Environmental Engineering Reference
In-Depth Information
variable
z
k
. In this case, the SDP analysis then shows that,
paradoxically at first, the state dependency is in terms of
the measured flow variable (i.e.
z
k
=
y
k
: see later explana-
tion) and is limited to those parameters
b
i
,
i
accounts for that part of the output flow measurement
that is not explained by
x
(
t
k
); and the estimates of the
parameters are as follows, where the figures in parentheses
are the estimated standard errors:
,
m
associated with the rainfall input
r
k
(i.e. the parameters
a
i
,
i
=
1, 2,
···
n
associated with the past flow variables
are all constant over time). Moreover, the SDP nonlin-
earities can be factored out of the model, so that they
become a single nonlinear operation on the rainfall input
r
k
(referred to as a 'Hammerstein' model in the systems
literature).
In the second stage of the analysis, the nonparametric
(graphical) estimate of the SDP nonlinearity obtained at
the first stage is parameterized in the simplest manner
possible based on the shape of the nonparametric esti-
mate; in this case a power law in
y
k
is found to be sufficient.
As in previous DBM modelling of rainfall-flow behaviour,
the constant parameters of this parameterized nonlinear
model are then estimated using a nonlinear optimization
procedure. Here, the power law parameter
β
in equation
(7.2a), below, is optimized by exploiting the Matlab opti-
mization routine LSQNONLIN, with the parameters of
the linear transfer function estimated concurrently within
the optimization routine, using the continuous-time
modelling routine RIVCBJ in the CAPTAIN Toolbox.
At every optimization step, these RIVCBJ estimates are
based on the 'effective' rainfall, as defined by the latest
optimized update of the power-law parameters
=
1, 2,
···
a
=
0
.
4421(0
.
033)
;
a
2
=
0
.
0208(0
.
0052)
;
b
0
=
0
.
0596(0
.
0032)
;
b
1
=
0
.
02577(0
.
0069)
;
b
2
=
0
.
0208(0
.
0051)
(7.3)
ˆ
with
β
=
0
.
819(0
.
013). Note that the model (7.2a) can be
written in the following equivalent differential equation
form:
d
2
x
(
t
)
dt
2
+
0
.
4421
dx
(
t
)
dt
+
0
.
0208
x
(
t
)
0596
d
2
u
(
t
)
dt
2
2577
du
(
t
)
dt
=
0
.
+
0
.
+
0
.
0208
u
(
t
)
(7.4)
which describes the variation of the modelled flow out-
put
x
(
t
) in fully continuous time terms. In other words,
although the model (7.2a) is identified and estimated
from discrete-time, sampled data, it can be used com-
pletely in continuous-time terms following estimation.
The advantages of such a differential equation model are
discussed later.
The effective rainfall observation in (7.2c) shows that
the effective rainfall input variable is a nonlinear function
in which the measured rainfall is multiplied by the mea-
sured flow raised to a power
ˆ
, with the normalization
parameter
c
simply chosen so that the steady state gain
of the linear TF between the effective rainfall and flow
is unity.
3
In other words, the SDP analysis shows, in a
relatively objective manner, that the underlying dynamics
are predominantly linear but the overall response is made
nonlinear because of a very significant input nonlinearity.
The coefficient of determination
R
T
associated with
this deterministic output is defined by:
β
,and
the measured flow. This approach results in the follow-
ing nonlinear, continuous-time, SDP transfer function
(SDTF) model (e.g. Young, 2000):
Deterministic Output :
β
b
0
s
2
+
b
1
s
+
b
2
x
(
t
k
)
=
u
(
t
k
) (7.2a)
s
2
+
a
1
s
+
a
2
Noisy Output Observation :
y
(
t
k
)
∈
k
)
var
(
y
(
t
))
,
var
(
R
T
=
1
−
(7.5)
=
x
(
t
k
)
+
ξ
(
t
k
)
(7.2b)
Effectiove Rainfall Observation :
u
(
t
k
)
where
x
(
t
) and var(.)
denotes the variance of the enclosed variable. In this case
R
T
=
the
model
error
ε
k
=
y
(
t
)
−
y
(
t
k
)
β
·
=
c
·
r
(
t
k
)
(7.2c)
958: i.e. 95.8% of the variance of the measured
output
y
(
t
) is explained by the model output
x
(
t
). It
is clear from this high value of the
R
T
that this model
explains the data well. More importantly, it is well
validated on data not used in the identification and
estimation analysis, as shown in Figure 7.2,
R
T
=
0
.
where the argument
t
k
indicates the sampled value of an
underlying continuous time variable: for instance
y
(
t
k
)is
the variable
y
(
t
) sampled at the
k
th
sampling instant. In
this transfer-function model,
s
is the differential operator,
i.e
S
r
y
(
t
k
)
d
r
y
(
t
k
)
dt
r
y
(
t
k
);
(
t
k
) is output noise,
2
=
ξ
which
0
.
951
2
For theoretical and technical reasons (see Young, 2008) in the
RIVCBJ algorithm, this noise is modelled as a discrete-time autore-
gressive moving average (ARMA) process, with AR order 9 and
MA order 3; i.e. an ARMA(9,3) process.
3
This is an arbitrary decision in this case. However, if the rainfall
and flow are in the same units, then this ensures that the total
volume of effective rainfall is the same as the total flow volume.
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