Environmental Engineering Reference
In-Depth Information
good introduction to the EKFand relatednonlinear,
minimum variance, state estimation filters, in-
cluding the iterated EKF, is given in Gelb (1974,
p. 182 et seq.), and a tutorial example in a hydro-
logical setting is given in Young (1984, p. 215 et
seq.). In the EKF, parameter updating is carried out
within a single, nonlinear, state space model, with
the parameters considered as adjoined state vari-
ables described by simple state equations, such as
the randomwalk (see 'Parameter updating' below);
and this nonlinear model is then 'relinearized' at
each recursive update.
The EKF is a simple, approximate solution to
the optimal, nonlinear estimation and filtering
problem, the general solution to which is infinite
dimensional (Kushner 1967). Unfortunately, as an
approximation, the EKF has various limitations,
such as problems with covariance estimation and
convergence for multi-dimensional, nonlinear
models. As Julier et al. (2000) conclude:
optimal Refined Instrumental Variable (RIV) algo-
rithm, which is used in the illustrative example
later in this chapter (Young 1976, 2008; Young and
Jakeman 1979); the Recursive Prediction Error
(RPE) algorithms of Ljung and Soderstrom (1983)
and Stigter and Beck (2004); the Bayesian Recur-
sive Estimator (BaRE) of Thiemann et al. (2001);
and DYNamic Identifiability Analysis (DYNIA)
proposed by Wagener et al. (2003).
Examples of recursive algorithms that include
the use of MCS methods and exploit ensemble
averaging are the Ensemble Kalman Filter (EnKF:
e.g. Evensen 2007; Vrugt et al. 2005); the Particle
Filter (PF: e.g. Gordon et al. 1993; Moradkhani
et al. 2005a; Smith et al. 2006); and the Unscented
Kalman Filter (UKF: e.g. Julier et al. 2000), all of
which are discussed under 'Large and highly non-
linear stochastic systems' below. Note also that
there are other 'Variational Data Assimilation'
methods that have been used mainly in weather
forecasting: these are not considered in this chap-
ter but a comparison of the '4DVar' variational
method and the EnKF, within a hydrological fore-
casting context, is available in Seo et al. (2003,
2009).
The KF is certainly the most famous recursive
algorithm and it has received much attention in
various areas of engineering, science and social
science. Within the present context, its main
limitation is that it only estimates and forecasts
the state variables in the system, under the as-
sumption that the parameters of the state space
model are known exactly. As Kalman (1960)
pointed out:
Although the EKF (in its many forms) is a widely
used filtering strategy, over 30 years of experience
with it
...
has led to a general consensus that it is
difficult to implement, difficult to tune, and only
reliable for systems that are almost linear on the
time scale of the update intervals structure.
Not surprisingly, other approximate solutions
are possible and some of these are discussed in the
earlybut very influential bookby Jazwinski (1970).
However, such algorithms are normally based on
higher order expansions and, while often theoret-
ically superior, they do not possess the attractive,
practical simplicity of the KF and EKF. It is for this
reason that subsequent research in this area has
taken the MCS-based route mentioned above.
This is exemplified by the EnKF algorithm, which
can be considered as a computationally more
expensive but algorithmically simpler and more
robust alternative to the EKF.
Within the specific flood forecasting context,
Refsgaard (1997) gives a review of different updat-
ing techniques used in real-time flood forecasting
systems, aswell as the comparison of two different
updating procedures applied to a conceptual hy-
drological model of a catchment,
...
[it is] convenient to start with the model and
regard the problemof obtaining themodel itself as
a separate question. To be sure, the two problems
should be optimized jointly if possible; the author
is not aware, however, of any study of the joint
optimization problem.
This challengewas taken up quickly and several
authors suggested approaches to recursive param-
eter estimation. Perhaps the best-known outcome
of this research effort is the ExtendedKalmanFilter
(EKF), first suggested by Kopp and Orford (1963). A
including
