Environmental Engineering Reference
In-Depth Information
governed by the equation:
26.3.2 Informalmodel exploration
As an illustration of the many types of data analysis that
we may carry out to explore the qualitative behaviour of
the model, we focus on the water discharged at hour 620,
and investigate its sensitivity to changes in a selection of
some of the 17 model input parameters.
We illustrate the process by observing in Figure 26.2
(left panel) how the logarithm of discharge at hour 620
varies over the range of
c
f
DP
for a selection of four values
of
p
f
DP
and in Figure 26.2 (right panel) how it varies with
b
AS
for four values of
c
s
AS
where in both illustrations the
other inputs were held fixed at their mid-range values.
As hour 620 is shortly after a large rainfall between 610
and 619 hours, peaking at hour 615, increasing
c
f
DP
from
its minimum value of 0.1 initially increases discharge, as
more water will flow out of the fast DP compartment
(see Figure 26.2). However, increasing
c
f
DP
past 0.2 leads
to a decrease in discharge, because lots of the water will
have drained away before 620, resulting in less flow.
Increasing
p
f
DP
increases the amount of water entering
the fast DP compartment (as opposed to the slow DP
compartment), which leads to a corresponding increase in
discharge.
Figure 26.2 (right panel) shows that, as
b
AS
is increased,
the system approaches saturation and more water is
directed into the fast and slow AS subcompartments,
with less going into the ineffective storage (IS) com-
partment. This behaviour leads to larger flows out of
the AS subcompartments, resulting in an increased dis-
charge, which tends to an asymptotic value. Increasing
c
s
AS
S
(
t
+
1)
=
S
(
t
)
+
RAIN(
t
)
−
AET(
t
)
−
F
soil
(
t
)
soil
c
s
ES
s
(
t
) are the flows out
of each soil-type compartment. Similarly, updating the
ineffective storage from time
t
to time
t
c
f
ES
f
(
t
)
where the
F
soil
(
t
)
=
+
+
1 is governed
by the equation:
RAIN(
t
)
1
r
soil
(
t
)
IS
(
t
+
1)
=
IS
(
t
)
+
−
−
AET(
t
)
soil
Hourly model outputs, discharge
D
(
t
), calcium
Ca
(
t
)
and silicon
Si
(
t
) are given by:
D
(
t
)
=
F
soil
(
t
)
soil
T
Ca
Ca
(
t
)
=
soil
F
soil
(
t
)
/
D
(
t
)
soil
T
Si
Si
(
t
)
=
soil
F
soil
(
t
)
/
D
(
t
)
soil
where the
T
tracer
soil
terms govern the tracer concentrations
of Ca and Si emanating fromeach soil-type compartment.
Thus,
f
(
x
) we need (i) a
computer-code implementation of
f
(
to run the model
y
=
); (ii) valid values
for the 17 components of
x
; (iii) the forcing functions
RAIN and AET; (iv) the initial conditions
ES
f
,
ES
s
and
IS
at
t
=
0; and (v) the values of the six tracer
concentrations
T
tracer
soil
·
.
p
DP
= 0.2
p
DP
= 0.4
p
DP
= 0.6
p
DP
= 0.8
C
s
AS
= 0.01
C
s
AS
= 0.04
C
s
AS
= 0.07
C
s
AS
= 0.1
0.10
0.15
0.20
0.25
C
f
DP
0.30
0.35
0.40
0.06
0.08
0.10
0.12
b
AS
Figure 26.2
Left panel: logarithm of discharge at hour 620 versus cfDP for four different values of pfDP; Right panel: logarithm of
discharge at hour 620 versus bAS for four different values of csAS.
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