Investigating species–environment relationships at multiple scales: Differentiating between intrinsic scale and the modifiable areal unit problem

Ecological Complexity 11 (2012) 91–102

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Ecological Complexity journal homepage: www.elsevier.com/locate/ecocom

Original Research Article

Investigating species–environment relationships at multiple scales: Differentiating between intrinsic scale and the modifiable areal unit problem Alex M. Lechner a,b,*, William T. Langford b,c, Simon D. Jones b, Sarah A. Bekessy c, Ascelin Gordon c a Centre for Mined Land Rehabilitation, Sustainable Minerals Institute, University of Queensland, Sir James Foots Building, Cnr Staff House and College Roads, St. Lucia, QLD 4072, Australia b School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia c School of Global Studies, Social Science and Planning, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia

A R T I C L E I N F O

A B S T R A C T

Article history: Received 26 September 2011 Received in revised form 18 April 2012 Accepted 18 April 2012 Available online 15 May 2012

In ecology, multi-scale analyses are commonly performed to identify the scale at which a species interacts with its environment (intrinsic scale). This is typically carried out using multi-scale species– environment models that compare the relationship between ecological attributes (e.g., species diversity) measured with point data to environmental data (e.g. vegetation cover) for the surrounding area within buffers of multiple sizes. The intrinsic scale is identified as the buffer size at which the highest correlation between environmental and ecological variables occurs. We present the first investigation of how the spatial resolution of remote sensing environmental data can influence the identification of the intrinsic scale using multi-scale species–environment models. Using the virtual ecologist approach we tested this influence using vegetation cover spatial data and a simulated species–environment relationship derived from the same spatial data. By using a simulation model there was a known truth to use as a benchmark to measure accuracy. Our findings indicate that by varying the spatial resolution of the environmental data, the intrinsic scale may be incorrectly identified. In some cases, the errors in the intrinsic scale identified were close to the maximum value possible that could be measured by this experiment. Consequently, multi-scale ecological analyses may not be suitable for distinguishing scale patterns caused by the relationship between an organism and its environment from scale patterns caused by the effect of changing spatial resolution: a phenomenon referred to as the modifiable areal unit problem (MAUP). Thus, observed scale-dependent ecological patterns may be an artefact of the observation of ecological data, not the ecological phenomenon. This study concludes with some suggestions for future work to quantify the effect of the MAUP on multi-scale studies and develop generalisations that can be used to assess when multi-scale analyses have the potential to produce spurious results. ß 2012 Elsevier B.V. All rights reserved.

Keywords: Multi-scale Modifiable areal unit problem Spatial resolution Spatial uncertainty Remote sensing Species–environment relationships Virtual ecologist Simulation modelling

1. Introduction 1.1. Scale Determining the scale of landscape processes and the scale dependency of ecological responses to the resulting landscape patterns are central questions in landscape ecology (Turner, 1989; Wu and Li, 2006). At the same time, spatial uncertainty arising from changes of scale is a major concern in the spatial sciences such as remote sensing research (Marceau and Hay, 1999; Openshaw, 1984). The aim of our study is to examine how the

* Corresponding author at: Centre for Mined Land Rehabilitation, Sustainable Minerals Institute, University of Queensland, Sir James Foots Building, Cnr Staff House and College Roads, St. Lucia, QLD 4072, Australia. Tel.: +61 401 233 019; fax: +61 733 464 021. E-mail addresses: [email protected], [email protected] (A.M. Lechner). 1476-945X/$ – see front matter ß 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecocom.2012.04.002

effects of scale in remote sensing data may lead to misidentification of scale-dependent ecological patterns. In this study, we distinguish three important components of scale from the spatial sciences and ecology literature: observation scale, analysis scale and intrinsic scale (Dungan et al., 2002; Wu and Hobbs, 2007; Wu and Li, 2006). Observation scale describes the size, shape, extent and distance between observational units used to sample a phenomenon (e.g. spatial resolution). Analysis scale refers to the units used in analysing data and is usually coarser than the observation scale. For example, when analysing remote sensing data in ecology, the observation scale is the spatial resolution of the image and the analysis scale employed by an ecological analysis may be the size of a circular buffer around point locations at which ecological data is sampled. Within each buffer multiple pixels of the remote sensing imagery are found. Intrinsic scale is the scale at which ecological phenomena interact with or perceive the environment. It is an emergent property of an organism’s relationship with its environment and is measured indirectly through using multiple observation scales and/or analysis scales.

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92 Table 1 Descriptive nomenclature. Descriptors

Definition

Scale Observation scale Spatial resolution

Generic term used to describe all three components of scale. Sampling unit used to measure the environment. Smallest identifiable feature within an image. In this study we only consider how pixel size and the application of a smoothing filter affect spatial resolution. Size of spatial units in remote sensing data. Remote sensing processing method that is used to enhance images and image classification quality, but results in a coarsening of spatial resolution. Size of spatial unit used in an ecological analysis. Area in which vegetation fraction cover is measured. Scale at which an ecological process interacts with the environment – measured indirectly with observation and analysis scale. Studies which are conducted at multiple analysis or observational scales. In some cases to identify the intrinsic scale.

Pixel size Smoothing filter Analysis scale Buffer Intrinsic scale Multi-scale studies Landscape Correct landscape Apparent landscape

Remote sensing dataset that is defined as correct – used to derive the correct ecological response. Remote sensing dataset with pixel sizes that differ from the correct landscape and/or has a smoothing filter applied. Used to derive the apparent ecological response.

Ecological response

Dependent variable in the simulated species–environment relationship, analogous to ecological attributes such as population abundance. Derived from vegetation fraction cover found in a specific buffer size using the correct landscape.

