Geoscience Reference
In-Depth Information
We now describe how the choice of static, but full-rank, versus flow-dependent,
but reduced-rank error covariance could impact all-sky radiance assimilation.
Forecast error covariance has a fundamental role in data assimilation as it defines
the subspace where the analysis correction can be defined (Appendix 1). Following
the relations ( 19.26 )and( 19.30 ) from Appendix 1, one can represent a generic
analysis increment as
D X
i
x a x f
ˇ i u i
(19.3)
where
i defined in Appendix 1, and u is a
singular vector. Therefore, an arbitrary analysis increment can be represented as
linear combination of forecast error covariance singular vectors.
The implication of ( 19.3 ) is that a well-defined forecast error covariance is
critical for successful data assimilation. Therefore, the quality of data assimilation
can be assessed by examining the structure of forecast error covariance used in
assimilation. In weather, climate, hydrology, and other geoscience applications the
structure of true forecast error covariance can be very complex, since it incorporates
relations between various state variables. Of special interest for all-sky radiance
assimilation is the structure of forecast error covariance with respect to cloud
microphysical variables since cloud variables are input to radiative transfer model.
In general, there are various processes that imply cross-correlation between cloud
variables, approximately described by the cloud microphysics component of a
forecast model. One can also anticipate correlations between cloud and standard
dynamical variables, such as temperature, pressure and wind. It is convenient to use
a block matrix form to represent forecast error covariance
ˇ i is a coefficient equal to
i and
P dd P cd
P cd P dd
P f D
(19.4)
where index
d
refers to dynamical variables, and index
c
to cloud variables (e.g.,
cloud ice, snow, rain, etc.).We also use the fact that
P f is symmetric matrix, thus
P cd D P dc . For simplicity, assuming that dynamical variables include temperature,
pressure and wind, and that cloud variables include cloud ice, snow and rain, the
block matrices in ( 19.4 )are
2
4
3
5 ;
P T;T P T;p P T; v
P T;p P p;p P p; v
P T; v P p; v P v ; v
P dd D
(19.5)
2
4
3
5 ;
P ice ; ice P ice ; snow P ice ; rain
P ice ; snow P snow ; snow P snow ; rain
P ice ; rain P snow ; rain P rain ; rain
P cc D
(19.6)
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