Geoscience Reference
In-Depth Information
Figure B8.1.1
Comparison of adaptive and non-adaptive five-hour-ahead flood forecasts on the River
Nith (after Lees et al., 1994, with kind permission of John Wiley and Sons).
where
e
t
is the scaled prediction error
y
t
−
y
t
1
e
t
=
(B8.1.9)
+
P
t
y
2
t
n
can be used to estimate prediction limits for the forecasts based on the
uncertainty in the time variable estimates of
The variance of
Further
details of this algorithm, and its extension to recursive updating of multiple parameters, is
provided by Young (2011a).
Note that in the predictor-corrector equations, only the error
y
t
−
y
t
at the current time
G
t
+
n
as reflected in the magnitude of
P
t
+
n
.
t
can be known, but that this can be fixed in extrapolating the estimates of
G
and
P
forwards using
the predictor-corrector equations to time
. In general, the magnitude of the prediction
limits increase with increasing discharge and with extrapolation further into the future. In
the Dumfries forecasting system described by Lees
et al.
(1994) it was found that reasonable
forecasts could be made for five hours ahead (see Figure B8.1.1), even though the natural time
delay in the river network was of the order of three hours.
The time variable gain in this type of model is a way of compensating for both data errors and
any nonlinearities that are not properly represented in the model structure. Adaptive methods
can also be used with nonlinear models (Beven, 2009). Time variable parameter estimation
can also be used to investigate the nature of the nonlinearities of an input-output system and
is the basis of the state-dependent parameter approach to model identification (see Box 4.3
and Young, 2011a).
t
+
n