Response correlation curve (RCC)

Function that describes correlation between the ecological response and vegetation cover for multiple buffer sizes. The shape of the curve is used to identify scale patterns, i.e. intrinsic scale. Function derived using correct landscapes. Function derived using apparent landscapes.

Correct RCC Apparent RCC Identified intrinsic scale Correct intrinsic scale Apparent intrinsic scale Base buffer

Buffer size at which the correct RCC has the highest correlation value. Buffer size at which the apparent RCC has the highest correlation value. Buffer size used to generate the RCC. Corresponds with the correct intrinsic scale.

In this study we will consider limited aspects of observational and analysis scales, and will refer to these respectively as spatial resolution, and buffer size, respectively to describe our experimental method. While terms like spatial resolution represent just one aspect of the observation scale, the terms buffer size and intrinsic scale make the discussions less abstract and easier to follow (Table 1). Finally, the term scale when used on its own refers generically to all three scale components. 1.2. Scale questions in ecology According to ‘hierarchy theory’ (O’Neill et al., 1989), ecological phenomena interact with or perceive the environment at relatively isolated, distinct intrinsic scales (O’Neill et al., 1989). For example, mobility, home range and proportion of habitat available are thought to be key drivers for the generation of patterns that vary across different scales (Addicott et al., 1987; Krawchuk and Taylor, 2003; Pearman, 2002; Soderstrom and Part, 2000; Wheatley and Johnson, 2009). Relationships found at one analysis and/or observation scale are not necessarily observable at other scales, so phenomena need to be measured at the appropriate scales (Levin, 1992; Turner et al., 2001; Wiens, 1989; Wu et al., 2006). This suggests that phenomena need to be analysed at multiple scales to detect scale-dependent ecological patterns (Wiens, 1989; Wu and Li, 2006). A particular focus among ecological investigations of scaledependent relationships is the determination of the scale at which a given ecological phenomenon operates (i.e. the identification of intrinsic scale(s)). This is often investigated by measuring environmental attributes at multiple buffer sizes, and then correlating these attributes to ecological measurements (Fig. 1). A common experimental strategy is to sample ecological attributes such as species diversity at random points across a landscape and compare these to environmental measurements made over a circular area centred at the point, commonly referred to as a buffer (e.g. Coreau and Martin, 2007; Cushman and McGarigal, 2004; Davis et al., 2007; Holland et al., 2004; Suorsa et al., 2005). The

environmental measurements within the buffer are most commonly derived from remote sensing data of a single spatial resolution (often the finest spatial resolution available) and are usually area based measurements such as percentage cover of single or multiple vegetation classes. However, other environmental variables describing vegetation characteristics such as forest edge may also be measured within the buffer area (e.g. Taki et al., 2007). This type of analysis is performed at multiple buffer sizes to test the strength of the relationship at each size (henceforth known as multi-scale buffer sampling design) (Fig. 1). The buffer size where the relationship between the ecological attribute and environmental measurement is strongest is identified as the intrinsic scale (Fig. 1b) (e.g. Coreau and Martin, 2007; Holland et al., 2004; Suorsa et al., 2005). At different buffer sizes the relative importance of environmental variables such as percentage cover changes as the amount and composition of these variables change. For example, the relationship between home range size and the proportion of habitat available at different buffer sizes is thought to be a key ecological factor in determining scale dependent ecological patterns (Pearman, 2002; Soderstrom and Part, 2000). Suorsa et al. (2005) found that the relationship between forest cover and probability of occupancy for the Eurasian tree creeper was the strongest at buffer sizes with a similar area to territory size. This type of analyses is more commonly conducted for bird species (Coreau and Martin, 2007; Pearman, 2002; Soderstrom and Part, 2000; Suorsa et al., 2005), but has also been conducted on other species, such as bees (Taki et al., 2007) and beetles (Holland et al., 2004). 1.3. Scale questions in the spatial sciences While the previously described form of analysis is intuitively appealing, a similar form of multi-scale experiment has been used to demonstrate something completely contradictory in the spatial sciences. In particular, similar experiments have been used to test the sensitivity of a statistical analysis to variations in the spatial

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Fig. 1. Example of an ecological analysis conducted using a multi-scale buffer sampling design. (a) In this example the ecological response (e.g. species diversity) is measured at point locations and vegetation cover fraction is measured within a circular buffer of multiple sizes surrounding the point samples. (b) Correlation (r) between species diversity and vegetation cover are calculated for each buffer size to derive the response correlation curve (RCC). The intrinsic scale is identified as 125 m, which is the buffer size at which the highest r value occurs.

resolution of remote sensing data (Marceau and Hay, 1999; Marceau et al., 1994). These studies investigate what is referred to as the modifiable areal unit problem (MAUP) (Openshaw, 1984). In the MAUP view, the same base data describing an ecological process can be aggregated or bounded in different ways before analysis, for example, using 1 m pixels versus 30 m pixels. Statistical relationships derived from these areal units with different boundaries and levels of aggregation can lead to different statistical conclusions about the same ecological process. Consequently, studies investigating the effect of the MAUP using spatial data from a range of sources, such as remote sensing and census data, have found that MAUP effects can render the results of some spatial statistical analyses meaningless (Jelinski and Wu, 1996; Nelson, 2001; Wu et al., 1997). This is important in ecological studies of scale since some ecological measurements derived from the arbitrarily sized pixels of a remote sensing image can be viewed as a specific case of the MAUP. Since such imagery, at various pixel sizes, is commonly used to map environmental covariates, intrinsic scale effects may be indistinguishable from effects of the MAUP. Indeed, tests for the MAUP often use multi-scale experimental designs that are similar to studies attempting to identify the scale of ecological interactions (i.e. intrinsic scale). For example, studies testing for the presence of the MAUP use differences in correlation coefficients to infer whether a statistical analysis is not robust to changes in analysis scale. In contrast, ecological studies searching for the correct analysis or observation scales at which to measure a phenomenon infer that differences in correlation coefficients are related to the response of an ecological phenomenon to particular scales. While the inferences made from these studies differ, there is no substantive difference in experimental design for studies with these two distinct goals. The primary focus of this study will be to give examples of scale-dependent patterns that result from the MAUP and the intrinsic scale. It is important to note that the MAUP is the result of the many ways in which non-overlapping units can be used to divide a study area for the purposes of analyses. It can be divided into two related components described by Openshaw (1984) as the: (i) scale problem and (ii) the aggregation or zoning problem. The scale problem affects analyses as a result of changes in the spatial units through the aggregation of small units into progressively larger units, and vice versa (i.e. changing spatial resolution). The zoning

