Geoscience Reference
In-Depth Information
Tabl e 8. 2
Relative CPU
times required by the MC and
HM methods to achieve the
accuracies
MC/LH0
MC/LH1
HM/LH0
HM/LH1
B
g
755
1205
680
520
(0.19)
(0.09)
(0.19)
(0.09)
of the LH0 and
LH1 methods (shown in
brackets)
h
"
i
B
2
780
490
850
330
(0.17)
(0.10)
(0.17)
(0.10)
Another series of experiments are performed with the varying scaling parameter
to find an optimal fit to
d
. Computations were made for
0
1
.The
best result for
B
g
was obtained for
2
D
0:30
which is fairly consistent with the
value (
2
D
0:33
)givenby(
8.50
). In the case of the
B
2
operator, the optimal
value is
2
D
0:24
, still in a reasonable agreement with (
8.50
), given the strong
inhomogeneity of
and deviation of the
B
2
operator from the Gaussian form. A
somewhat smaller value of
can be explained by the sharper shape of the
respective correlation function at the origin (Fig.
8.1
), which renders
d
0
to be less
dependent on the inhomogeneities in the distribution of
2
.
B
2
/
, and, therefore, requires
less smoothing in the next approximation.
8.3.4
Numerical Efficiency
Tab le
8.2
provides an overview of the performance for the tested methods. For
comparison purposes we show CPU requirements by the smoothed MC and HM
methods after they achieve the accuracies of the LH0 and LH1 methods. It is
seen that both MC and HM methods are 300-1,000 times more computationally
expensive than the LH technique. In fact, for the 2d case considered, the compu-
tational cost of the stochastic diagonal estimation method is similar to the cost of
the 3dvar analysis itself, which required several hundred iterations. The remarkable
CPU savings are due to the fact that the LH methods explicitly take into account
information on the local structure of
B
which can be derived by analytical methods.
Comparison of the spatial distributions of the approximation error h
"
i
.
(Figs.
8.5
b
and
8.6
b) favor the LH methods as well: They show significantly less small-scale
variations and may have a potential for further improvement. Comparing Figs.
8.5
b
and
8.6
b also shows that, in contrast to the statistical methods, LH0 errors tend
to increase in the regions of strong inhomogeneity, but decrease substantially after
smoothing by the LH1 algorithm. At the same time, the LH1 errors tend to have
relatively higher values near the boundaries. The effect is less visible in the HM
pattern (Fig.
8.5
b). This feature can be partly attributed to certain inaccuracy in
estimation of the near-boundary elements. However, there is certainly room for
further improvement with the issue.
x
/
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