Environmental Engineering Reference
In-Depth Information
increases the flow out of the slow AS compartment, which
results in a small increase in discharge. Several additional
plots were considered and they all demonstrated sensible
model behaviour.
a convenient way to sample a value of
η
,istosample
log
from the normal distribution with this mean and
variance, and then exponentiate the result.
Now suppose we (i) fix values for the 17 input param-
eters
x
; (ii) sample a value
η
η
1
of
η
; and (iii) run the
η
1
w
and inputs
x
.Let
D
1
(x),
...
,
D
839
(
x
) denote the resulting discharge output time
series: actually, we take the logarithm of discharge to be
the model output
y
. Now repeat the above with each of
another
K
26.3.3 Internalmodel discrepancy
model with initial condition
We consider assessment of the internal model discrepancy
contribution to overall model discrepancy for the runoff
model. Todo so, we perturbdifferent features of themodel
and focus on how they perturb the discharge output
D
(
t
).
There are several distinct model features that we consider
perturbing, including the six input parameters for each
soil-type compartment, the initial flow conditions and
the output tracer concentrations, the transfer functions
r
soil
(
t
) and the two forcing functions RAIN and AET.
To illustrate our approach, we focus on perturbing
(a) the initial conditions; (b) the forcing function RAIN;
(c) the parameters
a
soil
in the transfer functions
r
soil
(
t
)
that influence the amount of water entering each com-
partment; and (iv) the input parameters
c
soil
and
c
soil
governing the flow rates out of the three compartments.
Note that (c) is a simple example of perturbing the prop-
agation step in the equations for the system state, while
retaining the water-conservation constraint.
We adopt a similar formulation for each of the four
perturbations.
k
with
initial condition
η
kw
we have discharge model outputs
D
1
(
x
),
−
1 independent
η
values, so that for
η
,
D
839
(
x
)for
k
,
K
. In our implementa-
tion, we set the components of
x
to be equal to the middle
of the ranges specified by Iorgulescu
et al
. (2005):
p
...
=
1,
...
=
0.1
and
K
400.
Next, for each hour
t
, we calculate the sample vari-
ance
V
t
(
x
)of
D
t
(
x
),
=
,
D
t
(
x
). The 839
...
×
839 diagonal
matrix
V
INIT
x
,
V
839
(
x
)
is an estimate of the initial condition contribution to
the overall internal model discrepancy variance. To sim-
plify the discussion, we have chosen not to estimate the
off-diagonal covariance terms, setting them to be zero
instead. Figure 26.3 plots the standard deviations against
t. Notice that the effect of perturbing the initial condition
eventually decreases to a constant value.
We repeated the above perturbation exercise for a few
other fixed values of the inputs and discovered that the
pattern and magnitude of the initial condition contri-
bution was essentially the same, the biggest differences
occurring at essentially infeasible input combinations.
with diagonal elements
V
1
(
x
),
...
26.3.3.1 Initial condition contribution
First, consider the condition specified by Iorgulescu
et al
.
(2005) that the initial flow out of the slow groundwater
sub-compartment equals the observed initial discharge
and the other initial flows are all zero. This pattern implies
that the initial storage of water in each of the seven
subcompartments is zero except for the slow, ground
water sub-compartment (
ES
s
GW
(
t
26.3.3.2 RAIN contribution
We treat the forcing function RAIN similarly, except
we perturb RAIN(
t
) for each hour
t
, 839 and
also introduce a dependency between the perturbations
as follows. Write
=
1,
...
0)), which is chosen
to ensure that initial flow matches the observed flow.
This is not an unreasonable specification, as there was an
extensive dry period prior to the study. We will perturb
the initial slow groundwater content
ES
s
GW
(
t
=
ξ
(
t
)
=
log
η
(
t
), where the perturbation
is
η
(
t
)RAIN(
t
) and, as before, we assume E[
η
(
t
)]
=
1,
SD[
η
(
t
)]
=
p
and
ξ
(
t
) has a normal distribution with
p
2
) and variance
2
=
0), which
mean
+
p
2
), the same values for each hour
t
. We now need to
model the distribution of the collection
µ
=−
0
.
5 log(1
+
σ
=
log(1
we write as
w
.Wedosobyreplacing
w
by
η
w
, where
η
is a positive random quantity with expectation E[
η
]
=
1
η
(1),
...
,
η
(839)
or equivalently the collection
(839).
The simplest assumption would be to treat
ξ
(1),
...
,
ξ
and standard deviation SD[
p
corresponding to a
small percentage, such as 100
p
=
5%. Thus, E[
η
w
]
=
w
and SD[
η
]
=
the
pw
. We further assume, for convenience,
that
η
has a log-normal distribution; that is, log
η
has
a normal distribution with some mean
η
w
]
=
ξ
-collection as independent normal random quantities
and proceed as for the initial condition perturbation.
However, it makes sense to introduce a time dependency
which we do here by assuming the
µ
and variance
2
. It is reasonably straightforward to show that our
expectation and standard deviation conditions on
η
imply
that
ξ
σ
-collection to have
a multivariate normal distribution with a correlation
between
p
2
)and
2
p
2
). Thus,
µ
=−
.
+
σ
=
+
ξ
ξ
0
5 log(1
log(1
(
s
)and
(
t
) for any two hours
s
and
t
of
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