Variational Data Assimilation for the Global Ocean - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 300

to obtain the correction field in grid point space. A pre-conditioned conjugate gradi-
ent descent algorithm is used to solve ( 13.2 ) using block diagonal pre-conditioners.
The blocks are defined by decomposing the analysis grid into non-overlapping
partitions of a regular quilt laid over the analysis domain in model grid point
(

blocks rather than latitude-longitude blocks allows
the analysis to be completely grid independent. The flexibility of this approach is
shown in Fig. 13.1 for the global HYCOM Atlantic basin analysis (see Sect. 13.6
and Fig. 13.9 for a discussion of the HYCOM basins). A total of 1,935, 2,436,
and 1,147 blocks are defined for the global HYCOM Atlantic, Indian, and Pacific
analysis regions, respectively, which use Mercator grid projections. Observations
are sorted into the blocks and the pre-conditioning matrix is formed from a
Choleski decomposition of the correlations between observations in the same
block. The Choleski decomposed block matrices are calculated once and stored
before application of the conjugate gradient descent algorithm. Solution of the pre-
conditioned conjugate gradient for the vector z n( 13.2 ) typically converges in 6-10
iterations. Determination of convergence is based on the norm of the gradient of the
cost function estimated at each iteration step. This gradient is a vector the size of the
number of observations and the norm is the square root of the sum of the elements,
which are the residuals of the fit of the analysis to the innovations. In practice,
convergence is reached when the norm of the gradient is reduced by 2 orders of
magnitude. This is considered to be sufficient because an increase in the number
of iterations only fits small-scale features. This may appear to be beneficial, but it
must be noted that the post-multiplication step is a spatially smoothing operation
when the background error correlations are applied. Thus, the extra iterations in the
solver required to resolve small-scale features in the observations do not have much
effect on the final analyzed increment field because of the smoothing effect of the
post-multiplier.
Observation space 3DVAR algorithms converge quickly because very good pre-
conditioners can be developed. In fact, the pre-conditioner used in NCODA is
perfect. For example, NCODA is configured such that when the data count is less
than 2,000 the observation data block is defined as the entire analysis domain.
When this global pre-conditioned data block is presented to the conjugate gradient
solver the algorithm converges in a single iteration. No further work by the solver is
necessary. This sparse data pathway through the code is often encountered when
NCODA 3DVAR is executed on nested grids in the relocatable coupled model
system where the innermost grid represents a small geographic area.
As noted by Daley and Barker ( 2001 ), partitioning of the observations into
blocks has no effect on the final solution. The NCODA 3DVAR formulation is
guaranteed to include correlations between all observations in all blocks, thereby
achieving a global solution. After the vector z is obtained it is post-multiplied
by P b H T to create the analysis correction fields for each analysis variable. This
step is performed using blocks in grid space that are defined differently from the
observation blocks used to compute the solution vector z . To accommodate high
resolution ocean model forecast grids and minimize computer memory resource
requirements for the analysis, the grid space blocks are defined smaller by simply

i;j ) space. The use of i;j