Geoscience Reference
In-Depth Information
Vrugt
et al.
, 2003; Smith
et al.
, 2008b); and the Probability Distributed Moisture (PDM) and Grid to
Grid (G2G) models described in Section 6.2). The advantage of the IHACRES approach is that the data is
allowed to suggest the form of the transfer function used rather than specifying a fixed structure before-
hand. A recent software implementation of IHACRES (Classic Plus) is described by Croke
et al.
(2006)
and a freeware version can be downloaded from the Centre for Ecology and Hydrology (see Appendix A).
The IHACRES model has the right sort of functionality to reproduce hydrological responses at the
catchment scale with about the right number of parameters for those parameters to be identifiable given
a period of calibration data, at least for some environments. The parameters required to apply the model
are essentially the two time constants of the fast and slow pathways in the parallel transfer function,
the proportion of the effective rainfall following each pathway, and the
c, τ
w
and
f
parameters of the
effective rainfall filter. Jakeman and Hornberger (1993) suggest that these parameters can be considered
as “dynamic response characteristics” (DRCs) of a catchment and that it might be possible to relate these
DRCs to physical catchment descriptors to allow the prediction of the hydrological response of ungauged
catchments. This is discussed further in Chapter 10.
4.3.2 Data-Based Mechanistic Models Using Transfer Functions
The data-based mechanistic (DBM) approach of Young and Beven (1994), as far as possible, makes no
prior assumptions about the form of the model other than that a general linear transfer function approach
can be used to relate an effective rainfall input to the total discharge. In the spirit of letting the data
determine what the structure of the model should be, rather than making ad hoc prior decisions about
model structure, they make use of time variable parameter estimation to determine the form of the effective
rainfall nonlinearity. Their results suggest that the nonlinearity can often be approximated by the form
u
t
∝
Q
t
R
t
(4.9)
where
u
t
is the effective rainfall,
R
t
is the rainfall input,
Q
t
is discharge,
n
is a parameter and
t
is
time. Here, discharge is being used very much as a surrogate variable for the antecedent moisture status
of the catchment. In general, the measured discharge is the best readily available index of antecedent
conditions in a catchment but its use in this way does mean that discharge is being used in the prediction
of discharge. This is not a problem in model calibration at gauged sites; it is a problem in prediction or
real-time forecasting
but does not, in fact, turn out to be a difficult problem to overcome. In the original
application of Young and Beven (1991), the bilinear model (
n
=
1) was used, but more recent results
using time variable parameter estimation (see Box 4.3) have suggested values for
n
of between 0 and 1
(a bilinear power law model). The application of the DBM approach in this form is demonstrated in the
case study discussed in Section 4.4.
Young and Beven (1994) extended this idea to the case of a time variable analysis of the transfer
function gain to allow the data to suggest what the form of this nonlinear function should be. At that time,
this was achieved by tracking the gain on the transfer function over the simulation and then showing
that, when the system was forced by rainfall inputs, the gain showed a strong relationship with catchment
wetness (in that case, a power law function with discharge, see Section 4.5 and Box 4.3). Peter Young later
suggested filtering these time variable gains on the inputs (which define the “effective” rainfall) directly
in terms of the ordering of some indicator variable, rather than in time, in a state dependent parameter
(SDP) analysis (see Box 4.3). In this case, the indicator might be some index of wetness (such as the
discharge) and this approach has been used to define a nonlinear filtering in a number of applications of
the DBM approach (e.g. Young, 2000, 2002, 2003; McIntyre
et al.
, 2011), including flood forecasting
(see Section 8.3). Young (2011a) provides an extensive introduction to the theory and practice of transfer
function modelling techniques in both continuous and discrete time.
In a further application of these techniques to a longer period of data for a catchment in the
eastern USA, Young (2000) has demonstrated that there is a seasonal pattern in the time variable