problem can affect analyses as a result of varying the boundary of areal units while keeping the number of units constant. In remote sensing, the MAUP is a particular case where the units are controlled by spatial resolution. In this study we focus on the effects of the MAUP arising from the differences in spatial resolution. This is tested through a combination of two factors: pixel size and the application of a smoothing filter. In many studies, the effects of spatial resolution on analyses are investigated only through differences in pixel size. Spatial resolution is often considered equivalent to pixel size (Atkinson, 2004), however, the information content of a pixel and thus its spatial resolution is also affected by landcover found in neighbouring pixels (Cracknell, 1998; Fisher, 1997). In order to test for the influence of other factors affecting spatial resolution we applied smoothing filters. They are a commonly used remote sensing processing technique that increases the influence of neighbouring pixels and thus their application may potentially result in the MAUP. Smoothing filters can be used in remote sensing to increase global classification accuracy by decreasing the salt and pepper effect caused by per-pixel based landscape classification methods or to remove noise caused by sensor error in raw remote sensing data (Ivits and Koch, 2002; Zukowskyj et al., 2001). 1.4. Aims We present the first investigation of the degree to which the MAUP might influence the reliability of multi-scale analysis methods used to identify intrinsic scale. We use real vegetation cover spatial data and a simulated ecological response (i.e. species diversity) derived from these spatial data to give examples where the MAUP is problematic. A critical distinction between our work and previous work on intrinsic scale is the use of the virtual ecologist approach whereby we simulate ecological data and model the effects of the observation of that data on statistical analyses (Zurell et al., 2010). By using simulated data this study can distinguish between true effects of the simulated species and environment relationship and effects due solely to the spatial resolution of the underlying data. While the simulated ecological response model represents a simplified version of reality, if this relationship cannot be recovered accurately using a simulated response, there can be little confidence in differentiating ecological

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Fig. 2. Tree presence (grey)/absence (black) landscapes with percentage tree cover indicated below the landscape title. 30 sampling locations for the ecological model are indicated on the map by grey circles (500 m diameter).

effects from the MAUP effects in real data. We conclude with a discussion of the implications of our findings for future multi-scale studies in ecology. 2. Method 2.1. Experimental approach To test for the effects of the MAUP on the identification of intrinsic scale, we simulated an ecological response by applying a predefined species–environment relationship to a set of real landscapes. The simulation model was comprised of 3 parts: (i) landscape definition, (ii) sensor simulator and (iii) ecological response simulator. We first provide a general overview of our method and then go into more detail of the three parts. First, we defined a remote sensed dataset as being the correct landscape. In our simulation this represents what is actually present on-ground and this information would not normally be available when using real-world data. The remote sensing data were aggregated to an alternative pixel size and processed to create a range of realistic geographic representations of the same landscape. These multiple representations of the same landscape are designated as apparent landscapes and simulate the use of imagery derived from satellites with different spatial resolutions and different remote sensing processing techniques. For example, 10 m pixels instead of 30 m pixels. Next, we defined the correct response correlation curve (RCC), which describes correlation (r) between the ecological response and vegetation cover fraction for multiple buffer sizes derived using a multi-scale buffer sampling design with a specific (correct) intrinsic scale (e.g. Fig. 1b). By comparing the RCCs generated using the apparent and correct landscapes, we were able to determine whether it was possible to recover the correct RCC and the correct intrinsic scale when using apparent landscapes. If the recovered RCC or intrinsic scale changed when landscapes with different spatial resolutions were used, this would indicate the presence of the MAUP and therefore, an unreliable ecological inference. 2.2. Define landscapes The landscapes used in the simulation model were a stratified subset from a random sample of sub regions of the regional Tree25 presence/absence tree cover dataset (DSE, 2006). The Tree25 dataset covers most of the state of Victoria, Australia, comprising a wide range of land uses including conservation areas, dryland pasture, broad acre cropping and crop pasture. These differing land uses resulted in a wide range of landscape patterns. The dataset was derived from SPOT panchromatic imagery with a 10 m pixel

