On the mathematical properties of spatial Rao’s Q to compute ecosystem heterogeneity

Spatio-ecological heterogeneity has a significant impact on various ecosystem properties, such as biodiversity patterns, variability in ecosystem resources, and species distributions. Given this perspective, remote sensing has gained widespread recognition as a powerful tool for assessing the spatial heterogeneity of ecosystems by analyzing the variability among different pixel values in both space and, potentially, time. Several measures of spatial heterogeneity have been proposed, broadly categorized into abundance-related measures (e.g., Shannon’s H) and dispersion-related measures (e.g., Variance). A measure that integrates both abundance and distance information is the Rao’s quadratic entropy (Rao’s Q index), mainly used in ecology to measure plant diversity based on in-situ based functional traits. The question arises as to why one should use a complex measure that considers multiple dimensions and couples abundance and distance measurements instead of relying solely on simple dispersion-based measures of heterogeneity. This paper sheds light on the spatial version of the Rao’s Q index, based on moving windows for its calculation, with a particular emphasis on its mathematical and statistical properties. The main objective is to theoretically demonstrate the strength of Rao’s Q index in measuring heterogeneity, taking into account all its potential facets and applications, including (i) integrating multivariate data, (ii) applying differential weighting to pixels, and (iii) considering differential weighting of distances among pixel reflectance values in spectral space.