Geoscience Reference
In-Depth Information
Fig. 6.6
Spherical model (
top left
); exponential model (
top right
); Gaussian model (
bottom left
); and power-law models (
bottom right
)
the variance of a linear combination can be written in terms
of the variogram (or covariance) values as:
based on Fourier transforms, could be used. The additional
work and complications arising from attempting to use other
functions is rarely worthwhile: traditional parametric models
allow achieving a good fit in practice and also allow consid-
eration for the geological information commonly available;
and it also allows for a straightforward transfer into existing
estimation and simulation codes (Deutsch and Journel
1997
).
n
�
“
∑
Var Z
{
*
( )}
u
=
Var
λ
α
(
u
)
�
”
α
�
•
α
=
1
nn
∑∑
=
λλγ
(
uu
−≥
)0
αβ α β
αβ
==
11
6.3.1
Commonly Used Variogram Models
The variance must be non-negative for any choice of loca-
tions (
u
) and weights (
λ
) within the domain. The variogram
functions that yield non-negative variance are called semi
positive definite. If a covariance is used, then the condition
is restricted to positive variance only.
The positive definite condition implies that practitioners
typically use specific known positive definite functions; the
more commonly used are spherical and exponential, among
others described below (Journel and Huijbregts
1978
; Cressie
1991
; Christakos
1984
). Other arbitrary functions can be
used, but they must first be proven positive definite (Myers
1991
), for which Bochner's theorem (Reed and Simon
1975
),
Figure
6.6
shows the most common variogram shapes used
in mining. These shapes are all parameterized by a scalar
vector h and a range parameter. The use of anisotropy with
these variogram model shapes is discussed below. The first
is the spherical model, for which the spherical covariance,
1—Sph(h), is the volume of two intersecting spheres.
�
�
3
1.5( / )
ha
−
0.5( / ) ,
ha
h a
≤
Sph h
()
=
1, otherwise
