Inter-scale behavior of balanced entropy for soil texture

The balanced-entropy index – recently proposed by Martín et al. [Martín, M.A., Rey, J.-M., and Taguas, F.J., 2005. An entropy-based heterogeneity index for mass–size distributions in Earth science. Ecological Modelling, 182, 221–228.] as a soil texture indicator – can be computed for any arbitrary non-uniform particle size partition provided that associated data are available. In this paper, properties of the index with respect to refinements of the partition are derived. In particular, we analyze how the index value responds when a partition is refined. Variations in the values of the index are shown to be related to the mass splitting in the finer partition. Fine textural data – representing soil volume–size distribution – obtained by laser diffractometry from 70 different types of soils in the Iberian Peninsula are used as a case study to illustrate the theory. The evenness of the underlying distributions is explored at different (finer) size resolutions by computing the balanced-entropy index for the associated partitions. It is observed that, in general, the index values increase when the partition is refined. This is shown to be consistent with the spreading of the mass becoming more uniform when refining the scale. Also, the relative orderings induced by the index computations at different scales do not differ qualitatively, which is significant for classification purposes. Further, for distributions with a continuous probability density function, the index values are shown to approach one when the partition size goes to zero. This is a theoretical property of the index that can be used to test continuity of soil particle size distributions. In our case study, the continuity of the processed distributions cannot be discarded from the analysis.