Geology Reference
In-Depth Information
Tabl e 2 . 1
Degrees of freedom as a function of the number of overlapping
segments for 75% overlap.
Segments
Degrees
1
2.0099183
2
3.2179875
3
4.5260402
4
5.8522880
5
7.1850360
6
8.5208433
7
9.8583334
8
11.1968478
9
12.5360320
10
13.8756782
freedom is given by (2.391). These are given, as a function of the number of over-
lapping segments, in Table 2.1.
As a specific example of the computation of the cumulative distribution func-
tion (2.397) we consider the case where four separate spectral estimates enter
the product. With the change of integration variables to y
1
=
x
1
/2, y
2
=
x
2
/2,
y
3
=
x
3
/2, the cumulative distribution function (2.397) becomes
∞
0
y
ν
1
/2
−
1
e
−
y
1
∞
1
0
y
ν
2
/2
−
1
e
−
y
2
F
(
z
)
=
1
2
Γ
(ν
1
/2)
Γ
(ν
2
/2)
Γ
(ν
3
/2)
∞
0
y
ν
3
/2
−
1
e
−
y
3
P
(ν
4
/2,
z
ξ
3
/2)
d
y
3
d
y
2
d
y
1
,
·
(2.414)
3
where
z
z
16y
1
y
2
y
3
.
z
ξ
3
/2
=
·
x
1
·
x
2
·
x
3
=
(2.415)
2
e
−
y
i
brings the three integrals to
the range (0,1) and the cumulative distribution function takes the form
A further change of integration variables to
t
i
=
1
log
t
1
ν
1
/2
−
1
1
1
0
−
0
−
log
t
2
ν
2
/2
−
1
F
(
z
)
=
Γ
(ν
1
/2)
Γ
(ν
2
/2)
Γ
(ν
3
/2)
1
log
t
3
ν
3
/2
−
1
P
ν
4
dt
3
dt
2
dt
1
.
0
−
z
16 log
t
1
log
t
2
log
t
3
−
·
2
,
(2.416)
The triple integrations are evaluated numerically by breaking each (0,1) interval
into subintervals, to improve sampling at large values of y, with boundaries
1
4
cos
π
j
20
−
,
j
=
0,1,...,10
(2.417)
and using five-point Gaussian integration on each subinterval.
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