size and was classified by automated segmentation, and then the presence or absence of woody vegetation was labelled by a human interpreter. For the purposes of the simulation model, these images were declared to represent vegetation cover with 100% accuracy. Initially, 20 random 10 km  10 km sub-regions of the Tree25 datasets were extracted and then six of those sub regions (henceforth, called Landscape A, B, . . ., F) were chosen so that a range of tree-cover proportions and patterns were sampled (Fig. 2). The sub-regions included a range of ecologically important landscape elements, such as small remnant patches and linear strips. 2.3. Sensor and processing simulator Next, a sensor and processing simulator was used to create different representations of the original six landscapes. This involved two steps. First, the 10 m pixels of the Tree25 classified landscapes were aggregated to 30 m pixels using a majority rule reducing the number of pixels to 1/9 of the original total. This pixel size matches the Landsat TM/ETM+ sensors’ pixel size and is commonly used in landscape ecology (Table 2). In the second step, the dataset was smoothed using a majority filter creating an additional representation of the landscapes. The majority filter uses a 3  3 moving window filter to replace the value of a pixel in the centre of the window based on the majority value of all pixels in the window. This operation leaves the pixel size unaltered. Thus for each of the 6 landscapes, A, B, . . ., F, there were four variants: 10 m and 30 pixel sizes, which are both smoothed and unsmoothed (Fig. 3). This results in 24 different landscapes. 2.4. Simulating the ecological response and testing for the MAUP The crux of the simulation is to reproduce common elements of analyses with real ecological data that use a multi-scale buffer design. To make the explanation easier to follow, we will first describe the methods using a concrete but fictitious example rather than abstract names. Suppose that in a real situation, the normalised abundance of a particular bird species at any location was equal to the fraction of the landscape that was vegetated within 125 m of that point. In other words, there is a very simple linear, scale-dependent relationship, y = x, between an ecological response y (abundance; normalised from 0 to 1) and the independent variable x (vegetation cover fraction) at a given intrinsic scale (125 m). An ecologist could try to identify that intrinsic scale by 1) conducting a bird survey at 30 randomly chosen fixed locations on the ground to estimate the abundance

Table 2 Pixel sizes tested in the simulation model and equivalent satellite sensor. Correct landscapes did not include smoothing filter (*). Data source

Pixel size (m)

Smoothing

Total true landscapes

Total apparent landscapes

Sensor equivalent

Landscape A, B, . . ., F

10, 30

Yes/no*

12

24

SPOT XS, Landsat

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Fig. 3. Example of remote sensing simulator outputs: landscape C aggregated from 10 m to 30 m with a smoothing filter applied.

2) choosing 10 different analysis scales (buffer sizes) and determining the vegetation cover fraction within each buffer size at the same locations using a classified remote sensed image, and finally 3) determining the buffer size whose 30 vegetation cover fractions has the highest correlation with the 30 measured values of abundance. The above experiment can be mimicked for a particular map of a landscape by arbitrarily choosing an intrinsic scale (buffer size) and an equation to declare as our ecological response. Because the central premise of this study is that the identification of the intrinsic scale of the underlying ecological process may be affected by the observation scale of the map, we compared the results of doing the same experiment on different maps of the same place. Additionally, by using different intrinsic scales to generate the correct ecological response any effects due to the idiosyncrasies of one particular scale choice could be assessed. So, mimicking the structure of a real world experiment, each single experiment in our study is uniquely specified by 1) one intrinsic scale designated as correct, 2) one landscape, 3) one realisation of that landscape designated as correct. The full set of experiments consists of the combinations of all possible values of those three attributes chosen as follows: 1) possible base buffer radii of [25 m (1964 m2), 125 m (49 087 m2) and 225 m (159 040 m2)] 2) possible landscapes are the six Tree25 landscapes: [A, B, C, D, E, F] 3) possible correct landscapes are: [(10 m pixel, unsmoothed), (30 m pixel, unsmoothed)] which results in 3  6  2 = 36 experiments in total. Fig. 4 shows a schematic representation of the process of simulating the ecological response and testing for the MAUP after the application of the sensor simulator. This process involved four steps, which we describe below. Step 1a. In each experiment, the first step is to declare the correct value of the ecological response and the intrinsic scale. To generate the ecological response for any given landscape the following procedure was used. Thirty sample locations are selected randomly within the landscape. This number is similar to many landscape-level studies (e.g. Pearman, 2002; Taki et al., 2007). The locations were constrained to be far enough apart so that the largest buffer placed around them (250 m radius) would not overlap with the buffer of any other point. This is a common practice in ecological experiments to reduce the effect of spatial autocorrelation and avoid violating the assumption of independence for regression models.

The proportion of vegetation cover within these buffer areas is recorded for each location at each of three base buffer sizes [25 m (1964 m2), 125 m (49 087 m2) and 225 m (159 040 m2)] to use as the basis for three different experiments with three different intrinsic scales. For each experiment, the base buffer size used is declared to be the correct intrinsic scale for that experiment. To simulate the ecological response that would be measured at each location, we arbitrarily choose the simplest possible linear form commonly used in species–environment studies, that is, y = x, where y is the ecological response and x is vegetation cover fraction at one of the three base buffer radii specified above (25 m, 125 m, 225 m). The ecological response itself abstractly represents commonly measured ecological attributes, such as occupancy (Suorsa et al., 2005), population abundance (e.g. Heikkinen et al., 2007; Holland et al., 2004) and species diversity (e.g. Lawler et al., 2004; Pearman, 2002). In studies of this kind, measurements of vegetation cover fraction are commonly used as an explanatory variable to assess the impact of vegetation cover on ecological attributes. Because we use the simple y = x form in our simulations and the vegetation fraction recorded at that buffer size is used as the input variable x, the vegetation cover fraction itself becomes the value of the ecological response for all of these simulations (e.g. Fig. 1a). In a real experiment, we would not know the intrinsic scale, but in the simulation model the intrinsic scale is determined by the base buffer size used in the computation of the ecological response. While, the identification of the intrinsic scale in real experiments is conducted by deriving the RCC. To derive the RCC, vegetation cover fraction was measured at 10 buffer sizes (25 m, 50 m, . . ., 250 m) and correlated with the original vegetation cover fraction values of the ecological response. Correlation was calculated using Spearman’s r as the assumption for linear regression was not met. The assumption was not met because the distributions of vegetation cover fraction values were often non-normal and plots of the residuals demonstrated that the data were often heteroscedastic. Step 2a. The correct RCCs are derived for each of the 6 landscapes A, B, . . ., F, at two different pixel sizes (10 m and 30 m) using only the unsmoothed versions of these maps. The correct RCCs are measured by undertaking Step 1 at three base buffer sizes (25 m, 125 m and 225 m), thus creating six functions for each of the landscapes A, B, . . ., F (2 pixel sizes  3 base buffer sizes). Step 2b. The apparent RCCs are derived using four variants for each of 6 landscapes A, B, . . ., F: smoothed and unsmoothed at pixel sizes of 10 m and 30 m. In Step 2a the correct RCC was derived only for unsmoothed versions of these maps. Now we create a set of apparent RCCs using the apparent landscapes to compare to the correct RCC using the following method: 1) First, a landscape is assigned as the correct landscape (C), leaving three apparent landscapes (A1, A2, A3). For example, if landscape

