Geoscience Reference
In-Depth Information
Fig. 7.8 The heavy line triangular region
indicates the validity range of the
Muskingum formulation. The
area inside the triangle is defined
by the conditions that 0
≤
X
≤
0.5 and that the coefficients
c
0
,
c
1
and
c
2
be positive, as indicated
in (7.41);
c
m
denotes the celerity
of the wave. (After Miller and
Cunge, 1975.)
c
m
Δ
t/
Δ
x
2
c
2
<0
X>0.5
X<0
Valid
region
1
c
0
<0
0
0
0.5
1.0
X
(iii)
Several studies have dealt with the optimal length of the length of the reach
x
,but
there is no general consensus in the literature. As can be seen in Equation (7.20),
x
and
K
are related, so that the magnitude of
x
plays a role in the calculations
with (7.37) and (7.38). In the original applications of the Muskingum method
x
was taken as the total length of channel through which the routing was applied. In
more recent applications the total length of channel is usually subdivided in a number
of subsections of length
x
, and the calculations are carried out for each of these
subsections. One simple constraint, which would obviously avoid negative outflows
and which was suggested by Miller and Cunge (1975), is that the coefficients in
Equation (7.37) should all be positive. Hence, from (7.38) and (7.20) one can deduce
that, strictly speaking, the following should be satisfied.
x
t
≤
c
m
2
X
and
(7.41)
x
t
≥
c
m
2(1
−
X
)
With Equation (7.39) this allows construction of the diagram shown in Figure 7.8.
In practical applications the flood hydrographs needed for calibration may not be
available at the desired distances along the reach. However, a coarse spatial resolution
in available hydrograph data can sometimes be remedied by interpolation, to generate
a smaller step
x
. Laurenson (1959) has shown an example how such interpolation
can be implemented to achieve improved results.
In some situations it may prove impossible to adhere strictly to these conditions (i), (ii)
and (iii). Fortunately, however, the Muskingum method can be fairly robust and violation of
criteria like (7.41) does not necessarily lead to useless results. For instance, Weinmann and
