Geoscience Reference
In-Depth Information
τ
τ
τ
Fig. 12.24 Illustration of the convolution
operation
y
(
t
)
=
0
x
(
τ
)
u
(
τ,
t
− τ
)
d
τ
,
with a nonstationary unit response
u
=
u
(
τ,
t
). (The values of
y
and
u
are
not drawn to scale.) The variable
t
denotes the time for which the runoff
y
is calculated and
τ
is the time since the
start of the precipitation input
x
. The
unit response, which is shown at only
three instants of time
τ
, continually
changes its shape as the precipitation
continues. Compare this with the
stationary case of Figure 12.6.
1
2
3
τ
x
τ
−
τ
u(
1
,t
1
)
τ
−
τ
u(
2
,t
2
)
τ
−
τ
u(
3
,t
3
)
−
τ
t
3
y
y(t)
t
(see Equation (12.25)). In a similar vein, Mandeville and O'Donnell (1973) consid-
ered different combinations of time-variant linear channel and time-variant linear stor-
age elements, one among them being a cascade of equal storage elements. Diskin and
Boneh (1974) developed numerical least squares procedures to derive more general, i.e.
not with a preconceived mathematical form, response functions
u
i
−
k
+
1
,
k
, as shown in
Equation (12.60), from available rainfall-runoff data. Chiu and Bittler's (1969) study is
probably the only one that considered both fine-scale and coarse-scale non-stationarity as
formulated in Equation (12.57). The unit response function was obtained by routing the
input through a single linear storage element, as given by (12.25) with a time-dependent
storage coefficient
K
=
K
(
τ
) in the form of a power function
τ
−
b
K
=
a
(12.61)
in which
a
and
b
were assumed to be functions of the coarse time variable
. Rainfall-
runoff data obtained in Pennsylvania indicated that
a
was a function of
b
and that
b
could
be described well by a sine function of
χ
. Equation (12.61) indicates that in this study the
storage coefficient
K
was observed to decrease as the precipitation continued; because
K
is the time of travel through the system (cf. Equations (7.19), (7.31) and (12.42)), this
means that the unit response tended to become faster with storm duration. The approach
was later extended by Chiu and Huang (1970) to include nonlinear effects by replacing
(12.25) by its nonlinear analog (12.48).
χ
