Modeling experimental cross-transiograms of neighboring landscape categories with the gamma distribution

Effectively fitting the major features of experimental transiograms (or variograms) is crucial in characterizing spatial patterns and reproducing them in geostatistical simulations. Landscape patterns usually tend to contain neighboring structures. The experimental cross-transiograms of frequent neighboring landscape categories normally demonstrate apparent peaking features at low lag distances – they first quickly increase to a peak and then gradually flatten out. The flattening process may be smooth or may be through one or more alternate attenuating troughs and peaks. While alternate peaks and troughs may be a reflection of the cyclic occurrence of landscape categories, the single peak or the first peak at low lag section should be a signal of the neighboring structure of two related categories. This is further proved by the peaking features of some idealized transiograms calculated from single-step transition probability matrices. To effectively fit the first peak, especially when it is the single one, we propose using the gamma distribution function and the commonly used variogram models to form additive composite models. Cases of fitting experimental cross-transiograms of landscape data (here soil types and land cover classes) show that the additive gamma-exponential composite model works very well and may closely fit the single-peak features. Although it has multiple parameters to set, model fitting can be performed manually by trial and error. Other two composite models may provide alternatives for fine fitting of the root section (i.e., the left side of the peak). These models may also be applicable to fitting experimental variograms with similar features. We also reintroduce the multiplicative composite hole-effect models proposed for variogram modeling by earlier researchers, and test them on experimental cross-transiograms. It is found that composite hole-effect models are not sufficiently flexible to effectively fit the peak shapes of experimental cross-transiograms of neighboring categories, unless multiple peaks and troughs appear in regular shapes and rhythms.