Theoretical Semivariogram Models

The KRIGE2D Procedure

Theoretical Semivariogram Models

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.

Figure 34.3: Flowchart for Variogram Selection

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 $\gamma_z(h)$ , 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 { $Z(r), r \in D \subset \mathcal{R}^2$ }. The dimension of a₀ is length with the same units as h.

These three model forms are now examined in more detail.

The Spherical Semivariogram Model

The form of the spherical model is

$\gamma_z(h) = \{ c_0[\frac{3}2\frac{h}{a_0}-\frac{1}2(\frac{h}{a_0})^3], & {for h \le a_0} \c_0, & {for h \gt a_0}.$

The shape is displayed in Figure 34.4 using range a₀=1 and scale c₀=4.

Figure 34.4: Spherical Semivariogram Model with Parameters a₀=1 and c₀ = 4

The vertical line at h=1 is the "effective range" as defined by Duetsch and Journel (1992), or the ``range $\epsilon$ '' defined by Christakos (1992). This quantity, denoted $r_{\epsilon}$ , 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 $r_{\epsilon}$ 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, $\gamma_z(h)$ actually reaches this value. For the other two model forms, the sill is a horizontal asymptote.

The Gaussian Semivariogram Model

The form of the Gaussian model is

$\gamma_z(h)=c_0[1-\exp(-\frac{h^2}{a_0^2})]$

The shape is displayed in Figure 34.5 using range a₀=1 and scale c₀=4.

Figure 34.5: Gaussian Semivariogram Model with Parameters a₀=1 and c₀=4

The vertical line at $h=r_{\epsilon}=\sqrt{3}$ is the effective range, or the range $\epsilon$ (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; $\gamma_z(h)$ approaches the sill asymptotically.

The Exponential Semivariogram Model

The form of the exponential model is

$\gamma_z(h) = c_0[1-\exp(-\frac{h}{a_0})]$

The shape is displayed in Figure 34.6 using range a₀=1 and scale c₀=4.

Figure 34.6: Exponential Semivariogram Model with Parameters a₀=1 and c₀ = 4

The vertical line at $h=r_{\epsilon}=3$ is the effective range, or the range $\epsilon$ (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.

The Power Semivariogram Model

The form of the power model is

$\gamma_z(h) = c_0h^{a_0}$

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.

Figure 34.7: Power Semivariogram Model with Parameters a₀=0.4 and c₀=4

Nested Models

For a given set of spatial data, a plot of an experimental semivariogram may not seem to fit any one of the theoretical models. In such a case, the covariance structure of the spatial process may be a sum of two or more covariances. This is common in geologic applications where there are correlations at different length scales. At small lag distances h, the smaller scale correlations dominate, while the large scale correlations dominate at larger lag distances.

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

$\gamma_{z,1}(h) = c_{0,1}\exp(-\frac{h}{a_{0,1}})$

and

$\gamma_{z,2}(h) = \{ c_{0,2}[\frac{3}2\frac{h}{a_{0,2}}-\frac{1}2(\frac{h}{a_{0,2}})^3], & {for h \le a_{0,2}} \c_{0,2}, & {for h \gt a_{0,2}}\}$

with c_0,1=1, a_0,1=2.5, c_0,2=2, and a_0,2=1. If both of these correlation structures are present in a spatial process { $Z(r), r \in D$ }, then a plot of the experimental semivariogram would resemble the sum of these two semivariograms. This is illustrated in Figure 34.8.

Figure 34.8: Sum of Exponential and Spherical Structures at Different Scales

This sum of $\gamma_1(h)$ and $\gamma_2(h)$ 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 c_0,2=2.

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

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