All-Sky Satellite Radiance Data Assimilation: Methodology and Challenges - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 474

NA10NES4400012.We would also like to acknowledge high-performance computing support
provided by NCAR's Computational and Information Systems Laboratory, sponsored by the
National Science Foundation.

Appendix 1

Analysis Increment and Forecast Error Covariance

Forecast error covariance is one of the major components of the analysis update
equation. In this section we derive the analysis update as a function of forecast error
covariance. It is convenient to begin by defining the singular value decomposition
(SVD) of a square root forecast error covariance (e.g., Golub and van Loan 1989 )

D X
i

P 1=2
f

i u i v i

(19.21)

where f u i g and f v i g are singular vectors and f i g are singular values. This leads to
eigenvalue decomposition (EVD) in the form

D X
i

P f D P 1= f P T=2 i u i u i (19.22)

f

i D i

with .

1 Kalman Filter and Related Methods

The analysis update is given by the KF analysis equation

D P f H T .HP f H T C R/ 1 y h.x f / x a x f

(19.23)

where the superscript a denotes analysis, R is the observation error covariance, and

h are the nonlinear observation operator and its Jacobian, respectively. After

using ( 19.22 )in( 19.23 ), and denoting
i D i u i H T ŒHP f H T C R 1 y h.x f / ;

and H

(19.24)

the KF analysis update ( 19.23 ) becomes

D X
i

x a x f i u i (19.25)

i.e. it can be represented as a linear combination of the forecast error covariance
singular vectors.