Geoscience Reference
In-Depth Information
Instead of supplying crisp weights
(w
i
)
to sub-parameters, importance of each
subparameter can be assessed using fuzzy variables. An alternative to assessing each sub-
parameter quality is to use IF-THEN rules showing the relationships between the parame-
ter and sub-parameter qualities set by the experts. Readers are referred to
Sundararajan (1994 & 1998) for more details about this approach.
Combining Quality and Importance
The contribution of a given parameter to the total uncertainty is proportional to its
Im-
portance,
#. The higher the
Importance
of a parameter, the greater the uncertainty in the
output. Conversely, the higher the
quality
of a parameter, the lower the resulting
uncertainty. The contribution of a parameter to the total uncertainty is therefore
proportional to the so-called “mirror image” "! of ". If the membership function of " is
defined by couples {(
x
1
,
s
1
),…,
(x
n
, s
n
)
}, the membership of its mirror image is given by
{(1!
x
1
,
s
1
),…(1!
x
n
, s
n
)}, where
x
1
,…,
x
n
are base values defined in the scale of 0 to 1
and
s
1
,…,
s
n
are belief levels (i.e.
s
i
=
"(x
i
)
).
The combined effect,
U,
of " and #, is then given by
U
j
="!
j
($)#
j
(3.24)
where the subscript
j
stands for the
j
th
parameter, and the operator ($) denotes fuzzy
number multiplication.
Computing total uncertainty in the output
The total uncertainty in the output due to the uncertainty contributions of individual
parameters is given by
(3.25)
where
U
O
is the total uncertainty in the output in the form of a membership function.
When the evaluation is made by more than one expert, say
m
experts, then the fuzzy
uncertainty from different experts given by Equation (3.25) are averaged (Eq. (3.26)) to
get the final total uncertainty.
(3.26)
Interpretation of the output uncertainty
The output uncertainty given by Equation (3.25) or (3.26) is the fuzzy (or possibilistic)
uncertainty represented by a membership function. The length of the support and the
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