HYDROLOGY 477
characteristics are obtained with propagation velocities of
ugydxdt . At the other end of the spectrum is kine-
matic wave theory in which the bedslope term is assumed to
overpower the other three terms on the left hand side, namely,
depth slope, convective and local acceleration. Hence kine-
matic wave are friction controlled. Equation (36) is therefore
reduced toSv(^0) cR
2
2.^ (38)
To combine Eqs. (36) and (38) into the simplest character-
istic form, it is assumed that R, the hydraulic radius, may be
replaced by y. Then Eq. (38) yields
QvByBCyS
..^32
0
(^12) (39)
Substituting for in (37) by differentiating Eq. (39)
3
2 0
BCS12 12y y
t
B
y
t
∂
∂
∂
∂
d (40)
The left hand side is the total differential d y /d t if the term
3
2 0
CS y12 12 x
t
d
d
wave velocity. (41)
The important fact that monoclinical waves can only propagate
down-stream with a wave velocity given, in the simple case,
by Eq. (41), is well known, as is best discussed by Lighthill
and Witham.^27 As previously mentioned, as an empirical
result, Eq. (41) dates back to Seddon^25 and his study of fl ood
wave movement.
Comparing kinematic wave theory with channel routing,
Eq. (37) can be recast in exactly the same form as Eqs. (29)
and (30) if the lateral in fl ow term is ignored. The difference
in the two methods depends on the second equation chosen,
which is basically a stage–discharge relationship. Channel
routing, as illustrated by the Muskingum method, assumes
a linear relationship between stage and discharge to arrive
at Eq. (30) for storage. Kinematic theory uses a simplifi ed
equation of motion which is usually either the Chezy or
Manning equation.
In a further development of kinematic theory research-
ers such as Hyami^29 have shown that the infl uence of chan-
nel storage can be included and this results in an extra term
which is effectively a diffusion term:
∂
∂
∂
∂
∂
∂
y
t
c
y
x
K
y
t
2
2 (42)
Solutions of this equation are approximately of the form:
yye 0 axcos(btax) (43)
This solution assumes a sinusoidal wave profi le, and the solu-
tion is seen to be delayed as it moves downstream and also it
decays in amplitude. It is seen that effectively an exponential
damping term is introduced by the inclusion of diffusion.
Such relationships are complex and cumbersome,
demanding detailed data on the channels which often is not
available. Inspection of the resulting equations reveals a
similarity to the old lag and route method. The author has
used a simplifi ed method based on these equations which
has proved itself to be very successful.^46 The method uses a
variable travel time calculated from the velocity-stage curves
for the upstream end of the given channel reach. The kine-
matic result of approximately 1.5 times channel velocity is
used, although a best fi t result can be found by optimization.
The fl ow is then routed through a simple linear reservoir, the
size of which is related to the channel storage. This reservoir
allows a check to be kept on continuity and also it provides
an exponential decay term as demanded by the diffusion term
in the full kinematic theory. For relatively steep rivers like
the Fraser River in British Columbia, one linear reservoir is
adequate. Less steep streams may require two reservoirs, but
two reservoirs should be adequate in that a two-parameter
exponential will fi t almost any curve imaginable.
The simple channel routing procedure has been found to
be very powerful because it approximates the true physical
behavior very closely. It yields a simple non-linear behavior
and models the well documented fl ood wave subsidence. On
the Fraser River system the method proved very successful in
separating out the very large lateral fl ows which are charac-
teristic of that particular river system, and this separation was
really the key to the modeling of the whole system behavior.
ARTIFICIAL GENERATION OF STREAMFLOW
If streamfl ows can be considered to be statistically distributed,
then it may be possible to establish statistical measures of
their distribution. Such statistical measures will be the mean
fl ow, the variance and perhaps some correlation between suc-
cessive fl ows. It may also be possible to determine the nature
of the distribution, such as normal, or log normal. If such
parameters can be found, then these parameters can be used
to regenerate data with the same statistical patterns.
The reason for such data regeneration is not apparent
until we examine the purposes for which such data can be
used. If we are given some 30 years of streamfl ow data, it
is possible to make reasonable estimates of the statistical
parameters of, say, monthly or annual fl ows. The 30 years of
record will contain only a very few extreme events, such as
fl oods or prolonged droughts, and will therefore not impose
very severe tests on any water resource system which is
storage dependent, such as a hydro system or an irrigation
project. Generated data could presumably be constructed
for very long periods of time and would, if the procedure is
justifi ed, reproduce many more extreme fl ows and consecu-
tive run-off patterns which will test the proposed resource
system more thoroughly. The statistical risk of failure can
then be evaluated.
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