Geoscience Reference
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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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