Ocean Ensemble Forecasting and Adaptive Sampling - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 390

Thus, the ETKF allows non-linear cost functions without the need for the first
derivative (Jacobian) of the non-linear verification time functions of interest.
Equation 16.19 gives the forecast error covariance of the user specified functions
of interest for the ith deployment of adaptive observations. Often, users will
reduce the information in this matrix to a single cost function by, for example,
evaluating the trace of the matrix. To find which of all feasible deployments
of adaptive observations minimizes the user specified cost function, one simply
evaluates ( 16.19 ) for all feasible deployments of adaptive observations and chooses
the deployment which minimizes the cost. Since the transformation matrix ( 16.2 )
associated with the routine observational network and the

v

X v

matrix only
need to be evaluated once, the main computational expense associated with each
deployment is the K K eigenvector decomposition ( 16.12 ). For ensemble sizes
smaller than 100, this is a trivial expense on today's CPUs and thousands of
networks can be evaluated in a matter of minutes on moderate computing resources.
To highlight and predict the impact of the targeted observations, it is also of
interest to predict the covariance of the distribution of changes to the forecast that
would be imparted by the ith observational network given an infinite sampling of
the distributions of observation and forecast. As shown in Bishop et al. ( 2001 ), at
the observation time this covariance is given by
D x i x r x i x r T E

ŒH . /

H i P r H aT

i C I 1 H i P r

D P r H aT

i

/ 1 C i

X o TC i i . i C I T T X oT

D (16.21)

K 1

where x r represents the minimum error variance state estimates at the observation

time given routine observations while x i

represents the minimum error variance
state estimates at the observation time given routine observations and the ith
deployment of adaptive observational resources. Thus, it represents the covariance
of changes to the state estimate due to adaptive observations. The changes due to the
adaptive observations are called signals and the covariance of these changes is called
the signal covariance. The expression for the signal covariance at the verification
time is

D x v

T E

/ 1 C i

x v

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

i x v i x v

D (16.22)

r r

K 1

As can be seen by comparing ( 16.21 ) with ( 16.8 ) and as was discussed in Bishop
et al. ( 2001 ), for an optimal data assimilation scheme, the signal variance is precisely
equal to the reduction in forecast error variance due to the observations that created
the signals. Comparison of geographical plots of the diagonal elements of ( 16.21 )
and ( 16.22 ) with actual changes in forecasts due to targeted observations can give
a good indication of whether the ETKF signal variance predictions are reasonable
or not.