Geoscience Reference
In-Depth Information
Fig. A4
Illustration of the (Heaviside) unit step function
H
=
H
(
t
−
t
0
).
H
1
0
t
0
t
is valid for all constants
a
and
b
and for all
x
and
y
, such that
T
[
x
(
t
)] and
T
[
y
(
t
)]
exist. A transformation or operation is said to be
stationary
or
invariant
, if for the same
conditions at
t
0, it produces the same result under a coordinate translation; thus the
transformation
z
(
t
)
=
=
T
[
x
(
t
)] is stationary if
−
=
−
z
(
t
t
0
)
T
[
x
(
t
t
0
)]
(A10)
remains valid for any value of
t
0
.
The
unit response
of a linear system
u
=
u
(
t
) is its response to the unit impulse
function
(
t
). The unit response is also variously called the
impulse response
, the
Green's
function
, and the
influence function
of the system. If these response characteristics are
invariant in time or space (whatever domain
t
denotes), the response is
u
(
t
0
−
δ
t
), when
the input is
δ
(
t
0
−
t
). Because the system is linear, it follows that the response becomes
t
). By the same token, upon multiplication of
this response and of this input by
dt
and integration of both, comparison of the resulting
input with Equation (A7) shows that when the input is
x
(
t
), the response or output of the
system is given by
x
(
t
)
u
(
t
0
−
t
) when the input is
x
(
t
)
δ
(
t
0
−
+∞
y
(
t
)
=
x
(
τ
)
u
(
t
−
τ
)
d
τ
(A11)
−∞
τ
where
is a dummy variable of integration. The operation shown in (A11) is called the
convolution integral
.
The upper and lower limits of Equation (A11) indicate that the output from the
system is affected by input values of
x
(
t
) for
t
all the way from minus to plus infinity.
Time dependent hydrologic systems are causal and non-anticipatory; this means that
they only depend on the values of present and past (but not future) values of the input
function; such systems are also referred to as
hereditary
(Volterra, 1913). Thus whenever
t
denotes time, in hydrologic applications the upper limit of the integral in (A11) should
be
t
, and one can write
t
y
(
t
)
=
x
(
τ
)
u
(
t
−
τ
)
d
τ
(A12)
−∞
