Geoscience Reference
In-Depth Information
they can be represented as a stochastic model, then the result is generally to overestimate the information
content in a set of observation data. This is because epistemic errors are expected to be non-stationary in
nature (varying over time) so that the assumption of an error model of constant form and parameters does
not hold. Ignoring the nonstationarity of error characteristics also leads to biased parameter inference
and certainly over-conditioning in the sense of an underestimation of parameter variances.
This suggests that a more flexible and application-oriented approach to model calibration is required.
There are certainly many other performance measures that could be used that are not related to specific
statistical assumptions about the errors. Some examples, for the prediction of single variables, such as
discharges in hydrograph simulation, are given in Box 7.1. It may also be necessary to combine goodness-
of-fit measures for more than one variable, for example both discharge and one or more predictions of
observed water table level. Again a number of different ways of combining information are available; some
examples are given in Box 7.2. Some of the more interesting recent developments are based on a set theo-
retic approach to model calibration (see Section 7.9). However, something is also lost in using these other
models: the theoretical advantage that it is possible to estimate the probability of a new observation condi-
tional on a model. However, this advantage
only
holds when all the assumptions required can be justified.
7.8 Dealing with Input Uncertainty in a Bayesian Framework
There has been an important development in the last decade in trying to allow for input uncertainties
within the Bayesian statistical framework. It uses a hierarchical Bayes approach in which the model
inputs, parameters and observations are all treated as random (aleatory) variables. This recognises that,
in general, our information about the true inputs to a catchment area during a time step or a rainfall
or snowmelt event are generally poor because of the limitations of both raingauge and radar rainfall
techniques. The nature of the errors might, however, vary from event to event, and an obvious way of
dealing with this type of variability is to treat the error in the inputs as a multiplier, drawn from a random
distribution (equivalent to an additive error on the logs of the rainfalls). The prior expectation is that the
multiplier is positive, centred on a value of one and with a moderate variance.
Estimation of the multipliers for each event then becomes part of the Bayesian calibration process.
Each event requires the estimation of an appropriate multiplier which now serves as a parameter. Where
there are many events in the calibration data, there are often many more multiplier parameters than model
parameters to be identified, but this can be done sequentially event by event since the multiplier for an
event cannot have an effect on the predictions from previous events (though the value of a multiplier can
have an effect on subsequent events, albeit gradually dying away). The results of this type of calibration
methodology can lead to very good results in model calibration, with accurate predictions of discharge,
small predictive uncertainties and very well-defined model parameter values (Thyer
et al.
, 2009; Renard
et al.
, 2010; Vrugt
et al.
, 2009). The users claim that this provides an objective way of separating sources
of error in the modelling process.
I do not believe that this claim is justified. The good results are certainly, in part, a result of compensating
for the deficiencies in the estimation of catchment inputs but we should be very careful not to believe
that the resulting estimates of corrected inputs represent the true inputs. This is because the rainfall
multipliers are correcting not only for errors in estimating the true inputs but also for deficiencies in the
model structure. For example, it is very often the case that hydrological models are poor at estimating
observed discharges during the first few events of the wetting-up period after a long dry period. Let us
say that a model underestimates the runoff generation in the first event (this is not uncommon in this
situation); this can be corrected in calibration by increasing the rainfall multiplier. That, however, also has
the effect of putting more water into storage prior to the next event. The model might then overestimate
the runoff generation for the second event, which is corrected by decreasing the event multiplier for the
second event, and so on.
