Geoscience Reference
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Fig. 12.10 Sketch of a time-area function A r = A r ( t ), as an
extension of the rational method. The dashed
lines on the catchment map represent lines of
equal travel time or isochrones.
A r
t
Fig. 12.11 A linear translation element, as a
mechanistic metaphor for the runoff
derived by convolution (or routing) of the
instantaneous input δ ( t ) through a
time-area function.
δ
(t)
u(t)=A r (t)
.
.
the time-area curve represents outflow resulting from an instantaneous unit input applied
uniformly over the catchment area. Hence, the time-area function A r =
A r ( t ) is the unit
response function of this type of catchment (see Figure 12.11), and the outflow resulting
from a rainfall input x ( t ) is given by Equation (12.2), or
t
=
τ
τ
τ
y ( t )
x (
) A r ( t
) d
(12.24)
0
Note with Dooge (1973) that, while this approach made use of an instantaneous unit
hydrograph in the form of the time-area function, it predated the formal invention of the
unit hydrograph by Sherman (1932a,b) by about a decade. But the time-area approach
never gained wide acceptance, probably because it takes insufficient account of storage
mechanisms in the basin. In natural drainage basins precipitation cannot be immediately
translated to the outlet, but a portion of it first must build up some water as storage on
the vegetation and on the soil surface and in the pores of the soil profile, before any
flow can take place. Therefore, it can be expected that, when the travel times in a basin
are estimated on the basis of known velocities of overland flows and channel flows, the
calculated peak outflow rates will tend to be severely overestimated. This realization led
to increased efforts to include storage effects in subsequent developments.
In recent years the concept of translation, underlying the time-area-concentration
function, has continued to be studied and used in a more formal way. This is done by
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