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time distribution through the system). This is not generally the case, so that input flux residence time or
travel time distributions are different from output flux residence time or travel time distributions; they are
different from the residence time distributions of storage in the catchment; and they all change over time.
As we see in the following sections, it is often a convenient assumption that residence time distributions
do not change over time but this is, in fact, an assumption difficult to justify.
11.8.2
Introducing Some Simplifying Assumptions
Where long time series of input and output fluxes and concentrations are available we might be able to
ignore the variations in residence time distributions over time and look only at the dominant filtering of
the input time variability in producing the outputs by assuming that the distribution
f
() is a constant. The
analysis is further simplified by assuming that the mean flux rate through the storage is a constant. This
means that variations in flux can be ignored so that only the input and output concentration time series
are needed to determine a residence time distribution. It has been argued that this might be an adequate
simplification if the variation in storage in the catchment over time is small relative to the total volume
of active storage (McGuire and McDonnell, 2006), but this is also an argument from quasi-steady-state
principles. It may not be a compelling argument when small changes in storage result in very large
changes in the fluxes and velocities of discharge and tracer. The importance of this has perhaps been
neglected in the past because, as well as the convenience in simplifying the analysis, there have been
very few datasets that have adequately characterised the mass flux of tracer during storm periods.
The identification of a residence time distribution is simpler still if we can further assume a parametric
form for the residence time distribution. Then only a small number of parameters need to be estimated.
A number of different distributions have been used in the literature including the exponential and gamma
distributions (see Box 11.2). From a fitted distribution, characteristics such as the mean travel time of
the tracer in the system can be estimated. Although the value derived is affected by the simplifying
assumptions that have been made, it may still be a useful comparative measure across catchments, giving
an indication of the nature of the hydrological response. Indeed, it may be possible to get a good estimate
of these mean travel times from catchment characteristics (McGuire
et al.
, 2005; Soulsby
et al.
, 2010;
Lyon
et al.
, 2010). This can give some idea of the residence times to be expected in ungauged catchments,
albeit with significant uncertainty (Figure 11.7).
Figure 11.7
A comparison of mean travel times estimated from observations (MTT
GM
) and those estimated
from catchment characteristics (MTT
CC
) for catchments in Scotland (from Soulsby et al., 2010, with kind
permission of John Wiley and Sons). Error bars represent 5 and 95% percentiles.
