This Subsection examines the relationships between spatial covariance, semi-variance and correlation (including cross-correlation), under the assumption of second-order stationarity, leading to the concept of cross-correlation. This assumption (Subsection 13.2.1) may be restated as follows:
• The first moment (mean of points i) of the variable exists:
Its value does not depend on position in the study area.
• The second moment (covariance, numerator of eq. 13.1) of the variable exists:
The value of C(d) depends only on d and on the orientation of the distance vector, but not on position in the study area. To understand eq. 13.12 as a measure of covariance, imagine the elements of the various pairs yh and yi written in two columns as if they were two variables. The equation for the covariance (eq. 4.4) may be written as follows, using a final division by n instead of (n - 1) (maximum-likelihood estimate of the covariance, which is standard in geostatistics):
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