Geoscience Reference
In-Depth Information
estimates of the transfer function
a
parameter (and hence in the mean residence time) as well as a
similar correlation of the time variable estimates of the gain parameter
b
with discharge. It was found
that the seasonal pattern in
a
could be correlated with daily mean temperatures. The correlation for this
site is, however, interesting in that it suggests that the
a
parameter changes in an inverse relationship with
temperature. This implies that the higher the temperature, the longer the mean residence time. This is also
physically reasonable if higher temperatures are an indication of greater summertime evapotranspiration
and therefore lower levels of moisture storage and consequently slower response times. In another study
using temperature as an input, Young
et al.
(2007) have shown how a
hysteretic
input filter can be derived
in predicting periods of snowmelt runoff (see Box 4.3). The filter is different if the temperature is rising
or falling. The result is a form of degree-day method for predicting snowmelt, but one in which the coef-
ficient is determined from the observations and is variable with time. The DBM approach has also been
applied to predict surface and subsurface runoff in tropical forest catchments (Chappell
et al.
, 2006).
The DBM approach results in a minimal model relating inputs to outputs involving a nonlinear filter
(identified from the observations) and normally only one or two time constants for the transfer func-
tion. As will be apparent from the use of a nonlinear filter and temperature to account for “effective
rainfall”, seasonal and snowmelt effects, the approach does not necessarily require that the variables are
dimensionally consistent or that water or energy balances are conserved. The DBM approach has been
used, for example, to provide models directly from rainfall inputs to river stage (water level) and from
upstream stage to downstream stage in routing flood waves (e.g. Leedal
et al.
, 2008; Romanowicz
et al.
,
2008). Neither would be expected to maintain a water mass balance, even though this would normally
seem to be a requirement for a hydrological or hydraulic model. In fact, modelling water levels directly
can be advantageous in that level is easily measured and does not require the use of a rating curve to
estimate discharge, which might be very uncertain for flood flows. Also, for flood forecasting and incident
management, it is very often water level that is the variable of interest, not discharge. Not maintaining
mass balance can also be an advantage under flood conditions when knowledge of the pattern of rainfall
inputs might be poor. This makes the approach very suitable for use with data assimilation techniques in
forecasting (see Chapter 8).
The parameters of the DBM model may be considered to be bulk “physical” characteristics at the
catchment scale but, as in the regionalisation studies of the dynamic response characteristics of the
IHACRES model noted above, it is not yet clear how well these parameters might be related to catchment
characteristics nor the range of catchments for which such a simple model might be appropriate. However,
what is clear is that good estimates of the parameters can be obtained from only short periods of rainfall-
runoff data so that a period of field measurement of rainfalls and discharges at a site of interest might be
the best way of calibrating the parameters (see Seibert and Beven, 2009).
It is worth noting that one of the features of the linear time series analysis used to derive the transfer
function is that standard errors can be estimated for the transfer function parameters (see, for example,
Taylor
et al.
, 2007; Young, 2011a). These standard errors can be used to evaluate the physical interpreta-
tion of the model. Young (1992; see also Young
et al.
, 2007), for example, has investigated the sensitivity
of the proportion of effective rainfall going through the fast and slow flow pathways to error in the
estimated parameters. The results suggest that there will be considerable uncertainty in these propor-
tions, limiting any interpretation in terms of fast runoff contributing areas. At least in this type of model,
these uncertainties can be made explicit; there are similar implications for the more complex models that
are discussed in Chapters 5 and 6, but their sensitivities are seldom evaluated.
One interesting use of the DBM methodologies has been to emulate the functionality of much more
complex models. Beven
et al.
(2008b) have shown how the outputs from a hydrodynamic flood routing
model can be reproduced with great accuracy, including the hysteresis in the depth-discharge relationships
at a site. Tych and Young (2011) have similarly applied the DBM approach to emulating the outputs from
the OTIS transient storage model for pollutant transport in rivers. This could be useful in forecasting
either flood wave propagation or pollutant transport to a point because the run times of the DBM emulator
