Geoscience Reference
In-Depth Information
Box 11.2 Analysing Residence Times in Catchment Systems
The analysis of residence times of water in catchment systems has received much attention
in the last decade, in part because of the improvements of measurement techniques for envi-
ronmental tracers in terms of both cost and reliability, and in part because of the additional
information provided about flow sources and pathways in understanding how the catchment
works. The presentation here builds upon a number of review papers where more detail may
be found (see particularly Kendall and McDonnell, 1998; Ozyurt and Bayari, 2005a, 2005b;
McGuire and McDonnell, 2006).
As discussed in the main text of Chapter 11, difficulties of adequate observations and the
strong simplifying assumptions required in the analysis of residence times remain. The interpre-
tation of the results must bear these limitations in mind. Sources of uncertainty in the analysis
should be considered as a matter of good practice.
The first and most important requirement for a successful analysis is that there needs to be a
clear signal, a difference in concentrations between inputs and outputs. This is quite obvious
for the simple mixing models of Section 11.3, for which the calculation cannot be carried out
if there is no difference in concentration between input and output. The same holds for longer
term input and output signals. In this case, certainly for the isotopes of the water molecule,
we expect the variability in the isotope signal in the input to be smoothed out in the output
signal. The catchment acts as a low-pass filter for the higher frequency variations in the input.
However, if there is no variation at all in the output, we also cannot infer much about the
processes since the input variability has then been integrated out completely. We can say that
there must be a very large effective storage in the catchment relative to the input fluxes but
the mean residence time of the storage would be indeterminate. Thus, the ideal case is to have
some structured variability in the input that can still be distinguished as a signal in the output
after the smoothing in the catchment.
B11.2.1 Forms of Residence Time Distribution
Analysis of residence time distributions has classically been based on treating the output con-
centrations as a convolution of the input concentrations and the residence time distribution
such that:
t
C
Q
(
t
)
=
C
P
(
)
f
(
t
−
)
d
(B11.2.1)
0
where
C
Q
(
t
) is the sequence of output concentrations,
C
P
(
t
) is the sequence of input con-
centrations and
f
(
t
) is the residence time distribution. Note that this form is not as general as
Equation (B11.1.2) in that it assumes linearity and stationarity of the residence time distribution
in relating the inputs to the outputs.
Given an identification of the residence time distribution
f
(
t
), the mean residence time
T
can then be calculated as:
0
tf
(
t
)
dt
T
=
0
(B11.2.2)
f
(
t
)
dt
The form of residence time distribution of water in a catchment is not known
a priori
.Inone
of the first analyses of this type, Eriksson (1971) argued for an exponential distribution:
1
K
f
(
t
)
=
exp (
−
t/K
)
(B11.2.3)
where the parameter
K
has units of time and is equal to the mean residence time. As shown
in Box 4.1, this is the output from a simple well-mixed linear store, or what is known in the
