Environmental Engineering Reference
In-Depth Information
linear KF equations, which, as we have seen, are
clearly an alternative in this case because the only
nonlinearity in theHYMODmodel is the effective
rainfall nonlinearity at the input. Of course, in
other situations, where the selected model is large
and there are high levels of internal nonlinearity,
then computationally intensive algorithms, such
as the EnKF and the similar numerically intensive
methods, as discussed earlier (see Large and highly
nonlinear stochastic systems), provide a sensible
approach.
5 Finally, the continuing verification of the fore-
casting performance is important in practice. This
is aided by the adaptive nature of the recursive
algorithms employed here, where the user is con-
tinually informed of the updated parameters and
states and is able to assess performance. This can
be enhanced by the computation and presentation
of related performance measures: for example,
plots of statistics such as Root-Mean-Square Error
(RMSE), Mean Absolute Percentage Error (MAPE)
and other measures of skill against multiple fore-
cast lead-times.
One possibility is to estimate themodel directly in
the form used for forecasting, with the false time
delay included in the model and no instantaneous
term: this naturally reduces the explanatory power
of the model, in relation to the model
(Equations 9.7a and 9.7b), and makes it less real-
istic in physical terms, but it can sometimes im-
prove forecasting performance (Lees et al. 1994;
Young 2002). In this case, however, the overall
forecasting performance is not improved.
2 One limitation of the KF-based forecasting
scheme used in this example is that the forecast
does not take into account the uncertainty of the
parameter estimates, so that the uncertainty
bounds are theoretically a little too narrow. Al-
lowance for such uncertainty could be introduced
but the additional uncertainty is very small in the
case of the statistically efficient DBMmodel used
here, so it was not considered necessary.
3 Theoretically, the innovations sequence pro-
duced by the KF should be a zero mean, serially
uncorrelated white noise sequence. In the present
example, there is a just significant autocorrelation
of 0.19 at a lag of 1 day, which could be corrected by
adding a stochastic state variable to account for
this, based on an AutoRegressive AR(1) model.
However, the correlation is quite small and this
modification makes little difference to the fore-
casting performance. So, as in point 2, above, there
seems little reason to complicate the algorithm in
this case.
4 The computational cost of the adaptive forecast-
ing implementation described here is very small:
each daily state/parameter update takes only a few
microseconds on a standard desktop computer and
even this could be improved considerably by effi-
cient programming. In contrast, the numerically
intensive alternatives, such as the EnKF and PF,
are inherently more computationally expensive,
requiring a specified number of Monte Carlo rea-
lizations within each update ensemble: for exam-
ple, Moradkhani et al. (2005b) cite an ensemble of
40 realizations, each requiring solution of the
model equations. It is not clear, therefore, what
is gained by this additional computational load in
the present example because the EnKF only pro-
vides an approximate, numerical solution of the
Conclusions
This largely tutorial chapter has considered the
state of the art in the real-time updating of states
and parameters in flood forecasting models. Given
the enormous number of publications on this
topic, there has been no attempt to review all of
the available techniques that are currently avail-
able. Rather the chapter has concentrated on those
techniques that have come into prominence dur-
ing the last few years, and has addressed some
important topics raised by these developments,
including: the problem of model identifiability
and its effect on time-variable parameter estima-
tion; the relativemerits of joint and separate state-
parameter estimation in real-time updating; and
the choice between analytic or computationally
intensive methods of recursive state estimation
and forecasting.
The chapter concentrates on lumped parameter
models that can be used in the development of
quasi-distributed flood forecasting and warning
