Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 30

Fig. 1.18 Probability that
random square cell
measuring (2 n +1) 259 m
on a side contains one or
more deposits ( n denotes
number of dilatations of
set C ) (Source: Agterberg
and Fabbri 1978 , Fig. 5)

30

20

10

0 0

5 10
NUMBER OF DILATATIONS

15 20

14 deposit points. Hence the probability that an arbitrary pixel in this shell is a
deposit point is equal to 14/5,360 0.00261. This probability is one of the pro-

portions for separate shells shown in the histogram of Fig. 1.17 . The pattern of
Fig. 1.14b consists of 9,990 pixels and measures mes (

¼

Α ʘ

2 ) \ C ¼ 10 pixels. This

Β

indicates that 36-10 26 of the 40 deposits (or 65 %) occur in the zone identified as

acidic volcanic rocks on the geological map and within (2

¼

) 733 m
from a contact between acidic volcanics and other rocks on this map. This zone may
be favorable relatively for the occurrence of volcanogenic massive sulphide
deposits. The probability that a random point in the zone is a deposit point amounts
to 26/(18,843

2 259 m ¼

√

9,990) ¼ 0.00294. This is about eight times greater than the proba-

bility ( 0.00039) that a random point in the entire study area is a deposit point. On

the other hand, it is only about 1.5 times greater than the probability (

¼

0.00191)
that an arbitrary black pixel of the original pattern (Fig. 1.14d ) is a deposit point.
As mentioned before, 36 of the 40 deposit points (or 90 %) coincide with the
acidic volcanics of Fig. 1.14d . A generalized form of this ratio for a cell with side
(2 n +1)

¼

259 m is:

mes A \ ð C nB Þ

M d 1 ¼

mes C nB

For the patterns shown in Fig. 1.16 , M d 1 amounts to 0.646 (Fig. 1.16d ), 0.537
(Fig. 1.16e ), and 0.456 (Fig. 1.16f ). It represents a weighted average proportion of
acidic volcanics per cell for cells centered about the deposits. The preceding
probabilities and ratios follow directly from Minkowski operations on sets. Other
problems cannot be solved by measurement only but need a combination of
measurement and statistical modeling. An example of such a problem is the
determination of the frequency distribution of the random variable X with