Geology Reference
In-Depth Information
Substitution in (2.386) gives the variance of the final spectral density estimate as
1
var
S
gg
(
f
)
S
2
2
ρ(1)
+
ρ(3)
gg
(
f
)
κ
=
+
+
ρ(2)
3ρ(3)
ρ(1)
2
κ
−
+
2ρ(2)
+
1.490961
S
2
gg
(
f
)
κ
0
.
495895
κ
=
−
.
(2.389)
For a record of total length
T
, with 75%window overlap, the number of segments is
4
T
3
κ
=
M
−
.
(2.390)
The 75% overlap of the segments inflates the variance, for long records, by just
under 50%. This is more than compensated for by the fourfold increase in the
number of segments. The equivalent number of degrees of freedom, taking account
of the variance inflation due to segment overlap, is
2κ
1.490961
0
.
495895
κ
ν
=
−
.
(2.391)
As in the case of the average over multiple discrete segments (2.355), a confid-
ence interval of (1
S
gg
(
f
k
)isgivenby
−
α)
×
100% for
S
gg
(
f
k
)
x
(1
S
gg
(
f
k
)
x
(α/2)
,
ν
ν
−
α/2)
≤
S
gg
(
f
k
)
≤
(2.392)
where
x
(ζ) is the value of the random variable below which lies a fraction ζ of the
area under the cumulative χ
2
ν
distribution. On plots of the logarithm of the spectral
density, the confidence interval is the fixed length log[
x
(1
log[
x
(α/2)],
independent of frequency. The vertical confidence interval can then be parallel
transported on plots of the logarithm of the spectral density. The lengths of the con-
fidence intervals, on logarithmic plots of spectral density, are shown in Figure 2.17,
as functions of the number of degrees of freedom, for confidence levels of 90%,
95% and 99%.
−
α/2)]
−
2.5.4 The product spectrum
Often the power spectral density is found from observations at a number of obser-
vatories. These, typically, will contain uncorrelated systematic and random errors
Search WWH ::
Custom Search