Geoscience Reference
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Box 6.6 (continued)
Z 1
Z 1
Þ ¼2 ˀðÞ
n
2
ʾ x ;
ðÞ ¼
t
d
y
G
ð
x ; y ; ˄
Þ E x, t
ð
˄
Þ
d
˄
where G
ð
x ; y ; ˄
1
0
. The spatial covariance function is isotropic and satisfies:
2
fʱ˄ x y
j
j
exp
Z 1
2 ˄
Z 1
ʓ
(s)
¼
cov[
ʾ
(x, t ),
ʾ
(x + s, t )], and
ʓ ðÞ ¼
d
y
G
ð
x ; y ; ˄
Þ
G
ð
x þ s
,
y
,
˄
Þ
Z 1
1
0
e 2 ʱ˄s 2
1
4 ˀðÞ
= 4 ˄ d
d
˄ ¼
˄
where s is distance along a line in any direction.
n
=
2
2
0
p
e s
2 ʱ
ʱ
This expression can be evaluated as follows:
representing
the semi-exponential also generated from the first-order stochastic differential
equation;
ʓ
ðÞ ¼
s
ð
n
¼
1
Þ
p
2
p
2
n
1
2
representing the well-known 2-D result
first independently derived in 1948 by von K´rm´n( 1948 ) and Mat´rn ( 1981 ;
English version of topic published in Swedish in 1948);
ʓ
ðÞ ¼
s
K 0 s
ʱ
ð
¼
2
Þ
ˀ
2 ʱ
p
2 ˀs
e s
ʓ
ðÞ ¼
s
ð
n
¼
Þ
3
representing Whittle's relatively unknown 3-D solution.
Table 6.1 Pulacayo Mine variogram model
h , m.
˃ h 2
Experimental
Exponential
f ( L , h )
ʲ (h)
Deviation
2
0.303
0.325
2.891
0.105
0.286
0.017
4
0.402
0.367
3.580
0.112
0.354
0.048
6
0.436
0.401
3.985
0.109
0.394
0.042
8
0.465
0.429
4.273
0.109
0.422
0.043
10
0.408
0.452
4.496
0.091
0.444
0.036
12
0.412
0.471
4.678
0.088
0.462
0.050
14
0.464
0.486
4.832
0.096
0.477
0.013
16
0.452
0.499
4.966
0.091
0.491
0.039
18
0.472
0.510
5.083
0.093
0.502
0.030
20
0.545
0.518
5.189
0.105
0.513
0.032
Source: Agterberg ( 2012 , Table 1)
Experimental values from Matheron ( 1962 , p. 180); Lag distance ( h ) in m; Experimental values
from model of Fig. 6.16 ; f (
2 h
) is extension
variance of 50 cm line segments; Deviation is difference between columns 2 and 6. The small
deviations indicate good fit of Matheron's variogram model
ʸ
) as in text;
ʲ
( h ) ¼Experimental value/ f (
ʸ
);
˃
¼ ʲ
f (
ʸ
As explained in Box 6.5 , Whittle ( 1962 ) derived another theoretical equation for
a space series along a line in 3-D. It differs from the semi-exponential correlogram
shown in Fig. 6.14 in that distance (lag) occurs as a linear term of the denominator.
Suppose that both sides of the equation are multiplied by distance. Then the term on
the left-hand side (called xy in Fig. 6.15a ) becomes semi-exponential. However,
application of this simple curve-fitting method to the Pulacayo zinc values results in
a line that is approximately horizontal suggesting
ʱ ¼
0. Consequently, the resulting
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