Ocean Ensemble Forecasting and Adaptive Sampling - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 386

16.2.1 Analysis Error Covariance for the Routine Observations

with the ET Technique

To be consistent with the ET technique of ensemble generation, we need to utilize
a guess of the analysis error covariance matrix P g associated with the routine

observational network. Let the columns of the nxK matrices X o and X v list the raw
ensemble perturbations at the observation and verification times, respectively, of the
ensemble forecast initialized at the initialization time.
The forecast perturbations X o can be transformed into a set of perturbations X r
that are consistent with P g using

X r D X o T (16.1)

where

ƒ 1=2 B T

T D B — (16.2)

and where B D Πb 1 ; b 2 ;:::; b K is a K K orthogonal matrix containing the

eigenvectors of the symmetric matrix X oT P a 1
g

.Inotherwords,

X 0 =N

X oT P a 1
g

X o

ƒ K K B T : D B (16.3)

N

where ƒ D diag . 11 ; 22 ;:::; KK / is a K K diagonal matrix listing the eigen-

values of X oT P a 1
g

. Since the sum of the forecast perturbations is equal to
zero, one of these eigenvalues will be equal to zero. Consequently, provided each
ensemble contains K1 linearly independent perturbations,

X 0 =N

ƒ can be written in the

form,

ƒ .K 1/ .K 1/ 0
0

ƒ K K D (16.4)

0

where ƒ .K 1/ .K 1/ is a (K1)(K1) diagonal matrix whose diagonal elements

are all greater than zero. The —

ƒ used in ( 16.4 ) is obtained from ƒ by setting its zero

eigenvalue equal to 1, in other words,

ƒ .K 1/ .K 1/ 0
0

— ƒ K K D (16.5)

1

Note that while — does not exist. This adjustment of

the eigenvalue matrix is permissible because it does not affect the sample covariance
matrix of initial perturbations implied by ( 16.3 ). To see this, first note that pre and
post multiplying ( 16.5 ) by the eigenvector b K corresponding to the zero eigenvalue
K D 0

ƒ has an inverse, the inverse of ƒ

shows that