Geology Reference
In-Depth Information
0.5
0.4
0.3
0.2
1
v
=
23
0.1
0
0
2
4
6
8
10
12
Random variable
x
2
ν
Figure 2.16 The χ
(
x
) probability density function for degrees of freedom ν
=
1,2,3,10. For large ν it approaches that of a normal distribution.
such that
T
=
κ
M
. The final spectral density estimate is formed by taking the mean
of the κ independent estimates. It is
κ
1
κ
S
gg
(
f
k
)
S
gg,
l
(
f
k
).
=
(2.353)
l
=
1
This averaged spectral density estimate has 2κdegrees of freedom, and the variance
of an individual estimate is diminished by the same factor. Hence,
S
gg
(
f
k
)
S
gg
(
f
k
)/2κ
(2.354)
2
2κ
2
is χ
distributed. The probability density for the χ
ν
(
x
) distribution is shown in
Figure 2.16 for degrees of freedomν
=
1,2,3,10 as a function of the random vari-
able
x
. As the number of degrees of freedom increases without limit, this distribu-
tion approaches a normal distribution, as is true of all distributions by the
central
limit theorem
.
A confidence interval for the averaged spectral density
S
gg
(
f
k
) can be established
from the χ
2
2κ
(
x
) probability distribution. For
a
confidence
interval of
(1
−
α)
×
100%, the range is (Jenkins and Watts, 1968, pp. 254-255)
S
gg
(
f
k
)
x
(
1
S
gg
(
f
k
)
x
(
α/2
)
,
2κ
2κ
−
α/2
)
≤
S
gg
(
f
k
)
≤
(2.355)
where
x
(ζ) is the value of the random variable below which lies a fraction ζ of the
area under the cumulative χ
2
2κ
distribution.
2.5.3 Overlapping segment analysis
Although the average over multiple discrete segments allows the reduction of the
variance of the spectral density estimate, it is very wasteful of the data. Here we
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