Geoscience Reference
In-Depth Information
can take advantage of the linearity assumption to apply it in an inverse sense to estimate a pattern of
effective rainfalls given a (previously separated) storm runoff hydrograph (see Box 4.2). The aim is then
to interpret this pattern to understand the nonlinearity of the catchment response to rainfall.
There have been earlier attempts to deal with the nonlinearity of the catchment response using transfer
function approaches. Early concerns about the linearity of hydrological responses resulted in a number
of attempts to formulate a nonlinear transfer function (e.g. Amorocho and Brandstetter, 1971; Diskin and
Boneh, 1973). These early attempts were based on the use of Volterra series (see also the more recent
studies of Ahsan and O'Connor, 1994, and Liang
et al.
, 1994). More recently, a new methodology based on
an extension of the class of generalised linear models to the nonlinear case using nonlinear autoregressive
moving average with exogenous inputs (NARMAX) models has been applied to hydrological problems.
Tabrizi
et al.
(1998) demonstrate how NARMAX models can be used for both single input and multiple
input cases.
Another ongoing issue in TFM modelling is the use of multiple rainfall input series in modelling
discharges, derived either from multiple raingauges in a catchment area or from radar rainfall data.
Multiple input transfer functions have been proposed, for example, by Liang
et al.
(1994), Tabrizi
et al.
(1998) and Kothyari and Singh (1999). A major problem with these approaches is the correlation to be
expected amongst the multiple inputs. In the general case, there may be no unique solution to the multiple-
input, single-output problem (Cooper and Wood, 1982) and a robust identification of the parameter values
may be difficult. One solution to this problem has been proposed by Cooper and Wood (1982) using
canonical correlation to identify an appropriate model structure and maximum likelihood estimation to
identify the required parameters. For a modern account of multiple-input transfer function models, see
the work of Young (2011a).
4.3.1 The IHACRES Model
The identification of unit hydrographs and component flows from rainfall, evaporation and streamflow
data (IHACRES) model of Jakeman
et al.
(1990) derives from the work of Young (1975) and Whitehead
et al.
(1979) , which attempted to avoid the problem of hydrograph separation in classical unit hydro-
graph models by relating total rainfall to total discharge. Recent developments have been the result of a
collaboration between the UK Institute of Hydrology (IH) at Wallingford and the Centre for Resource
and Environmental Studies (CRES) in Canberra, Australia, resulting in an IHACRES package for PC
computers. The model uses a particular set of functions to filter the rainfall to produce an effective rain-
fall that is then related to total discharge using a generalised linear transfer function. The rainfall filter
introduces a soil storage variable and, for longer period simulations, uses temperature as an index of
evapotranspiration. A number of different forms of rainfall filter have been used in different IHACRES
applications. One form is as follows (see also Croke and Jakeman, 2004). If the rainfall input at time step
t
is denoted as
R
t
, while the effective rainfall is denoted as
u
t
, then
u
t
=
R
t
(
S
t
+
S
t
−
1
)
/
2
(4.6)
1
−
S
t
−
1
1
τ
(
T
i
)
S
t
=
R
t
+
(4.7)
τ
(
T
i
)
=
τ
w
exp(10
f
−
T
i
f
)
(4.8)
where
S
t
is the storage variable at time
t
,
τ
(
T
i
) is a mean residence time for the soil storage depending on
mean daily temperature
T
i
,
c
controls the proportion of rainfall contributing to catchment storage,
τ
w
is
the mean residence time for soil storage at 10
o
C
and
f
is a scaling parameter to allow for the relationship
of evapotranspiration effects to this temperature difference. In many respects, this part of the IHACRES
model represents a simplified form of explicit soil moisture accounting (ESMA) model (see Section 2.4).
The effective rainfall,
u
t
then forms the input to a transfer function analysis based on the generalised
