Variational Data Assimilation for the Global Ocean - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 302

algorithm is used to spread the work related to the block-dependent calculations
out evenly across the processors. In the conjugate gradient descent step, the work
load for an observation block is calculated as the sum of the observation-observation
interactions. In the post-multiplication step, the work estimate is based on the sum
of the observation-grid point interactions. Observation and grid point blocks are
determined to be close enough to contribute to the solution if the block centers
are within 8 correlation length scales. Thus, for a given block size, the number of
observation-observation and observation-grid point block interactions varies with
the horizontal correlation length scales and will be more numerous where length
scales are long. Further efficiency is achieved by keeping communication among
the processors minimal. To do this matrix elements are calculated, stored, and
used on each processor, they are never passed between processors. Only elements
of the solution and correction vectors scattered across the processors have to be
communicated and reassembled and, in the case of the solution vector, broadcast
for the next iteration. Note that memory utilization for the conjugate gradient solver
in the 3DVAR is reduced as the number of processors is increased. This feature
allows the 3DVAR to scale very well across many processors on large machines,
and run equally well on small platforms with limited memory.

13.3 Error Covariances

Specification of the background and observation error covariances in the assim-
ilation is very important. As previously noted, the background error covariances
control how information is spread from the observations to the model grid points and
model levels, but they also ensure that observations of one model variable produce
dynamically consistent corrections in the other model variables. The background
error covariances in the NCODA 3DVAR are similar to the error covariances defined
for the MVOI, but with some notable exceptions. As in the MVOI, the error
covariances in the 3DVAR are separated into a background error variance and a
correlation. The correlation is further separated into a horizontal (C h ) and a vertical
. component. Correlations are modeled as either second order auto-regressive

(SOAR) functions of the form,

C v /

C h D .1 C s h / exp . s h /

C v D .1 C s v / exp . s v / (13.4)

or Gaussian functions of the form,

. s h / C h D exp

. s v / C v D exp (13.5)

where s h and s v are the horizontal and vertical distances between observations or
observations and grid points, normalized by the arithmetic mean of the horizontal or