Geoscience Reference
In-Depth Information
follows directly from (3.10) with (3.11), namely
D
L
i
=
c
Pdt
for
S
≤
S
ic
and
D
≤
t
0
(3.14)
0
For long events Equation (3.6) can be rewritten as
t
0
D
L
i
=
E
i
dt
+
S
ic
+
E
i
dt
for
S
=
S
ic
and
D
>
t
0
(3.15)
t
0
0
or, upon substitution of the first two terms on the right by (3.9) with (3.11), and of the third
by (3.13),
⎛
⎞
t
0
D
⎝
⎠
L
i
=
c
Pdt
+
E
po
dt
for
S
=
S
ic
and
D
>
t
0
(3.16)
0
t
0
Equations (3.14) and (3.16) can be readily solved numerically by also keeping track of
S
by means of (3.9).
Lumped kinematic solution
The assumption that the vegetation, as a hydrologic flow system, can be represented by a
storage element governed by Equation (3.8) (with
O
=
0), and with a storage-outflow rela-
tionship given by Equation (3.13), is a perfect example of the lumped kinematic approach.
Gash (1979; Gash
et al
., 1995) made use of this simple structure to derive a closed form solu-
tion for the evolution of
S
with time; by assuming constant (or averaged) values of
P
and
E
po
during the precipitation event of duration
D
, he obtained
ln
1
−
S
ic
cE
po
E
po
S
PS
ic
D
=
(3.17)
The time to saturation is therefore
t
0
=
(
S
ic
/
cE
po
)ln[1
−
(
E
po
/
P
)]
(3.18)
which can be used immediately with (3.14) and (3.16) to estimate the interception loss. For
constant (or averaged) values of
P
and
E
po
during the precipitation event, (3.14) and (3.16)
can be written simply as
L
i
=
cPD
for
S
≤
S
ic
and
D
≤
t
0
(3.19)
and
L
i
=
c
[
Pt
0
+
(
D
−
t
0
)
E
po
]
for
S
=
S
ic
and
D
>
t
0
(3.20)
where (
PD
) is the cumulative precipitation at the end of the precipitation event, and (
Pt
0
)
is the cumulative precipitation needed to saturate the vegetation.
