Geoscience Reference
In-Depth Information
In view of
Eq. (13.9)
we can interpret the first of
Eqs. (13.12
)as
e
iκu(t)
.
f(κ
;
t)
=
(13.13)
Taking the
n
-th derivative of the first of
Eqs. (13.12
) with respect to
iκ
and evaluating
the derivatives at the origin yields the moments of
u
:
κ
=
0
=
∂
n
f(κ
t)
∂(iκ)
n
;
u
n
(t).
(13.14)
These moments determine the series expansion of
f
:
∞
1
n
u
n
(t)(iκ)
n
.
f(κ
;
t)
=
(13.15)
!
n
=
0
Thus knowledge of all the moments, or knowledge of
f
or of
β
, are equivalent.
13.2.3 Joint probability densities and distributions
The probability density
β(u
;
t)
gives information about
u(t)
at any given time but
not at two or more times. But we can use the indicator function
φ(u
;
t)
to write
(
1
;
=
;
≥
(
1
;
=
if
u(t
1
)<u
1
,
t
1
)
1
if
u(t
1
)
u
1
,
t
1
)
0
,
and it follows that the probability that
u(t
1
)<u
1
and
u(t
2
)<u
2
is given by the
average of
φ(u
1
;
t
1
)φ(u
2
;
t
2
)
:
φ(u
1
;
t
1
)φ(u
2
;
t
2
)
=
P
2
(u
1
,u
2
;
t
1
,t
2
).
(13.16)
This is called a
joint probability distribution function
. Its second derivative with
respect to amplitudes is a
joint probability density
:
∂
2
∂u
1
∂u
2
P
2
(u
1
,u
2
;
β
2
(u
1
,u
2
;
t
1
,t
2
)
=
t
1
,t
2
).
(13.17)
This can be interpreted as a probability:
β
2
(u
1
,u
2
;
t
1
,t
2
)u
1
u
2
=
Pr
{
u
1
≤
u(t
1
)<u
1
+
u
1
,u
2
≤
u(t
2
)<u
2
+
u
2
}
.
(13.18)
This has the normalization properties
+∞
β
2
(u
1
,u
2
;
t
1
,t
2
)du
1
du
2
=
1
,
(13.19)
−∞
