Environmental Engineering Reference
In-Depth Information
look briefly at a real example, where the para-
meters of a daily rainfall-flow model are being
updated each day, in real time, using a recursive
estimation algorithm. The exact nature of these
data, the associated model and the method of
recursive estimation are not important at this
time, sincewe are concerned onlywith the general
consequences of over-parameterization and poor
identifiability on real-time updating. However, in
order to place the results in context, they relate to
the analysis of daily effective rainfall-flow series
fromthe Leaf River inMississippi, USA, a segment
of which is shown in Figure 9.3, where the effec-
tive rainfall is the rainfall adjusted nonlinearly to
account for catchment storage effects. These se-
ries are part of a dataset that is used in the next
section, which presents an illustrative example of
forecasting and real-time state/parameter updat-
ing. Consequently, the identifiability results pre-
sented here are relevant to the selection of the
rainfall-flow model used in this subsequent
example.
Figure 9.4 compares the recursive estimates of
all the assumed time-invariant parameters, as ob-
tained over several years (a total of 1681 days) of
data. The results for a reasonably identifiable,
third-order model, are shown in the lefthand pan-
el; and those for an over-parameterized and poorly
identifiable seventh-order model, are plotted in
the righthand panel. We see that the recursive
estimates in the lefthand panel converge rapidly
and, although they fluctuate to some degree, as
one would expect with estimation from noisy
data, this is far less than that encountered in the
righthand panel. In order to examine this more
closely, Figure 9.5 shows the estimation results for
a typical parameter in each model, where the
recursively updated estimate is displayed, togeth-
er with its estimated standard error bounds. This
illustrates how the estimated parameter in the
higher ordermodel not only fluctuatesmuchmore
widely, but also has much wider uncertainty
bounds.
In spite of the extremely volatile behaviour seen
in the righthand panels of Figures 9.4 and 9.5,
however, the model defined by the final estimates
of the parameters explains the flow data to
is poorly identifiable, but also it can affect the
identifiability by attenuating the information con-
tent in the data.
4 Prior knowledge about model structure and pa-
rameter values: While it is clear that exact prior
information on the model structure, as well as the
value of certain parameters that characterize this
structure, can enhance the model identifiability
by reducing the number of parameters that have to
be estimated, it is also clear that such a desirable
situation never exists in the real world. For in-
stance, the majority of assumed known para-
meters in such models have to be defined from a
priori information, such as soil types and vegeta-
tion cover. But, unlike the situation in other areas
of engineering, catchment hydrology modellers
are not dealing with a well-defined 'man-made'
system, so that the relevance of such parameters
within the assumedmodel structure is often ques-
tionable. Consequently, the parameters normally
have to be adjusted fromtheir 'measured' values in
order that the model is able to explain the data
satisfactorily. This implies that they are not 'well
known' when utilized in this modelling context
and so, instead of being constrained to fixed va-
lues, they need to be considered as inherently
uncertain.
Fortunately, with the increasing recognition of
these inherent problems and the advent of com-
puter-based, numerical stochastic techniques,
such as MCS and sensitivity analysis, it has be-
come increasingly common for hydrological mod-
ellers to assume that both themodel structure and
the associated parameters are uncertain and so
defined by probability distributions, rather than
point values (Beven2009). Evenwith thiswelcome
development, however, the problem of identifia-
bility remains: as we see in the following illustra-
tive example, if the model is over-parameterized,
then the assumption of uncertainty in the para-
meters can tend to conceal rather than cure any
underlying lack of identifiability.
An illustrative example of identifiability
In order to illustrate the concept of identifiability
and its importance in real-time updating, let us
