Geoscience Reference
In-Depth Information
τ
=t
Fig. 12.6 Convolution operation with an instantaneous
unit hydrograph
u
=
u
(
t
), as the analog of
the summation shown in Figure 12.2, in the
limit as
D
u
→
dt
→
0. (The values of
y
and
u
are not drawn to scale.) (See also Figure A5.)
τ
τ
x(
) d
τ
d
τ
u
−
τ
u(t
)
−
τ
t
y(t)
t
hence, for a streamflow rate
Q
=
Q
(
t
) at the outlet of a catchment of area
A
, the function
x
(
t
). This means
that the dimensions of the unit response function in Equation (12.2) are [
u
]
y
represents
Q
/
A
, so that it has the dimensions [L
/
T], just like
x
=
[T
−
1
],
corresponding, for example, with the units of cm per hour of runoff per cm of rainfall
input. In the operation represented by the convolution integral (12.2),
t
=
=
0 is defined
as the start of the input rate
x
x
(
t
). At any given value of time
t
, the total output rate
y
is the result of all past inputs from the start of the input at
=
τ
=
0 until
τ
=
t
, weighted
at each instant
τ
with the unit response, as indicated in Figure 12.6, with the argument
(
t
is the dummy time variable of integration, and
t
is treated
as a constant. (A more mathematical illustration of this
convolution
or
folding
is given
in Figure A5.)
−
τ
). In the integration
τ
Relationships between these different response functions
The instantaneous unit hydrograph
u
(
t
) can be used to derive the finite duration unit
hydrograph
u
(
D
u
;
t
) by applying Equation (12.2) with the following input
1
D
u
x
=
for 0
≤
t
≤
D
u
(12.3)
x
=
0
for
t
>
D
u
This yields, say, for
t
>
D
u
D
u
1
D
u
u
(
t
u
(
D
u
;
t
)
=
−
τ
)
d
τ
(12.4)
0
−
τ
=
τ
=−
After putting (
t
)
s
, so that
d
ds
, Equation (12.4) becomes
t
1
D
u
u
(
D
u
;
t
)
=
u
(
s
)
ds
(12.5)
t
−
D
u
This indicates that the finite duration unit hydrograph at time
t
is the average of the
instantaneous unit hydrograph taken over the period between (
t
−
D
u
) and
t
. This is
