Environmental Engineering Reference
In-Depth Information
To obtain an order of magnitude assessment of the
external model discrepancy
the residual plot
z
t
−
σ
t
limits as in the
upper panel and three-sigma limits for both the internal
and the external discrepancy. Note how frequently the
external discrepancy is zero.
A large external model discrepancy standard deviation
σ
E
t
indicates that the model fails to predict well for
reasons not explained by measurement error or internal
error. These errors occur here mainly when either the
model reacts too quickly or too slowly during heavy
rainfalls, for example around 490 hours, or when the
model overreacts to smaller rainfall events, such as at 395
and 690 hours. We might expect such deficiencies in a
simple compartment model of a complex physical runoff
system.
f
t
, the same 3
σ
E
t
, we might choose a small
number n of the least implausible inputs
x
1
,
,
x
n
of
the 100 000 runs, and consider the corresponding model
outputs
f
t
(
x
i
). We then choose
...
E
t
σ
in Equation 26.8
so that:
max
1
≤
t
≤
N
≤
z
t
−
f
t
(
x
)
σ
t
3
.
(26.11)
E
t
can be zero. Figure 26.5 shows
the results when we choose
n
=
8. The upper panel shows
plots f
o
r the logarithm of observed discharge
z
t
,the
mean
f
t
of the corresponding eight model outputs and
the 3
Note that this choice of
σ
σ
t limits about that mean. The lower panel shows
Observed discharge
Total discrepancy plus measurement error
Mean discharge
0
200
400
Time in hours
600
800
Total discrepancy plus measurement error
Internal discrepancy
External discrepancy
Observed minus mean discharge
0
200
400
Time in hours
600
800
Figure 26.5
The upper panel shows the logarithm of observed discharge, the mean of the corresponding eight 'best' model outputs
and overall three-sigma limits (3
t) about that mean; while the lower panel shows the residual about that mean, the same
three-sigma limits as in the upper panel and three-sigma limits for both the internal and the external discrepancy. See the text for the
definition of each sigma.
σ
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