Environmental Engineering Reference
In-Depth Information
Updated steady−state gain
Nominal steady state gain
1.1
1
0.9
0.8
1953.1
1953.15
1953.2
1953.25
1953.3
Date
600
95% confidence bounds
Updated model forecast
Measured flow
Nominal model forecast
400
200
0
1953.1
1953.15
1953.2
1953.25
1953.3
Date
Fig. 9.14
Leaf River example: the lower panel is a short part of Figure 9.13 showing more clearly the forecasting
performance. The changes in the continually updated estimate of the model steady-state gain (top panel) make
little difference to the model forecast (full-line) in comparison with that of the constant parameter nominal model
(dash-dot line).
straightforwardly from the changes in these esti-
mated transfer function parameters, and the same
estimation behaviour is reflected in the associated
changes in the partition percentages, shown in the
lower middle panel. Here, for clarity, the quick-
flow percentage is obtained by aggregating the
percentages of the two estimated quick-flow path-
ways. The overall steady-state gain, plotted in the
lower panel, also shows little change after the
learning period is complete.
Figure 9.13 presents a more detailed view of the
real-time updated forecasting over the first year,
showing estimated 95% confidence bounds and
running mean values of the coefficient of deter-
mination R
T
(Nash-Sutcliffe efficiency) for the
updated (blue line) and fixed parameter, nominal
model (red line) forecasts. The improvement is
very small because the nominal model clearly
provides a good representation of the rainfall-flow
dynamics over the whole of the time period and
there is little need for significant adaption: in
effect, both fixed and adaptivemodels are perform-
ing in a quite similarmanner overmost of the time
period.
Figure 9.14 presents a stillmore detailed viewof
the adaptive forecasting performance. The lower
panel is a short segment of Figure 9.13 showing
more clearly the effect that real-time model pa-
rameter updating has on the forecasting perfor-
mance. The estimated 95% confidence interval is
consistent with the forecasts and captures the
heteroscedastic behaviour of the forecasting er-
rors, except during the upward part of the hydro-
graph, where the flowmeasurement is sometimes
marginally outside this interval. This is a direct
consequence of the forecasting problems, men-
tioned previously, caused by the absence of any
advective time delay between the rainfall and flow
in this example: the forecasting system has no
prior warning of the rainfall and it is impossible
