Environmental Engineering Reference
In-Depth Information
inadequacy function (e.g. Kennedy and
O'Hagan 2001) and a proper approach to such
a formulation should include the identification of
an appropriate statistical model. Where the resi-
duals have a non-Gaussian distribution it is also
possible to use Box-Cox or Meta-Gaussian trans-
forms so that the theory of Gaussian residuals can
be used (e.g. Montanari 2005; Beven et al. 2008).
This does not change the assumption that the final
series of residuals is purely random and that every
residual is informative. This may, in some cases,
be a useful approximation for practical applica-
tions but in hydrological models, when epistemic
uncertainties aswell as randomuncertaintiesmay
be significant, it will rarely be a correct approxi-
mation. Epistemic uncertainties are those associ-
ated with lack of knowledge. This term is
sometimes used in respect of model structural
error alone, but we should also expect epistemic
(non-random) uncertainty in model inputs and
boundary conditions. One result of epistemic
uncertainty is that different periods of observation
data (or, theoretically, different input error reali-
zations) will lead to quite different models having
apparently high likelihood (see Beven et al. 2008).
Since epistemic uncertainty is generic in real rain-
fall-runoff modelling applications, we might wish
to be more circumspect about over-conditioning
in these (quite normal) circumstances.
evaluation, a choice thatmay be justifiedwhen the
strong assumptions required are justified.
At the centre of this debate is the issue of the
information content of observations in informing
decisions about which models are better (have
higher likelihood) than others. Formal statistical
likelihood measures effectively assume that the
residual errors between models and observations
can be treated as random variables, or can be
transformed to a form inwhich they can be treated
as simple random variables. When this is the case,
then every residual can be treated as informative
in the model evaluation or conditioning process.
The longer the series of residuals, then the more
constrained should be the models with high like-
lihood so that, given enough residuals, equifinal-
ity should not be a problem. This argument is
made strongly, for example, by Mantovan and
Todini (2006) but the example used is a hypothet-
ical case where the model is known to be correct
and the residuals are constructed so as to have
simple form, and prior knowledge of that structure
is used in formulating the correct likelihood
function.
When hydrological models are applied to real
catchment data the residuals are not expected to be
of simple form and the model may not be identi-
fiable from the data. The residual series are often
highly structured, usually showing autocorrela-
tion, heteroscedasticity (variance that changes
with the magnitude of the prediction) and with
non-stationary bias because of either model struc-
tural effects or non-stationary bias in the inputs. In
this case, making simple statistical assumptions
about the errors can lead to unrealistic condition-
ing of the likelihood surface (i.e. stretching the
surface until only one small part of the model
space has high likelihood). An example of this is
provided by Thiemann et al. (2001). They assumed
that the residuals were Gaussian, of constant var-
iance and not autocorrelated - assumptions that
were obviously not correct for their application
(see discussion by Beven and Young 2003).
It is, of course, quite possible to define more
realistic statistical likelihood functions, by taking
account of spatial or temporal autocorrelation, or
heteroscedastic variance, or a simple bias ormodel
Model rejection issues
A further issue that arises in this debate has to do
with model rejection. When a hydrological or
hydraulic model is calibrated using statistical
likelihood methods, no model realization is ever
rejected. The fitting is always carried out condi-
tional on an assumption that the model is correct
and that the (perhaps transformed) residuals can be
treated as random. If the performance of amodel is
poor, relative to other models, in terms of the
likelihood measure used, then that model will be
given a low likelihood. The model with the high-
est relative likelihood, however, will survive even
if its performance is actually poor (though the
model as hypothesis might be reconsidered if
the predictions do not appear to be presentable).
