Geoscience Reference
In-Depth Information
from which one obtains the main result
n
(
n
−
1)
m
y
,
n
=
m
x
,
n
+
nm
x
,
n
−
1
m
u
,
1
+
m
x
,
n
−
2
m
u
,
2
+···+
m
u
,
n
(A27)
2!
The commas have been introduced in the subscripts in this expression merely for clarity
of notation. The first moments about the mean are zero; hence, for the smaller values of
n
, which are the ones of practical importance, one obtains from (A27)
m
y
2
=
m
x
2
+
m
u
2
m
y
3
=
m
x
3
+
m
u
3
(A28)
m
y
4
=
m
x
4
+
m
u
4
+
6
m
x
2
m
u
2
Equations (A22) and (A27) (or (A28)) are jointly sometimes referred to as the theorem of
moments; it was introduced in the hydrologic literature by Nash (1959) for the purpose
of deriving the moments of the unit response function from observed records of effective
rainfall and storm runoff.
A3
THE GENERAL RESPONSE OF A NONLINEAR SYSTEM
The response of a nonlinear system can be described by a generalization of the convolu-
tion integral operation. This type of approach goes back to the work of Volterra (1913;
1959; also Barrett, 1963) who showed that a hereditary system can be described by a
convergent series of integrals,
t
y
(
t
)
=
F
(
x
=
0)
+
u
1
(
t
,τ
)
x
(
τ
)
d
τ
−∞
t
t
1
2!
+
u
2
(
t
,τ
1
,τ
2
)
x
(
τ
1
)
x
(
τ
2
)
d
τ
1
d
τ
2
+···
−∞
−∞
t
t
n
1
n
!
···+
···
u
n
(
t
,τ
1
,...,τ
n
)
x
(
τ
i
)
d
τ
i
+···
(A29)
i
=
1
−∞
−∞
in which the
u
i
( ) terms are the kernels of the integrals and the subscripts indicate their
order. As before,
y
(
t
) and
x
(
t
) are the output and input of the system, as functions of
time. If the system does not generate any output, when the input is zero, the first term
on the right of (A29) can be omitted. Also, when the system is time invariant, it can be
shown that the kernels must be of the form
u
1
(
t
), etc. The lower limit of
(A29) indicates that the system has an infinite memory. Thus, assuming that the system
has, first, no zero input response, second, time-invariant response characteristics, and
third, a finite memory
m
, one can rewrite Equation (A29) (by analogy with (A13)) as
,τ
)
=
u
1
(
t
−
τ
