(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

8

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

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

erally done because computations in the kriging

range value *r*. The range is qualified as practical

algorithms are more efficiently performed using a

covariance function.

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

fit. A final calibration of the variogram param-

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-

linity in groundwater systems (Myers et al. 1980).

tical analysis. Whatever the fitting method used,

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-

combinations of these forms also are acceptable as

mates and the subsequent fit evaluated. Although

theoretical variograms (see section 4-5*b*).

the spherical variogram is one of the most often

4-12

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