Geoscience Reference
In-Depth Information
The exponential model is also common, and is similar to
the spherical, except that it rises more steeply and reaches
the sill asymptotically. The practical range is where the
variogram value is 95 % of the sill. Some old definitions
of this variogram do not include the “3” and use a range
parameter 1/3 of the practical range; modern practice is to
consider the practical range:
Decide on the variable to be modeled. Conventional mineral
resource estimation requires that grade variables be mod-
eled. There are estimation methods that also require trans-
formation, including the log-normal (not in common usage),
and Gaussian or Indicator-based methods. Most simulation
methods require that either the Gaussian or Indicator trans-
forms is used. The transformed variables are generally easier
to model, but the characteristics of the continuity models are
generally different than the original variables.
Non-transformed variables may require more exploratory
analysis and clean-up work using h -scatterplots to obtain
reasonable models; but the practice of modeling a trans-
formed variable to obtain a variogram model and then back-
transforming the variogram mode to obtain a model for the
original variable is discouraged. This idea has been applied
using Gaussian and logarithm transformations. For a given
dataset, it can be shown that the variogram models obtained
from the original data often have significant differences with
the back-transformed models derived from using the Gauss-
ian or log transforms.
Exp h
( )
=− −
1 exp( 3 / )
h a
The Gaussian model exhibits a parabolic rather than linear
behavior at the origin, which implies more short scale con-
tinuity. It is suitable for slowly-varying variables, since the
increase in variance is very gradual with distance; examples
of such variables are elevations, water-table measurements
in hydrogeology, or thicknesses. The practical range is where
γ(h) is 95 % of the sill:
(
)
Gaus h
( )
=− −
1 exp
3(
h a
/
)
2
The power law model is associated to self affine random
fractals. The parameter ω of a power model is related to the
fractal dimension D. The variogram model is defined by a
power 0 < ω < 2 and a positive slope, c.
Find a good estimate of the nugget effect. The nugget effect
is a variance that results from measurement errors and geo-
logical short-scale variance. It can sometimes be obtained
from repeat assays of the same sample, subdivided several
times. This is done as part of the general quality assurance
and quality control (QA/QC) of the sample database, but
the number of repeat samples and its spatial distribution
is generally insufficient to estimate a nugget effect for the
variogram model. More common is to use a direction where
there is large amount of data at short spacing, as for example
in the down the hole direction of the drill hole. Typically,
there is an order of magnitude difference between the lat-
eral drill hole spacing and the sampling frequency down the
hole. Since by definition the nugget is the variance at dis-
tance zero, then it has to be isotropic, that is, the same in
all directions. Therefore, it is licit to estimate it from data
in any direction; and closely spaced data will provide the
better estimates. The nugget effect obtained from down the
hole variograms should always be compared to the sampling
variance resulting from the QA/QC program available and
heterogeneity or other sampling studies. This comparison
should be made always mindful of the stationary domains
used in variogram modeling.
γ
( h ch ω
= i
Other important models, although not frequently used, are
the hole effect models: (a)
sin
r
γ =− and (b)
() 1
h
r
γ =− The sinusoidal model (a) is valid in three
dimensions, with r specified in radians, while model (b) is
only valid in one dimension, particularly useful when a
strong hole-effect needs to be modeled in a particular
direction.
An important notion in variogram modeling is the use of
nested structures. The variogram can be fit with a positive
sum of valid variogram models—called nested structures.
So, for example, the final variogram could be a sum of a
spherical variogram explaining part of the variance and an
exponential variogram explaining the remaining variance.
Both structures should typically have different ranges.
( )
h
1 cos
h.
6.3.2
Basic Variogram Modeling Guidelines
Determine the best anisotropy directions. The anisotropy
is found by looking at the variogram in multiple directions.
These directions can be pre-determined based on geologic
knowledge, fixing the directions that may be reasonable can-
didates for best representing the anisotropy. Sometimes, the
software tool used allows specifying multiple directions, with
no pre-conceived anisotropy model. If more data is available,
Virtually all experimental variograms can be modeled using
these various types of models. The specific steps and the
order in which they are completed may vary with the soft-
ware tool used, variogram modeling entails making some-
times subjective decisions on several issues:
 
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