Environmental Engineering Reference
In-Depth Information
prediction step and linear updating in the correc-
tion step. A definitive account of the EnKF
appeared with the publication of Evensen's topic
on the subject (Evensen 2007), and the recent paper
by Clark et al. (2008) provides a comprehensive
and critical evaluation of both the EnKF and the
related Ensemble Square Root Filter (EnSRF)
when applied to the distributed hydrological mod-
el TopNet of the Wairau River basin in New Zeal-
and. There are various ways in which joint state/
TVP estimation can be carried out within an EnKF
framework, but a relevant one in the present con-
text is that suggested byMoradkhani et al. (2005b)
and tested on data from the Leaf River in the USA
(the same data as those used for the example given
in subsequent sections of the present chapter).
The basic implementation of the EnKF for state
estimation and forecasting is quite simple because
the correction step in the recursions is the same as
the standard KF. The Monte Carlo sampling and
ensemble averaging is only required in the predic-
tion step, which simply involves the computation
of the ensemble mean and its associated covari-
ance matrix, computed from the deviations of the
ensemble members from the mean (acting as a
surrogate for the true state, which is unknown, of
course). However, Moradkhani et al. develop a
'dual EnKF', which requires separate state-space
representation for the state variables and para-
meters through two linked algorithms (filters)
running in parallel. Here, the parameters are trea-
ted in a similar manner to the state variables, with
the parameters assumed to follow a stochastic RW
process, exactly the same as that used for the
implementation of recursive TVP estimation dis-
cussed above. However, the implementation of
the dual recursions could be accomplished in var-
ious ways, and this is not all that clear from the
description in the paper.
The EnKF results obtained with the Leaf River
data are promising but they suggest the need for
further research on the practical implications of
the filter in relation to real-time state/parameter
updating. For example, the flow forecasts appear
good but it may be that these are estimates rather
than one-day-ahead forecasts (see 'Comments'
following the illustrative example given below).
Hybrid continuous-discrete time updating
Although the previous subsections have assumed
that themodel is formulated in discrete-time, con-
tinuous-time models can be handled in a similar
manner. As regards state updating, the most obvi-
ous approach is to utilize the continuous-discrete
time formof the KF (see, e.g., Young 1984, p. 215 et
seq.). Here, themodel prediction step is carried out
in continuous-time, normally by the numerical
integration of the continuous-time model equa-
tions; while the correction step, which is likely to
involve only discrete-time sampled data, is re-
tained in the same form as Equations 9.4c
and 9.4d. This formulation has the additional ad-
vantage of allowing for irregularly sampled data.
Large and highly nonlinear stochastic systems
As we shall see later, normally the nonlinear rain-
fall-flow model can be decomposed into a serial
connection of an 'effective rainfall' input nonline-
arity feeding into a purely linear system that de-
fines the unit hydrograph properties of the system
(termed a 'Hammerstein' model in the Systems
literature). In this situation, the linear methods of
updatingoutlinedintheprevioussubsectioncanbe
amended simply for use in nonlinear rainfall-flow
modelling and flow forecasting. However, if the
nonlinearity is internal to themodel and themodel
is complex, then it is necessary to consider more
general methods that are not as restricted as the
modified linear procedures. Here, we will consider
briefly twosuchmethods thathavebeenutilized in
a hydrological context and are relevant to the il-
lustrative practical example described later: the
EnKF and the PF. These have been compared re-
cently byWeerts and Serafy (2006) when applied to
the conceptual rainfall-runoff model HBV-96 for
flood forecasting purposes. The related and rather
quaintly named Unscented Kalman Filter (UKF)
method is also discussed briefly.
The Ensemble Kalman Filter (EnKF)
The EnKF is an adaptation of the standard, analytic
KF algorithm to nonlinear systems using Monte
Carlo sampling and ensemble averaging in the
