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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Measuring heterogeneity in satellite imagery is an important task to deal with. Most measures of spectral diversity have been based on Shannon Information theory. However, this approach does not inherently address different scales, ranging from local (hereafter referred to alpha diversity) to global scales (gamma diversity). The aim of this paper is to propose a method for measuring spectral heterogeneity at multiple scales based on rarefaction curves. An algorithmic solution of rarefaction applied to image pixel values (Digital Numbers, DNs) is provided and discussed.

Measuring heterogeneity in satellite imagery is important, since heterogeneity in an image represents the degree of diversity of objects reflecting within a landscape. In fact, since the IFOV (Instantaneous Field of View) of an image represents a spatially implicit representation of reality, each pixel is expected to represent reality at a certain resolution.

Despite the attribute being considered, the diversity of that attribute has been proven to change as a function of scale [

The aim of this paper is to propose a method for measuring spectral heterogeneity at multiple scales simultaneously based on ecological theory.

In ecology, there is a long history of dealing with species diversity over space or time. In particular, given

alpha or local diversity (α), i.e. the number of species within one plot

gamma or total diversity (γ), i.e. the number of species considering

beta or between-plots diversity (β), i.e. the diversity deriving from the complementarity of the species composition considering pairs of plots [

In this view, accumulation curves, showing the number of accumulated species given a certain number of sampled plots, have long been used for estimating the expected number of species within a study area given a specific sampling effort. Since the order that samples are added to an accumulation curve accounts for its shape [_{s}

On the other hand, an analytical solution (i) may be formalized as:

Let _{s}_{i}

Generally, the steeper the curve, the greater the increase in species richness as the sample size increases [

From a landscape perspective, rarefaction curves are directly related to the environmental heterogeneity of the area sampled. In fact, it is expected that the greater the landscape heterogeneity, the greater the species diversity, including both fine-scale and coarse-scale species richness (i.e. α- and γ-diversity, respectively), and compositional variability, or β-diversity [

Computing β-diversity deals with looking at the difference between pairs of plots in terms of species composition [_{j}_{j}

An alternative definition of β-diversity has been provided by Whittaker [

In this paper, different “species” will be replaced by different “DNs” (Digital Numbers, i.e. spectral values).

Consider a satellite image with a radiometric resolution of 8 bit. This means that the reflectance values of the pixels, i.e. the Digital Numbers (DNs), may range from 0 to 255.

Subsampling the image by means of _{DN}

Given the matrix _{DN}_{i}

Therefore, the same concepts introduced for species diversity may thus be applied to satellite imagery diversity. Applying rarefaction theory to DNs rather than species leads to consider three different components of pixels diversity:

alpha or local diversity (α_{DN}), i.e. the number of different DNs within one plot

gamma or total diversity (γ_{DN}), i.e. the number of different DNs considering

beta or between-plots diversity (β_{DN}), i.e. the diversity deriving from _{DN} = _{DN} − _{DN} [

_{DN}_{DN}_{DN}

the plots are rows

the DN values are columns

the cells composing the matrix are presence/absence values, i.e. they are dummy coded as 1s and 0s.

For instance, _{DN}_{DN}

Thus, before building rarefaction curves one should choose a single band to work with. Following biological theory, an infrared waveband should be used when working with vegetation based on its intrinsic capability of discriminating different vegetation types [

Once the rarefaction algorithm (_{DN}

Considering the ecologically heterogeneous area (upper curve of _{DN} equals 55 and 24 respectively, i.e. there are on average 55 and 24 distinct reflectance values for each plot (spatial window).

Meanwhile γ turns out to be 253 and 50, for heterogeneous vs. homogeneous area, respectively.

This means that the spectral value diversity β_{DN} as calculated by γ_{DN}-α_{DN} is 198 and 26, respectively.

Notice that in this worked example, the rarefaction algorithm (

Rarefaction and additive partitioning of diversity, which are often used in ecology with reference to species diversity [

In summary, for each plot (spatial window), containing a number

The approach proposed for measuring spectral heterogeneity is robust but straightforward and consists of three main tasks: (i) selecting within the image adjacent or random windows containing a given number of pixels; (ii) choosing one band (

Of course, other techniques rather than spectral rarefaction could account for the spatial variability of DN values as well, e.g. semivariograms [

I strongly acknowledge Root Gorelick, Carlo Ricotta and a third anonymous referee for precious insights on a previous draft of the paper. I am particularly grateful to Brian S. Cade and Daniel J. McGlinn for additional comments on the manuscript.

Additive partitioning of diversity. γ-diversity is represented by the sum between α and β. This leads to consider β in the same unit of measurement (i.e. number of species) of α and γ.

The presence/absence matrix _{DN}

A worked example of spectral rarefaction.. Once differently heterogeneous areas are sampled by the same number of plots (windows) containing the same number of inner pixels, the rarefaction curves computed by