Environmental Engineering Reference
In-Depth Information
have access to a one-day-ahead rainfall forecast,
which is not available in this case. As a result, it is
necessary to change the sampling index on the
effective rainfall in Equation 9.7a to k
(see 'Parameter updating' above), configured spe-
cially for the state space model (Equations 9.7a
and 9.7b).
Finally, note that the separate parameter esti-
mation approach used here can be contrasted with
that used by Moradkhani et al. (2005b) and Smith
et al. (2006) in their research on the Leaf River,
where the state of the EnKF is extended to include
model parameters and both are estimated concur-
rently and interactively (see previous discussion
on this under 'Parameter updating' above).
1 and
ignore the second term in Equation 9.7b. In this
manner, the KF is producing a true one-day-ahead
forecast based only on the rainfall-flow measure-
ments available at the time the forecast is com-
puted. On the other hand, the parameter updating
is based on the nominal estimated model form,
without these changes, so ensuring that themodel
is always able to explain the data as well as
possible.
State updating by the Kalman Filter (KF)
Given the model (Equations 9.7a and 9.7b) and the
associated stochastic hyper-parameter definitions
in Equations 9.11 and 9.12, the KF described by
Equations 9.4a to 9.4f provides an obvious starting
point for the design of a real-time forecasting
engine. In order to utilize this, it is necessary to
quantify the various hyper-parameters that con-
trol the KF forecasting performance; namely, the
observation noise variance
s
Parameter updating by recursive RIV
estimation
The model (Equations 9.7a and 9.7b) constitutes
the 'nominal' model introduced in an earlier
section, with the state space matrices F
0
and G
0
defined as:
2
4
3
5
2
4
3
5
ð
9
:
13
Þ
f
11
000
0
f
22
00
00
f
33
f
34
0010
^
g
1
g
2
g
3
0
k
in Equation 9.11 and
the diagonal elements of the stochastic input co-
variance matrix Q in Equation 9.12. And if param-
eter updating is required, then the parameter
d
that defines the covariancematrix Q
for recursive
parameter estimation, as discussed above, is also
required.
Although all of these hyper-parameters could be
optimized, this was not attempted in this case so
that the ease of manual selection could be dem-
onstrated. The diagonal elements of Q were se-
lected on the basis of the empirically estimated
diagonal elements of the state variable covariance
matrix, computed over the same data as those used
for the nominal model estimation. These were
then normalized around the second state variance
and Q defined as follows:
F
0
¼
and
G
0
where the parameter estimates are those given in
Equation 9.10. In order to allow for parameter
updating, we need to introduce a capacity for
updating the parameters of this model as addition-
al daily rainfall-flow data arrive. This can be in-
troduced in various ways, as outlined earlier (see
'Parameter updating' above). Here, however, the
recursive form of the RIV algorithm used at the
nominal model parameter estimation stage is em-
ployed to update the parameters of the DBMtrans-
fer function model; the parameters in the state
space model are then obtained from these by
transformation, on a continuing basis. The recur-
sive RIV estimation could include the continual
updating of the parameter
c
in the SDP effective
rainfall nonlinearity (Equation 9.7c) but this did
not affect the forecasting ability very much and
was maintained at its nominal value. An alterna-
tive recursive estimation approach, in this case,
would be to use the Lin and Beck RPE algorithm
Q
¼ diag d
q
0:634
½
1:03:245
0
ð
9
:
14
Þ
where
d
q
is now the only hyper-parameter to be
determined.
At this point it is necessary to consider the
relative levels of uncertainty in the system. If we
assume that the random errors in the flow
