Geoscience Reference
In-Depth Information
∞
∞
m
y
1
=
t
x
(
τ
)
u
(
t
−
τ
)
d
τ
dt
(A20)
−∞
−∞
=
−
τ
=
Inversion of the order of integration, and substitution of
s
t
, so that
dt
ds
,
changes Equation (A20) into
∞
∞
m
y
1
=
x
(
τ
)
d
τ
(
τ
+
s
)
u
(
s
)
ds
−∞
−∞
∞
∞
∞
∞
=
τ
x
(
τ
)
d
τ
u
(
s
)
ds
+
x
(
τ
)
d
τ
su
(
s
)
ds
(A21)
−∞
−∞
−∞
−∞
Because the zeroth moments are equal to one, this yields the following relationship
between the first moments about the origin
m
y
1
=
m
x
1
+
m
u
1
(A22)
The
n
th moment of the output
y
(
t
) about its center of area
m
y
1
can be written, after
substitution of the convolution integral (A11) and of (A22), as follows
∞
∞
m
x
1
−
m
u
1
)
n
x
(
m
yn
=
(
t
−
τ
)
u
(
t
−
τ
)
d
τ
dt
(A23)
−∞
−∞
Again, inverting the order of integration, and putting
s
=
t
−
τ
, so that
dt
=
ds
, one can
rewrite (A23) as
∞
∞
m
x
1
)
m
u
1
)]
n
u
(
s
)
ds
m
yn
=
x
(
τ
)
d
τ
[(
τ
−
+
(
s
−
(A24)
−∞
−∞
The term with the square brackets can be expanded as follows
m
x
1
)
m
u
1
)]
n
m
x
1
)
n
m
x
1
)
n
−
1
(
s
m
u
1
)
[(
τ
−
+
(
s
−
=
(
τ
−
+
n
(
τ
−
−
−
n
(
n
1)
m
x
1
)
n
−
2
(
s
m
u
1
)
2
m
u
1
)
n
+
(
τ
−
−
+···+
(
s
−
(A25)
2!
This allows Equation (A24) to be written as
∞
m
x
1
)
n
m
x
1
)
n
−
1
m
u
1
m
yn
=
x
(
τ
)[(
τ
−
+
n
(
τ
−
−∞
−
n
(
n
1)
m
x
1
)
n
−
2
m
u
2
+···+
+
τ
−
τ
(
m
un
]
d
(A26)
2!
