Geoscience Reference
In-Depth Information
1
Fig. 12.20 The unit response
function, obtained
with the gamma
density function
(12.40) or (12.41) as
u
=
u
(
t
/
K
), for
different values of the
parameter
n
.
0.8
u
0.6
n=1
0.4
2
3
4
6
0.2
8
10
0
0
10
15
5
t/K
For example, Nash (1960) used Equation (12.41) to determine the instantaneous unit
hydrographs for a number of British catchments by means of the method of moments. It
can readily be shown (cf. Equation (13.9)) that its first moment about the origin is
m
u
1
=
nK
(12.42)
and that its second moment about the mean, or center of gravity (cf. Equations (13.10)
and (13.12)), is
nK
2
m
u
2
=
(12.43)
Note that with
n
0 these are the same as Equations (7.31) and (7.34)
for the Muskingum formulation. Because the moments of
u
(
t
) can be calculated from
available rainfall and streamflow records by means of the theorem of moments, as given
by Equations (A22) and (A28), Nash (1959) was able to relate the parameters
K
and
n
directly with relevant basin characteristics; in this case, these were found to be drainage
area, mean slope and length of the main channel. A similar study was carried out by Wu
(1963) with catchments in Indiana. Actually, prior to its conceptual derivation by Nash,
the incomplete gamma function had already been used by Edson (1951) on different
grounds, to describe finite duration unit hydrographs. It was subsequently also used for
this purpose by Gray (1961).
Several features of the tank cascade may help to explain, perhaps, why the inte-
grand of the incomplete gamma function has been used so widely in applied hydrology.
First, consider the case where
n
is allowed to increase indefinitely. As indicated by
Equation (12.42), the center of gravity of the flood wave will then occur at a finite value
of the time
t
, only if the storage coefficient of each tank in the cascade,
K
, is made to
become very small. But if
K
is made to become very small, the second moment (12.43)
indicates that the duration of the flood wave will become very short. At the same time, if
the area under the wave curve is to maintain a magnitude of one, the magnitude of the peak
must become very large. This is illustrated in Figure 12.21, for the case
nK
=
1 and
X
=
=
1. Thus, in
the extreme case of
n
→∞
,
K
→
0
,
but finite
nK
, the unit response function (12.41)
