Effects of sample design and landscape features on a measure of environmental heterogeneity

Methods in Ecology and Evolution 2016, 7, 770–782 doi: 10.1111/2041-210X.12539 Effects of sample design and landscape features on a measure of environmental heterogeneity Danielle S. Christianson 1 * and Cari G. Kaufman 2 Energy and Resources Group, University of California, 310 Barrows Hall, Berkeley, CA 94720-3050, USA; and Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA, 94720-3860, USA Summary 1. Environmental heterogeneity, an important inﬂuence on organisms and ecological processes, can be quanti- ﬁed by the variance of an environmental characteristic over all locations within a study extent. However on land- scapes with autocorrelation and gradient patterns, estimating this variance from a sample of locations may lead to errors that cannot be corrected with statistical techniques. 2. We analytically derived the relative expected sampling error of sample designs on landscapes with particular gradient pattern and autocorrelation features. We applied this closed-form approach to temperature observa- tions from an existing study. The expected heterogeneity diﬀered, both in magnitude and direction, amongst sample designs over the study site’s likely range of autocorrelation and gradient features. 3. We conducted a simulation study to understand the eﬀects of (i) landscape variability and (ii) design variabil- ity on an average sampling error. On 10 000 simulated landscapes with varying gradient and autocorrelation fea- tures, we compared estimates of variance from a variety of structured and random sample designs. While gradient patterns and autocorrelation cause large errors for some designs, others yield near-zero average sam- pling error. Sample location spacing is a key factor in sample design performance. Random designs have larger range of possible sampling errors than structured designs due to the potential for sample arrangements that over- and under-sample certain areas of the landscape. 4. When implementing a new sample design to quantify environmental heterogeneity via variance, we recommend using a simple structured design with appropriate sample spacing. For existing designs, we recom- mend calculating the relative expected sampling error via our analytical derivation. Key-words: autocorrelation, gradient, monitoring, sampling, spatial, variability Introduction Environmental heterogeneity, a measure of variability in abiotic and biotic conditions, is important to many areas of ecological study, such as species diversity via niche theory (Holdridge 1947; Whittaker 1956), microbial biogeochemical processes (e.g. Fierer et al. 2006; Sierra et al. 2011) and population dynamics (e.g. Garc ia-Carreras & Reuman 2013). Increasingly, ecologists use environmental heterogene- ity to understand how organisms and ecological processes may respond to a warmer future (e.g. Jentsch et al. 2011; Clark et al. 2013; Thornton et al. 2014). For example, heterogeneity in climate over small spatial scales results in climate refugia that may be of critical importance to the persistence of species with limited options for range expan- sion (Dobrowski 2011; Keppel & Wardell-Johnson 2012). Additionally, while many organisms optimize for average conditions, heterogeneity is important for understanding when and how likely thresholds (e.g. frost tolerances, heat stress) may be breached (Meehl et al. 2000; Jentsch & Beierkuhnlein 2008). *Correspondence author. E-mail: dsvehla@berkeley.edu There are many deﬁnitions of heterogeneity. Some authors simply deﬁne a range of values, while others include aspects of scale, quantify spatial features such as clumping, or qualitatively describe composition (Kolasa & Rollo 1991; Wiens 2000). Quantitative metrics include coef- ﬁcient of variance (standard deviation/mean), variance/mean ratio, variograms and others (see overview by Downing 1991). To be useful in many ecological applications, such a metric must describe the range of available environmental conditions, either in space or time. For example, consider a population of trees whose seeds disperse on the landscape. To maintain a viable population, there must be an adequate number of locations on the landscape through time that have environmental conditions suitable for the seeds to ger- minate and the resulting seedlings to survive. By quantifying the available set of environmental conditions, we can deter- mine whether the environments for successful germination and establishment exist and thus predict whether early life- history stages will limit population viability. Accurately estimating heterogeneity, however, is not straightforward when sampling autocorrelated quantities that may also have gradient patterns, a gradual directional change. To account for autocorrelation or gradients, ecologists employ © 2016 The Authors. Methods in Ecology and Evolution © 2016 British Ecological Society