Ocean Ensemble Forecasting and Adaptive Sampling - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 389

observational resources will want to use them to minimize the error variance of
some -vector function f v of some subset(s) of the forecasted variables.

A perfect raw ensemble would provide

q

draws from the distribution of verifying
functions given the forecast. In particular, the jth ensemble member gives

K

v x v
j

v h

x v

j x v i

f v x v

j D H

D H C

/ C H v x v

j x v

v

. x v

' H (16.17)

v is the non-linear
function of interest and H v is the derivative of the non-linear function with respect to
the model variables about the mean of the ensemble forecast x v . Thus, the estimate
of the qxq forecast error covariance matrix of the vector function f associated with
the forecast upon which targeting decisions is made is given by

x v

where is the mean of the ensemble forecast and where H

H

T

D f f t f f t T E

v x v
j

v x v
j

v x v
j

v x v
j

X

1
K 1

' H H H

j D 1

H v x v

j x v x v j x v T H v T

X

1
K 1

'

j D 1

/ T H v X v H v X v .

D : (16.18)

K 1

denotes the mean of the ensemble of vector functions. Using ( 16.18 )
and ( 16.16 ) leads to the following estimate of forecast error covariance matrix

v x v
j

where H

D

f f t / T E f f t /.

for the vector function f given routine observations and the ith
deployment of adaptive observations.

.

i

D f f t f f t T E

H v X v
i

H v T
K 1

X v T
i

i

/ 1 C i

H v X v TC i . i C I T T X v T H v T

D

K 1

/ 1 C i / T

v

X v T T ŒH

v

X v

ŒH

. / TC i . i C I .

(16.19)

K 1

v

X v

where the qxK matrix ŒH . / is given by

v

X v ŒH . /

D h

i

v x v
1

H

v x v
2

H

v x v
K

H

H

v

.

x v

/ ; H

v

.

x v

/ ;:::; H

v

.

x v

/ :

(16.20)