Variational Data Assimilation for the Global Ocean - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 319

where X f is the matrix of ensemble forecast perturbations about the ensemble
forecast mean, P a is the analysis error covariance matrix, n is the number of model
variables (state vector), and C are the eigenvectors and the eigenvalues of the left

hand side of ( 13.10 ). Superscript T indicates matrix transpose. Given the eigenvector
decomposition the transformation matrix T is given by T 1=2 C T ,which

is used to transform a matrix of forecast perturbations to a matrix of analysis
perturbations according to X a D X f T . If the ensemble size is large enough it can
be shown that the covariance of the analysis perturbations equals the prescribed
analysis error covariance P a ( McLay et al. 2008 ). Thus the analysis error covariance
is an effective constraint in the ET, ensuring that the ensemble generation system is
consistent with the data assimilation system.
The NCODA ET is multivariable and computes the transformation matrix for
temperature, salinity, and velocity simultaneously. As a result the NCODA ET
perturbations are balanced and flow dependent. In an ET ensemble generation
scheme the control run is the only ensemble member that executes the 3DVAR. This
results in a considerable savings in computational time as compared to a perturbed
observation approach where the analysis must be executed by all of the ensemble
members. Given a 3DVAR control run analysis and its corresponding analysis
error covariance estimate, the system calculates the ET analysis perturbations and
adds the perturbations to the control run to form new initial conditions for each
ensemble member. The forecast model is then integrated creating a new set of
ensemble forecasts for the next cycle of the ET. The NCODA ET and 3DVAR have
been successfully implemented in a coupled ocean atmosphere mesoscale ensemble
prediction system ( Holt et al. 2011 ).

D C

13.5.4 Residual Vector

The residual vector is very useful in assessing the fit of the analysis

to specific observations or observing systems. It is usually calculated at the end
of the analysis after the post-multiplication step by horizontally and vertically
interpolating the analysis vector

Πy H . x a /

to the observation locations and application
of the nonlinear forward operators H to obtain H

. x a /

in observation space. This
result is then subtracted from the observations to form the residual vector. The
problem here is that horizontal and vertical interpolations of the analysis grid to
the observation locations and subsequent application of the H operator introduces
error into the residual vector, which may change interpretation of the quality of
the fit of the analysis to an observing system. A better approach is to estimate
the analysis result, and the residual vector, while still in observation space, that
is, before application of the post-multiplication ( 13.3 ). Daley and Barker ( 2000 )
show that a good approximation of the true residuals while in observation space
can be obtained from y a D y Rz ,where y is observation vector, y a the residual
vector, R is the observation error covariance matrix, and z is defined in ( 13.2 ). Using

. x a /