Geoscience Reference
In-Depth Information
Tabl e 5. 3
Numerical experiment results
Erroneous control
Correct control
x
0
.0/;
0
;
0
x.0/; ;
2.0, 10.0, 3.0
1.0, 11.0, 2.5
Experiment 1: Observations at
k
D
15
, 16, 17, and 18
r
J
-method
FSM
Adjusted Control
(
x.0/
,
,
/
(2.00, 10.93, 3.1)
(1.23, 10.99, 2.0)
1:74
W
8:3
10
4
1:74
W
6:1
10
2
J (initial): J (final)
0:010; 0:42; 3:99
0:002; 0:56; 4:99
Eigenvalues of H
Experiment 2: Observations at
k
D
1
, 2, 17, 18
r
J
-method
FSM
Adjusted control
(
x.0/
/
,
,
(1.39, 11.13, 2.2)
(1.05, 10.98, 2.3)
1:56
W
2:0
10
2
1:56
W
6:0
10
2
J (initial): J (final)
Eigenvalues of H
0.06, 1.82, 5.06
0.06, 1.94, 5.12
J (initial): Initial value of the cost function
J (final): Final value of the cost function
H: Hessian matrix at the point of adjusted control
surface is also relatively flat along the direction of the turbulent transfer coefficient.
Not only is the surface flat in these directions, the negative gradient with respect to
the initial condition and turbulent transfer coefficient will push the correction toward
higher values of these elements whereas ideal corrections are toward lower values.
Nevertheless, it is the end result of the search for the minimum that is critical to
examine. Four iterations of the conjugate gradient method are executed before the
empirical criterion for termination is satisfied (a change in the cost function less
than
10
4
from one iteration to the next).
A summary of results for both r
J
-method and FSM are shown in the top portion
of Table
5.3
. Although the value of the cost function has significantly decreased
via the
-method, the corrections to initial condition and turbulent transfer are
minimal. The reduction is obviously due to the SST correction alone. The search
moves steadily toward the correct value of SST because of a significant negative
gradient in that direction, but moves nary a bit in the other directions because of
flatness of the surface in those directions. In short, the value of the cost function can
be reduced significantly due to correction of SST alone—poor estimates of initial
condition and turbulent exchange coefficient have little influence on the fit. What
is the consequence of the poor fit to these elements? The consequence that can be
determined in this idealized experiment is a poor forecast of temperature at the early
times—based on observations that are known although unused in the functional.
Eigenvalues of the Hessian fundamentally determine the adequacy or inadequacy
of observations in this case. The eigenvalue set for the r
J
r
J
-method is of mixed sign
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