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The KRIGE2D Procedure |

PROC VARIOGRAM computes the sample, or experimental semivariogram. Prediction of the spatial process at unsampled locations by techniques such as ordinary kriging requires a theoretical semivariogram or covariance.

When you use PROC VARIOGRAM and PROC KRIGE2D to perform spatial prediction, you must determine a suitable theoretical semivariogram based on the sample semivariogram. While there are various methods of fitting semivariogram models, such as least squares, maximum likelihood, and robust methods (Cressie 1993, section 2.6), these techniques are not appropriate for data sets resulting in a small number of variogram points. Instead, a visual fit of the variogram points to a few standard models is often satisfactory. Even when there are sufficient variogram points, a visual check against a fitted theoretical model is appropriate (Hohn 1988, p. 25ff).

In some cases, a plot of the experimental semivariogram suggests that a single theoretical model is inadequate. Nested models, anisotropic models, and the nugget effect increase the scope of theoretical models available. All of these concepts are discussed in this section. The specification of the final theoretical model is provided by the syntax of PROC KRIGE2D.

Note the general flow of investigation. After a suitable choice is made of the LAGDIST= and MAXLAG= options and, possibly, the NDIR= option (or a DIRECTIONS statement), the experimental semivariogram is computed. Potential theoretical models, possibly incorporating nesting, anisotropy, and the nugget effect, are computed by a DATA step, then they are plotted against the experimental semivariogram and evaluated. A suitable theoretical model is thus found visually, and the specification of the model is used in PROC KRIGE2D. This flow is illustrated in Figure 70.10; also see the "Getting Started" section in the chapter on the VARIOGRAM procedure for an illustration in a simple case.

Four theoretical models are supported by PROC KRIGE2D: the
spherical, Gaussian, exponential, and power models.
For the first three types, the parameters
*a _{0}* and

In particular, the dimension of *c _{0}*
is the same as the dimension of the variance
of the spatial process {}.
The dimension of

These three model forms are now examined in more detail.

The shape is displayed in Figure 34.4 using range *a _{0}*=1 and
scale

The vertical line at *h*=1
is the "effective range" as defined by Duetsch
and Journel (1992), or
the ``range '' defined by Christakos (1992). This
quantity, denoted , is
the *h*-value where the covariance is approximately zero.
For the spherical model, it is *exactly* zero; for
the Gaussian and exponential models, the definition of
is modified slightly.

The horizontal line at 4.0 variance units (corresponding to
*c _{0}*=4) is called
the "sill." In the case of the
spherical model, actually reaches this
value. For the other two model forms, the sill is
a horizontal asymptote.

The shape is displayed in Figure 34.5 using range *a _{0}*=1 and
scale

The vertical line at is
the effective range, or the range (that is, the *h*-value where the covariance
is approximately 5% of its value at zero).

The horizontal line at 4.0 variance units (corresponding to
*c _{0}*=4) is the sill; approaches the
sill asymptotically.

The shape is displayed in Figure 34.6 using range *a _{0}*=1 and
scale

The vertical line at is
the effective range, or the range (that is, the *h*-value where the covariance
is approximately 5% of its value at zero).

The horizontal line at 4.0 variance units (corresponding to
*c _{0}*=4) is the sill, as in the other model forms.

It is noted from Figure 34.5 and Figure 34.6 that the
major distinguishing feature of the Gaussian and
exponential forms is the shape in the
neighborhood of the origin *h*=0. In general,
small lags are important in determining an
appropriate theoretical form based on a sample
semivariogram.

For this model, the parameter *a _{0}* is a dimensionless
quantity, with typical values 0 <

As an illustration, consider two semivariogram models, an exponential and a spherical.

with *c _{0,1}*=1,

This sum of and in
Figure 34.8 does not resemble any *single*
theoretical semivariogram; however, the shape
at *h*=1 is similar to a spherical. The asymptotic
approach to a sill at three variance units, along
with the shape around *h*=0, indicates an exponential
structure. Note that the sill value is the sum of
the individual sills *c _{0,1}*=1 and

Refer to Hohn (1988, p. 38ff) for further examples of nested correlation structures.

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