Environmental Engineering Reference
In-Depth Information
to their range constraints and a sum-to-one restriction
for three of them, leaving 17 inputs that can be varied
independently.
deviation between model output and observed data
was small.
Our aim is to 'rule out' any input
x
for which
I
(
x
)is
'too large' when compared to a threshold based on a rea-
sonable calibration for
I
(
x
). One such calibration is based
on assuming independent standard normal distributions
for the signed standardized deviations in Equation 26.7,
deeming an input
x
to be implausible if say
I
(
x
) exceeds
the upper 5% point of its distribution in the null case
when
x
=
x
∗
. Then, the distribution of
I
(
x
) is such that:
p
=
P
[
I
(
x
)
≥
m
|
x
=
x
∗
]
=
1
−
[2
(
m
)
−
1]
N
26.3.4.1 Implausibility
We have structured our uncertainty specification in gen-
eral form, in which:
f
t
(
x
∗
)
ε
t
=
y
t
−
(26.3)
is the model discrepancy at time
t
, where
x
∗
is taken to be
the appropriate model representation of the actual system
properties, and which correspond to the actual unob-
served system output
y
t
. We regard the values of
x
∗
and
the
y
t
as random quantities, as their values are unknown.
Next, we write:
(26.9)
where
(
·
) is the cumulative distribution function of the
standard normal distribution. Hence, we want to choose
m
so that the probability
p
in Equation 26.10 is 'small';
that is, choose
m
such that:
p
)
1
/
N
+
−
z
t
=
y
t
+
e
t
(26.4)
1
(1
(
m
)
=
(26.10)
2
where
z
t
is the measurement of
y
t
and
e
t
is the associated
measurement error. Furthermore, we have decomposed
the overall model discrepancy into the sum of internal
and external components, which we write as:
=
.
=
=
.
38.
At the other extreme, when the signed standardized
deviations in Equation 26.7 are completely dependent,
corresponding to
N
When
p
0
01 and
N
839 we find that
m
4
=
1 in Equation 26.10, we find that
m
01. The actual result will be some-
where between these two extremes. The corresponding
values of
m
for
p
=
2
.
58 when
p
=
0
.
ε
t
=
ε
I
t
+
ε
E
t
.
(26.5)
Putting these relationships together, we obtain:
05 are 4.01 and 1.96. We adopt the
conservative, stringent independence assumption with
p
=
0
.
f
t
(
x
∗
)
z
t
=
+
ε
I
t
+
ε
E
t
+
e
t
.
(26.6)
=
0.01. Thus, we deem an input
x
implausible if
I
(
x
)
38.
We applied this implausibility criterion to the log-
arithm of discharges from 100 000 runs of the runoff
model, where the inputs were from a subset of a Latin
hypercube design chosen to accommodate the sum-to-
one restriction. The
>
4
.
ε
E
t
and
e
t
to be uncorrelated random (uncertain) quantities each
with expectation zero and respective variances
σ
We regard the discrepancy and error terms
ε
I
t
,
I
t
,
σ
E
t
and
σ
e
. We will assume that the value of the measurement
error variance
E
t
need
to be carefully assessed, preferably in conjunction with
a system expert, taking into account the limitations of
the model in describing the actual system. We define the
implausibility
I
(
x
)ofamodelinput
x
to be:
e
I
t
and
σ
σ
is known, whereas
σ
t
in Equation 26.7 were modified
to be the sum of the measurement error variance and
the internal model discrepancy variance contribution to
the overall component-wise model discrepancy variance;
that is,
σ
e
. The intention was to see if we
could find some non-implausible inputs (without intro-
ducing any external model discrepancy) to help assess the
external discrepancy variance contribution to the overall
model discrepancy. However, we found that without the
external discrepancy, every one of the 100 000 inputs were
implausible (given zero external model discrepancy): the
lowest implausibility is about 4.7 with only two runs less
than 5.0. In fact, we observed that, for all 100 000 runs, the
model consistently overreacted to short periods of rain
and reacted too quickly (or too slowly) to major peaks in
rain, demonstrating that its predictive adequacy may be
regarded as questionable for such rainfall patterns.
t
I
t
σ
=
σ
+
σ
max
1
≤
t
≤
N
z
t
−
f
t
(
x
)
σ
t
I
(
x
)
=
(26.7)
where:
t
I
t
E
t
e
σ
=
Var
[((
z
t
)
−
f
t
(
x
))]
=
σ
+
σ
+
σ
(26.8)
t
do not depend
on
x
. Other definitions of implausibility are possible; for
example, the average of the deviations in Equation 26.7
or the average of their squares. The definition in
Equation 26.7 is more stringent than these two: imposing
a constraint upon
I
(
x
) would demand that the maximum
Note that
I
(
x
) is scale-free and the
σ
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