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(2000) on the observations showed that the catchment model based on TOPMODEL did not have the
right sort of filtering of the signal to reproduce the catchment behaviour. It was getting reasonably correct
results (given the uncertainties in the inputs) but not, it seems, for the right reasons. This might be because
of the particular mixing assumptions incorporated for this study but also because of the lack of a space
dimension in the TOPMODEL process representations, which Kirchner
et al.
(2001) suggested might be
important in explaining the mixing characteristics inferred from the chemistry (see also the discussion
in Chapter 9).
However, it is becoming increasingly possible to obtain more frequent measurements of concentrations
and this may gradually become less of an issue in the future (e.g. for environmental isotopes, see the
work of Berman
et al.
, 2009). It might then become more feasible to relax more of the simplifying
assumptions and consider the time variability in the residence time distributions. Botter
et al.
(2010) and
Rinaldo
et al.
(2011) have produced a framework that does allow the time variability in the residence
time distributions to be predicted as the catchment wets and dries. It remains to be seen how well this
dynamic residence time distribution formulation can reproduce the behaviour of different catchments.
Earlier work on transient residence time distributions was reported by Niemi (1977), Zuber (1986) and
Foussereau
et al.
(2001).
11.8.5 Residence Time Distributions and Rainfall-Runoff Model Calibration
There has been some discussion in the literature about how the use of environmental tracer information
might help in calibrating rainfall-runoff models so that we do get the right results for the right reasons.
There have also been a number of model applications that have been checked against both flow and
environmental tracer concentrations (Vache and McDonnell, 2006; McGuire
et al.
, 2007; the case studies
in Sections 11.7 and 11.9). The application of these models is also aimed at understanding the “paradox”
of Kirchner (2003) although, as noted earlier, this is perhaps not such a paradox after all.
However, nearly all the models used in simulating both flow and tracer concentrations might be getting
reasonable results for the wrong reasons. The storage models often use a complete mixing assumption for
the local concentrations and an instantaneous response of discharge to changes in storage regardless of the
length scale of the calculation element. With these assumptions, such models cannot adequately represent
the difference between the pressure wave celerity and the distribution of flow velocities (see Chapter 9).
They are relying on uncontrolled numerical dispersion interacting with the mixing assumptions to get
the right sorts of answer. Even distributed models, such as that of Sayama and McDonnell (2009), which
include some routing between elements, but with explicit time stepping, could be subject to solution error
if their numerical algorithms are not properly implemented. In that case, the tracer dispersion also relies
on numerical dispersion. We should be wary of inferences derived from models with numerics of this
type (Clark and Kavetski, 2010; Kavetski and Clark, 2010). The processes and storages involved in the
hydrograph response and tracer response are certainly closely linked but they are different.
11.9 Case Study: Predicting Tracer Transport at the Gardsj on
Catchment, Sweden
An interesting alternative to this storage-based approach has recently been suggested by Davies
et al.
(2011), although it is still very much in its early stages of testing (see also Section 9.6). This methodology
represents the water and associated tracer as discrete particles that move through the flow domain. Such
particle tracking models have been used widely in the past to predict the transport of tracer and solutes in
a specified velocity field, determined by a flow model, particularly in groundwater modelling where there
is interest in the movement of solutes over long periods of time. The USGS MODFLOW groundwater