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Fig. 4. A schematic diagram that describes the relationship between Figs. 5 and 7. See Section 2 for further details. Step 1. Generate the ecological response for any given landscape using vegetation cover fraction at 30 points. Step 2. Derive correct RCCs for each of the 6 landscapes A, B, . . ., F, at two different pixel sizes (10 m and 30 m) and three buffer sizes (25 m, 125 m and 225 m), using only the unsmoothed versions of these maps (2 pixel sizes  3 buffer sizes). Next, derive the apparent RCCs using the apparent landscapes (A) to compare to the correct RCC (T) using the following method. Step 3. Compare shape of correct and apparent RCCs and buffer size of correct and apparent intrinsic scales. Step 4. Compare buffer radii for the smallest and largest apparent intrinsic scales to the correct intrinsic scale. Note that each circle in example Fig. 6 corresponds to the maximum correlation value on a different RCC.

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B at 10 m was selected, this becomes the correct landscape, then the three associated apparent landscapes are: unsmoothed at 30 m, smoothed at 10 m, and smoothed at 30 m. 2) As described in Step 2a select the correct landscape and measure vegetation fraction using one of three buffer sizes (25 m, 125 m and 225 m). The vegetation cover fraction value at the 30 locations becomes the dependent variable, the ecological response. Next, measure vegetation cover fraction at 30 locations using a particular buffer size and the apparent landscape. Vegetation cover fraction is measured at the 30 locations using the apparent landscape and correlated with the ecological response measured in step 1a. This is repeated for multiple buffer sizes (25 m, 50 m, . . ., 250 m) to derive the apparent RCCs. 3) The previous step is then repeated for each of the three buffer sizes (25 m, 125 m and 225 m) used to derive the ecological response. 4) Finally, applying the previous two steps on each of the three apparent landscapes using the ecological response derived for buffer sizes: 25 m, 125 m and 225 m and pixel sizes: 10 m or 30 m. This results in six sets of one correct RCC and three apparent RCCs for each of the six ecological responses (3 buffer sizes  2 pixel sizes). Step 3. The buffer size at which the correct ecological responses are derived is the correct intrinsic scale as it will have a correlation coefficient value of 1 and thus the highest correlation coefficient. At this buffer size we are using the same data for both dependent (ecological response) and independent variables. If the buffer size is changed the correlation coefficient decreases. The buffer size identified as having the highest r value using the apparent RCC is termed apparent intrinsic scale. Step 4. The effects of the MAUP are measured by comparing the differences between the correct and apparent RCCs. The peaks and troughs of the RCC correspond to the response of ecological phenomenon to buffer size as in other ecological studies (e.g. Holland et al., 2004; Pearman, 2002). From the apparent RCC, multiple intrinsic scales may be identified if multiple peaks occur, however, the correct RCC was derived with a single intrinsic scale. For cases where two or more buffer sizes have the maximum value (i.e. in cases where the shape of the curve plateaus and does not peak), the intrinsic scale is identified as the average of the maximum range of these values. Differences in the intrinsic scale identified between the correct RCC and the apparent RCC as well as the shape of the function indicate the presence of the MAUP.

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3. Results The strength of relationships between vegetation cover and the ecological response changed with buffer size and pixel size. For example, Fig. 5 (landscape D) presents 2 out of 144 sets of scatter plots (6 landscapes (Landscape A, B, . . ., F)  6 buffer sizes and pixel combinations  3 apparent and 1 correct landscapes) produced in this study used to describe the relationship between vegetation cover fraction and the ecological response. Fig. 5a shows the correlations generated for a range of buffer sizes to produce the correct RCC. The function was generated by correlating vegetation cover fraction at 125 m with vegetation cover fraction at all buffer sizes. In this case the ecological response was derived from a buffer size of 125 m and thus the correlation coefficient was 1 and there was a perfect linear relationship. This buffer size is identified as the intrinsic scale. When buffer size increased or decreased the correlation between vegetation cover fraction and the ecological response became weaker. Changing the pixel size by using an apparent landscape with a pixel size of 30 m compared to a pixel size of 10 m at which the ecological response was derived and applying a smoothing filter resulted in incorrectly identifying the intrinsic scale at a buffer size of 175 m (Fig. 5b). At this radius the highest correlation coefficient of 0.90 was recorded. The scatter plots presented in each row of Fig. 5 can be summarised as a single curve by plotting the correlation coefficient versus buffer size for the correct or apparent landscapes to derive the RCC. Curve a in Fig. 6 refers to the scatter plots of Fig. 5a and curve b in Fig. 6 refers to scatterplots of Fig. 5b. This is a common method of depicting the relationship between buffer size and correlation (e.g. Holland et al., 2004; Pearman, 2002). However, previous studies use only a single spatial resolution and test multiple buffer sizes which is equivalent to a single curve in Fig. 6. The effect of the MAUP is illustrated by comparing curves describing the correlation coefficient versus buffer radii for apparent landscapes to curves for correct landscapes. This effect was tested for multiple RCCs and multiple landscapes (Fig. 7 Landscape B; plots for other landscapes can be found in the supplementary material). Using apparent landscapes usually decreased correlation coefficients across most buffer sizes (e.g. Fig. 7b, c, e, f). Furthermore, the intrinsic scale was misidentified in one or more apparent landscapes in every case except when the RCC was derived with a pixel size of 30 m and buffer size of 25 m (e.g. Fig. 7b). Not only did correlation coefficient values change with the use of apparent landscapes, but so did the shapes of the apparent RCC. In

