Geoscience Reference
In-Depth Information
2
K/
t
c
=0.1
u
×
t
c
0.2
1
0.5
1
2
0
2
3
0
1
t/t
c
Fig. 12.14 Unit response function resulting from a time-area function with the shape of an isosceles triangle
A
r
(
t
)
routed through a linear storage element for different values of the time-scale ratio
K
/
t
c
. Both
u
(
t
) and
t
are made dimensionless with
t
c
. The time scale
t
c
is the time of concentration of the
time-area function (dashed line) and the time scale
K
is the coefficient of the linear storage element.
Example 12.6
Consider a hypothetical diamond-shaped catchment with the stream channel running
along one of the diagonals; in the present context this produces a triangular time-area
function (or width function), which can be formulated as follows
4
t
t
c
A
r
=
for 0
≤
t
≤
t
c
/
2
A
r
=
−
4
t
t
c
4
t
c
(12.30)
+
for
t
c
/
<
≤
2
t
t
c
A
r
=
0
for
t
c
<
t
where
t
c
is the time of concentration. Observe that the area under
A
r
=
A
r
(
t
) equals
unity, as it should. The unit response is calculated by applying (12.29) with (12.30).
Thus, one has for
t
≤
t
c
/
2
t
4
t
c
K
e
−
(
t
−
τ
)
/
K
d
u
(
t
)
=
τ
τ
(12.31)
0
which upon integration results in
4
t
c
K
(
e
−
t
/
K
u
(
t
)
=
(
t
+
−
1))
(12.32)
Similarly for
t
c
/
2
<
t
≤
t
c
, one can write
t
c
/
2
t
t
4
t
c
K
4
t
c
K
4
t
c
K
e
−
(
t
−
τ
)
/
K
d
e
−
(
t
−
τ
)
/
K
d
e
−
(
t
−
τ
)
/
K
d
u
(
t
)
=
τ
τ
−
2
τ
τ
+
τ
(12.33)
0
t
c
/
t
c
/
2
