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related to accurate and efficient assimilation of all-sky radiances are fundamentally
related to each other, one could try to distinguish the challenges related to (1) data
assimilation, (2) simulation and prediction, and (3) computation. Simulation and
prediction of clouds is related to the ability of prediction models to represent clouds,
and the complexity of the employed microphysics. Although this clearly impacts
data assimilation, it is typically assumed an input to data assimilation and thus it will
not be discussed here. Computational requirements for all-sky radiance assimilation
we refer to are caused by high spatiotemporal resolution of cloud microphysical
processes, as well as by a necessity to include cloud scattering processes in forward
radiative transfer model. Computational restrictions will impact the choices one
could have regarding methodology and algorithms used in data assimilation. In this
paper we will focus on data assimilation issues related to all-sky satellite radiances,
and discuss model prediction and computational issues only in context of data
assimilation.
Data assimilation challenges of all-sky radiance assimilation are defined as the
aspects of data assimilation that are especially exposed by assimilation of all-sky
radiances and related cloud-resolving scales. They can be all traced back to clouds,
and range from methodological to computational: (1) forecast error covariance,
(2) correlated observation errors, (3) nonlinearity and non-differentiability, and (4)
non-Gaussian errors.
19.3.1
Forecast Error Covariance
Forecast error covariance is typically used as a measure of uncertainty of the
forecast, and could be defined as
P
f
D
˝
.x
f
x
t
/
T
˛
x
t
/.x
f
(19.2)
x
f
and
x
t
are the first-guess forecast and the (unknown) truth, respectively,
Where
hi
denotes a transpose.
It also represents one of the main differences between variational and ensemble
based data assimilation methodologies:
denotes mathematical expectation and the superscript
T
P
f
is modeled in variational methods,
while computed from a forecast ensemble in ensemble methods. This implies time-
independent covariance in variational methods, while ensemble methods produce
a flow-dependent structure. Time-dependence is in principal an advantage for
applications at cloud-scales with characteristic fluctuating dynamical processes.
One can think of cloud microphysical processes in a hurricane, or in severe storm
outbreaks. However, one should also be aware that ensemble data assimilation
at such high resolution implies a low-rank approximation to the forecast error
covariance, with the number of ensembles much smaller than the state dimension.
Although this issue can be considerably improved by error covariance localization
techniques (e.g.,
Hamill et al. 2001
;
Houtekamer and Mitchell 2001
), this is still a
limitation.
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