Environmental Engineering Reference
In-Depth Information
2.5
2
Updated steady−state gain
Nominal steady state gain
1.5
1
1953.1
1953.15
1953.2
1953.25
1953.3
Date
1000
95% confidence bounds
Updated model forecast
Measured flow
Nominal model forecast
800
600
400
200
0
1953.1
1953.15
1953.2
1953.25
1953.3
Date
Fig. 9.15
Leaf River example: repeat of Figure 9.14 but with the measured flow doubled in magnitude so that the
nominalmodel nowhas a substantial steady state gain error. The continually updated gain estimate correctly identifies
the effective doubling of the gain and the adaptive forecast is better than the nominal model forecast, although
the difference is still not substantial.
for it to forecast the associated change in flow any
better than this unless a one-day-ahead rainfall
forecast is also available.
Finally, Figure 9.15 shows what happens if the
rainfall-flow dynamics are not captured well by
the nominal model. In particular, the flow mea-
surements are doubled in magnitude, so that the
nominal model gain is now considerably in error.
It is clear that the recursively estimated steady-
state gain has responded to the change in the flow
data and is about double the estimate in
Figure 9.14. Perhaps more surprisingly, at first
sight, the resulting difference in the adaptive and
nominal model forecasts is still not all that sig-
nificant: the adaptive forecast is better but the
differences are only visible around the peaks of
the hydrograph. This demonstrates the robustness
of the KF, particularly in this situation when the
observational errors are assumed to be low (with a
small associated hyper-parameter
b¼
0.01 relative
to the stochastic input hyper-parameter
d
q
¼
10,000), so that the output estimate is brought
back closely to the region of the flow measure-
ment at each corrective update.
Comments
1 Given the nature of the unit hydrograph plots of
Figure 9.11, with the absence of any advective
time delay and a significant instantaneous effect
of rainfall on flow, it is not surprising that the
immediate forecasts on the upward part of the
hydrograph, following a rainfall event, are poorer
than those on the recession part of the hydrograph.
