The second approach is that of the sequential data assimilation analyses, which move forward in time from a previous analysis using a numerical simulation of the evolving temperature and other variables produced by an ocean general circulation model.
The third approach is 4D-Var, which in the implementation described uses the initial conditions and surface forcing as control variables to be modified in order to be consistent with the observations as well as a numerical representation of the equations of motion through iterative solution of a giant optimization problem.
ISHII uses and alternative 3DVAR approach to do an objective mapping with a smaller decorrelation scale in midlatitudes (300 km) that elongates in the zonal direction by a factor of 3 at equatorial latitudes.
LEVITUS begins similarly to ISHII, but uses the technique of Cressman and Barnes with a homogeneous scale of 555 km to objectively map the temperature innovation onto a uniform grid.
This approach faces daunting computational challenges, but provides some interesting benefits including satisfying some conservation laws and the construction of the ocean model adjoint.