Periodicity in spatial data and geostatistical models: autocorrelation between patches

Several recent studies in landscape ecology have found periodicity in correlograms or semi-variograms calculated, for instance, from spatial data of soils, forests, or animal populations. Some of the studies interpreted this as an indication of regular or periodic landscape patterns. This interpretation is in disagreement with other studies that doubt whether such analysis is valid. The objective of our study was to explore the relationship between periodicity in landscape patterns and geostatistical models. We were especially interested in the validity of the assumption that periodicity in geostatistical models indicates periodicity in landscape pattern, and whether the former can characterize frequency and magnitude of the latter. We created maps containing various periodic spatial patterns, derived correlograms from these, and examined periodicity in the correlograms. We also created non-regular maps that we suspected would cause periodicity in correlograms. Our results demonstrate that a) various periodic spatial patterns produce periodicity in correlograms derived from them, b) the distance-lags at which correlograms peak correspond to the average distances between patch centers, c) periodicity is strongest when the diameter of patches is equal to the distance between patch edges, d) periodicity in omni-directional correlograms of complex spatial patterns (such as checkerboards) are combinations of several waves because inter-patch distances differ with direction; multiple directional correlograms can decompose such complexity, and e) periodicity in correlograms can also be caused when the number of patches in a study site is small. These results highlight that correlograms can be used to detect and describe regular spatial patterns. However, it is crucial to ensure that the assumption of stationarity is not violated, i.e., that the study area contains a sufficiently large number of patches to avoid incorrect conclusions.