Environmental Engineering Reference
In-Depth Information
Rainfall
Initial Condition
Structural
Flow Rate
0
200
400
Time in hours
600
800
Figure 26.3
Standard deviations of the logarithm of discharge for four contributions to internal model discrepancy: initial flow
condition, the RAIN forcing function, structural inflow (green) and parameter outflow.
the form exp
2
where the number of hours
−
s
−
t
θ
a
soil
parameters in a similar way as we did for RAIN.
Specifically, we used the same perturbation process for
η
t
θ
is to be chosen. Notice that, for any given choice
for
a
DP
,
a
AS
and
a
GW
with
p
100, reflect-
ing slowly varying changes in the physical system. As
previously noted, this is a simple example of perturb-
ing the propagation step in the equations for the system
state, while retaining the water conservation constraint.
Figure 26.3 shows the standard deviation of the loga-
rithm of discharge for each hour for this internal error
contribution.
=
0
.
1and
θ
=
of
θ
, the correlation decreases as the time difference
|
increases. On the other hand, the correlation
decreases as
θ
decreases when the time difference is held
fixed. In our implementation, we set
p
s
−
t
|
=
.
θ
=
5
hours, reflecting the belief that the correlation in rainfall
measurement error will not persist over the duration of
an average storm. The 839 values of
ξ
(
t
), hence those of
η
0
1and
(
t
), can be simulated, for example, using the function
mvrnorm
in the R libraryMASS (see Venables and Ripley,
2002). We now run the model at some input
x
, using the
original initial condition and perturbed forcing function
values
26.3.3.4 Parameter outflow contribution
The flow out of each soil compartment is governed by
c
soil
and
c
soil
. We perturb these six parameters as we did for
the
a
soil
parameters in Section 26.3.3.3 using the same
(839)RAIN(839) and record
the perturbed discharge series. We repeat this procedure
K
times and, exactly as we did with the perturbation of
the initial condition above, estimate a diagonal variance
matrix
V
RAI
x
. Figure 26.3 plots the standard deviations
(the square roots of the diagonal elements of
V
RAI
x
)
against
t
when the components of
x
arechosentobethe
mid-range values specified by Iorgulescu
et al
. (2005).
η
(1)RAIN(1),
...
,
η
t
process for each of them. Figure 26.3 shows the standard
deviation in the logarithm of discharge for each hour
for this internal error contribution. Overall, the patterns
of the RAIN, structural and flow contributions to inter-
nal model discrepancy shown in Figure 26.3, are similar
with flow lagging a few hours behind the other two: they
all increase significantly during periods of heavy rain-
fall. Figure 26.4 shows three traces: (i) the logarithm of
observed discharge; (ii) three standard deviation intervals
of observed error in the logarithmof discharge (whichwas
chosen to be 5%); and (iii) three standard deviation inter-
vals of internal model discrepancy, where the standard
η
26.3.3.3 Structural inflow contribution
The amount of water flowing into each soil subcompart-
ment at each hour
t
is governed by its transfer function
r
soil
(
t
)and
p
soil
. There are many possible perturbations:
for illustrative purposes we chose to perturb the three
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