Geoscience Reference
In-Depth Information
that is, a quadratic equation in
a
. For cases with real roots, this may be solved using the
standard quadratic formula (
x
4
AC
/
2
A
with
A
±
√
B
2
=
−
B
−
=
1
,B
=−
a
1
,C
=−
a
2
). The
two solutions will be the two coefficients,
a
and
a
.
Once
a
and
a
are known, for a parallel model (
m
2) the relative amounts of flow pass-
ing through each component can also then be determined since it is easy to show from the
expressions
b
o
=
=
b
and
b
1
=−
b
a
+
b
a
that
b
+
b
=−
b
1
+
a
b
o
/
a
+
a
b
=
b
b
o
−
Factorisation is also sometimes possible for higher order models with real roots, but the
number of possible combinations of serial and parallel connections increases rapidly with
model order.
B4.1.5 Determination of the Time Constant for a First-Order Model Component
By setting the impulse response of the continuous time and discrete time models to be equiva-
lent, it can be shown that the
a
coefficient of a first-order model is related to the mean residence
time of the continuous time equivalent approximately as
T
=−
t/
ln(
a
)
(B4.1.21)
where
t
is the time step of the discrete time model. The mean residence time, with units of
time, has a direct physical interpretation. Young and Beven (1991), for example, demonstrate
that for a discrete time model fitted to hourly data for the CI6 catchment at Llyn Briane,
Wales, a transfer function model suggests two parallel pathways (see Section 4.3), with mean
residence times of 3.1 hours and 66.4 hours, though it must be remembered that there will be
some uncertainty associated with these values. Determination of the
b
coefficients,
b
and
b
for the two components suggests that some 28% of the effective rainfall takes the faster
pathway in the parallel model and the remaining 72% the slower pathway. The fit of the model
for this case is shown in Figure 4.8 (
R
t
0
.
991). This application used the bilinear power filter
applied to the total rainfall to create an effective rainfall series for use as an input to the transfer
function modelling.
This box has described the discrete time step form of the general linear transfer function since
in hydrology both input and output data are generally specified for discrete time increments.
Where the sampling time step is short relative to the response time of the catchment, it might
be more appropriate to use the continuous time form. Young (2011b) discusses the background
to continuous time transfer functions. Routines for fitting both discrete and continuous forms
are included in the MATLAB CAPTAIN Toolbox (Taylor
et al.
, 2007; see Appendix A).
=
B4.1.6 Summary of Model Assumptions
The assumptions made in applying the generalised linear transfer functionmodel are as follows:
A1 The transfer function uses an effective input (such as a suitably transformed
effective rain-
fall
) to predict an output such as stream discharge, although differences in total volumes of
input and output can be scaled by the calibrated coefficients of the numerator (
b
o
,b
1
,...
).
A2 A transfer function is made up of one or more linear storage elements with different time
constants combined in series or parallel, choosing the simplest structure that is consistent
with the observations.
