Geoscience Reference
In-Depth Information
A useful technique for the study of cross-correlation, which had not yet been
developed at the time of the varve analysis project of Agterberg and Banerjee (
1969
),
is wavelet analysis (see Hubbard
1996
). Applications of wavelet analysis to sedimentary
sequences include Prokoph and Agterberg (
1999
) and Prokoph and Bilali (
2008
).
6.1.3 Stochastic Modeling
Agterberg and Banerjee (
1969
) discuss how the theoretical autocorrelation function
of a second order stochastic process was fitted experimentally to the silt and clay
thickness correlograms shown in Figs.
6.7
and
6.8
, respectively. The corresponding
theoretical power spectra are shown in Fig.
6.11
. The corresponding theoretical
phase difference is given in Fig.
6.12
. For periods greater than 130 years, the clay
leads the silt. This corresponds to the frequency bands where most of the power in
concentrated in Fig.
6.11
for both silt and clay. The zero crossing points in the
cross-correlogram of Fig.
6.9
occur at 56
years indicating that the
major oscillations for the silt are not in phase with those of the clay, but that, on the
average, clay leads silt by about 15
and
25
½
½
years. This phase lag of the silt is explained, at
least in a qualitative manner by the second-order stochastic model (Fig.
6.12
).
½
Box 6.4: Time-Dependent Stochastic Processes
The continuous
m
-th order autoregressive process is represented by the
stochastic differential equation:
a
m
d
m
x
a
m
1
d
m
1
x
dt
m
1
þ
þ
...
þ
a
0
xt
ðÞ
¼
E
ðÞ
t
dt
m
where
(
t
) is a white-noise function driving the stationary random variable
x
(
t
). Its autocorrelation function is of the type:
E
A
1
e
ʻ
1
jxj
þ
A
2
e
ʻ
2
jxj
ˁ
ðÞ
¼
x
A
m
e
ʻ
m
jxj
where
A
1
,
A
2
,
þ
...
þ
ʻ
m
represent the real or complex roots of a polynomial. A pair of conjugate
complex roots can be combined into a single term of
,
A
m
are constants, and
ʻ
1
,
ʻ
2
,
,
...
...
the type
e
k
1
jxj
cos
k
2
þ ˆ
ð
Þ
where
k
1
,
k
2
and
ˆ
are constants. The first order process
results in
ˁ
(
x
)
¼
exp (
a
·
j
x
j
). The second order can be written in the form:
xt
d
2
x
dt
2
dx
0
2
0
þ
2
ʱ
dt
þ ˉ
þ ʱ
ðÞ
¼
E
ðÞ
t
.If
ˉ
>
0,
ʻ
1
and
ʻ
2
form a pair of conju-
h i
. The latter result can only
be applied to discrete data if the sampling interval is sufficiently small
(
cf
. Yaglom
1962
; Jenkins and Watts
1968
; Kendall and Stuart
1958
). Fourier
þ
ʱ
sin
ˉ
0
jxj
ˉ
0
ˁ
ðÞ
¼
x
e
ʱjxj
cos
ˉ
0
x
gate roots, and:
transformation of the second order autocorrelation function gives:
f
ðÞ
¼
ð
Þ
=ˀ
2
0
þʱ
2
2
ʱˉ
2
where
ˉ
represents angular frequency. The corresponding
phase lag satisfies:
2
ð
Þ
ˉ
2
ʱ
2
ˉ
2
0
þ
4
ʱ
2
ˉ
ˆðÞ
¼
2
ʱˉ
ˉ
2
(
cf
. Parzen
1962
, p. 112; Sommerfeld
2
0
þʱ
2
ˉ
1949
, p. 101).
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