Geoscience Reference
In-Depth Information
Uncertainty limits are related to the changes in the predicted variable in the parameter space or, more
precisely, if a predicted variable (rather than the performance measure) is represented as a surface in
the parameter space, to the
gradient
or slope of the surface with respect to changes in the different
parameter values. If the slope is steep then the uncertainty in the predictions is large. If the slope is quite
small, however, then the uncertainty is predicted as small since the predicted variable changes little if
the parameter is considered to be uncertain. Recalling Equation (7.1), the slopes are an indication of the
local sensitivity of the predictions to errors in the estimation of the parameter values.
7.7 Model Calibration Using Bayesian Statistical Methods
Forward uncertainty estimation can be a useful technique but it is essentially a form of sensitivity analysis
since the outputs depend totally on the assumptions about the prior distributions that represent the sources
of uncertainty. We might try to make those assumptions as realistic as possible, but very often there is
little information on which to justify either the form of a distribution or the values of its characteristic
parameters (the assumptions are themselves subject to epistemic uncertainties).
The whole process becomes much more interesting when there are some observations available to try
and condition the uncertainty estimation. In principle, the more observations that can be made avail-
able about the system response, the more we should be able to constrain the uncertainty in the model
predictions. As we will see in the following discussion, this is also generally the case in practice, but
not all the observations might be as informative as we might have hoped or expected. The observations
are themselves subject to error, are not always hydrologically consistent with other data and might be
incommensurate
. This is a technical term that describes the case when observed and predicted variables
might have the same names and appear to be comparable, but in fact the observations are at a different
time and space scale to the model predicted variable. Thus, a soil moisture measurement, for example,
is usually a point sample. A model might predict soil moisture, but as a bulk measure over some grid
element or hydrological response unit. The two have the same name but are not the same quantity. Similar
commensurability issues arise with model parameters. We can go into the field and measure hydraulic
conductivity at a point, stomatal resistance for a collection of leaves or the roughness coefficient for
a channel cross-section, but the model needs values at the scale of a grid element, vegetation stand or
channel reach. Again these are not commensurate quantities.
Thus, we cannot assume that every observation is equally informative in any analysis. But the ob-
servations do provide information that can be used both in model identification or calibration and in
constraining uncertainty in the predictions. This has led many hydrologists to adopt formal statistical
methods, where both parameter values and observations are treated as random variables and uncertainty
estimation is an intrinsic part of the analysis. In most cases, the methods adopted have been within a
Bayesian statistical framework. The Bayes equation (see Box 7.2) provides a formalism to combine
prior distributions that allow input of prior knowledge about the problem with a likelihood based on
the model predictions of the observations to form posterior distributions of parameters and model er-
rors that can be used to estimate the probability of predicting the next observation conditional on the
model. The equation derives from an article written by the Reverend Thomas Bayes (1702-1761) and
found amongst his papers after his death by his friend Richard Price. The paper was read at and pub-
lished by the Royal Society of London (Bayes, 1763) and formed the basis of statistical methods for some
200 years. The development of frequentist statistics was an attempt to provide a basis for statistics that did
not involve the subjectivity inherent in defining the prior distributions. More recently, however, Bayesian
methods have come to dominate statistical analysis again, primarily as a result of the more sophisticated
and powerful computer methods that can be brought to bear in estimating the posterior distribution and
resulting prediction uncertainties for nonlinear models.
