Geoscience Reference
In-Depth Information
It is important to note that this is a different convolution to the unit hydrograph convolution of
h
() in
Section 2.2 and transfer functions of Box 4.1. Here, the convolution represents the travel times of inputs
to the outlet rather than the hydrograph response. It is therefore representing the integrated effect of
velocities in the system rather than the (faster) celerities that control the hydrograph response. Following
Botter
et al.
(2010), we can write a similar general function for the loss of water to evapotranspiration as:
t
E
(
t
)
=
P
(
τ
)
g
(
t
−
τ
|
t
)
dτ
(11.5)
0
where
g
(
t
t
) is the residence time distribution for inputs to be lost from the system, again conditional
on
t
as evapotranspiration is also expected to vary with wetting and drying and the energy available to
drive the evapotranspiration process which varies with time on both daily and seasonal time scales. Both
f
(
t
−
τ
|
t
) can also be thought of as conditional probability distributions for particles of
water entering the system at time
τ
to reach the outlet at time
t
.
The difficulty in estimating these residence time distributions from input and output fluxes and con-
centrations is then readily apparent. We have no
a priori
knowledge of the form of the distributions
f
()
and
g
(), nor how they vary with time. It is also known that, even in simple cases where the distributions
can be assumed constant and linear, estimation from time series of inputs and outputs (a “deconvolution”
operation) is known to be poorly posed and highly sensitive to errors in the data series. In the case of
catchments, not only are the residence time distributions expected to vary with time, they may not be lin-
ear, and observations of both the inputs and output fluxes and concentrations may be subject to significant
uncertainty. In one sense, therefore, the identification of
f
() and
g
() is an impossible problem. However,
significant insights into how the catchment system is responding can be obtained if some assumptions
are made about the nature of the distributions in order to simplify the identification. More detail on the
estimation of residence time distributions is given in Box 11.2.
−
τ
|
t
) and
g
(
t
−
τ
|
11.8.1 Which Residence Time Distribution?
As defined above, a residence time distribution is a summary of the times that water molecules have spent
within the flow domain of interest. There are, however, a number of different residence time distributions
that are of interest (Rinaldo
et al.
2011). Consider a simple watertight catchment system in which the
rainfall inputs become either discharge or evapotranspiration. Then a given increment of rainfall will
have a
residence time
(or
travel
(
transit
)
time
) distribution in the system that is different for discharge and
evapotranspiration fluxes. A given increment of discharge will have a residence time (or travel (transit)
time) distribution that (except for some very special circumstances) is different from the residence time
distribution for rainfall increments (and the same is true for an increment of evapotranspiration). These
distributions will also be different from the residence time distribution for water stored in the system at
any time (travel (transit) time distributions are generally used only for input or output fluxes and not for
storage). Thus, in referring to the residence time distribution for a catchment, it is necessary to be clear
about what is actually meant. A theorem of Niemi (1977) can be used to define the relationship between
these different residence times analytically under certain simplifying assumptions (Botter
et al.
, 2010;
Rinaldo
et al.
, 2011). In the general time variable case, if the fraction of rainfall reaching the discharge
outlet at time
t
is
θ
t
then
f
(
t
−
τ
|
t
)
Q
t
=
P
t
θ
t
f
(
t
−
τ
|
t
)
(11.6)
where
f
(
t
t
) is the residence time distribution for water making up the increment of the discharge
at time
t
. The input and output flux increment residence time distributions can only be the same if the
system is completely dry before the increment of rainfall input (so that no displacement of pre-event
water forms the discharge) or if the system is at steady state (with linear partitioning of inputs between
discharge and evapotranspiration (
θ
t
constant) such that all increments of input have the same travel
−
τ
|
