Geoscience Reference
In-Depth Information
exist if possible. However, intrinsic stationarity with infinitely large variance will
be assumed to hold true as an alternative approach to describe strictly local
variability of element concentration values in rocks (Sect.
6.2.6
).
Box 6.1: Autocovariance and Power Spectrum
The autocovariance of an ordered sequence of random variables is defined as
ʓ
h
¼
E
[(
X
k
μ
)(
X
k
+
h
μ
)]. The sample autocovariance for
n
values can be
h
P
nh
n
P
k
¼1
x
k
.
1
1
estimated by:
C
h
¼
ð
x
h
x
Þ
ð
x
kþh
x
Þ
where
x
¼
n
k
¼1
2
and the sample auto-
The autocorrelation function satisfies
ˁ
h
¼
ʓ
h
/
ʓ
0
¼
ʓ
h
/
˃
correlation coefficient is
r
h
¼
C
h
/
C
0
. Together the
r
h
values form a so-called
“correlogram”. Wiener's (
1933
)
theorem of autocorrelation states that
¼
R
1
1
ˁ
h
e
iˉh
dh
. Since
ˆ
ˉ
ˁ
h
is even,
the power spectrum satisfies:
(
)
¼
R
1
1
ˁ
h
cos
ˆ
hdh
. In the next section the power spectral density
P
(
f
)
will be computed as:
P
(
f
)
ˉ
ˉ
(
)
m
¼
C
0
+2
k
¼1
W
(
k
)
C
k
cos 2
ˀ
kf
with
W
(
k
)
¼
∑
k
/
m
). The weighting function
W
(
k
) defines a cosine-shaped lag
window for the autocovariance function whose use is called “hanning”.
(1 + cos
ˀ
½
The purpose of hanning is that individual values
P
(
f
) can be considered to
estimate a smoothed version of the underlying true spectrum. For more explana-
tions of why this procedure can be useful, see Blackman and Tukey (
1959
). The
power spectrum
P
(
f
) can be regarded as a decomposition of all variability in a
series in terms of components of the variance for narrow frequency bands. Smooth-
ing operations such as hanning significantly improve their estimation from the
autocovariances by eliminating distortions. An alternative approach of constructing
the power spectrum (Sect.
6.2.7
) consists of averaging adjoining values in the
periodogram over equal intervals.
6.1.1 Spectral Analysis: Glacial Lake
Barlow-Ojibway Example
Glacial varves, because of their presumed annual nature have attracted the attention
of geologists as a geochronological tool. Anderson and Koopmans (
1963
) were
among the first to apply spectral analysis to varves. Theory of cross-correlation and
cross-spectral analysis was treated by Goodman (
1957
), Amos and Koopmans
(
1963
), and Kendall and Stuart (1966), with applications by Hamon and Hannan
(
1963
), Koopmans (
1967
), Anderson (
1967
) and Agterberg and Banerjee (
1969
).
Lake Barlow-Ojibway is a late-glacial water body that formed approximately
11,000 years ago during the retreat of the Late Wisconsin ice-sheet in northern
Ontario and western Quebec. The lake, in its maximum extent, measured about
960 km in the east-west and 240 km in the north-south direction (Fig.
6.1
inset).
The most extensive deposit formed in it was the sheet-like body of varves which
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