Environmental Engineering Reference
In-Depth Information
1
0.9
95% confidence bounds
Updated model forecast
Measured flow
Nominal model forecast
R
T
0.8
2
, Updated model forecast
0.7
2
, Nominal model forecast
R
T
0.6
0.5
0.4
0.3
0.2
0.1
0
1952.6
1952.7
1952.8
1952.9
1953
1953.1
1953.2
1953.3
1953.4
1953.5
Date
Fig. 9.13
Leaf River example: more detailed view of the real-time updated forecasting over the first year showing
estimated 95% confidence bounds and running mean R
T
values for updated and fixed model forecasts, based on
the innovation errors. (See the colour version of this Figure in Colour Plate section.)
numerator coefficients in the transfer function,
with the denominator coefficients maintained at
their nominal estimated values. To ensure this,
the initial covariance matrix for the recursive RIV
parameter estimation P
;0
, is set to reflect some
considerable uncertainty in the numerator para-
meters but no uncertainty in the denominator
parameters (i.e. the relevant elements of P
;0
are
set to zero). In effect, this is informing the algo-
rithm that we are confident in the constrained
eigenvalues of the nominal model and the associ-
ated residence times of the flow pathways, but we
are not sure that the steady-state gains and the
consequent partition percentages of flow in these
pathways will not change over time.
Figure 9.12 shows three years of real-time up-
dating following initiation after 50 days. Here,
since the initial covariance matrix for the recur-
sive RIV parameter estimation is set to reflect
some considerable uncertainty in the parameters,
the estimates are rather volatile when the first
large rainfall and flow events occur. In particular,
the recursive estimates of the four updated model
parameters (i.e. the coefficients of the numerator
polynomial in the transfer function), as plotted in
the upper middle panel, vary quite a lot while the
RIV estimation algorithm is 'learning' the model
parameters from the rainfall-flow data. However,
after this is completed early in 1953, they then
settle down to become fairly stable when suffi-
cient information has been processed to engender
confidence in the estimates. Note that the asso-
ciated changes in the parameters of the state
space model
(Equation 9.7a) can be inferred
