## Introduction

The causes of patterns of biodiversity are a topic of considerable research interest in ecology and conservation science, and many studies seek to explain observed spatial variation in diversity in various landscapes through correlation with a range of different factors such as species pool, land cover types, disturbance regime, heterogeneity, patch geometry and area and anthropogenic influences (Pärtel & Zobel 1999; Münzbergová 2004; Pierce *et al.* 2007; Klimek *et al.* 2008; Marini *et al.* 2008; Reitalu *et al.* 2008). There is a growing awareness in such studies that spatial scale is a critical determinant of which drivers are found to influence diversity (Gering & Crist 2002; Økland, Rydgren & Økland 2008; Lang *et al.* 2009), and that conservation methods seeking to maintain diversity within regions or landscapes must target management at the appropriate scale. Functional explanations for diversity, such as niche differentiation, disturbance, or neighbourhood recruitment limitation are also expected to have characteristic spatial scales in different systems (Hurtt & Pacala 1995; Tilman 1999; Økland, Rydgren & Økland 2008; Murrell 2010).

### Measures of diversity

Conventionally, measures of biodiversity (Whittaker & Woodwell 1969; Whittaker 1972) recognize a distinction between local, or alpha (α-) and regional, or gamma (γ-) diversities. Beta (β-) diversity is then the diversity between habitats or sampling units, a conceptual bridge between these two discrete scales. β-diversity is caused by the tendency of individuals of the same species to be aggregated together in space, whether individualistically or in clearly defined communities. It is also clearly a function of both the grain size at which α-diversity is measured and the degree of ‘patchiness’ of the sampled domain within which γ-diversity is measured. Jurasinski *et al.* (2009) have suggested the term ‘inventory diversity’ for both α- and γ-diversity, as they are identical measures differing only in the spatial scales at which they are applied. These scales are usually the smallest sampling unit of a survey, or ‘grain’ as defined by Dungan *et al.* (2002) and the entire sampled domain or ‘extent’. In most studies, α-diversity is assumed to be sampled comprehensively at the grain scale, whereas γ-diversity is approximated by pooling samples within the extent. For example, Martin, Moloney & Wilsey (2005) measured plant species α-diversity in 40 × 100 cm quadrats in prairie remnants and restoration sites and pooled data from eight quadrats from each site to estimate γ-diversity for each site.

Despite the widespread use of the concept, methods for quantifying β-diversity vary and the appropriateness of different methods, and of the concept itself, is much debated (Veech *et al.* 2002; Jost 2006; Jurasinski *et al.*, 2009). Jurasinski *et al.* (2009) recommend the terminology ‘differential diversity’ to describe measures of compositional difference between samples, and ‘proportional diversity’ to describe comparisons between scales. As species distributions in natural environments tend to aggregate both in space and along environmental gradients, β-diversity may have two components:

- 1environmental variation, because of species aggregation along environmental or disturbance gradients within the domain, which may or may not have spatial structure at the given grain size;
- 2spatial aggregation that is independent of environmental variation, due to the inherent patchiness of species distributions, for example, determined by biotic processes such as competitive exclusion and recruitment limitation, clonality or stochastic population processes.

In the original definitions of Whittaker (1972)β-diversity was defined as the ratio β* = *γ/α. An alternative definition of γ = α + β has become widely used, allowing the additive partitioning of components of diversity (Lande 1996; Veech *et al.* 2002). This approach has the advantage that β shares the same units as α and γ, but suffers from the disadvantage that β is not independent of α, and therefore cannot be compared between sites (Jost 2007), and that β-diversity defined in this way does not recognize the degree of difference or similarity between samples. Diversity in this context is conventionally measured as species richness (simply the total number of species recorded in a sample), or with indices such as the Gini–Simpson index (Simpson 1949) or Shannon index (Weaver & Shannon 1949). However, both of the latter indices are constrained to values between 0 and 1, and may give non-intuitive results (Jost 2006). It has been argued that transforming such indices into true measures of diversity, with units commensurate with species richness, as ‘Hill numbers’, or ‘numbers equivalent’ (Hill 1973; Jost 2006) provides a more intuitive measure of diversity. Following Jost (2007), the Gini–Simpson index (and its paired-sample spatial equivalent) is here referred to as a diversity index and denoted by the symbol *H*; Jost’s numbers equivalent, or ‘true diversity’ (and its paired-sample equivalent) is denoted by the symbol *D*.

