Geoscience Reference
In-Depth Information
δ
(t)
A
r
(t)
.
.
u(t)
Fig. 12.13 A linear translation element placed in series with a linear tank element, as a mechanistic metaphor for
the runoff derived by successive convolution (or routing) of the instantaneous input
δ
(
t
) through a
time-area (or width) function and through a concentrated storage function.
terms of powers of basin characteristics, such as
L
S
a
and
A
. Note that both (12.21)
and (
12.
22) suggest that
t
c
should be proportional to a power of the combined variable
(
L
,
/
√
S
a
).
To derive the unit response function of this model, it is necessary to consider first the
unit response of a concentrated storage element. The flow through a linear storage element
can be represented by the storage equation (1.10) (or (7.11)), or in the catchment-scale
notation with an input
x
(
t
), an output
y
(
t
) and a storage per unit area
S
(
t
), as
dS
dt
x
−
y
=
(12.26)
After substitution of the concentrated storage function (12.25), this can be rearranged as
d
dt
+
y
x
K
which, upon multiplication of both sides by exp(
t
1
K
=
/
K
), yields the solution
x
(
t
) exp(
t
−
/
exp(
t
K
)
y
=
/
+
K
)
dt
constant
(12.27)
K
With a delta function input, that is
x
(
t
)
(
t
), the output of (12.27) is the unit response
function for a single storage element; in light of (A7) this has the form
=
δ
exp(
−
t
/
K
)
=
u
(
t
)
(12.28)
K
As could be expected, this is the same as the Muskingum response function (7.28) for
X
0.
The model of Clark (1945), O'Kelly (1955), and others (Dooge, 1973), in which
the storm runoff is derived by successively routing the rainfall input through trans-
lation and storage, can thus be formulated by simply putting
A
r
(
t
) in sequence with
Equation (12.28). Hence,
A
r
(
t
), which is the output from the translation operation,
becomes the input into the storage element, whose unit response is (12.28). This is
illustrated in Figure 12.13. The routing is accomplished by a convolution operation,
which produces immediately the unit response function of this combined system,
=
t
u
(
t
)
=
A
r
(
τ
) exp[
−
(
t
−
τ
)
/
K
]
d
τ/
K
(12.29)
0
