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MINIREVIEW

Quantifying Landscape SpatialPattern: What Is the State of the Art?

Eric J. Gustafson

North Central Forest Experiment Station, 5985 Highway K, Rhinelander, Wisconsin 54501–9128, USA

ABSTRACTLandscape ecology is based on the premise thatthere are strong links between ecological patternand ecological function and process. Ecological sys-tems are spatially heterogeneous, exhibiting consid-erable complexity and variability in time and space.This variability is typically represented by categori-cal maps or by a collection of samples taken atspecific spatial locations (point data). Categoricalmaps quantize variability by identifying patches thatare relatively homogeneous and that exhibit arelatively abrupt transition to adjacent areas. Alter-natively, point-data analysis (geostatistics) assumesthat the system property is spatially continuous,making fewer assumptions about the nature ofspatial structure. Each data model provides capabili-ties that the other does not, and they should beconsidered complementary. Although the conceptof patches is intuitive and consistent with much ofecological theory, point-data analysis can answertwo of the most critical questions in spatial patternanalysis: what is the appropriate scale to conduct

the analysis, and what is the nature of the spatialstructure? I review the techniques to evaluate cat-egorical maps and spatial point data, and makeobservations about the interpretation of spatial pat-tern indices and the appropriate application of thetechniques. Pattern analysis techniques are mostuseful when applied and interpreted in the contextof the organism(s) and ecological processes of inter-est, and at appropriate scales, although some may beuseful as coarse-filter indicators of ecosystem func-tion. I suggest several important needs for futureresearch, including continued investigation of scal-ing issues, development of indices that measurespecific components of spatial pattern, and efforts tomake point-data analysis more compatible withecological theory.

Key words: spatial pattern; index; indices; spatialheterogeneity; patchiness; landscape ecology; scale;geostatistics; autocovariation; spatial models.

INTRODUCTION

The quantification of environmental heterogeneityhas long been an objective in ecology (Patil andothers 1971; Pielou 1977). Efforts to develop meth-ods to quantify the spatial heterogeneity of land-scapes began more recently (Romme 1982; Bur-rough 1986; Krummel and others 1987; O’Neill andothers 1988a), but have accelerated so that there are

now literally hundreds of quantitative measures oflandscape pattern that have been proposed to quan-tify various aspects of spatial heterogeneity (Bakerand Cai 1992; McGarigal and Marks 1995). In spiteof this, it has been argued that the quantification ofspatial heterogeneity remains problematic becauseof the complexity of the phenomena (Kolasa andRollo 1991) and because the components of hetero-geneity are ill-defined (Li and Reynolds 1994).

There is a rapidly growing demand for measure-ment and monitoring of landscape-level patternsand processes. This demand is driven by the premisethat ecological processes are linked to and can be

Received 3 October 1997; accepted 18 November 1997.e-mail: [email protected]

Ecosystems (1998) 1: 143–156 ECOSYSTEMSr 1998 Springer-Verlag

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predicted by some (often unknown) ecological pat-tern exhibited at coarse spatial scales. Althoughthere has been broad acceptance of this premise, thedifficulties associated with predicting the responseof ecological entities to spatial pattern has led to fewdefinitive tests. For example, in the United States,an ambitious program to sample and monitor land-scape patterns nationwide (EMAP) has been pro-posed (Overton and others 1990; Hunsaker andothers 1994), but it remains in a pilot phase due touncertainty over what to measure and what it mightmean (Kepner and others 1995). The US NationalForests are currently involved in major planningefforts that include consideration of landscape com-position and structure both within each forest andin the surrounding areas (Jensen and others 1996).However, planners often do not have the knowl-edge to make valid projections of the response ofecological systems to the patterns that will beproduced by management activities. Furthermore,information about the historical range of landscapepatterns may be difficult to derive. Nevertheless,many conservation and land management organiza-tions, both public and private, now view a landscapeperspective as essential for sound resource manage-ment (Wallinger 1995; Wigley and Sweeney 1993).

