Geoscience Reference
In-Depth Information
Figure 4.6.
Three options for derivative assessment. The bold line represent the
original function
y
and the straight lines represent the approximate
functions
f
obtained using the backward
(B),
centred
(C)
and forward
(F)
difference methods.
4.2.2
Problems attached to FOSM method
Several theoretical and/or conceptual problems in the application of the FOSM method
are:
• The function
f
aims to approximate a function
y
that is nonlinear in most
cases. Therefore, the computed value
f
(
X
1
,…,
X
n
) may depart substantially
from the actual value
Y
when the standard deviation of the input variable is
large compared to the interval over which the function
f
is assumed to be linear.
• When the function
y
is highly nonlinear, the method poses problems in that
(i) the backward, centred or forward estimates often result in very different com-
puted values, and (ii) the result is very sensitive to the size of the perturbation
'
.
• Eventually, a problem that has not been discussed widely in literature occurs when the
mean value of the input variable is very close to a local or global extremum of the
function. In this case, the computed standard deviation (and consequently the
uncertainty in the variable) given by the FOSM method may be very different from the
real value. Figure 4.7 illustrates such a situation. In this example, the function
y
has an
extremum close to the average
Assume now that the range of the uncertainty
in
X
is given by the interval [
x
1
,
x
2
]. This results in the range of uncertainty [
y
1
,
y
2
] for
Y
. However, the FOSM method, which consists of using the
tangent of the linearised function
f
around would give an interval
[
f
1
,
f
2
] different from the real one [
y
1
,
y
2
]. In particular,
f
2
is larger than
y
2
because
the real function
Y(X)
takes its maximum within the interval [
x
1
,
x
2
]. Moreover,
f
1
is larger than
y
1
because the function
Y(X)
is convex.
Search WWH ::
Custom Search