Continuous soil maps — a fuzzy set approach to bridge the gap between aggregation levels of process and distribution models

Soil maps as multi-purpose models of spatial soil distribution have a much higher level of aggregation (map units) than the models of soil processes and land-use effects that need input from soil maps. This mismatch between aggregation levels is particularly detrimental in the context of precision agriculture. It is argued that, in order to bridge the gap, soil distribution modelling should be based on a new classification paradigm: that of fuzzy set theory. In geographic space, this enables representation of gradual as well as abrupt transitions, i.e., soil distribution models that can predict variables at pedon level. In a case study we used fuzzy k-means with extragrades to derive a continuous classification from data on thicknesses of 25 layers measured in 552 soil profiles. For interpolation of the class memberships we developed a new method, Compositional Kriging, which takes into account that the memberships have the structure of compositional data: they must be positive and add up to a constant (1) for each individual. These conditions were added to the regular Kriging equations. For cartographic representation of the continuous soil distribution models we developed a new technique, the Pixel Mixture technique, by which we generated a large number of small coloured pixels in each raster cell of the map. The colours of the pixels symbolize the classes, and the proportions of iso-coloured pixels in a cell symbolize the grades of the class memberships as predicted for that cell. The combination of continuous classification and Compositional Kriging convincingly bridged the gap between aggregation levels, and with the aid of the Pixel Mixture technique the resulting soil distribution model could also be visualized at the appropriate level of aggregation. The continuous soil map showed both the general landscape structure, as well as the varying degree of variability within the study area. Based on this multi-purpose continuous soil model, functional models of soil processes and land-use effects can be developed.