The integration of landscape ecology into re-source management has not been easy, nor is itcomplete. Many persons without special training inspatial analysis are attempting to calculate andinterpret pattern indices. This has been facilitated bythe availability of software capable of calculatingliterally hundreds of indices of landscape patternfrom digital maps. The distinction between theheterogeneity that can be mapped and measuredand the heterogeneity that is ecologically relevant tothe resource being managed is sometimes blurred(Turner 1989). In an attempt to clarify some of thesecomplexities, I have prepared this review with thefollowing objectives: (a) to describe the state of theart of spatial pattern quantification, (b) to discussthe interpretation of pattern indices, (c) to describethe appropriate application of spatial analysis tech-niques, and (d) to suggest lines of research thatmight be particularly useful to meet the needs ofresearchers and resource managers.

DEFINITIONS AND APPROACHES

A useful discussion of the state of the art of land-scape pattern analysis must begin with definitionssince there is considerable inconsistency of terminol-ogy in the literature. Spatial heterogeneity can bedefined as the complexity and variability of a systemproperty in time and space (Li and Reynolds 1994).

A system property in this case can be nearly anymeasurable entity, such as the configuration of thelandscape mosaic, plant biomass, annual precipita-tion, or soil nutrient concentrations. Some impor-tant measures of heterogeneity are clearly not spa-tially explicit (for example, number of land typesand proportions of these types), but have importantspatial effects. Spatial structure is a major subset ofthe concept of spatial heterogeneity, usually refer-ring to the spatial configuration of the systemproperty. The term spatial pattern has been usedextensively in the landscape ecological literature,primarily to describe both the composition andstructure of landscapes [for example, see Turner(1989)]. In this review, I use the terms spatialheterogeneity and spatial pattern synonymously torefer comprehensively to the composition, configu-ration, and temporal aspects of heterogeneity, andwill use the terms structure and configuration whenreferring solely to the spatial components of hetero-geneity.

Pioneers in the quantification of spatial heteroge-neity have suggested that heterogeneity can beconceived as an environmental mosaic [see refer-ences in Turner (1989)], a patterning of ecosystemtypes in space (Forman and Godron 1986; Krummeland others 1987; O’Neill and others 1988a), orpatchiness (Wiens and others 1993). Spatial hetero-geneity is also not static, and neither are ecologicalprocesses. Landscapes are usefully visualized as‘‘shifting mosaics’’ [sensu Bormann and Likens(1979)], and ecological systems are characterized bydynamics, disturbance, and change (Huston 1979;Reice 1994). Thus, spatial heterogeneity also has atemporal component, although much less work hasbeen done to capture this aspect of heterogeneityquantitatively [see Dunn and others (1991) andFahrig (1992)].

Heterogeneity is also a function of scale (Wiens1989; Allen and Hoekstra 1992). The spatial andtemporal variation of a system property that can bedetected will depend on the spatial and temporalscale at which the property is sampled and the sizeof the mapping unit. If you were to study anagricultural landscape, for example, you wouldnotice relatively little heterogeneity in vegetation ifyou stood within a planted field, more heterogene-ity if you flew over the fields, woodlots and villagesat low altitude, and relatively little if you lookeddown from a spacecraft. Two primary scaling factorsaffect measures of heterogeneity: grain is the resolu-tion of the data (minimum mapping unit, pixel size,and time interval) and extent refers to the size of thearea mapped or studied, or the time period over

144 E. J. Gustafson

which observations were collected (O’Neill andothers 1986).

The quantification of spatial heterogeneity re-quires a way to describe and represent variability inspace and time. In practice, spatial heterogeneity issampled and represented in a number of differentways, each useful for different types of data andhaving specialized methods for quantification ofheterogeneity (Table 1).

