Geoscience Reference
In-Depth Information
about the origin is (see Chapter 13)
0
∞
tudt
m
u1
=
(7.29)
0
∞
udt
With the unit response (7.28), one can check that the integral in the denominator of (7.29),
i.e. the zeroth moment, equals one, as it should; moreover, the first moment of the delta
function
δ
(0) is zero. Therefore, after insertion of (7.28), Equation (7.29) can be rewritten
as
∞
e
−
t
/
K
(1
−
X
)
K
(1
−
X
)
2
dt
m
u1
=
t
(7.30)
0
After integration by parts, one finds that the first moment of the unit response is simply
equal to the Muskingum parameter
K
,or
m
u1
=
K
(7.31)
The second moment of the unit response about the origin is
0
∞
t
2
udt
m
u2
=
(7.32)
0
∞
udt
Proceeding in the same way as for the first moment, one finds that this second moment
about the origin can be written in terms of the Muskingum parameters as
m
u2
=
2
K
2
(1
−
X
)
(7.33)
Since the second moment about the mean is related to the first two moments (see Equation
(13.12)), one obtains finally
m
u2
=
K
2
(1
−
2
X
)
(7.34)
The higher moments can be derived in the same way.
As an aside, comparison of Equations (7.31) and (7.34) with (7.19) and (7.25), respec-
tively, reveals that
m
u1
=
m
i2
). This is not unexpected.
Indeed, the Muskingum channel reach is a linear system, to which the theorem of moments,
as given by Equations (A22) and (A28) should be fully applicable.
(
m
e1
−
m
i1
) and
m
u2
=
(
m
e2
−
7.2.3
Standard implementation
Numerical calculations
Although the analytical solution provides insight into the structure of the Muskingum
formulation, it is difficult to use with observed streamflow data. In hydrologic practice,
the Muskingum method is normally applied over finite time increments
t
; for this
