Geoscience Reference
In-Depth Information
2
3
P
ice
;T
P
ice
;p
P
ice
;
v
P
snow
;T
P
snow
;p
P
snow
;
v
P
ice
;T
P
snow
;p
P
rain
;
v
4
5
:
P
cd
D
(19.7)
The diagonal blocks (
19.5
)and(
19.6
) are symmetric matrices, while the off-
diagonal block matrix (
19.7
) is not symmetric.
The forecast error covariance structure defined by (
19.4
)-(
19.7
) is indicative of
the complexity of relations that this matrix represent. Given that these matrices
represent the uncertainty of model variables, one can quickly realize that elements of
these matrices are fundamentally time dependent. The natural formation and decay
of clouds will have a profound impact on the elements of matrices (
19.6
)and(
19.7
).
In clear skies these matrices have all elements essentially equal to zero. When
clouds begin forming, the block matrices describe how various variables impact
each other in the process. Unfortunately, there is a limited capability of current data
assimilation methodologies to accurately address the structure (
19.4
)-(
19.7
).
Variational methods include modeling of forecast error covariance and typically
do not represent time-dependent information in its definition. One should note
that 4d-Var method includes time-dependence through tangent linear and adjoint
model integration, which does have some impact on the uncertainties at the end of
assimilation interval. However, the forecast error covariance defined at initial time
of assimilation is modeled as in 3d-Var data assimilation. For dynamical variables
one can identify simplified relations such as hydrostatic, geostrophic, and similar
balance constraints that are commonly used in modeling cross-variable interactions
(e.g.,
Parrish and Derber 1992
). Unfortunately, this approach is much more difficult
to apply at cloud scales due to poorly known or unknown balance constraints. This
apparently creates a difficulty for variational data assimilation to represent cloud
variable cross-correlations in (
19.6
), as well as dynamical-cloud correlations in
(
19.7
). Although in principle it may be possible to successfully model cross-variable
correlations, this still has not been done for cloud variables. A more feasible solution
applicable to current variational methods is to assume a regular (i.e., isotropic
and homogeneous) correlation for the diagonal blocks in (
19.6
), i.e. to pre-define
correlation function for
P
rain
;
rain
and thus avoid modeling
more complex cross-correlations. In this case the matrices
P
ice
;
ice
,
P
snow
;
snow
,and
P
cd
and
P
cc
become
2
3
P
ice
;
ice
0
0
4
5
;
P
cc
D
0P
snow
;
snow
0
(19.8)
0
0
P
rain
;
rain
2
3
000
000
000
4
5
:
P
cd
D
(19.9)
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