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discussion by Sorooshian and Gupta, 1995). The irregularity of the surface might be dependent both
on the choice of performance measure and other numerical issues. Again, different starting points for
a hill-climbing algorithm might lead to very different final values. Most such algorithms will find the
nearest local optimum, which may not be the global optimum. This is not just an example of mathematical
complexity: there may be good physical reasons why this might be so. If a model has components for
infiltration excess runoff production, saturation excess runoff production or subsurface stormflow (we
might expect more than two parameters in that case), then there are likely to be sets of parameters that
give a good fit to the hydrograph using the infiltration excess mechanism; sets that give a good fit using
a saturation excess mechanism; sets that give a good fit by a subsurface stormflow mechanism; and even
more sets that give a good fit by a mixture of all three processes (see Beven and Kirkby (1979) for an
example using the original TOPMODEL). The different local optima may then be in very different parts
of the parameter space.
The types of behaviour shown in Figure 7.2 can make finding the global optimum difficult, to say the
least. Most parameter optimisation problems involve more than two parameters. To get an impression of
the difficulties faced, try to imagine what a number of local optima would look like on a three-parameter
response surface; then on a four-parameter response surface .... Some advances have been made in
computer visualisation of higher dimensional response surfaces but trying to picture such a surface
soon becomes rather taxing for bears of very little brain (or even expert hydrological modellers). The
modern hill-climbing algorithms described in Section 7.4 are designed to be robust with respect to such
complexities of the response surface.
There is, however, also another way of approaching the problem by designing hydrological models to
avoid such calibration problems. A model could be structured, for example, to avoid the type of threshold
maximum storage capacity parameter that is activated only for a small number of time steps. Early work
on this type of approach in rainfall-runoff modelling was carried out by Richard Ibbitt using conceptual
ESMA-type models (Ibbitt and O'Donnell, 1971, 1974) and, as noted in Section 6.2, the PDM model
was originally formulated by Moore and Clarke (1981) with this in mind (see also Gupta and Sorooshian,
1983). Normally, of course, hydrological models are not designed in this way. The hydrological concepts
are given priority rather than the problems of parameter calibration, particularly in physics-based models.
However, for any model that is subject to calibration in this way, these considerations will be relevant.
Care should also be taken with the numerical methods used in solving the model equations. It has
been suggested that some complexities of the response surface might be induced only because of poor
approximations in solving the model equations (Kavetski
et al.
, 2006; Kavetski and Kuczera, 2007;
Kuczera
et al.
, 2010a; Kavetski and Clark, 2010; Schoups
et al.
, 2010).
There are particular problems in assessing the response surface and sensitivity of parameters in dis-
tributed and semi-distributed models, not least because of the very large number of parameter values
involved and the possibilities for parameter interaction in specifying distributed fields of parameters.
This will remain a difficulty for the foreseeable future and the only sensible strategy in calibrating dis-
tributed models would appear to be to insist that most, if not all, of the parameters are either fixed (perhaps
within some feasible range, as in Parkin
et al.
, 1996) or calibrated with respect to some distributed ob-
servations and not catchment discharge alone (as in the work of Franks
et al.
, 1998; and Lamb
et al.
,
1998b). Claims have been made that optimisation of many thousands of parameter values in a semi-
distributed model is possible (e.g. Arabi
et al.
2007; Whittaker
et al.
, 2010) but, if this is done only on the
information contained in observations on the outputs from a catchment area, it cannot be a well-posed
problem. Rather obviously, an increase in runoff generation in one part of the domain can be offset by a
decrease in another part (or vice versa) and it is impossible to tell which is the correct spatial representa-
tion. Thus it might be possible to obtain an
acceptable
fit to some set of observations, but this solution
might be neither optimal nor unique. This has been one reason why Beven (1993, 2006a) has argued
for hydrological modellers to consider the
equifinality
thesis: that many different model representations
(both parameter sets and model structures) might be found that demonstrate acceptable fits to any set of
