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Fig. 6.4 Twenty-three data
and 16 pairs that meet the
tolerance sketched in the
lower left of the figure. There
are some gaps where the data
are too close
In some cases there may be enough data to calculate the
variogram values for many different distances and directions
and plot a two- or three-dimensional map of variogram val-
ues. This is true for both gridded data and dense irregular
data. These maps are useful for detecting directions of an-
isotropy, and to avoid imposing pre-conceived ideas on the
variogram.
rately for each pair of values, using the average of the two
values:
2
1
[ ( )
z
u uh
−+
z
(
)]
h
γ
()
h
=
PR
2
2 ()
N
h
z
()
u uh
++
z
(
)
N
()
2
The variable used should be strictly positive because of the
denominators in both relative variogram's definition. Rela-
tive variograms were originally proposed to remove the
proportional effect commonly found in positively-skewed
mineral grade distributions. The standard deviation of the
samples within sub-zones of our “stationary” domain is like-
ly proportional to their mean grade. Experience has shown
that the relative variograms, and in particular the pairwise
relative variogram, are significantly more structured and
easier to model than the traditional variogram. This makes
it a suitable experimental variogram estimator in presence
of sparse, clustered and erratic data. A concern is that the
sill is not clearly defined—it depends on the shape of the
histogram and the coefficient of variation. One useful imple-
mentation approach is to transform the data to a lognormal
distribution with a mean of 1 and a variance of 1, then the
sill of the pairwise relative variogram is 0.44 (Babakhani
and Deutsch 2012 ).
The covariance defined Eq. 6.2 assumes that the domain
is stationary, and thus the mean of the data is the same at both
ends of the separation vector. A more general spatial cova-
riance definition is the non-ergodic covariance (Srivastava
1987 ) that does not assume that the averages of the tail and
head of the separation vector are the same:
6.2.1
Other Continuity Estimators
The experimental variogram is often calculated as in Eq. 6.4;
this is the traditional tool used to assess variability. However,
the presence of a small number of extreme data values can
cause the variogram to become very noisy. Normal scores,
indicator data, or log-normal transformations often make the
variogram more robust, but the nature of the variability is
changed. Several other continuity estimators have been pro-
posed to make the variogram function more robust (Journel
1988 ; Isaaks and Srivastava 1988 ).
The general relative and the pairwise relative variograms
were popularized by M. David ( 1977 ). The general relative
variogram standardizes the traditional variogram using the
squared average of the data points for each lag:
γ
(
= hhh
)
γ
( )/
m
(
)
2
GR
with the average of data for each lag being
mm
+
h
h
m
()
h
=
2
The pairwise relative variogram often produces a more clear
spatial continuity function than the general relative vario-
gram. The difference is that the adjustment is done sepa-
1
C
()
h
=
[() (
z z
u uh
i
+−
)]
mm
i
hh
N
() N
h
()
h
 
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