Geoscience Reference
In-Depth Information
Fig. 6.5
Common variogram shapes (variograms are standardized, but with no specific distance scale). The figures illustrate trends (
top left
),
cyclicity (
top right
), geometric anisotropy (
bottom left
), and zonal anisotropy (
bottom right
)
Geometric anisotropy is when the variance (sill) is reached
at different distances (lags) for different directions. Zonal
anisotropy is when for any distance considered in the vario-
gram calculation the variogram never reaches the expected
sill variance. Zonal anisotropies can also be considered geo-
metric anisotropies if it assumed that the same sill variance
is reached at distances much larger than used to calculate the
variogram.
used to obtain the variogram points, these need to be
interpreted and interpolated into a γ(
h
) function for all
h
values. The modeled γ(
h
) function should carry all the
geological information derived from the experimental
model, including anisotropies, trends, nugget effects, etc.
A smooth interpreted function also allows filtering arti-
facts of data spacing, location, and sampling practices.
2. The covariance values C(
h
) derived from γ(
h
) using
Eq. (6.3) must have a mathematical property called
posi-
tive definiteness
. A positive definite model ensures that
the kriging equations used can be solved, that this solu-
tion is unique, and that the kriging variance is positive. A
positive definite function is a valid measure of distance.
The positive definite condition is required because the krig-
ing estimates are weighted linear combinations of samples:
6.3
Modeling 3-D Variograms
The experimental variogram points are not used directly in
subsequent calculations; rather, a parametric function is ad-
justed to those points to obtain a three-dimensional model
(Armstrong
1984
). The two most important reasons for mod-
eling variograms are:
1. Most subsequent geostatistical calculations, including
estimation and simulation methods, require a variogram
or covariance value for all possible distances and direc-
tions. Since only specific distances and directions are
n
=
∑
z
*
()
u
λ
α
( )
u
α
α
=
1
The variance of the estimates
Z
*
(
u
) must, by definition, be
positive. It can be shown (Journel and Huijbregts
1978
) that
