Geoscience Reference
In-Depth Information
Table 12.4 Relationship
between
i
0
1
2
3
2
i
and
f
1
(
z
2
)
(Source: Agterberg
1984
,
Table 3)
ˈ
ˈ
0
0.04757
0.10983
0.18072
2
i
f
i
(
z
2
)
0.29078
0.37437
0.22493
0.08366
F
i
(
z
2
)
0.29078
0.66515
0.89008
0.97374
Table 12.5 Asymmetrical bivariate binomial model: comparison of observed and theoretical
frequencies for 40-km cells (Source: Agterberg
1984
, Table 4)
x
2
Observed frequency
Theoretical frequency
0
12
13.96
0-0.4757
17
17.97
0.4757-0.10983
12
10.80
0.10983-1.0
7
5.28
(
i
x
1
(
F
i
)+
x
1
(
F
i
1
). These values are slightly too large
because curvature of the frequency density function was neglected. The three
values of
¼
1, 2, 3) satisfy 2
ˈ
1
i
¼
ˈ
1
i
in Table
12.3
, together with
N
¼
15 and
q
1
¼
097333, were used as
2
N
j
¼ 1
c
j
2
2
j
W
(
q
1
) was tabulated as a
input to a computer program in which
σ
2
¼
∑
ˁ
. By means of this special-purpose table, the variance s
2
2
function of
ˁ
¼
0.00386
p
2
q
q
it follows that
p
2
¼0.07905 and
q
2
p
1
was converted into
r
¼0.565. From
ˁ
¼
q
2
¼
ˈ
2
i
derived in Box
12.6
can be used resulting in
the values shown in the first row of Table
12.4
in relation to
f
1
zðÞ
¼
0.92095. Now the equation for
q
N
2
p
2
.
The resulting theoretical frequencies are compared to the observed frequencies for
Fig.
12.48
in Table
12.5
. The degree of correspondence is fairly good. In a
goodness-of-fit test,
N
i
0
:
05
(1)
3.84. More general applica-
bility of the asymmetrical bivariate binomial model remains to be investigated. This
method can be useful for modeling arbitrary frequencies of empty cells as a
function of different cell size. The approach taken in Sect.
12.8
relies strongly on
application of the method of diagonal expansion applied to bivariate distributions.
Kotz (
1975
, p. 259) considered this method to provide the most powerful unified
attack on the structure of bivariate distributions (cf. Hutchinson and Lai,
1991
,p.
222). In Sect.
12.8
this concept was adopted to quantify the spatial distribution of
rock types represented on geological maps.
ˇ
2
¼
1
:
02 is less than
ˇ
¼
References
Agterberg FP (1970) Autocorrelation functions in geology. In: Merriam DF (ed) Geostatistics.
Plenum, New York, pp 113-142
Agterberg FP (1973) Probabilistic models to evaluate regional mineral potential. In: Proc Symp
Mathematical Methods in the Geosciences, P
ˇ
ibram, Czechoslovakia, pp 3-38
Agterberg FP (1974) Geomathematics. Elsevier, Amsterdam
Agterberg FP (1977) Frequency distributions and spatial variability of geological variables. In:
Ramani RV (ed) Application of computing methods in the mineral industry. American Institute
of Mining, Metallurgical, and Petroleum Industry, New York, pp 287-298
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