Geoscience Reference
In-Depth Information
Box 6.5 (continued)
Z
Lh
Z
L
Cov
0
h
1
1
L
ðÞ
¼
½
fx
ð
þ
h
Þ
f
L
½
fx
ðÞ
f
L
dx
where
f
L
¼
fx
ðÞ
dx
.
L
h
0
0
It can be shown that
the expected value of this pseudo-covariance is
¼
ʻ˃
2
h
2
E
Cov
0
h
2
L
4
. Such pseudo-parabolic
behavior completely distorts the real behaviour of the random variable at
the origin.
ðÞ
3
3
h
þ
3
L
Þ
ð
0
h
L
Þ
Behavior of covariance and semivariogram near the origin and at very great
distances (
h
) generally is difficult to determine in practice, because it must
be based on extrapolation from observations at sampling intervals that, for
practical reasons, cannot be very small nor very large. Additional problems may
arise in practice when the data have frequency distributions that are positively
skewed with relatively few very large values that strongly affect estimations of
covariance and semivariogram. The purpose of the following example is to
consider extrapolations for
h
!1
!
0and
h
!1
before and after logarithmic
transformation.
Suppose that the semivariogram of
X
and log
e
X
are written as
ʳ
*(
h
) and
ʳ
(
h
)
ʳ
*(
h
)
¼
½
E
(
X
i
X
i
+
h
) and
ʳ
(
h
)
¼
½
E
(log
e
X
i
log
e
X
i
+
h
), respectively.
with
If it can be assumed that the mean
EX
¼
μ
exists and the variance (to be written
as
var
) is finite, the covariance satisfies
cov
(
h
) ¼
2
ʳ
*(
h
).
This covariance is related to the semivariogram for logarithmically transformed
values by means of (
cf
. Matheron
1974
; Agterberg
1974
, p. 339) as
cov
(
h
)
E
(
X
i
·
X
i
+
h
)
μ
¼
var
¼
var
· exp[
ʳ
(
h
)] provided that log
e
X
is approximately normal (Gaussian) with
2
variance
(
h
). An example is as follows.
Estimated covariances for the 118 Pulacayo zinc values of Table
2.4
are shown
in Fig.
6.13a
. The scale used for the covariance is logarithmic. Also shown is the
straight line corresponding to the signal-plus-noise model for
cov
(
h
)
˃
>> ʳ
¼
c
·
var
· exp
(
0.5157) previously shown in Fig.
2.10
. The exponential
covariance model of Fig.
6.13a
corresponds to a linear semivariogram for
logarithmically transformed zinc values as follows immediately from substitution
of
ch
)(
a
¼
0.1892;
c
¼
(
h
)]. Consequently, the semivariogram of
logarithmically transformed zinc values plotted on log-log paper should be
according to a straight line with slope equal to one. In Fig.
6.13b
it is shown that
this model is approximately satisfied. It can be concluded that the logarithmically
transformed zinc values approximately have a linear semivariogram. This conclu-
sion is in accordance with the linear semivariogram originally fitted by Matheron
(1964) to the logarithmically transformed zinc values.
ʳ
(
h
)
¼
3
A
· h into
cov
(
h
)
¼
var
· exp[
ʳ
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