Spatial dependence – The tendency for observations close together in space to be more highly correlated than those that are further apart. Also called spatial autocorrelation. Spatial dependence imputes that up to some distance apart from each other, two observations at different locations are not statistically independent.

Semi-variance is a measure of the spatial dependence between two observations as a function of the distance between them.

Semivariogram – a graph of how semivariance changes as the distance between observations changes. Semivariograms are used for measuring the degree of dissimilarity between observations as a function of distance. Based on the “first rule of geography” that things close together are more similar than things far apart, semi-variance is generally low when two locations are close to each other (i.e. observations at each point are likely to be similar to each other:. Typically, semi-variance increases as the distance between the locations grows until at some point the locations are considered independent of each other and semi-variance no longer increases.

A variogram is usually characterized by three measures. The nugget refers to the variability in the field data that cannot be explained by distance between the observations. Many factors influence the magnitude of the nugget including imprecision in sampling techniques and underlying variability of the attribute that is being measured. Also, the minimum spacing between observations can influence the nugget because if there are no observations located close to each other, it is impossible to estimate “close-range” spatial dependence. The sill refers to the maximum observed variability in the data. In theory, the sill corresponds to the variance of the data as normally estimated in statistics. The difference between the sill and the nugget represents the amount of observed variation that can be explained by distance between observations. An ideal situation would consist of a small nugget and a large sill (i.e., there is much spatial dependence and a lot could be inferred about an unobserved location based on its distance from an observed site). Finally, the range is the point at which the semivariance stops increasing. The range represents the distance at which two observations are unrelated (i.e., independent). Often a model is fit to the empirical variogram to aid in interpretation and in order to make use of the spatial dependence in other statistical techniques.

## Application References

**Using Semivariograms to Determine Ecological Scale**

- Karl,J.W. and B.A. Maurer. 2010. Spatial dependency of predictions from image segmentation: a variogram-based method to determine appropriate scales for producing land-management information.
*Ecological Informatics*5; 194-202.