Geoscience Reference
In-Depth Information
The corresponding non-ergodic correlogram (Srivastava and
Parker
1988
) is commonly used as a robust alternative to the
traditional variogram. The sample correlogram is calculated
with:
emerge from short-scale data, compared to the larger, do-
main-wide scale.
Decisions as to whether clustering is truly imparting an
artifact to the variogram model should be made considering
geologic information. A few possible solutions may be to (1)
remove the clustered data, leaving an underlying grid; (2)
discredit unusually high variogram points for the short dis-
tances; or (3) using more robust measures such as the sample
correlogram or pairwise relative variogram rather than the
traditional variogram.
The interpretation of the variogram consists of explaining
the variability over different distance scales as a function of
known geological and mineralogical factors. Experimental
variograms should always be reconciled with known geol-
ogy. Potential artifacts introduced by data configurations
and sampling practices should be discussed with geologists
familiar with the deposit. The degree of continuity observed,
the anisotropies observed, and the relative variances of
each structure should be discussed. The variogram function
should be representative of the expected geologic variability.
Discrepancies that may arise between perceived geologic
knowledge and interpretations and inferences from the ex-
perimental variograms should be resolved before proceeding
with the resource evaluation process.
There are four common cases in variogram interpretation:
trends, cyclicity, zonal (or areal) anisotropy, and geometric
anisotropy (Fig.
6.5
). Trends are common in mineral deposits,
and often question the definition of stationarity chosen. For
example, Cu grades tend to decrease towards the periphery
of a typical Porphyry Cu deposit, while the grades in some
base metal mines are a function of the porosity of the host
rock; if it is sedimentary or pseudo-sedimentary, a clear trend
across the direction of deposition will be found. These trends
cause the variogram to increase above its sill variance
σ
2
, or
stationary variance, and show a negative correlation at large
distances.
Cyclicity can be a result of the mineral depositional
phenomena occurring repeatedly over geologic time and
leading to repetitive en echelon variations in the resulting
mineral grades. Cyclical behaviors are more common, but
not exclusive to, sedimentary or strata-bound deposits. This
behavior is observed in the variogram as cycles of positive
and negative correlation at the length scale of the geologic
cycles. These cyclic variations often dampen out over large
distances as the size or length scale of the geologic cycles
is not perfectly regular. These cycles are usually modeled
with sinusoidal functions, often referred to as a
hole effect
. In
early mining geostatistics, cyclicity was observed in “down-
hole” variograms; hence the name
hole effect
.
Anisotropic variograms are extremely common in mining
and other geostatistical applications. Occasionally the pat-
tern of continuity will be similar in all directions, and thus
the variogram is said to be isotropic. But by far the most
common occurrence is for variograms to be anisotropic.
ρ= ⋅
h
() C()/(
hh
σσ
−
)
h
where C(
h
) is defined above and:
1
1
∑
∑
m
=
z
( ),
u
m
=
z
(
uh
+
)
−
h
h
N
()
h
N
()
h
N
()
h
N
()
h
1
∑
σ
=
[ ( )
z
u
−
m
] ,
2
−
h
−
h
N
()
h
N
()
h
1
∑
and
σ
=
[(
z
uh
+−
)
m
]
2
h
h
N
()
h
N
()
h
This measure is robust because of the use of lag-specific
mean and variance values. In practice, it has become the
most popular option when dealing with untransformed vari-
ables.
6.2.2
Inference and Interpretation
of Variograms
The single biggest problem in variogram interpretation is a
lack of data to calculate a reliable variogram. Too few data
for reliable variogram interpretation does not mean there is
no variogram; it is just hidden or masked by data paucity.
Geological analogues or expert judgment may be required.
In general, inference is affected by data density; the use of
different data types (drill holes, blast holes, trench samples,
etc.); influence of outliers; and trends. Also, a high relative
variability of the samples as measured, for example, by the
coefficient of variation (CV), is an indication that robust
measures of continuity are necessary.
Inference is an iterative process and exploratory in nature.
Inference generally starts with the initial sample collection,
which results in early geologic interpretation and estimation
domain definition.
Spatial clustering is common in regions of high grades
that will likely be mined sooner. Geologists would naturally
seek to confirm and carefully delineate such areas. These
clustered data, however, can cause the variogram at short
lags to be too high or too low, which could lead to a misin-
terpretation of the variogram structure. The most common
misinterpretations would be a too-high a nugget effect or
short-scale structure. There can also appear to be a short-
scale cyclic characteristic. The clustered data could provide
an improved representation of the higher grades (and higher
variance if a proportional effect exists) sub-zone within the
stationary domain. Different patterns of anisotropy may
