Geoscience Reference
In-Depth Information
then scaling the remainder hydrograph to the desired unit volume, that is by dividing it by
the observed excess rainfall depth. Trial and error may be required in the determination
of the baseflow and of the precipitation losses, to ensure that the excess rainfall volume
equals the storm runoff volume over the catchment area. Several unit hydrographs derived
from different storm events of a similar duration can be averaged in order to obtain a
more representative result. If necessary, a common duration can also be obtained by
means of their respective S hydrographs.
Complex storm events
More often than not, precipitation events do not have a uniform intensity in time and
the resulting runoff hydrograph may be quite irregular. The unit hydrograph must then
be estimated by considering its mathematical operation in some detail. Precipitation
and also streamflow data are usually given with discrete time steps. According to
Equation (A18), the discrete analog of the convolution integral is
n
y
(
t
)
=
x
(
k
τ
)
u
(
τ
;
t
−
k
τ
)
τ
(12.8)
k
=
0
τ
=
τ
≤
in which
n
(
t
) is the time of the last input pulse prior to the designated response
τ
=
τ
time
is literally taken as one, and the output and input times
are discretized with the same resolution, this can be rewritten as
t
. If the unit period
i
y
i
=
x
k
u
i
−
k
+
1
(12.9)
k
=
1
Note that by analogy with Equation (A16) this can also be written as
i
y
i
=
x
i
−
k
+
1
u
k
(12.10)
k
=
1
It should further be noted that
y
i
and
u
i
can designate either their respective average
values over the
i
th time period, or their actual values at the end of that same period,
depending on how the discretization is specified. Equation (12.9) (or (12.10)) leads to
the following set of equations
y
1
=
x
1
u
1
y
2
=
x
1
u
2
+
x
2
u
1
y
3
=
x
1
u
3
+
x
2
u
2
+
x
3
u
1
.
y
i
=
x
1
u
i
+
x
2
u
i
−
1
+···+
x
i
u
1
.
(12.11)
y
p
−
1
=
x
1
u
p
−
1
+
x
2
u
p
−
2
+···+
x
p
−
1
u
1
y
p
=
x
1
u
p
+
x
2
u
p
−
1
+···+
x
p
u
1
.
y
m
−
1
=
x
p
−
1
u
n
+
x
p
u
n
−
1
y
m
=
x
p
u
n
