Geoscience Reference
In-Depth Information
exactly those that are most important for the inverse, making the decision about the
threshold difficult.
Additional computational issues arise due to the use of radiative transfer operator
for all-sky radiances. Inclusion of cloud and precipitation scattering processes
required for all-sky radiance calculations adds considerably to the computational
cost of data assimilation (e.g.,
Stephens 1994
). Coupled with a significant increase
of the number of radiance observations, the cost of all-sky radiance calculations can
be much larger than the cost of clear-sky radiances. This directly impacts the cost
of data assimilation and needs to be taken into account.
19.3.3
Nonlinearity and Non-Differentiability
Nonlinearity of cloud microphysical processes and the radiative transfer operator
for all-sky radiances is a well-known issue (e.g.,
Errrico et al. 2007b
;
Steward
et al. 2012
). The approach to address nonlinearity may be to choose fundamentally
different methodology, such as particle filters (e.g.,
Gordon et al. 1993
;
Xiong
et al. 2006
;
van Leeuwen 2009
), or to improve minimization conditioning (e.g.,
Axelsson and Barker 1984
;
Axelsson 1994
;
Zupanski et al. 2008
). Also, there are
so-called linear channels (e.g., frequencies) that do not have strong nonlinearity
and thus can be treated using linear or weakly nonlinear methods. Variational
methods are generally equipped to address nonlinearity using iterative minimization
of the cost function. Standard ensemble Kalman filtering methods do not address
the observation nonlinearity specifically, which prompted a development of hybrid
variational-ensemble Ensemble methods (e.g.,
Zupanski 2005
;
Wang et al. 2007
),
ensemble iterative Kalman filters (e.g.,
Gu and Oliver 2007
), or a refinement of the
ensemble Kalman filter (e.g.,
Evensen 2003
).
Since majority of practical data assimilation algorithms today use iterative
minimization to solve nonlinear problems, we will discuss this approach in more
detail. These minimization algorithms are typically unconstrained algorithms, most
often a nonlinear conjugate-gradient algorithm or quasi-Newton algorithms (e.g.,
Luenberger 1989
). One of the minimization algorithm components most relevant for
all-sky satellite radiance assimilation is Hessian preconditioning (e.g.,
Axelsson and
Barker 1984
;
Axelsson 1994
;
Yang et al. 1996
;
Zupanski 1995
;
Steward 2012
). Its
general role is in speeding up minimization by a change of variable that effectively
reduces the condition number of Hessian matrix (e.g., second derivative of the cost
function). The ideal impact of preconditioning is illustrated in Fig.
19.3
,which
shows a change of a quadratic cost function from an elongated ellipse to a circle.
Starting minimization from an arbitrary point will lead to numerous minimization
iterations for the original ellipsoidal cost function (Fig.
19.3
a), whilst a single
iteration will be sufficient for a preconditioned minimization problem (Fig.
19.3
b).
Another role of preconditioning, of special importance in practical applications,
is to provide a “balanced” reduction of cost function. This means that minimization
should produce a change of all control variables that is in agreement with actual
Search WWH ::
Custom Search