The VARIOGRAM Procedure

Empirical Semivariograms and Surface Trends

It was stressed in the beginning of the section Theoretical and Computational Details of the Semivariogram that if your data are not free of nonrandom surface trends, then the empirical semivariance $\hat{\gamma }_ z(\bm {h})$ you obtain from PROC VARIOGRAM represents a pseudo-semivariance rather than an estimate of the theoretical semivariance $\gamma _ z(\bm {h})$.

In practice, two major difficulties appear. First, you might have no knowledge of underlying surface trends in your SRF $Z(\bm {s})$. It can be possible to have this information when you deal with a repetitive phenomenon (Chilès and Delfiner, 1999, p. 123), or if you work within a subdomain of a broader region with known characteristics; often, though, this is not the case. Second, even if you suspect the existence of an underlying nonrandom trend, its precise nature might be unknown (Cressie, 1993, p. 114, 162).

Based on the last remark, the criteria to define the exact form of a surface trend can be subjective. However, statistical methods can identify the presence and remove an estimate of such a trend. Different trend forms can be estimated in your SRF depending on the trend estimation model that you choose. This choice can lead to different degrees of smoothing in the residual random fluctuations. It might also have an effect on the residuals spatial structure characterization, because trend removals with different models are essentially different operations acting upon the values of your original observations. Following the comment by Chilès and Delfiner (1999, section 2.7.3), there are as many semivariograms of residuals as there are ways of estimating the trend. The same source also examines the introduction of bias in the semivariance of the residuals as a side effect of trend removal processes. This bias is small when you examine distances close to the origin $\bm {h}=0$, and it can increase with distance.

Keeping in mind the preceding remarks, an approach you can take is to use one of the many predictive modeling tools in SAS/STAT software to estimate the unknown trend. Then you use PROC VARIOGRAM to analyze the residuals after you remove the trend. If the resulting model does not require too many degrees of freedom (such as if you use a low-order polynomial), then this approach might be sufficient. The section Analysis with Surface Trend Removal demonstrates how to use PROC GLM (see Chapter 42: The GLM Procedure,) for that purpose.

Apart from the standard semivariogram analysis, you can attempt to fit a theoretical semivariogram model to your empirical semivariogram if (a) either the analysis itself or your knowledge of the SRF does not clearly suggest the presence of any surface trend, or (b) the analysis can indicate a potentially trend-free direction, along which your data have a constant mean.

For example, you might observe overall similar values in your data. This can be an indication that your data are free of nonrandom trends, or that a very mild trend is present. The case falls under the preceding option (a). A very mild trend still allows a good determination of the semivariance at short distances according to Chilès and Delfiner (1999, p. 125), and this can be sufficient for your spatial prediction goal. An analysis of this type is assumed in the section Preliminary Spatial Data Analysis.

If you observe similar values locally across a particular direction, this an instance of option (b). Olea (2006) suggests recognizing a trend-free direction as being perpendicular to the axis of the maximum dip in the values of $Z(\bm {s})$. If you suspect that at least one such direction exists for your data, then run PROC VARIOGRAM for a series of directions in the angular vicinity. The trend-free direction, if it exists, coincides with the one whose pseudo-semivariogram exhibits minimal increase with distance; see Analysis without Surface Trend Removal for a demonstration of this approach. However, you cannot test $Z(\bm {s})$ for anisotropy in this case, because you can investigate the semivariogram only in the single trend-free direction (Olea, 1999, p. 76). Chilès and Delfiner (1999, section 2.7.4) suggest fitting a theoretical model in a trend-free direction only if the hypothesis of an isotropic semivariogram appears reasonable in your analysis.

As a result, you need to be very cautious when you choose to perform semivariogram analysis on data you have not previously examined for surface trends. In this event, both of the options (a) and (b) that were reviewed in the preceding paragraphs rely mostly on empirical and subjective criteria. As noted in this section, a degree of subjectivity exists in the selection of the surface trend itself. This fact suggests that a significant part of the semivariogram analysis is based on metastatistical decisions and on your understanding of your data and the physical considerations that govern your study. In any case, as shown in the section Theoretical and Computational Details of the Semivariogram, your semivariogram analysis relies fundamentally on the use of trend-free data.