Regionalised Variables: Conceptual BackgroundBackground
Random Functions Random Functions
The observed value at each data pointx can be considered as the outcomez(x) of a random variable (RV)Z(x). Figure 5.2 illustrates the relationship involved. The mean of the RVZ(x)at the pointx is called the driftm(x).
At locations in space where no samples are available, the values ofz(x) are well defined, even though they are unknown. The values ofz(x) at these locations may be viewed as outcomes of random variablesZ(x). In mathematical terminology, the family of all such RV's,Z(x) is called aRandom Function (RF)15.
A random function bears the same relationship to one of its realisations as a random variable does to its outcome. Note that the realisation of an RF is a function, whereas the outcome of an RV is a number. A random function is characterised by the joint distribution of a set of random variables, i.e.
k k x x x x Z x Z x
Z ( ), ( ),K ( ) forall points , ,K 2 1 2
1
15Synonyms for RF include ‘stochastic process’ and ‘random field’.
Noise Noise
The ‘random’ or chaotic component of grade variability cannot be ignored
if we are to have ef ficient estimation.
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For our probabilistic model to be useful we need to make some assumptions about the characteristics of these distributions. Specifically, as discussed earlier, we only have one realisation available (in general). This is a general problem ofstatistical inference : when only a single realisation is available we require further assumptions. These additional assumptions, or hypotheses, reduce the number of parameters upon which the RF depends.
The whole point is to introduce the minimum number of hypotheses to enable our model to cover the widest range of practical situations.
Note that no estimation methodology is devoid of assumptions, and that some of the assumptions behind ‘classical’ methodologies are really very strong. For example, in a polygonal estimate, we assume that the grade isconstant over the area of influence defined by a polygon!
Figure 5.2 Summary of Concepts (Random Variables & Random Functions)
Strong or Weak? Strong or Weak?
In science and mathematics, ‘weaker’ hypotheses are usually preferred because they admit more general cases.
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Stationarity
Stationarity
It is common is many statistical applications to assume that a variable can be considered as beingstationary . In other words, that the distributional law of the variable is invariant (does not change) under translation. A stationary random
function is homogeneous and self-repeating in space. The assumption of stationarity makes statistical inference possible.
Strict Stationarity Strict Stationarity
In its strictest sense, stationarity requires all the moments of the distribution to be invariant under translation, i.e. exactly the same distribution at every point in the field considered. This cannot be verified from the limited sampling usually available. In any case, such a strong assumption is not necessary to enable statistical inference in geostatistical applications.
Weak or 2nd Order Stationarity Weak or 2nd Order Stationarity
In geostatistics we usually require only that the first two moments of the distribution—the mean and the covariance—be invariant under translation (i.e. constant). This is called‘weak’ orsecond order stationarity For this hypothesis we assume:
1. That theexpected value (or mean) of the RFZ(x) is constant for all pointsx , i.e.
E Z x ( )
=
m x ( )=
mfor anyx.
2. That the covariance function C(h) between any two points x and (x+h), where (x+h) is separated from x by a vector distance h, is independent of the location of the points x and (x+h). This is expressed mathematically as:
E Z x Z x h ( )
⋅ + − =
( ) m 2 C h( )In other words the covariance between any two points depends only on the distance and direction between the two points, not on the specific locations of the points themselves.
In particular, whenh=0 the covariance comes back to the ordinary variance of
Z(x), which must also be constant under the assumptions of weak stationarity. The Intrinsic Hypothesis
The Intrinsic Hypothesis
In practice, it is often the case that the assumptions of weak stationarity are not satisfied. Clearly, when there is a marked trend in the mean (for example a pronounced and systematic increase in grades towards the core of a mineral deposit) the mean value cannot be considered constant, so assumption (1) above will not be valid. Likewise, there are situations where definition of a constant
Lotteries Lotteries
The probability of getting a particular number in a lottery Intrinsic Intrinsic Hypothesis Hypothesis
This is the practical definition for stationarity used in mining geostatistics.
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covariance is problematic. Refer to Journel and Huijbregts (1978) for a more expanded discussion.
So, on both theoretical and practical grounds it is convenient to be able to further weaken our stationarity hypothesis.
Under theintrinsic hypothesis we suppose that the increments of the function are weakly stationary. This means that the mean and variance of the increments, i.e.
Z x h (
+ −
) Z x( )are independent of the specific location of the pointx . The intrinsic hypothesis can be summarised:
E Z x h Z x Var Z x h Z x h ( ) ( ) ( ) ( ) ( )
+ −
=
+ −
=
0 2γThis is the intrinsic hypothesis with zero mean increment.
Using the intrinsic hypothesis means that we have decided that it is appropriate to pool sample pairs, separated by (approximately) the same distance vector, in the domain of interest.
The function γ h( ) is called thesemi-variogram (we usually say thevariogramfor short). The variogram is the basic tool of geostatistical structural analysis and it is employed for subsequent estimation. Given its importance to geostatistics, we will consider the variogram in some detail in this and the next chapter. However, before we continue, we will first discuss the practical aspects of stationarity a little further.
The Stationarity Decision The Stationarity Decision
It is important to understand that stationarity is a property of the model, not of the phenomena we are considering. Furthermore, the correctness of the decision to assume stationarity in our model cannot be refuted or provena priori . The decision of stationarity is a decision we make based on all the information at hand, for example:
Geological zonation. Weathering domains
Statistical characteristics (especially variability)
Spatial variation (as characterised by the variogram or equivalent) Assumptions about zones of ‘homogeneous’ mineralisation, etc.
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The decision of stationarity is thus an ‘expert decision based ona piori assumptions about the homogeneity of the zones over which averaging is to take place’ (Journel, 1987).
Note also that in practical estimations, the variogram is only used up to a certain distance. This limit is generally the diameter of the search neighbourhood to be used in kriging. Consequently, stationarity need only apply to separations up to this distance: the further limitation of the stationarity hypothesis to distances less than this is called the hypothesis ofquasi-stationarity.
Accepting the assumption of quasi-stationarity means that we can consider a series of sliding neighbourhoods within which stationarity applies. As such the decision of quasi-stationarity is scale-dependent.