30 Jun 97
(3) The sill determines the maximum value of
Geometric relationships to aid in obtaining param-
a variogram and approximates the variance of the
eters for the four variogram forms are described in
data. However, the points defining ( take prece-
the following sections and are illustrated in Fig-
dence over the sample variance in locating the sill.
ure 2-3 for reference.
Some variograms are unbounded, and others may
b. Exponential variogram.
only reach a sill value asymptotically. A defined
sill allows conversion of the variogram to a covari-
ance function using Equation 2-27, which is gen-
The exponential variogram (Equation 2-23) is
specified by the nugget g, sill s, and a practical
range value r. The range is qualified as practical
algorithms are more efficiently performed using a
because the sill is reached only asymptotically.
The initial behavior of the exponential variogram is
(4) Fitting a function to the sample variogram
different from the behavior of the spherical vario-
values can range from a visual fit to a sophisti-
gram in that the convex behavior extends to the
cated statistical fit. A statistical fit is an objective
nugget value (Figure 2-3). Again, a nugget value
and a sill value are first specified based on the 8
method as long as the choice of bins and weighting
of the sample variogram points remain fixed.
points. The practical range is chosen so that the
However, because the inputs will vary with investi-
value of the resulting exponential function evalu-
gators, inherent subjectivity persists as in a visual
ated at the practical range lag is 95 percent of the
sill value. The specified exponential function
eters would be based on the kriging algorithm and,
would mesh with the sample variogram points at
thus, either of the initial fitting methods at this
least through the practical range lag. An initial
stage would suffice.
estimate of the practical range can be made by
checking if the intersection of the sill value with a
(5) Because the initial part of the variogram
line tangent to the variogram at the nugget is at a
has the most effect on subsequent kriging output, a
lag value equal to one-third of the assumed prac-
good estimate of the nugget value becomes a most
tical range value as illustrated in Figure 2-3.
important first step. The range and the sill, in that
Examples of the exponential variogram may be
order, complete the ranking of the influence of
found in spatial studies of sulfate and total alka-
variogram parameters on the output of a geostatis-
tical analysis. Whatever the fitting method used,
c. Spherical variogram. The spherical vario-
the theoretical variogram needs to be supported by
the sample variogram values. For variograms with
gram parameters (Equation 2-24) are a nugget
value g, a range r, and a sill s. At smaller lag
a range parameter, this support should extend to
the range. Journel and Huijbregts (1978) suggest
values the sample variogram points indicate linear
that this support should be through one-half the
behavior from the nugget that then becomes con-
dimension of the field or essentially through one-
vex and reaches a sill value at some finite lag
half the maximum lag distance of the sample data.
(Figure 2-3). A sill is estimated, and a line drawn
through the points of the initial linear part of the
(6) Most geostatistical studies can be success-
variogram would intersect the sill at a lag value
fully completed using the following four singular
approximately equal to two-thirds of the range.
theoretical variogram forms: exponential, spheri-
With these estimates of the parameters, a spherical
cal, Gaussian, and linear functions (Figure 2-3).
variogram is defined that should be supported by
For the example variogram determination
the sample variogram points. If the spherical plot
described in this section, only one of these singular
does not fall near the sample variogram points,
forms will be selected; however, positive linear
adjustments need to be made to the parameter esti-
mates and the subsequent fit evaluated. Although
theoretical variograms (see section 4-5b).
the spherical variogram is one of the most often