Fig. 5. Scatter plots of synthetic ecological response versus vegetation cover fraction (%) for landscape D. The line of best fit plotted in dark grey and Spearman’s r is presented in the top left corner. The ecological response was derived at a pixel size of 10 m and the buffer size of 125 m for both row a and b. (a) Landscape D, correct landscape, with a 10 m pixels and no smoothing. (b) Landscape D, with 30 m pixels and smoothing applied (apparent landscape). Each plot represents results at the buffer size for that column. Each point inside a plot represents a single sampled location on the map at the given buffer size.

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Fig. 6. Landscape D. Correlation (Spearman’s r) versus buffer radius for a range of correct and apparent RCCs for an ecological response derived at a pixel size of 10 m and the buffer size of 125 m. Curve (a) in bold is the correct RCC based on the correct landscape used to derived the ecological response. This curve is the equivalent of Fig. 5a. Curve (b) is derived using an apparent landscape with 30 m pixels and smoothing applied is the equivalent of Fig. 5b. The intrinsic scale is identified as the buffer size with the highest r value. The buffer size of the true intrinsic scale is indicated by the dotted vertical line. The apparent intrinsic scales identified for each of the apparent RCCs is indicated with a circle.

some cases the shape of the functions changed greatly, from a simple exponential to a complex polynomial resulting in the appearance of multiple peaks in the function indicating two or more intrinsic scales rather than a single intrinsic scale as represented by the correct landscape. For example, in Fig. 7a, an ecological response was derived for a pixel size of 10 m and a buffer size of 25 m for landscape B. The correct intrinsic scale was 25 m, but when using the corresponding apparent landscapes with a pixel

size of 30 m and no smoothing filter applied, there were two peaks in correlation coefficient values which indicate intrinsic scales at buffer sizes 100 m and 200 m. The effect of applying the smoothing filter was dependent on the pixel size of the correct landscape. Pixel size had a greater impact on the RCC than the application of the smoothing filter. In most cases, applying a smoothing filter decreased correlation coefficients. In some cases, it changed the intrinsic scale identified when the pixel size of the apparent landscapes did not match the correct pixel size (e.g. Fig. 7f; Pixel size 10 m and smoothing filter applied). The effects of using apparent landscapes varied between landscapes and depended on the buffer size and pixel size used to derive the RCC. For some landscapes, such as Landscape B, the effect on analysis of using apparent landscapes resulted in no differences in intrinsic scale identified for one combination of buffer and pixel sizes (30 m pixel, 25 m buffer). Meanwhile, errors of 125 m in the intrinsic scale were identified for another combination (30 m pixel, 125 m buffer) (Fig. 8b). Using one or more apparent landscapes for Landscape C resulted in erroneously identifying the intrinsic scale for all combinations of buffer and pixel sizes in at least one apparent landscape in (Fig. 8c). In contrast, the intrinsic scale was identified correctly for all combinations of buffer and pixel sizes in Landscape E (Fig. 8e). 4. Discussion 4.1. Overview The results of this study demonstrate that spatial uncertainty arising from the MAUP has the potential to produce misleading

Fig. 7. Landscape B. Correlation (Spearman’s r) versus buffer radius for a range of correct and apparent RCCs. Curves in bold are based on the correct landscape used to derive the RCC at a buffer radius and pixel size indicated above each plot. The other RCC describe r values versus buffer size relationship derived with apparent landscapes. The buffer size of the true intrinsic scale is indicated by the dotted vertical line.

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Fig. 8. Comparison of buffer radii (Y-axis) for the smallest and largest apparent intrinsic scales and the correct intrinsic scale identified for all landscapes and all derived RCCs combinations. For example, each of the 6 plots in Fig. 7 are summarised as six sets of three bars in plot (b). The first of the three bars describes the smallest intrinsic scale identified from the apparent RCC. The second bar describes the correct intrinsic scale. The third bar describes the largest intrinsic scale identified from the apparent RCCs. The specific pixel and buffer size at which the RCC was derived is described in the x axis. Differences between the second bar and the first and/or third bar represent the size of errors in identifying the intrinsic scale. While, if all three bars are of the same height there was no error in identifying the intrinsic scale.

results in ecological analyses. We found that the observed ecological responses repeatedly exhibited the MAUP and that these responses were not just a property of the buffer size but also of spatial resolution. More specifically, we detected a high level of variability in the effect of using apparent landscapes on the accuracy of the identified intrinsic scales between landscapes and for different combinations of pixel sizes and buffer sizes used to derive the RCC. In some cases, the differences in pixel size only affected the strength of the relationship between the ecological response and vegetation cover fraction. In other cases the intrinsic scale was misidentified altogether with error values ranging from 0 to 175 m, where 225 m was the maximum measured value. Furthermore, in other cases multiple intrinsic scales were identified when in reality there was only a single intrinsic scale. The effects of the MAUP could also be seen qualitatively in the large differences in the shape of the apparent and correct RCCs. Finally, changing pixel size appeared to have the greatest effect on the outcome of analyses and the smoothing filter only had a similar effect size for specific combinations of landscapes and derived ecological responses. This is interesting as both 30 m unsmoothed pixels and the 10 m smoothed pixels result from a similar level of degradation but using different methods. In light of such divergent findings demonstrating instances where the MAUP had large effects on ecological analyses, there is a need to assess whether scale patterns reported using ecological analyses result from scale-dependent ecological phenomenon (i.e. intrinsic scale) or the MAUP. We consider that the investigative strategy of using a simulation model was exemplary in detecting these apparently divergent effects of scale that were undetectable when using a multi-scale buffer sampling design.

