Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 228

Quantitative modeling of the nugget effect in KTB copper determinations has
yielded better results than could be obtained for our examples frommineral deposits
including the Pulacayo Mine. This is not only because the series of chemical
determinations is much longer but also because the nugget effect remains clearly
visible over lag distances between 2 m (

¼ original sampling interval) and 10 m.

6.3 Autocorrelation of Discrete Data

In order to visualize theoretical autocorrelation of discrete data, the example of
random succession of lithologies in Sect. 2.2.5 can be considered again. For a point
in a specific state (e.g., shale), there is a fixed probability that the next point along
the section will be in the same state or that another state will occur. Suppose that
instead of using letters to indicate the states, presence of a given state is coded as +1
and its absence as 1. The result is a series of discrete data that can be plotted as in

Fig. 6.25 . The main characteristic of this situation is that a transition from +1 to 1

or from 1 to +1 can occur any time with a probability that is independent of place

of occurrence along the series.
Assume that Fig. 6.25 represents a process that started at T

¼ 1 . For any

discrete point k , the probabilities of being in state +1 or 1, are equal to constants

adding to one. For simplicity, it is assumed that P ( X k ¼ 1) ¼ P ( X k ¼ 1) ¼ ½ .

Consequently, E ( X k ) 0. This series is stationary and has expectation equal to

zero. The autocovariance function satisfies:

¼

E( X k · X k + h ). This is the sum of
two terms: (1) the probability that the number of changes in state is even during the
interval [k,k + 1] multiplied by +1; and (2) probability of an odd number of changes
multiplied by

ʓ h ¼

1, or:

ʓ h ¼ P k ¼0

e ʻjhj

ʻ jð k
k

e 2 ʻjhj where P k ¼ denotes the Poisson-

type probability of having exactly k changes in an interval of length h .The
parameter

ð P 2 k P 2 kþ 1 Þ ¼

!

ʻ can be interpreted as the number of changes (both from +1 to 1

and 1.

The absolute value of lag h indicates that proceeding in the positive or negative

1 to +1) expected per unit of time or distance along the axis when j h j ¼

Fig. 6.25 Random
telegraph signal (After
Jenkins and Watts 1968 ).
Zero crossings between two
states +1 and 1 occur at

random along the line
(Source: Agterberg 1974 ,
Fig. 57)