A Question of Adequacy of Observations in Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 119

indicating a saddle point at the location of the final iteration. Positive eigenvalues
are indicative of upward turning of the cost function whereas a negative eigenvalue
is indicative of downward turning of the surface. In this circumstance of mixed signs
in the Hessian's eigenvalue set, we conclude that the observations are insufficient to
locate the minimum of J.
One iteration of the FSM for this case yields an adjusted control shown beside the
results for the r J -method (again in the top portion of Table 5.3 ). The cost function

is reduced but not as much as in the case of the r J -method. Further iterations with

FSM would improve the fit. Whereas flatness in the cost function surface is the
problematical aspect encountered by the

-method, the corresponding problem
for FSM is displayed as an absence of differences in the sets of sensitivities at the
last four times (evident through examination of the last four rows of the sensitivity
functions in Table 5.1 ). This lack of difference in the corresponding elements in
these rows indicates near singularity of the

r J

S T S

matrix. The near singularity is
measured by the largeness of the condition number of the

S T S matrix (D10 9 )—the

S T S

ratio of the largest to the smallest eigenvector. Inversion of matrix is essential

to finding corrections to control [See ( 5.12 )]. As was the case for the r J -method,

the FSM exhibits an eigenvalue set of mixed sign indicating insufficiency of the
observations to locate the minimum of the cost function.

5.4.4 Experiment 2: Sufficiency of Observations

In this experiment, we replace the observations at t D 15 and 16 with observations

at t and 2. We now have observations where the forecast is sensitive to initial

conditions and turbulent exchange coefficient (times 1 and 2) as well as sensitive to
SST (at all four times). As opposed to results from Experiment 1, the structure of the
cost function in the vicinity of

D 1

Y 0 is not flat in any direction. Further, the magnitudes
of the various elements of r J

are comparable and they have signs consistent with
ideal corrections to control (positive for SST and negative for initial condition and
turbulent transfer coefficient). The first-iteration components of r J are

@J
@x.0/

DC 0:971; @J
@

and @J
@

D 1:267; DC 1:436 (5.14)

Results from the optimization processes are found in the lower portion of Table 5.3 .
Adjusted controls for both FSM and -method are reasonably good and the

eigenvalue set consists of positive eigenvalues for both forms of assimilation. Thus,
the terminal points of the optimization process for both schemes are extremely close
to the minimum of the cost function and the observation set is sufficient to find the
minimum of

r J

J . Another indication of sufficiency is the small value of the condition

S T S number for the matrix. In this case the condition number is 60 .