Geoscience Reference
In-Depth Information
the variances of all input variables can be computed for multiple input variables using
Equation (3.8), assuming that the inputs are independent. Let the function of
y
in
Y
=
y(X)
be approximated by a parabolic function (Fig. 4.8), that is:
(4.18)
Y
=
f (X)
=
aX
2
+
bX
+
c
If there are multiple input variables, say
n
input variables, there will be
n
number of
equations like Equation (4.18). In which case, the coefficients
a, b,
and
c
of each of them
are the functions of the mean values of the remaining variables.
It is assumed here that the problem is solved numerically by applying a perturbation
on the input variable. Three points
x
C
, x
B
and
x
F
are defined as follows
x
B
=
x
C
!
'
(4.19)
x
F
=
x
C
+
'
(4.20)
(4.21)
where
is equal to the mean value of the variable.
Figure 4.8.
Approximation of the real function
y
(bold, dashed line) with a
parabolic function
f
(thin line) using the three points
x
B
, x
C
and
x
F
.
The first step in the method consists of identifying the coefficients
a, b
and
c
. This is
achieved by requiring the function
f
to take the same values as
y
at the points
x
B
, x
C
and
x
F
. This yields the following equalities:
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