4.2. Scale and ecological analysis The results of our study are relevant to ecological analyses using multiple and single analysis or observational scales. We tested commonly used pixel sizes in landscape ecology based on standard operational remote sensing sensors and demonstrated that the outcome of analyses will be different when using pixel sizes equivalent to either a SPOT or a Landsat sensor. Landscape level research, however, is typically only conducted at fixed observation and analysis scales (i.e. single pixel size and buffer size). These are often arbitrarily selected according to readily available generic datasets or sensors without any investigation of the sensitivity of analysis to different scales (Comber, 2008; Pontius et al., 2008; Schmit et al., 2006). Our study showed that it is critical to test for scale sensitivity and that studies that use fixed scales such as a single buffer or pixel size may unknowingly overlook a scaledependent ecological relationship or the MAUP. The most common recommendation in landscape ecology to test for intrinsic scale is to conduct studies at multiple analysis or observation scales (Levin, 1992; Wiens, 1989; Wu et al., 2006). In fact, multi-scale studies tend to use a small number of scales and test a single scale-dependent factor such as pixel size or buffer size. In contrast, our results suggest that the relationship between the intrinsic scale, buffer size and spatial resolutions may vary substantially along a continuum. The intrinsic scale may change with multiple buffer sizes, often abruptly, and thus the results of studies using a small selection of buffer sizes may lead to erroneous conclusions being drawn due to the absence of data at untested sizes. For example, in Fig. 7, plot b, correct RCC, there was a change in r values from 0.8 to 1.0 between buffers size 75 m to 125 m, where 0.7 was the lowest r value recorded by this curve.

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More robust studies that use a continuum of analysis scales to identify intrinsic scale(s) cannot guarantee that observed patterns at a particular analysis scale are not spurious due to the presence of the MAUP. This study utilised a simulation model and thus there was a known truth based on the simulated ecological response to use as a benchmark to measure accuracy. When conducting multiscale ecological analyses using real data, the intrinsic scale identified is a property of both how the intrinsic scale is measured and the underlying scale-dependent process, so there is no way of knowing whether scale-dependent patterns observed are affected by the MAUP. 4.3. Simulation modelling: simplifying complex ecological relationships A novel aspect of our study is the use of the virtual ecologist approach using a simulation model to investigate the effects of spatial resolution on measuring the intrinsic scale. Simulated data allows some elements of the complexity of real world phenomena to be studied. Our simulation model represents an idealized scenario for conducting ecological analyses, where no ‘unknown’ sources of uncertainty were present in the spatial data or in the ecological model. Consequently, there was no measurement error and all the parameters and all statistical relationships were known. Additionally, the correct landscapes represent the true geographic representation that perfectly describes how the species perceives the environment. By contrast, such relationships are typically never known with absolute certainty when using real data and ecologists can only make inference about causal processes based on empirical data. In studies using real data the effects of spatial uncertainty on ecological analyses are likely to be much greater and more difficult to distinguish than in simulation studies. A feature of simulation modelling is that if simple theoretical relationships cannot be derived correctly using simulation models, there can be little confidence in recovering these from more complex ‘real’ data (Austin et al., 2006). In comparison between this study and other studies, the measurements of correlation were much higher indicating a stronger relationship resulting from the absence of other unmeasured factors that commonly influence species–environment relationships. The shape of the curves of buffer size versus correlation coefficient reported here appear to be distributed more smoothly (i.e., less erratically), albeit having qualitatively similar trends to other studies that have conducted similar types of analyses on a continuum of buffer sizes using real data (e.g. Holland et al., 2004; Pearman, 2002). However, the derived correlation coefficients were much higher in our simulation model with a consistently smaller range than these other studies. The maximum range of r for any landscape tested in this study was approximately 1.0–0.57. Other studies investigating the relationship between an ecological attribute and vegetation cover had larger ranges and smaller r values. For example, Holland et al. (2004) calculated Pearson’s r values of 0.0–0.3, Pearman (2002) calculated r2 values of 0.35–0.7 and Taki et al. (2007) calculated r2 values of 0.045–0.165. Additionally, differences between buffer sizes and correlation coefficients tended to be smaller for the simulation model compared to real studies. For example, landscape D with an RCC derived using a pixel size of 10 m and buffer size of 125 m with the true landscape at a buffer size of 25 m the r value was 0.78 while for all the other buffer sizes values ranged from 0.95 to 1 (Figs. 5a and 6). Comparison between other studies, however, is made difficult due to the variety of correlation measures used such as r2 and Pearson’s r. Untested forms of uncertainty that are common in real data with more complex, sometimes multivariate relationships have the potential to amplify the impact of the MAUP on analyses. For

