scheme seeks a
vertical displacement of water masses which would correspond to the observed
change in sea level. Such a vertical displacement obviously satisfies the
Lagrangian redistribution requirement although it is not as general as a full 3D
rearrangement ofwater properties. The advantage is that a unique vertical displacement can easily be derived from an observed sea level anomaly with the imposition
of a simple additional constraint. This constraint is: no change is permitted to the
pressure at the ocean floor. Consider the hydrostatic equation for a given water column;
11Pog + l pgdz = p(-H)
-H
(1)
where 11 is the sea level above a reference level O, H is the depth ofthe ocean, p(z)
is the in situ water colwnn density and p( -H) is the pressure at the ocean floor. If
an observation requires a change in the model sea level, ,111, and we place the constraint that Ap( -H) = O then we get;
A11Pog + r Apgdz = O
-H
(2)
If Ap(z) is required to represent a vertical displacement ofthe existing water column then the amount of vertical displacement, 8h, is determined uniquely. When
the water column is lifted, the bottom of the water column is extended with water
of the same properties already present at the bottom, when the water column is
lowered the surface water properties are again those already present at the surface.
Fig. 7.2 illustrates this displacement, 8h. This process conserves water properties
and volumes except at the top and bottom ofthe column where the volume ofwater
masses is clearly not conserved locally. The surface and bottom water volumes
may still be conserved if the volume changes in all the surrounding water colurnns
119
consistent with these observations. We have a better chance that the 3D assimilation scheme will converge if we are confident that at each assimilation time the
post assimilation state could have been achieved by advection from the state prior
to assimilation, which would guarantee that the analysed states are consistent in the
important conserved properties of the ocean. Fig. 7.1 illustrates the nature of the
convergence of the 2 assimilation schemes. Although the sequential scheme does
not provide a good analysis of the ocean state before the end-time of the assimilation, it is possible to work backwards to obtain the solution at earlier times using
the 'Kalman Smoother' operation.
In Cooper and Haines (1996) a practical water redistribution assimilation scheme
was developed for use with altimeter data. It is well known that in mid-Iatitudes a
high sea level is usually correlated with a deeper thermocline, of ten with a warmer
water layer at the surface, as exemplified in a warm core ocean ring. Similarly a
low sea level usually indicates a raised thermocline, with cold water at the surface,
as in a cold core ring. The Cooper and Haines (1996) assimilation scheme seeks a
vertical displacement of water masses which would correspond to the observed
change in sea level. Such a vertical displacement obviously satisfies the
Lagrangian redistribution requirement although it is not as general as a full 3D
rearrangement ofwater properties. The advantage is that a unique vertical displacement can easily be derived from an observed sea level anomaly with the imposition
of a simple additional constraint. This constraint is: no change is permitted to the
pressure at the ocean floor. Consider the hydrostatic equation for a given water column;
11Pog + l pgdz = p(-H)
-H
(1)
where 11 is the sea level above a reference level O, H is the depth ofthe ocean, p(z)
is the in situ water colwnn density and p( -H) is the pressure at the ocean floor. If
an observation requires a change in the model sea level, ,111, and we place the constraint that Ap( -H) = O then we get;
A11Pog + r Apgdz = O
-H
(2)
If Ap(z) is required to represent a vertical displacement ofthe existing water column then the amount of vertical displacement, 8h, is determined uniquely. When
the water column is lifted, the bottom of the water column is extended with water
of the same properties already present at the bottom, when the water column is
lowered the surface water properties are again those already present at the surface.
Fig. 7.2 illustrates this displacement, 8h. This process conserves water properties
and volumes except at the top and bottom ofthe column where the volume ofwater
masses is clearly not conserved locally. The surface and bottom water volumes
may still be conserved if the volume changes in all the surrounding water colurnns