Geoscience Reference
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Fig. 6.3 Correlograms of log-thickness data, silt and clay, series 4. Best-fitting negative
exponential curves with
m
¼
10 and
m
¼
50 are also shown (Source: Agterberg and Banerjee
1969
, Fig. 6)
end results. The logarithmic transformation stabilizes the variance. For the example,
the ratio of the variances for the first and last sets of 100 silt values for series 4 is 33.6,
but after logarithmic transformation it becomes 1.5.
Box 6.2: Signal-Plus Noise Model
Suppose a series of observed values (
x
k
) is the sum of two series: signal (
s
k
)
and white noise (
n
k
) with
k
¼
1, 2,
,
n
. The autocovariance functions of
...
these series written as
ʓ
x
(
h
),
ʓ
s
(
h
) and
ʓ
n
(
h
) satisfy the relations:
ʓ
n
(
h
)
¼
0if
h
c
. Let
F
(
h
)
denote a filter to which the record must be subjected in order to obtain the
6
¼
0 and
ʓ
x
(
h
)
¼
ʓ
s
(
h
)+
ʓ
n
(
h
). Suppose
ʓ
x
(0)
¼
1 and
ʓ
n
(0)
¼
signal:
s
k
¼
R
1
1
¼
R
1
1
+
R
1
1
F
(
h
)
x
(
k
+
h
)
dh
. Then,
ʓ
s
(
h
)
F
(
h
+
˄
)
ʓ
s
(
˄
)
d
˄
F
¼
R
1
1
(
h
+
˄
)
ʓ
n
(
˄
)
d
˄
or
ʓ
s
(
h
)
F
(
h
+
˄
)
ʓ
s
(
˄
)
d
˄
+(1
c
)
F
(
h
). Fourier transfor-
mation of both sides gives:
G
(
ˉ
¼
ʦ
ˉ
)·
G
(
ˉ
c
)
ʦ
ˉ
)
(
)+(1
(
) where
¼
R
1
1
ʓ
h
cos
¼
R
1
1
G
(
ˉ
)
ˉ
hdh
and
ʦ
(
ˉ
)
F
h
cos
ˉ
hdh
. It follows that:
G ðÞ
ʦðÞ
¼
. The filter
F
(
h
) can be found by taking the inverse Fourier
G
ðÞþ
ð
1
c
Þ
¼
R
1
1
ʦ
transform:
F
(
h
)
(
ˉ
)cos
ˉ
hd
ˉ
. When
ʓ
x
(
h
)
¼
exp(
a
j
h
j
), then:
G
Z
1
1
2
ac
a
2
ac
ˀp
2
1
c
[
a
2
+2
ac
/
ðÞ
¼
þc
2
and
Fh
ðÞ
¼
1
cos
ˉ
hd
ˉ
where
p
¼
ð
Þ
ˉ
2
p
2
þ
1
c
)]
0.5
. After
(1
some manipulation it
follows
that:
F
(
h
)
¼
q
exp
(
p
j
h
j
) where
q
¼
ac
/[(1
c
)
p
](
cf
. Yaglom
1962
; Agterberg
1967
,
1974
).
Correlograms for log-thickness data, silt and clay, for series 4 are shown in
Fig.
6.3
. Best-fitting semi-exponential curves with
r
h
¼
c
·exp (
a
j
h
j
)where
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