Ocean Ensemble Forecasting and Adaptive Sampling - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 388

where X i

is a nKmatrixthen

/ 1=2 C i

X i D X r C i . i C I

(16.11)

where the KK orthonormal matrix C i and the K K diagonal matrix i is given

by the eigenvector decomposition

X aT
r

H aT
i

R 1
i

H i X r

D C i i C i : (16.12)

K 1

The columns of X i may be interpreted as transformed ensemble perturbations

whose covariance gives the analysis error covariance at the observation time
assuming that the ith deployment of adaptive observations had been assimilated.
To see the impact of the adaptive observations at the verification time, one needs to
be able to propagate each of the columns of X i

through time in a manner consistent
with the governing dynamical equations. A computationally expensive way of doing
this would be to define a tangent linear model M such that

x c C x ji

M x c Mx ji

M (16.13)

where is the non-linear dynamics propagator that maps state vectors from the

observation time to the verification time, x c

M

is the control forecast at the observation

time and x ji is the jth column of X i . If one had this operator in hand, then the

forecast error covariance matrix given the ith deployment of observations P v
i

would

be given by

MX i MX i T
K 1

P v

i D

(16.14)

However, using ( 16.11 )and( 16.1 )in( 16.14 )gives

MX i D . MX o / / 1=2 C T : TC . C I (16.15)

Now MX o represents a tangent linear approximation to the propagation of the raw
untransformed ensemble perturbations at the observation time to the verification
time. Of course, the non-linear equations map the observation time raw pertur-
bations X o to the verification time perturbations X v . These are directly available
from the raw ensemble without any additional computational expense. Hence, a
computationally inexpensive way of computing P v
i

that is more accurate than that

given by ( 16.14 )is

X v
i

X v T
i
K 1 ;

P v X v

i D X v TC i . i C I

/ 1=2 C i :

i D

where (16.16)

Equation 16.16 gives the forecast error covariance of the model variables given
the ith deployment of adaptive observations. Often the controller of adaptive