All-Sky Satellite Radiance Data Assimilation: Methodology and Challenges - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 469

Fig. 19.3 Impact of Hessian
preconditioning on
minimization: ( a ) physical
space, and ( b ) preconditioned
space. In this example of a
quadratic cost function it is
shown how an ideal
preconditioning can change
the cost function so that the
minimum is reached in single
minimization iteration
weather situation. In principle, one would like to achieve a similar percentage of
adjustment for each control variables, with the idea that although minimization may
not have sufficient time to reach mathematical convergence it will still produce an
acceptable physical solution. Consider temperature and wind as an example. Let the
initial guess have dynamically balanced fields, which is generally true given that
it is produced by a forecast. If the temperature component of the cost function is
perfectly preconditioned, while the wind component is not preconditioned at all, the
analysis after first minimization iteration would have fully adjusted temperature but
practically unchanged wind. Since wind and temperature were in dynamical balance
before minimization, the produced solution after first iteration would be unbalanced
and eventually create noise in the ensuing forecast. This situation can be visualized
from Fig. 19.3 with temperature converging according to Fig. 19.3 b, while the wind
slowly converging as shown in Fig. 19.3 a. This situation also illustrates the impact
of Hessian preconditioning on the utility of observations: temperature observations
will be efficiently used, while wind observations would have a marginal impact. The
important point is that this potential problem can be resolved by adequate Hessian
preconditioning.
Let now consider the impact of Hessian preconditioning on all-sky satellite
radiance assimilation. Assimilation of all-sky satellite radiance would be most
beneficial if cloud variables were defined as control variables since they have the
strongest impact on the radiative transfer operator. However, if the cloud variable
component of the cost function is not adequately preconditioned, all-sky radiances
will not be well utilized, eventually producing an unbalanced analysis. Assuming
that other dynamical variables were well preconditioned, it is likely that the ensuing
forecast will get rid of clouds and precipitation created by the analysis, simply as
a consequence of inadequate preconditioning. This problem is real and it can have
dire consequences for all data assimilation methods.
In case of variational methods one should recall that forecast error covariance is
used to precondition minimization. Unfortunately, since there are no practical ways
to include cross-variable correlations for cloud variables, while error covariance
of dynamical variables has a relatively well-defined cross-variable structure, it is
clear that the overall preconditioning will be off balance. This can also happen in