Stationarity

In the combined presence of the previous two assumptions—that is, when is constant and spatial increments are used to define —the SRF is characterized as intrinsically stationary (Cressie; 1993, p. 40).

The expected value is the first statistical moment of the SRF . The second statistical moment of the SRF is the covariance function between two points and in , and it is defined as

Note that when , the covariance expression provides the variance at .

The assumption of a constant means that the expected value is invariant with respect to translations of the spatial location . The covariance is considered invariant to such translations when it depends only on the distance between any two points and . If both of these conditions are true, then the preceding expression becomes

When both and are invariant to spatial translations, the SRF is characterized as second-order stationary (Cressie; 1993, p. 53).

In a second-order stationary SRF the quantity is the same for any two points that are separated by distance . Based on the preceding formula, for 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

which relates the theoretical semivariance and covariance. Note 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).