Geoscience Reference
In-Depth Information
The second type of approach uses a finer time variable to describe response changes
during the event itself, as a result of physical changes inside the catchment resulting
from continued rain, snowmelt or flooding. These two types can be combined formally
in a convolution integral, as an extension of Equation (12.2), namely
t
=
τ
χ,τ,
−
τ
τ
y
(
t
)
x
(
)
u
(
t
)
d
(12.57)
0
in which, as before,
y
(
t
) is the output resulting from an input
x
(
t
). In contrast to the
stationary case of Equation (12.2), here the unit response
u
(
χ,τ,
t
−
τ
) is a function
of three time variables; the first variable
χ
is the coarse time scale in terms of months,
seasons or years. The second,
τ
, is the dummy variable of integration, such that 0
≤
τ
≤
t
;
however, as argument in the unit response, it denotes the time of the input
x
(
), and thus
specifies the response for the input at that time. The third,
t
, is the time for which
the output is to be determined and (
t
τ
).
The convolution operation of Equation (12.57) describes superposition, which is the
essence of linearity. However, as pointed out by Diskin and Boneh (1974), in contrast
to the stationary case, here the convolution (or superposition) operation is generally not
commutable. This means, for example, that two non-stationary systems connected in
series, say A followed by B, will produce a different output when their order is reversed,
B followed by A.
For the first type of non-stationarity the unit response in Equation (12.57) depends only
−
τ
) is the time elapsed since the input
x
(
τ
χ
−
τ
τ
on
, but only, say, on the season or the
year. This means that during the input event the response can be considered stationary, and
the concepts discussed in Sections 12.1 and 12.2 are applicable. Therefore, the second
type is the one usually considered, when time variance effects are to be included. In this
case, the unit response depends only on
and (
t
); thus it does not depend on the time
τ
and (
t
−
τ
), and (12.57) can be simplified to
t
y
(
t
)
=
x
(
τ
)
u
(
τ,
t
−
τ
)
d
τ
(12.58)
0
This is illustrated in Figure 12.24. In discrete form (cf. Equation (12.9)) this can be
written as
i
y
i
=
x
k
u
k
,
i
−
k
+
1
(12.59)
k
=
1
or alternatively (cf. Equation (12.10)), as
i
y
i
=
x
i
−
k
+
1
u
i
−
k
+
1
,
k
(12.60)
k
=
1
where the first subscript of the response function refers to the time of the input, and the
second indicates its role in the numerical convolution.
In several applications of the nonstationary linear approach in the past, the form of
the unit response function has been assumed a priori. For example, Snyder
et al.
(1970)
derived the catchment response by routing, what was essentially a time-area diagram
through a linear storage element with time-dependent storage coefficient
K
=
K
(
τ
)
