Observing-System Research and Ensemble Data Assimilation at JAMSTEC - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 504

Kalman filter, and the unperturbed observations are assimilated to update the
ensemble mean. The analysis ensemble mean

a

x is calculated from the forecast

f and the linear combination of the ensemble forecast perturbation

ensemble mean x

matrix X f w a .

with the weight

a f

C X f w a

x Dx ; (21.1)

where superscripts a, f, and o denote analysis, forecast and observation, respectively,
and over bars indicate the ensemble mean. The th member) of X f

i th column ( i is

x f

f . The weight is determined by

i x

D P a Y f T R 1 y 0 y

f

w a (21.2)

where R the observation error covariance matrix and Y f is a forecast perturbation

f . The forecast

i

th column is y f

matrix in the observation space, whose

i y

observation vector y f
i

is calculated by

i D H x f

:

y f

(21.3)

i

where H is the observation operator. The analysis covariance is computed as

D .k 1/

T R 1 Y f 1

P a

Y f I C . / : (21.4)

The analysis ensemble perturbation matrix X a is obtained from the forecast
ensemble perturbation matrix X f using the transform matrix W a :

X a D X f W a (21.5)

where

D .k 1/ P a 1=2

W a (21.6)

( Bishop et al. 2001 ). Analysis in LETKF is performed in a local subspace of the
model, and different linear combinations of ensemble members in different regions
( 21.3 ) can be chosen. In this way, localization acts to reduce the sampling error by
making the global dimensions larger than the ensemble size, and to remove spurious
correlations between distant locations ( Hunt et al. 2007 ).
In the previous version of the LETKF code used in ALERA ( Miyoshi and
Yamane 2007 ), analysis is performed with a square-shaped local patch in the
model grid space. The zonal distance of a local patch decreases toward the poles
because of the convergence of meridians. As a result, significant discontinuity exists
in ALERA, especially in the analysis ensemble spread in the polar regions. In
ALEDAS2 the error covariance is localized by physical distance rather than by
model-space local patches ( Miyoshi et al. 2007b ). The weight w for covariance
localization diminishes with distance r from each grid point for analysis to an

observation in the form