All-Sky Satellite Radiance Data Assimilation: Methodology and Challenges - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 465

variables (e.g., perturbation pressure, perturbation height, perturbation potential
temperature, and winds) as well as cloud variables (e.g., cloud ice, cloud snow,
cloud water, graupel, and water vapor). In Fig. 19.1 a, b we show a horizontal map
of analysis increment for cloud snow and for the north-south wind component at
the level of the cloud snow observation. One can see that snow analysis has a strong
positive response to snow observation (Fig. 19.1 a), as expected. It is also interesting
to note that cloud snow observation impacts wind (Fig. 19.1 b), a dynamical variable,
corresponding to P snow ; v component of the forecast error covariance from ( 19.7 ).

This is important since it indicates that, with adequate forecast error covariance
structure, one can impact dynamical variables by all-sky radiance observations. One
can also note a relatively regular response that resembles modeled error covariance
structure (e.g., Parrish and Derber 1992 ; Wu et al. 2002 ), possibly suggesting that
such covariance components can be successfully modeled in variational methods.
In Fig. 19.2 we show cloud snow and rain analysis responses in the vertical,
corresponding to the components

P snow ; rain . One can see a well-
defined cloud snow response centered at the observation location (Fig. 19.2 a).
The response is confined to few levels above and below the observation, again
suggesting that modeling this covariance component may be possible. However, the
response of cloud rain (Fig. 19.2 b) exposes a potential difficulty in modeling cross-
variable correlations such as snow-rain. The first problem is to create a non-centered
response of rain to cloud snow observation. Although this may be mathematically
possible (e.g., Gaspari and Cohn 1999 ), it has not been done in practice and opens
several new problems. One such problem is to know what exactly needs to be
modeled, because there is a very limited knowledge about cloud-variable correlation
statistics. The most difficult problem may be related to flow-dependence of these
correlations. It is clear that the existence of cloud rain and snow depends on the
current temperature conditions that change with time and thus require additional
flow-dependent parameters to be introduced to the modeling function and eventually
estimated.
Therefore, forecast error covariance can have very different structure depending
on the methodology used. Even within same methodology one can choose different
parameters related to decorrelation length of correlation function in variational
methods, or to the covariance localization in ensemble methods, effectively imply-
ing a large number of possible choices for forecast error covariance. This apparent
variety of possible choices for the forecast error covariance creates a problem
since in light of ( 19.3 ) it implies a non-unique analysis solution. The “optimal”
choice of forecast error covariance may be good for overall data assimilation
performance, but may not adequately address all-sky radiance observations since
they are generally confined to a smaller local area of intense dynamical development
and thus their global impact is relatively small. One possible way to address the
problem of non-uniqueness of forecast error covariance is to reduce the number of
additional parameters, or at least to include their estimation in the data assimilation
algorithm. Another possibility may be to develop a new methodology that will be
less dependent on forecast error covariance and include fewer undefined parameters.

P snow ; snow and