All-Sky Satellite Radiance Data Assimilation: Methodology and Challenges - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 467

2. Observation thinning : If observation density is high, thin observations by
selecting every -th observation. The reduced number of observations could keep

the original error, or have some error adjustment.
3. Cut-off based selection ( Fertig et al. 2007 ): Based on an empirical estimate of
observation correlations one can design an algorithm to select which radiance
observations to assimilate.
4. Eigenvalue decomposition ( Parrish and Cohn 1985 ; Anderson 2003 ): For non-
diagonal

n

work in eigenvectors space where the errors are diagonal, i.e.
R D SƒS T . Introduce the change of variable

R

S T Œy h.x/ D S T "

to obtain

R s D ˝ .S T "/.S T "/ T ˛ D S T ˝ "" T ˛ S D S T RS

.
Note that new observation error is diagonal, thus a standard data assimilation
algorithm with diagonal observation covariance can still be applied.
5. Direct application of the inverse . This can be done directly by calculating
the matrix inverse, or indirectly by using a matrix-vector product. In both
cases one needs to assume the correlation properties, since there is insufficient
statistical information available from data. Given large number of observations
the former approach may be computationally prohibitive. The latter approach is
computationally feasible and can be described as follows. Here we assume that
R D

transformed error as D ƒ

DCD where D is the diagonal matrix of observation errors, and C is the

EE T using the unique symmetric

correlation matrix. One can decompose C D

square root E . The inverse square root of correlated observation error covariance

/ 1 D E 1 D 1 . Since the inverse of a symmetric positive
definite matrix is symmetric and positive definite,

R 1=2 is D . DE

E 1 can be modeled using
a simple correlation matrix such as Toeplitz (e.g., Golub and van Loan 1989 )
and thus avoid the calculation of the inverse

E 1 . In practice the method is

R 1=2 Œy h.x/

applied using a matrix-vector product such as ,whichmakes

the approach feasible even for large number of observations.

Note that the approaches (1) and (2) never assume non-diagonal , they only adjust

the observation errors (1), or the number of observations (2) to match the desired
observation impact. However, if observations errors are correlated, the approach
(1) is implicitly using a top-hat function instead of a true correlation function.
The approach (2) is implying that near-by observations have similar information
content (i.e. homogeneity) which may not be true for observations of clouds and
precipitation given that the quality of radiance observation depends on the scan
angle, for example. The approach (3) implicitly assumes non-diagonal R but it
employs this information only to select radiance observations to be assimilated, still
using the original diagonal-based assimilation framework.
The approaches (4) and (5), assume non-diagonal (e.g., correlated) observation
errors. The approach (4) is mathematically more general than (5), since it can be
applied to an arbitrary

R

correlations
in order to be practical. However, the approach (4) also needs an assumption about
an eigenvalue threshold. For example, the inverse square root of

R , while the approach (5) requires simplified R

R D SƒS T is

R 1=2 D Sƒ 1=2 S T . Calculation of ƒ 1=2 requires defining a threshold value in

order to avoid the division by zero. Unfortunately, the smallest values of

ƒ are