From zero to infinity: Minimum to maximum diversity of the planet by spatio‐parametric Rao’s quadratic entropy

Abstract

Aim The majority of work done to gather information on the Earth's biodiversity has been carried out using in‐situ data, with known issues related to epistemology (e.g., species determination and taxonomy), spatial uncertainty, logistics (time and costs), among others. An alternative way to gather information about spatial ecosystem variability is the use of satellite remote sensing. It works as a powerful tool for attaining rapid and standardized information. Several metrics used to calculate remotely sensed diversity of ecosystems are based on Shannon’s information theory, namely on the differences in relative abundance of pixel reflectances in a certain area. Additional metrics like the Rao’s quadratic entropy allow the use of spectral distance beside abundance, but they are point descriptors of diversity, that is they can account only for a part of the whole diversity continuum. The aim of this paper is thus to generalize the Rao’s quadratic entropy by proposing its parameterization for the first time. Innovation The parametric Rao’s quadratic entropy, coded in R, (a) allows the representation of the whole continuum of potential diversity indices in one formula, and (b) starting from the Rao’s quadratic entropy, allows the explicit use of distances among pixel reflectance values, together with relative abundances. Main conclusions The proposed unifying measure is an integration between abundance‐ and distance‐based algorithms to map the continuum of diversity given a satellite image at any spatial scale. Being part of the rasterdiv R package, the proposed method is expected to ensure high robustness and reproducibility.

About the link between biodiversity and spectral variation

“…The SV_categorical approach is closely related to the spectral-species approach originally suggested by Féret and Asner (2014), where class numbers (typically obtained by a spectral clustering approach) are considered proxies of species numbers (thus relating to the 'species SVH'). Recently, this approach has been scaled up to wider spatial extents and to a higher level of biological organization (e.g., vegetation types or habitats) (Rocchini et al, 2021). The approach has some obvious advantages: (1) consistency over time may increase as even if the optical traits of plant species change over the year, the landcover patches or landscape elements may be more persistent (e.g., a broad-leaved forest stand may look very different in a satellite scene acquired in summer and winter but may be detectable as a spectrally homogeneous spatial unit/patch in both scenes); (2) spectrally extreme pixels or land-cover types will not have unproportionally large influence on the spectral-variation metric but will rather represent individual discrete classes amongst a plethora of other classes; and…”

Integrals of life: Tracking ecosystem spatial heterogeneity from space through the area under the curve of the parametric Rao’s Q index

“…Increasing alpha in Eq. (3) will increase the weight of higher distances among different values until reaching the maximum distance value possible (Rocchini et al, 2021a). For this reason, spatio-ecological heterogeneity values of the parametric Rao's Q increase with each alpha progressively added to the calculation constructing a curve for every moving window built around each pixel (Rocchini et al, 2021b).…”

“…Both Shannon and Rao's Q indices are point descriptors of heterogeneity, namely they can only show part of the whole heterogeneity spectrum. Recently Rocchini et al (2021a) proposed an implementation of the Rao's Q index by parameterizing the original formula, and allowing the whole continuum of heterogeneity to be measured thanks to Rao's Q continuous profiles (see Section 2).…”

A generalizable normalization for assessing plant functional diversity metrics across scales from remote sensing

“…This approach is supported by the spectral variation hypothesis, which states that the spectral diversity in an RS image originates from the spatial heterogeneity of the environment, which influences the distribution of plant species and their functional traits (Palmer et al, 2002). PFD can be estimated from the variability of (1) spectral signals (spectral diversity) (Rocchini et al, 2021; Wang, Gamon, Cavender‐Bares, et al, 2018) or (2) plant traits (PT) estimated from spectral imagery (e.g. leaf pigments) (Schneider et al, 2017; Torresani et al, 2021).…”

“…PFD can be estimated from the variability of (1) spectral signals (spectral diversity) (Rocchini et al, 2021; or (2) plant traits (PT) estimated from spectral imagery (e.g. leaf pigments) (Schneider et al, 2017;Torresani et al, 2021).…”