Geoscience Reference
In-Depth Information
a larger number of directions can be run, using tighter toler-
ances, such that a more precise definition of the anisotropy is
possible. But even if the data is sparse such that wide angle
and lag tolerances must be used, the anisotropic model is
probably more realistic than an isotropic one. Although rare,
occasionally geologic phenomena can result in a 3-D isotro-
pic variogram over a relatively small distance scale.
The most common type of anisotropy is geometric an-
isotropy. This is the case when the directional variograms
present the same level of variance (sill) in all directions, but
the ranges are different. The variogram model is an ellipsoid
characterized by the three principal directions (axes) with
three different ranges. All other directional variograms can
be derived from this ellipsoid. A linear transformation of the
coordinates is sufficient to obtain an effective lag distance
h.
This transformation involves a rotation to make the ellipsoid
axes coincident with the main coordinate axes; and a transla-
tion, specified as an affinity matrix, to obtain the equivalent
effective ranges. This transformation allows inferring the
variogram value for any direction and any distance:
statistics are not representative of the entire domain, and (3)
outlier values, which can cause eratic and unstable estimates
of the variance. The main point being made by Journel and
Huijbregts (
1978
), and also Barnes (
1991
), is that the sample
variance
ˆ
σ
is
not
an estimator of the stationary variance σ
2
;
rather, it is an estimate of the dispersion variance of samples
of point support within the stationary domain
A
(
D
2
(
•
,A)
).
Only when the domain
A
approaches an infinite domain does
the sample variance
2
ˆ
σ
approach the stationary variance σ
2
.
However, the data used to estimate the variogram represent
the area of interest
A
and not an infinite domain. Thus, the
point where
Y(
u
)
and
Y(
u
+
h
)
are uncorrelated is the disper-
sion variance
D
2
(
•
,A).
Thus, the sample variance should be
used as the sill of the sample semivariogram understanding
that it is, in fact, a dispersion variance.
The second issue relates to the use of the naïve sample
variance or the sample variance accounting for decluster-
ing weights. The use of declustering weights is important,
but they are not used in the calculation of the variogram.
There are more pairs in areas of greater sampling density,
and therefore the variogram at shorter distances will reflect
more the local variance of the clustered data. Also, the sill is
reached at the naïve sample variance. Omre (
1984
) proposed
incroporating declustering weights into the variogram cal-
culation; however, they provide no better variogram and are
difficult to implement in practice. As a result, declustering
weights are not used in variogram calculation, interpretation,
and modeling. The sill should be taken as the naive equally
weighted variance.
The third issue that must be addressed is the influence
of outlier sample values. It is well known that the variance,
being a squared statistic, is sensitive to outlier values. For
this reason, the sample variance may be unreliable. This is
not a problem with correlograms, or transformed data: the
Gaussian and indicator transforms remove the sensitivity to
outlier data values. While some outliers may be removed (or
capped) in traditional variogram modeling, the correct vari-
ance for variogram interpretation is the naïve equal-weight-
ed variance. Outliers influence more than just the variogram
sill: as shown by Rossi and Parker (
1993
), the nugget effect
and short scale continuity modeled are also affected, and can
have a significant impact on the final variogram model. The
higher the variability, the more significant the impact of out-
liers is. Although we still recommend keeping all the data for
variogram modeling, there may be cases where the outliers
are so extreme and with little or no spatial influence that it
may be better to remove or reduce (cap) its value.
2
2
2
2
†‡
h
†‡
h
†‡
h
y
h
=
x
+
+
z
ˆ‰
ˆ‰
ˆ‰
Š‹
Š‹
a
a
Š‹
a
x
y
z
This would be applied for each structure separately. The
range of each structure would be one. Common geostatisti-
cal software requires the user to specify the orientation of the
ellipsoid (three angles) and the three ranges; there is no need
to explicitly calculate the scaled h distances.
Zonal anisotropy cannot generally be modeled using a
simple coordinate transformation; in this case, one option is
to add an additional structure in the specific direction where
the zonal component appears. But it is more common to con-
sider it a special case of geometric anisotropy, where the sill
is reached asymptotically at large distances. The zonal an-
isotropy thus appears as a very large range parameter in one
or more of the principal directions.
Define the sill at which the variogram reaches the zero cor-
relation distance.
There is often confusion about the correct
variance to use for variogram interpretation. It is important to
have the variance σ
2
to correctly interpret positive and nega-
tive correlation, as well as confirm trends in the data. Some
authors have discussed the issue about which is the correct
variance to use as sill variance for variogram interpretation
(Gringarten and Deutsch
1999
; Journel and Huijbregts
1978
,
p. 67; and Barnes
1991
).
There are three issues that may affect the decision about
which is the correct variance to use: (1) the dispersion vari-
ance, which accounts for the difference between our finite
domain and the infinite stationary variance; (2) declustering
weights, which account for the fact that our summary
Define how to deal with the trends.
If the variable being
modeled shows a systematic trend, that is, the data is non-
stationary over the study domain, it is generally better to
remove it before further geostatistical analysis. Trends in
the data can be identified from the experimental variogram,
