Geoscience Reference
In-Depth Information
Fig. 12.19 The tank cascade, proposed by Nash
(1957), consisting of
n
equal storage
elements placed in series, as a metaphor
for the response
u
=
u
(
t
) of a catchment
to an instantaneous input
x
=
δ
(
t
).
(
t
)
δ
1
2
.
.
.
3
u
(
t
)
n
When this is taken as input into the second storage element, the output from that second
storage is
t
exp(
−
τ/
K
)
exp[
−
(
t
−
τ
)
/
K
]
t
exp(
−
t
/
K
)
u
2
(
t
)
=
d
τ
=
(12.38)
K
K
K
2
0
This, in turn, is input into the third storage element and produces an output
t
τ
exp(
−
τ/
K
)
exp[
−
(
t
−
τ
)
/
K
]
t
2
exp(
−
t
/
K
)
u
3
(
t
)
=
d
τ
=
(12.39)
K
2
2
K
3
K
0
The same process can be continued to obtain the outflow from the last storage element,
K
)
n
−
1
exp(
(
t
/
−
t
/
K
)
u
n
(
t
)
=
(12.40)
(
n
−
1)!
K
which is the response function of the entire system. In order to allow the use of fractional
values of
n
, the factorial can be replaced by the complete gamma function. Finally, the
unit response of the entire catchment can be written as
K
)
n
−
1
exp(
(
t
/
−
t
/
K
)
u
(
t
)
=
(12.41)
K
(
n
)
Equation (12.41) is known as the integrand of the incomplete gamma function or as the
gamma density function. It has only two parameters, but it is quite flexible as it can
accommodate a wide variety of hydrograph shapes; as illustrated in Figure 12.20,
K
can
be considered a scale parameter and
n
a shape parameter. Equation (12.41) has been
applied widely in watershed hydrology to parameterize unit hydrographs in terms of
drainage basin characteristics.
