Geoscience Reference
In-Depth Information
6
Spatial Variability
Abstract
An essential aspect of geostatistical modeling is to establish quantitative measures of spatial
variability or continuity to be used for subsequent estimation and simulation. The model-
ing of the spatial variability has become a standard tool of mineral resource analysts. In the
last 20 years or so, the traditional experimental variogram has given way to more robust
measures of variability. Details of how to calculate, interpret and model variograms or their
more robust alternatives are contained in this chapter.
6.1
Concepts
dependent; hence, the notation Z(
u
), with
u
being the coor-
dinate location vector.
A random function (RF) is a set of RVs defined over some
field of interest, for example, {Z(
u
),
u
is an element of study
area A} also denoted simply as Z(
u
). Usually the RF defini-
tion is restricted to RVs related to the same attribute, hence,
another RF would be defined to model the spatial variability
of a second attribute, say {Y(
u
),
u
is an element of the study
area A}.
The use of Random Functions implies that the variables
are within a subset of the deposit or an area that is considered
stationary. The ability to apply the RF concept is based on
the belief that the locations
u
in A and variable Z belong to
the same statistical population. The purpose for conceptual-
izing a RF as {Z(
u
),
u
is an element of study area A} is never
to study the case where the variable Z is completely known.
If all the z(
u
)'s were known for all
u
in the study area A,
there would be neither any problem left nor any need for
the concept of a random function. The ultimate goal of a RF
model is to make some predictive statement about locations
u
where the true outcome z(
u
) is unknown.
Just as a RV Z(
u
) is characterized by its cumulative distri-
bution function (cdf), a RF Z(
u
) is characterized by the set of
all its N-variate cdfs for any number N and any choice of the
N locations
u
i
, i = 1,…, N within the study area A:
Mineral grades are generated through a succession of geo-
logical processes not always completely known or under-
stood. Necessary conditions for mineral deposition include
mineralization sources, pathways, and favorable geological
conditions for deposition. The right physical and chemical
processes can lead to significant mineral concentrations. The
characteristics of mineral deposition invariably impart pat-
terns of spatial correlation that are important for resource
evaluation and mine planning.
The description and modeling of these correlation pat-
terns allows better understanding of the depositional pro-
cesses and improves on the prediction of mineralization and
mineral grades at unsampled locations. Statistical tools can
be used to describe those correlations within an appropriate
theoretical framework.
The material in this section summarizes other geosta-
tistical texts such as Geostatistical Ore Reserve Estimation
(David
1977
), Mining Geostatistics (Journel and Huijbregts
1978
), An Introduction to Applied Geostatistics (Isaaks and
Srivastava
1989
), or Geostatistics for Natural Resources
Evaluation (Goovaerts
1997
).
Random Function Concept
The uncertainty about an
unsampled value z is modeled through the probability dis-
tribution of a random variable (RV) Z. The probability dis-
tribution of Z after data conditioning is usually location
F
(,
uu
… … =
,
;,
z
,
z
)
Prob Z
{()
u u
≤… ≤
z
,
,
Z
( )
z
}
1
N
1
N
1
1
N
N