The concept of additive partitioning of diversity has been used to compare diversity across discrete spatial scales, and for attributing determinants of species diversity within different sized sampling units (Klimek *et al.* 2008). In the additive partitioning framework, α-diversity across a hierarchy of nested spatial scales can be expressed as α_{x} = α_{x-1} + β_{x-1}, where diversity α_{x} measured at the *x*th level of a scale hierarchy is the sum of the mean α-diversity and the β-diversity between samples at the *x* - 1th level (Wagner *et al.* 2000). However, in addition to the theoretical drawbacks with the additive definition of β-diversity (Jost 2007), one clear drawback with this method is that the scales at which diversity is measured are subjective and defined by the sampling design. If species are spatially aggregated at a scale greater than the distance between samples at which α-diversity is measured, then samples may not be independent and pooled-sample estimates of γ-diversity will be biased. Furthermore, a sampling design in which samples is selected from within clearly identifiable communities or habitats will under-sample ecotones and edge habitats and potentially underestimate the average diversity of a landscape sector. This is particularly important as in many ecosystems transition zones between communities are regions of increased biodiversity (Smith *et al.* 1997; Kark *et al.* 2007).

The concept of β-diversity is often considered as a measure of species turnover along spatial or environmental gradients. Vellend (2001), however, has shown that the most frequently used methods of calculating it are independent of the distributions of species on either spatial or environmental gradients and recommends plotting similarity-distance graphs to show the rate of species turnover per unit distance.

Ecologists have long made use of spatial statistical methods including semivariogram analysis (Garrigues *et al.* 2006) and spatial autocorrelation (Fortin, Drapeau & Legendre 1989) to provide spatial equivalents of statistical properties of populations (in these cases correlation coefficients or variance), and to describe spatial structure and pattern in ecosystems. This approach has been extended to spatial patterns of diversity by plotting the spatial covariance of species richness (‘variogram of complimentarity’; Wagner 2003; Bacaro & Ricotta 2007). Condit *et al.* (2002) and Chave & Leigh (2002) calculate a ‘similarity function’ representing the probability that two trees separated by a given distance in a forest plot belong to the same species. This function is essentially a paired-sample version of the Simpson concentration, which is equal to unity minus the Gini–Simpson index of diversity. We propose a similar method for spatially explicit analysis of species diversity, based on a paired-sample version of the Gini–Simpson diversity index (Simpson 1949; Pielou 1969) and its numbers equivalent (Jost 2006). This method has the advantage that it is in units directly comparable to a commonly used measure of α- and γ-diversity. The method is suitable for systematic or random sampling designs and requires no prior assumptions about the spatial arrangement of communities other than the choice of appropriate grain size and spacing of the sampling unit and the domain of the sample. The method is also similar in approach to spatial statistical techniques such as variography (Fortin & Dale 2005), paired-quadrat variance methods (Schaefer & Messier 1994; Guo & Kelly 2004), similarity-distance graphs (Vellend 2001) and ‘variograms of complementarity’ (Wagner 2003) in that a characteristic of paired samples is plotted against the lag, or distance between samples in space and/or along a environmental gradient. Unlike the former approaches, the proposed method is based on analysing the spatial structure of diversity rather than patterns in the distribution of a single species or variable and measurements are in units commensurate with α- and γ-diversity.

### Study system

To demonstrate the potential of this method we apply it to a fine-scale vegetation survey and hydrological dataset collected for a separate study. Vegetation data were collected at 5-cm intervals along two 10-m transects at plots of contrasting hydrology at a raised bog in Cumbria, UK. The main environmental gradient at the site, variation in surface height above the water table, was measured at each plot using hydrological data loggers in conjunction with a terrestrial laser scanner to accurately measure surface topography (Anderson, Bennie & Wetherelt 2010a). Intact peat bogs typically show distinct patterns in hummock-hollow topography which is associated with fine-scale patterns in species composition, particularly of *Sphagnum* species (Nordbakken 1996; Økland, Rydgren & Økland 2008). Interactions between individual *Sphagnum* plants necessarily occur at a fine spatial scale (<10 cm) and experimental studies have shown that whether species are grown in monocultures or mixtures (and hence the relative importance of inter- and intra-specific interactions) has marked effects on height increment and cover changes (Robroek *et al.* 2007). Previous studies at this site have shown that the characteristic scale of remotely sensed surface (shrub canopy structure and *Sphagnum* microtopography) reflects hydrological status (Anderson *et al.* 2010b).