example, ecological relationships may be non-linear and difficult to derive statistically because of confounding factors such as spatial autocorrelation (Legendre et al., 2002; Wintle et al., 2005). Our study used a simple univariate linear model to describe the species–environment relationship commonly used by other authors (e.g. Holland et al., 2004; Pearman, 2002; Taki et al., 2007), however, many studies use multiple explanatory variables that may increase the potential for the MAUP. Our model used only a single spatial dataset as an explanatory variable, while many other studies utilise spatial data from multiple sources with different analysis and observation scales. Every spatial dataset included in a model has the potential to introduce MAUP effects. Furthermore, the MAUP can be the result of other scale-dependent factors not tested in this study, such as patch location, minimum mappable units and thematic resolution (e.g. Buyantuyev and Wu, 2007; Lechner et al., 2009; Wu et al., 2000). Our study only tested two scale-dependent factors (pixel size and the application of a smoothing filter) and found that the effect of applying the smoothing filter was dependent on the pixel size at which the ecological response was derived and the pixel size of the apparent landscapes. Thus, spatial uncertainty factors interact and potentially may be difficult to predict and/or magnify when occurring together, further complicating ecological analyses. While our study investigated spatial uncertainty arising from the MAUP, untested sources of spatial uncertainty which are not related to scale may further degrade analyses in unpredictable ways. There are many forms of error that result from the process of generating binary habitat maps such as those used in this study and commonly used in landscape ecology (Antrop, 2007). Spatial uncertainty can be the result of factors, such as classification error (e.g. Langford et al., 2006; Shao and Wu, 2008) and variations and ambiguity in landcover class definitions (e.g. Colson et al., 2009; Comber et al., 2005). For example, Langford et al. (2006) found that in some cases map classification error can cause a thousand-fold increase in error in the calculation of landscape metrics. Ecologists spend considerable effort testing uncertainty in ecological data (Chapman et al., 2005) while often ignoring the effects of spatial uncertainty. Users of spatial data often blindly accept them as error free (Adams and Gillespie, 2006; Evans, 1997), even though uncertainty in spatial data may in some cases be as important or more important than errors in other model parameters (Schmit et al., 2006). Previous studies comparing the impact of spatial uncertainty versus model or parameter uncertainty show that both may impact ecological analyses. However, whether spatial or non-spatial factors are a greater source of error may depend on the specific study. For example, a study by Ruckelshaus et al. (1997) found that their model was more sensitive to error in model parameters compared to spatial inputs. Conversely, Minor et al. (2008) found that the habitat map was the largest source of error for their spatially explicit population model. Thus, to ensure ecological studies are robust to uncertainty, there is a need for testing of spatial data to become as common as testing for uncertainty in non-spatial model parameters. 4.4. Recommendations Further research is needed to expand on our findings and develop methods for reducing the effects of MAUP. The current approach commonly used to address the MAUP involves testing analyses for scale sensitivity by altering observation and/or analysis scale. The research shown here demonstrates that this approach is still useful in detecting the MAUP, but it does not provide a solution to the MAUP, as the causal process cannot be determined using this method. There is a need to develop methods to determine the cause of scale-dependent patterns and develop methods to reduce

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the effects of the MAUP. Alternative approaches to simple binary map classification methods may reduce the effect of the MAUP. These include using more complex landcover classification methods rarely utilized in ecology such as fuzzy classifiers (Foody, 1998) that can describe sub-pixel landcover area (e.g. Robinson, 2007), new approaches to rescaling data (e.g. Gardner et al., 2008) and/or methods for performing multi-resolution analyses with categorical data (e.g. Pontius and Connors, 2009). The findings in this study demonstrate examples of where the MAUP is problematic, but do not identify if it is widespread in ecology. Neither do the findings provide generalisable results due to the sample size and the specific environments that the landscapes represent. The study design used was specifically chosen so that the deterministic mechanisms (i.e. changing pixel size) could be explicitly described and to demonstrate the use of the virtual ecologist approach for evaluating multi-scale models. In this study we used six landscapes with a range of fragmentation patterns to demonstrate that the MAUP effect is not confined to a specific landscape pattern. However, a broader range of studies using greater number of landscapes, controlling for influential factors is necessary to attempt to make generalisations that can predict how landscape pattern influence the effects of the MAUP. Further development of our model using synthetic landscapes to control for spatial patterns with large sample sizes can provide an understanding of the properties of the MAUP. Additional avenues of development required to create generalisations include testing other types of ecological models such as non-linear and/or multivariate models and testing the effect of other scaledependent factors which affect spatial resolution. These scaledependent factors may result from the application of remote sensing and GIS methods such as imposing a minimum mappable unit, applying a low pass filter on raw imagery and resampling for orthorectification. A key obstacle to developing generalisations is the wide range of landscape patterns and the idiosyncratic nature in which analyses may respond to these patterns and the way they are observed. Thus, generalisations using quantitative approaches will always be problematic due to the vast array of untested sources of uncertainty. 5. Conclusion This study used the virtual ecologist approach to investigate whether multi-scale ecological analysis methods can be reliably used to identify the intrinsic scale. It found that the common practice of conducting multi-scale studies to identify the intrinsic scale can in some cases produce flawed results because of the effects of the MAUP. Thus, multi-scale ecological analyses may not be able to distinguish scale-dependent patterns caused by the relationship between an organism and its environment and scaledependent patterns resulting from the MAUP. This is in contrast to the assumption that performing multi-scale analyses such as using multiple buffer sizes will address scaling issues. This study shows that scale-dependent ecological patterns identified in ecological studies may be an artefact of the way ecological phenomenon are observed, which results from a combination of scale-dependent factors such as pixel size, the application of a smoothing filter and buffer size. Acknowledgements This work was supported by the Australian Research Council’s Discovery grant DP0450889, the Australian Research Council’s Linkage grant LP0882780 and the Australian Commonwealth Environment Research Fund: Landscape Logic research hub. Thanks to Patrick Audet and special thanks to the anonymous reviewers for their comments.

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