Geoscience Reference
In-Depth Information
pendent of location, that is,
m
(
u
) =
m
and σ
2
(
u
) = σ
2
for all
locations
u
in the study area. The variogram is defined as:
γ
Sill
2()
γ = −+= −+
h
ar Z
[ ()
u uh
Z
(
)]
EZ
{[ ()
u uh
Z
(
)]}
2
(6.1)
In words, the variogram is the expected squared difference
between two data values separated by a distance vector
h
.
The
semi
variogram γ(
h
) is one half of the variogram 2γ(
h
).
To avoid excessive jargon we simply refer to the variogram,
except where mathematical rigor requires a precise defini-
tion. As with the mean and the variance, the variogram does
not depend on location; it applies for a separation vector that
is translated or scanned over all locations in the chosen area
of interest. The variogram is a measure of variability; it in-
creases as samples become more dissimilar. The covariance
is a statistical measure that is used to measure correlation or
similarity:
Nugget
Effect
h
Range
Fig. 6.1
Features of a variogram
2. The “range” is the distance at which this zero correla-
tion is reached. If the variogram reaches the sill multiple
times, it is common to consider the range as the first oc-
currence.
3. The “nugget effect” is the variogram value at a distance
just larger than the sample size, which characterizes the
very short scale variability. It is common also to use the
term short-scale variability when referring to variogram
distances less than the smallest spacing between sample
points.
The correlation between
Z
(
u
) and
Z
(
u
+
h
) is positive when
the variogram value is less than the sill, and the correlation
is negative when the variogram exceeds the sill. Figure
6.2
shows three
h
-scatterplots corresponding to three lags on
a typical semivariogram. Geostatistical modeling gener-
ally uses the variogram instead of the covariance for mainly
historical reasons, that is, it was considered more robust to
cases where the mean changes locally. In practice, second
order stationarity is almost always assumed and this advan-
tage is not practically important.
C
( )
h
= ⋅ +−
EZ
{[
( )
u uh
Z
(
)]}
m
2
(6.2)
The covariance
C
(
h
) is 0.0 when the values
h
-apart are not
linearly correlated. At
h
= 0 the stationary covariance C(0)
equals the stationary variance σ
2
, that is,
2
C(0)
=
E{Z(
u
+
0) Z( )} - [E{Z( )}]
u
u
2
2
=
=
E{Z(
u
) } - [E{Z(
u
)}]
Var{Z(
u
)} σ
=
2
In certain situations the standardized covariance, the correla-
tion coefficient, is preferred:
ρ
(
=
hh
)
C(
)/
C
( )
0
By further expanding Eq. 6.1, the following relation between
the semi-variogram and covariance is established for a sta-
tionary RF:
6.2
Experimental Variograms
and Exploratory Analysis
γ
() C(0) C() or C()
h
=
−
h
h
=
C
(0)
−
γ
()
h
(6.3)
A single variogram value γ(
h
) for a particular distance and di-
rection vector
h
is straightforward to interpret and understand.
Practical difficulties arise from the fact that we must calculate
reliable values and simultaneously consider many lag vectors
h
. The variogram is a measure of geologic variability versus
distance. This variability can be different in different direc-
tions; for example, in sedimentary formations there is typi-
cally much greater spatial correlation in the horizontal plane.
The pattern of spatial continuity is anisotropic when the vari-
able is more continuous in one direction than another.
Some important steps have to be completed before calcu-
lating experimental variograms: (1) the data must be visual-
ized and understood from a geological perspective, (2) an
appropriate coordinate system must be established, and (3)
This relation depends on the model decision that the mean
and variance are constant and independent of location. These
relations are important in variogram interpretation and in
providing covariances to kriging equations.
The principal features of the variogram are the sill, range,
and nugget effect. Figure
6.1
shows a variogram with these
three important parameters:
1. The “sill” of the variogram is the equal weighted vari-
ance of the data going into variogram calculation, which
is the variogram value that corresponds to zero linear cor-
relation. The variogram may flatten off at an apparent sill
below or above the sill variance.
