Recent Applications in Representer-Based Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 286

x 10 4

7.5

7

6.5

6

5.5

5

4.5

4

3.5

3

0 10 20 30 40 50 60

iteration count, p

Fig. 12.2 using GCR. The performance of the GCR solver as measured by the

value of the objective function for an ocean data assimilation problem is shown. J .

Reduction of J . x /

is computed
using ( 12.14 )and( 12.15 ) in the text. The application involves the assimilation of satellite altimetry
data into a three dimensional primitive equations ocean model encompassing the Hawaiian Ridge,
with the goal of estimating the tidal circulation around the Ridge

x p /

J .

x p /

Figure 12.2 shows the progress of for a data-assimilative three-

n D 400 300 30 5 D 18 10 6

dimensional ocean model with approximately

m D 17 10 4 observations (see Zaron et al. 2009 for a similar
application in a smaller computational domain). The figure shows that the decrease
in cost function is not monotonic, and increases can occur. This behavior does not
occur in smaller, exactly symmetric problems, and the working hypothesis is that the
non-monotonicity is caused by asymmetry or lack of positive-definiteness in either
the adjoint model or background covariance. Pointwise tests of the symmetry of B
and HBH T indicate that the former is symmetric to machine precision, while the
latter contains symmetry errors of 10 % of the diagonal elements. The computational
cost of evaluating Ax is approximately 100 cpu-hours, so there is a substantial need
for computational efficiency.
Further diagnostic information is available from the GCR iterates as well.
Qualitative assessment of the solution in the state space is available since the
solution x p is computed at each iterate. Because AU

state variables and

D C , with C orthogonal, the

of U approximate the singular values of A 1 ( Golub and Van
Loan 1989 ). Knowledge of the singular spectrum and orthogonal decomposition of
U may be used to better precondition subsequent outer iterations ( Giraud et al. 2006 ;
Parks et al. 2006 ).
Assuming the observation error is uncorrelated and constant, R

. /

singular values U

D

I , one can

R D HBH T

approximate the singular spectrum of the so-called representer matrix

/ g p
i D 1
denotes the ordered singular spectrum, the set of nonzero singular values of the
matrix U

. R / . / 1 . / Df i .

( Bennett 1992 ) with U . Here the notation U U

2 R m p ,where

i C 1 . / i . / p m

U U and are assumed, and the

. / 1 is defined as the set of reciprocals of

inverse of the singular spectrum U