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chain Monte Carlo (MCMC) analysis, sequential
importance sampling and sampling importance
resampling: see, for example, Moradkhani
et al. (2005a). The latter reference and the one by
Smithet al. (2006) are interesting because they also
analyze the Leaf River data, although only in an
estimation, rather than a forecasting sense.
The results obtained by Moradkhani et al. are
rather mixed: for example, some of the high flows
fall outside of the estimated uncertainty intervals
and there is high interaction between the estimat-
ed states and parameters, to the detriment of the
parameter estimates (also a common characteris-
tic of the EKF and perhaps an argument, once
again, for the separation of state and parameter
estimation). Similar comments to those made
about the EnKF in the last subsection apply to the
PF: namely, (i) it is not clear how well the PF
approach would work in the case of a large, highly
nonlinear model rather than the simple HYMOD
model used by Moradkhani et al.; and (ii) the
parameter estimation results show fairly slow
convergence and its time variable parameter track-
ing ability is questionable.
On the other hand, the results obtained by
Smith et al. make good sense and they allow the
authors to investigate the shortcomings in the
HYMODmodel structure. Also, the time variable
parameter tracking results seem quite reason-
able, although some of the estimated variations
are rather volatile when compared with the re-
cursive parameter estimates obtained in the ex-
ample given later in this chapter, using the same
data, and a similar complexity model.
Finally as regards the comparison of the EnKF
and PF, Weerts and Serafy (2006) (see earlier) con-
clude that: 'For low flows, [the] EnKF outperforms
both particle filters [the Sequential Importance
Resampling (SIR) filter; and Residual Resampling
filter (RR) variations], because it is less sensitive to
mis-specification of the model and uncertainties.'
Also the parameter estimation convergence is
fairly slowwhen compared to that of the recursive
RIV estimation algorithmapplied to the same data
(see later Fig. 9.12) and so some questions remain
about how useful an EnKF implementation, such
as this, would be in tracking time-variable para-
meters. Finally, the HYMOD model used in the
study is quite small and simple, so it is not clear
how well this EnKF approach would work in the
case of a large, highly nonlinear model for which
the EnKF is really intended (see Weerts and Ser-
afy 2006; Clark et al. 2008).
The particle filter (PF)
When interpreted in Bayesian terms, the KF can be
considered as a very special, analytically tractable
version of the general recursive Bayesian Filter, as
obtained when the state space model and obser-
vation equations are linear and the additive sto-
chastic disturbance inputs have Gaussian
amplitude distributions. The PF, on the other
hand, is a sequential, Monte Carlo-based approx-
imate mechanization of the prediction and correc-
tion stages of a fairly general recursive Bayesian
filter (Gordon et al. 2000; Doucet et al. 2001; Mor-
adkhani et al. 2005a; Smith et al. 2006) and so it
applies to general nonlinear models with nonlin-
ear observations and non-Gaussian stochastic dis-
turbances. In the PF, the underlying posterior
probability distribution function is represented by
a cloud of particles in the state space and the
samples automatically migrate to regions of high
posterior probability. Moreover, in theoretical
terms, convergence is not particularly sensitive
to the size of the state space.
On the basis of this description, the PF seems
extremely flexible and potentially very attractive.
As so often with general methods such as this,
however, there are practical drawbacks. It is natu-
rally very expensive in computational terms, and
practical restrictions on the number of particles
that canbe used in sampling the prior distributions
often lead to posterior distributions that are dom-
inated by only a fewparticles. This can introduce a
need for modifications, such as the use of techni-
ques that include residual resampling, Markov
The unscented Kalman filter (UKF)
TheUKF operates on the premise that it is easier to
approximate a Gaussian distribution than it is to
approximate an arbitrary nonlinear function (see,
