Regionalized Variable and Random Function

A mineralized phenomenon can be characterized by the spatial distribution of a certain number of measurable quantities called ``regionalized variables''. Examples are the distribution of grades in the three-dimensional space, the distribution of vertical thicknesses of a sedimentary bed in the horizontal space, and the distribution of market price of a metal in time.

Geostatistical theory is based on the observation that the variabilities of all regionalized variables have a particular structure. The grades $z(/boma{x})$ and $z(/boma{x+ h})$ of a given metal at points $/boma{x}$ and $/boma{x+ h}$ are auto-correlated. If it rains here then it may be probable that it is also raining at a certain distance $/boma{h}$. This auto-correlation depends on both the vector $/boma{h}$ (in modulus and direction) which separates the two points, and on the particular mineralization considered. The variability of gold grades in a placer deposit will differ from that of gold grades in a massive deposit. The independence of the two grades $z(/boma{x})$ and $z(/boma{x+ h})$ beyond a certain distance h is simply a particular case of auto-correlation and is treated in the same way.

We have used the word probable. If one observes the value of a variable, as grade, concentration, etc. at a certain point (or very small region) in space, then this value could be smaller or larger than the average value in a larger region, and this by random. We such have the value $z(/boma{h})$ at a certain point $/boma{h}$ connected with some random deviation. Therefore we interpret the observed value $z(/boma{h})$ as a realization of a random variable (a realization of random quantity with-eventually infinite-many possible values).

Without touching theoretically deep axioms in probability theory, we may understand intuitively a random variable as a function of random (elementary) events to the real numbers which, of course, needs to have certain well-defined properties.

Let us consider a simple example with throwing of dice. After the throwing of a ``true'' die, it will show a number (1, 2, 3, 4, 5 or 6) where it is supposed that each number has the same chance to appear. The probability of each number is $P=/frac{1}{6}$. We write

A special number of the die $z=i$ represents a realization of $Z$. Let us consider a little bit more complicated experiment with the throwing of $n$ $(>1)$ dice at the same time. We are interested only in the sum $Y$ of all shown numbers. The range of values of $Y$ is $n$, $n+1, /ldots, 6 /times n$ and the probability of these values can easily be calculated. E.g. for the probability of getting $n$ we have

The probability of each other value of $Y$ can be found by combination of the corresponding numbers (elementary events) shown by the dice, which probabilities are known.

Similar reasonings can be used in the much more complicated case of variables which appear in geosciences. The main difference may be stated as the impossible access to the source of the random mechanism. The grade of ore $z(/boma{x})$ at the point $x$ in space has been produced by a series of elementary events (as rock movements, chemical reactions, etc.), which are not known in detail. If the random mechanism acted differently, we had a different result of $Z(/boma{x})$. We see that the result of $Z(/boma{x})$ has been realized only with a certain probability. All possible (elementary) events are combined in the random variable $Z(/boma{x})$, and a certain probability distribution is associated with $Z$. Figure 1.1 represents the distribution of iron in % in a certain region. The polygon line represents the empirical values (a sort of histogram) and the smoothed curve ($f$) is a sort of stylized presentation. The area under the curve $f$, which is called probability density function, by definition is one. The probability that a value of $Z$ falls in the interval [a,b] can be easily determined by integration:

The function $P$ on events is also called probability measure.

Figure 1.1: Frequency Distribution of Iron in %.

Besides the density $f$ we frequently use the notion of the cumulative distribution function $F$ which is defined by summation:

Summarizing we see that a random variable $Z(/boma{x})$ is described by the probability measure $P$, by $F$ or $f$. A particular realization is denoted by the corresponding small letter $z(/boma{x})$. The value $z(/boma{x}')$ at another place $/boma{x}'$ essentially cannot be interpreted as a realization of the same random variable $Z(/boma{x})$, but as a realization of another random variable, namely $Z(/boma{x}')$. The function, which is defined at each point $/boma{x}$ of a certain domain $D$ as a random variable $Z(/boma{x})$ is called random function. Besides the probability distribution at each place $/boma{x}$ there still are connections (correlations) between the variables $Z(/boma{x})$ and $Z(/boma{x}')$ to be considered. The regionalized variable $z$ represents a realization of the random function $Z$.