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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.
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Four theoretical models are supported by PROC KRIGE2D: the
spherical, Gaussian, exponential, and power models.
For the first three types, the parameters
a0 and c0, corresponding to the
RANGE= and SCALE= options in the MODEL statement
in PROC KRIGE2D, have the same dimensions
and have similar affects on the shape
of , as illustrated in the following
paragraph.
In particular, the dimension of c0
is the same as the dimension of the variance
of the spatial process {}.
The dimension of a0 is length with
the same units as h.
These three model forms are now examined in more detail.
The shape is displayed in Figure 34.4 using range a0=1 and scale c0=4.
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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
c0=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 a0=1 and scale c0=4.
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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
c0=4) is the sill; approaches the
sill asymptotically.
The shape is displayed in Figure 34.6 using range a0=1 and scale c0=4.
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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 c0=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 a0 is a dimensionless quantity, with typical values 0 < a0 < 2. Note that the value of a0=1 yields a straight line. The parameter c0 has dimensions of the variance, as in the other models. There is no sill for the power model. The shape of the power model with a0=0.4 and c0=4 is displayed in Figure 34.7.
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As an illustration, consider two semivariogram models, an exponential and a spherical.
with c0,1=1, a0,1=2.5, c0,2=2, and a0,2=1.
If both of these correlation structures are present in
a spatial process {}, then
a plot of the experimental semivariogram would resemble
the sum of these two semivariograms. This is illustrated
in Figure 34.8.
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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 c0,1=1 and c0,2=2.
Refer to Hohn (1988, p. 38ff) for further examples of nested correlation structures.
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