Geoscience Reference
In-Depth Information
Fig. 6.7
Example of three
directions, experimental
correlograms with their fit,
Arsenic data
which keeps increasing above the theoretical sill. A “power”
or “fractal” variogram model could be fit to the experimental
variogram; however, these models do not have a sill value
(it is infinite), and thus they have no covariance counterpart,
which is a problem for most geostatistical applications.
If the trend is removed, variogram analysis and all sub-
sequent estimations or simulations are performed on the re-
siduals. The trend is added back to estimated or simulated
values at the end of the study. Although there are difficulties
in defining a robust trend model and removing the its deter-
ministic portion from the data, the only practical option is to
model trends deterministically. Removing a trend by estima-
tion from the data themselves can introduce a bias; however,
this bias may be less significant than the errors introduced
by leaving the trend alone. It is often suggested (Journel
and Huijbregts
1978
and others) that the variogram should
be computed in directions and/or areas where the trend does
not appear as significant. Directly considering the residuals
in variogram calculation can lead to erroneously high vari-
ability.
nuity and anisotropy, considering also that most of the clus-
tered samples in these types of deposits sample the higher
grade mineralization. A larger intrusive body or the geom-
etry of the host rock may be responsible for a larger-scale
anisotropic feature.
The use of nested structures provides enough flexibility
to model most combinations of geologic controls. However,
there is no gain in over-modeling and over-fitting. It is ex-
tremely rare that more than three nested structures would be
necessary; only in those cases where, for example, a zonal
anisotropy or a hole-effect are modeled as an additional
structure in a specific direction.
In summary, good practice is to pick a single
isotropic
nugget effect, choosing the same number of variogram struc-
tures for all directions based on the most complex direction;
zonal anisotropies are modeled as geometric anisotropies,
with one direction having a very long, unrealistic range to
account for the lower variance; the same sill parameter is
used for all variogram structures in all directions, and allow-
ing a different range parameter in each direction. An interest-
ing possibility afforded by at least one semi-automatic fitting
program is that different nested structures may present dif-
ferent anisotropies. While care must be taken not to over-
fit the models, these can lead to reasonable and defensible
models, particularly if there is sufficient data and geologic
knowledge to back them up.
Figure
6.7
show as an example of an experimental corre-
logram and its model for three main directions of anisotropy.
The correlograms are of Arsenic values, and the graph shows
the fit, the number of pairs used to calculate each variogram
point, and the model fitted, with the ranges corresponding to
each direction. Two exponential structures were used to fit
the models, with significant anisotropy in SE vs. NE direc-
tions, as well as in the vertical direction.
Fit the variogram models, and decide on the number and
type of nested structures.
Experimental variograms show
different behavior at different distances
h
. Other than the
discontinuity at the origin (the nugget effect), the variogram
may also present a long range structure superimposed on a
short distance structure. These different spatial continuity
structures are a reflection of different geologic controls; for
example, it is common for more than one geologic factor to
influence mineral deposition. Normally, in precious and base
metal deposits the higher grade mineralization is controlled
by fractures or veins, which tend to have a clear preferential
orientation, with a strong anisotropy and short range. These
will evidence themselves as a distinct the short-scale conti-
