Variational Data Assimilation for the Global Ocean - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 299

time, the forward operator in NCODA is spatial interpolation performed in three
dimensions by fitting a surface to a grid point target and evaluating the

surface at the observation location. Thus, HP b H T is approximated directly by the
background error covariance between observation locations, and P b H T directly by
the error covariance between observation and grid locations. For the purposes of
discussion, the quantity

4 4 4

Πy H . x b / is referred to as the innovation vector, Πy H . x a /

is the residual vector, and x a -x b is the increment (or correction) vector.
The observation vector contains all of the synoptic temperature, salinity and
velocity observations that are within the geographic and time domains of the
forecast model grid and update cycle. Observations can be assimilated at their
measurement times within the update cycle time window by comparison against
time dependent background fields using the first guess at appropriate time (FGAT)
method. An advantage of the FGAT method is that it eliminates a component of the
mean analysis error that occurs when comparing observations and forecasts not valid
at the same time. As will be described in Sect. 13.6 , the use of FGAT in real-time
HYCOM allows for assimilation of late receipt observations.
Equation ( 13.1 ) is the observation space form of the 3DVAR equation. A dual
form of the 3DVAR is the analysis space algorithm, which is defined by the
model state vector on some regular grid. Courtier ( 1997 ) has shown that the two
formulations are equivalent and give the same solution. However, as discussed by
Daley and Barker ( 2000 , 2001 ), there are advantages to the use of an observation
space approach in Navy ocean model applications. In the observation space
algorithm the matrix to be inverted C R) 1 is dimensioned by the number

of observations, while in the analysis space algorithm the matrix to be inverted is
dimensioned by the number of grid locations. Given the high dimensionality of
global ocean forecast model grids, and the relatively sparse ocean observations
available for the assimilation, an observation space 3DVAR algorithm will have
a clear computational advantage. Further, an observation space system is more
flexible and can more easily be coupled to many prediction models. As has
been discussed, NCODA must work equally well with multiple atmospheric and
oceanographic prediction systems in a wide variety of applications, as well as a
wave model prediction system. Finally, due to the local nature of the observation
space algorithm, the background error covariances are multivariate and can be
formulated and generalized in a straightforward manner. As will be shown, this
aspect of the 3DVAR is an important feature of NCODA. On the other hand, analysis
space systems typically restrict the background error covariances to be sequences of
univariate, one-dimensional digital filters, thereby ignoring the inherent multivariate
nature of the background error correlations.
Solution of the observation space 3DVAR problem is done in two steps. First, the
equation,

HP b H T .

HP b H T C R/

. z D Œy H.x b / (13.2)

is solved for the vector z . Next, a post-multiplication step is performed by left-
multiplying z using,

x a x b D P b H T z (13.3)