The VARIOGRAM Procedure

Stationarity

In the combined presence of the previous two assumptions—that is, when normal upper E left-bracket upper Z left-parenthesis bold-italic s right-parenthesis right-bracket is constant and spatial increments define gamma Subscript z Baseline left-parenthesis bold-italic h right-parenthesis—the SRF upper Z left-parenthesis bold-italic s right-parenthesis is characterized as intrinsically stationary (Cressie 1993, p. 40).

The expected value normal upper E left-bracket upper Z left-parenthesis bold-italic s right-parenthesis right-bracket is the first statistical moment of the SRF upper Z left-parenthesis bold-italic s right-parenthesis. The second statistical moment of the SRF upper Z left-parenthesis bold-italic s right-parenthesis is the covariance function between two points bold-italic s Subscript i and bold-italic s Subscript j in upper Z left-parenthesis bold-italic s right-parenthesis, and it is defined as

upper C Subscript z Baseline left-parenthesis bold-italic s Subscript i Baseline comma bold-italic s Subscript j Baseline right-parenthesis equals normal upper E left-parenthesis left-bracket upper Z left-parenthesis bold-italic s Subscript i Baseline right-parenthesis minus normal upper E left-bracket upper Z left-parenthesis bold-italic s Subscript i Baseline right-parenthesis right-bracket right-bracket left-bracket upper Z left-parenthesis bold-italic s Subscript j Baseline right-parenthesis minus normal upper E left-bracket upper Z left-parenthesis bold-italic s Subscript j Baseline right-parenthesis right-bracket right-bracket right-parenthesis

When bold-italic s Subscript i Baseline equals bold-italic s Subscript j Baseline equals bold-italic s, the covariance expression provides the variance at bold-italic s.

The assumption of a constant normal upper E left-bracket upper Z left-parenthesis bold-italic s right-parenthesis right-bracket equals m means that the expected value is invariant with respect to translations of the spatial location bold-italic s. The covariance is considered invariant to such translations when it depends only on the distance bold-italic h equals bold-italic s Subscript i Baseline minus bold-italic s Subscript j between any two points bold-italic s Subscript i and bold-italic s Subscript j. If both of these conditions are true, then the preceding expression becomes

upper C Subscript z Baseline left-parenthesis bold-italic s Subscript i Baseline comma bold-italic s Subscript j Baseline right-parenthesis equals upper C Subscript z Baseline left-parenthesis bold-italic s Subscript i Baseline minus bold-italic s Subscript j Baseline right-parenthesis equals upper C Subscript z Baseline left-parenthesis bold-italic h right-parenthesis equals normal upper E left-parenthesis left-bracket upper Z left-parenthesis bold-italic s right-parenthesis minus m right-bracket left-bracket upper Z left-parenthesis bold-italic s plus bold-italic h right-parenthesis minus m right-bracket right-parenthesis

When both normal upper E left-bracket upper Z left-parenthesis bold-italic s right-parenthesis right-bracket and upper C left-parenthesis bold-italic s Subscript i Baseline comma bold-italic s Subscript j Baseline right-parenthesis are invariant to spatial translations, the SRF upper Z left-parenthesis bold-italic s right-parenthesis is characterized as second-order stationary (Cressie 1993, p. 53).

In a second-order stationary SRF the quantity upper C left-parenthesis bold-italic h right-parenthesis is the same for any two points that are separated by distance bold-italic h. Based on the preceding formula, for bold-italic h equals 0 you can see that the variance is constant throughout a second-order stationary SRF. Hence, second-order stationarity is a stricter condition than intrinsic stationarity.

Under the assumption of second-order stationarity, the semivariance definition at the beginning of this section leads to the conclusion that

gamma Subscript z Baseline left-parenthesis bold-italic h right-parenthesis equals upper C left-parenthesis bold 0 right-parenthesis minus upper C left-parenthesis bold-italic h right-parenthesis

which relates the theoretical semivariance and covariance. Keep in mind that the empirical estimates of these quantities are not related in exactly the same way, as indicated in Schabenberger and Gotway (2005, section 4.2.1).

Last updated: December 09, 2022