Geology Reference
In-Depth Information
this approach. It is simply not easy to make dis-
tributional assumptions about model parameters,
even for models that can be interpreted physi-
cally. One reason is that what is required by the
model are effective values of parameters, which
might be different from, or might not be com-
mensurate with, values that could be measured
in the field due to scale effects (see Chapter 5),
non-linearities and non-stationarities (Beven,
2006a). It is then even more difficult to make
assumptions about the co-variation to be expected
between different effective parameters. We often
suspect that parameters might interact within
a model structure, but defining that interaction
a priori
will be difficult. Neglecting such
co-variation will overestimate the output uncer-
tainty. If the interaction can be defined, however,
there are simple ways of generating co-varying
parameter values, such as the use of copula sam-
pling (again, see Beven, 2009).
The results of any forward uncertainty analy-
sis will also be conditional on the decisions made
about model structure and input data. Where
there are no evaluation data available this will
have to be accepted, but it is far better if such
data can be used at least to check if the resulting
predictions are within the range of the observa-
tions. Experience suggests that this will not
always be the case.
variation needs to be considered (e.g. Kennedy &
O'Hagan, 2001).
In the simplest possible case, for a model that
performs very well and for observations that are
not biased in any way, then checking for struc-
ture in the residuals will show that they are inde-
pendent and identically distributed and can be
represented by a Gaussian distribution with zero
mean and constant variance. This then implies
that a simple Gaussian likelihood function can be
used to evaluate the predictions of any parameter
set in the model. Within a formal Bayesian frame-
work, Bayes' equation is then used to update the
parameter estimates as more data are added in
the form:
()
()
()
o
P
qµq q
P
L
O
O
(4.2)
or in words: the posterior probability of a param-
eter set conditional on a set of observations
O
,
P(
|
O
), is proportional to the product of the
assumed prior distribution of the parameters,
P
o
(
θ
), and the likelihood of predicting the observa-
tions with the model conditional on the parame-
ters, L(
O
|
θ
). The Bayes approach allows for the
subjective choice of the prior distribution of
parameters (and their co-variation) that is required
for a forward uncertainty analysis, but applica-
tion of Bayes' equation will normally reduce the
uncertainty in the predictions and make the pos-
terior parameter distributions better defined as
more data are added. In fact, it can be used iteratively
as each new set of observations is added, the pos-
terior distribution up to now becoming the prior
distribution for the additional conditioning pro-
vided by the new observations. Where we are not
too sure about parameter distributions in starting
off the process, we can choose non-informative
prior distributions.
There are two important issues in applying
Bayesian uncertainty estimation. The first is
choosing an appropriate likelihood function in
Equation 4.2; the second is integrating that likeli-
hood over the parameter space which generally
has to be done numerically. The difficulty in
choosing a likelihood function arises because
θ
4.2.2
Bayesian uncertainty estimation
Where some evaluation data are available then
it might be possible to use a formal statistical
framework to evaluate predictive uncertainty.
This requires assumptions about the nature of
the residual model error,
ε
r
in
OM
x, t
=
( )
q
+
e
(4.1)
x,t
r
where
O
is some observation,
M
(
) is the equiva-
lent output from a model with parameter vector
θ
θ
,
x
are space coordinates and
t
is time. In particu-
lar, it is necessary to assume that a model of the
residual errors can be found such that, having taken
out any structure in the residuals, only random
