Geoscience Reference
In-Depth Information
Linear translation in series with one linear storage element
The transformation of a rainfall input hyetograph into a streamflow output hydrograph
involves both a delay as a result of translatory effects and a deformation and attenuation
as a result of storage effects. In a linear context, the simplest way of incorporating both
effects is simply to add them. Thus it stands to reason that historically the next step
in the development of linear runoff routing procedures consisted of the superposition
of a linear time-area function, representing pure translation, with a linear reservoir,
representing pure storage. In one of the better known implementations of this idea,
Clark (1945) derived the instantaneous unit hydrograph from streamflow records by
numerically routing the time-area concentration function of the basin through a single
concentrated storage element by means of the Muskingum method with
X
=
0. Hence,
in light of Equation (7.15), this type of storage element is characterized by
S
=
Ky
(12.25)
where
y
is the outflow rate, and
S
the storage, both per unit catchment area, so that
[
y
]
=
[L
/
T], [
S
]
=
[L] and [
K
]
=
[T]; the parameter
K
is commonly referred to as the
storage coefficient.
A similar approach was also applied successfully in the development of large-scale
drainage schemes in a number of Irish catchments by O'Kelly (1955) and his fellow
engineers at the Office of Public Works. However, in the early stages of this work
it became clear that the routing through the concentrated storage element had such a
smoothing effect on the time-area function, that the exact shape of the time-area function
was not very critical, and that there was little loss in accuracy when it was replaced by
an isosceles triangle. The main parameters in the applications of this concept were the
time of concentration
t
c
, which is the time base needed to scale the triangular time-area
function, and the storage coefficient
K
, or the delay, in the routing procedure by means
of Equation (12.25). O'Kelly's report is noteworthy and it suggests that in many studies
the importance of the time-area function, and of the width function, may have been
exaggerated.
Various procedures have been used in the past to estimate the two parameters
t
c
and
K
.
In Clark's (1945) application, it was assumed that the direct effect of the rainfall ceases
at the inflection point of the recession limb of the outflow hydrograph, and that, from that
time on, the outflow is merely a release from storage in the basin; accordingly he took
t
c
as the time between the end of rainfall and the inflection point of the falling leg of the
hydrograph, and the storage coefficient from the recession after the inflection point with
K
=−
y
/
(
dy
/
dt
), which is obtained from Equation (1.10) (or (7.11)) with (12.25) and
0. In the procedure described by O'Kelly (1955) the shape of the instantaneous unit
hydrograph is uniquely determined by the ratio
K
x
=
t
c
(see also Example 12.6); thus the
general shape of the experimentally obtained unit hydrographs for the basin provided
an estimate of this ratio, which allowed then in turn the estimation of the value of
t
c
(or
K
) by matching the peak outflow rates. Because the shapes could not always be
fitted well, often various
K
/
t
c
ratios were tried yielding different
t
c
and
K
values. A
review of some of the earlier methods to estimate these parameters has been presented
by Dooge (1973, pp. 198-200). In several of these studies
t
c
and
K
were expressed in
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