Categorical map analysis involves mapping thesystem property of interest by identifying patchesthat are relatively homogeneous with respect to thatproperty at a particular scale and that exhibit arelatively abrupt transition (boundary) to adjacentareas (patches) that have a different intensity (orquality) of the system property of interest (Kotliarand Wiens 1990). However, the criteria for defininga patch may be somewhat arbitrary, depending on

how much variation will be allowed within a patch,on the minimum size of patches that will be mapped(minimum mapping unit), and the components ofthe system that are ecologically relevant to theorganism or process of interest. Furthermore, asingle landscape may exhibit many different patchstructures, depending on the system properties mea-sured. For example, very different maps may resultif one defines patches based on forest type, standage, or canopy closure. Thus, the spatial structuremay in part be determined by these arbitrary deci-sions, rather than strictly by the properties of thesystem itself. As an example, consider the study ofthe distribution of aspen in the Wisconsin North-woods. As we attempt to identify and draw bound-aries around aspen stands, we are immediatelyconfronted by the question of how much aspenmust be present to classify a stand as aspen. If thereis a small aspen clone within a hardwood stand,should we map the clone as an aspen patch? Themap we produce would show where aspen wasdominant, but would contain little informationabout the relative abundance of aspen in other partsof the landscape and would imply that all aspenstands were alike. Furthermore, the structure exhib-ited by the map would at least be partly an artifact ofthe mapping decisions made.

Alternatively we might wish to conduct a point-data analysis of this landscape. Point-data analysisassumes that the system property is spatially continu-ous, making fewer assumptions about the nature ofspatial structure. The landscape is sampled to gener-ate spatially referenced information about systemvariables and is analyzed using geostatistical tech-niques (so called because of their origin in thegeosciences). Here we might measure the abun-dance of aspen at sample points distributed ran-domly or systematically. The abundance would behighest in aspen stands, but there would be quanti-fication of the abundance of aspen in other foresttypes as well. In this approach, there are no explicitboundaries (that is, patches are not delineated), andfewer simplifying assumptions are made concerningthe spatial configuration of the system. However,real discontinuities that have ecological relevance(for example, a clear-cut boundary or agriculturaledge) are not as easily represented and studied, andthe density of sample points affects the spatialproperties perceived by analytical tools applied tothe landscape.

Because these two approaches to the descriptionof spatial heterogeneity are not combined in moststudies, there is an appearance that they are notcomplementary. However, each provides a validway to describe and analyze spatial phenomena,

Table 1. Methods to Represent the Heterogeneityof a System Property and the Techniques Used toQuantify That Heterogeneity [see also Li andReynolds (1995)]

SystemPropertyRepresentation Description Quantification

Categorical maps Qualitativevariablesmappedin space

Nonspatial Composition Number ofcategories

ProportionsDiversity (richness,

evenness)Spatial Configuration Patch-based indices

SizeShapePatch densityConnectivityFractal

dimensionPixel-based indices

ContagionLacunarity

Point data Continuousvariablessampledregularly orirregularly inspace

Trend surfaceCorrelogramSemivariogramFractal dimensionLacunarityAutocorrelation

indicesInterpolation (e.g.,

kriging)

Quantifying Landscape Spatial Pattern 145

and each has strengths and weaknesses that dependon the ecological system being studied. Patch-basedanalysis uses a simplified description of the systemthat is perhaps more amenable to interpretation andis more directly relevant to much of ecologicaltheory than is point-data analysis. However, point-data analysis can answer two of the most criticalquestions in the analysis of spatial heterogeneity:what is the appropriate scale at which to conductthe analysis, and what is the nature of the spatialstructure? In many studies, the identification ofpatches typically reflects a minimum mapping unitthat is chosen for compilation and not ecologicalreasons. Point-data analysis can provide insight intothe scale of patchiness, whether there are hierar-chies of scale, and whether the spatial distribution israndom, aggregated, or uniform. Many point-dataanalysis techniques also provide robust capabilitiesto conduct reliable statistical inference. One of thepurposes of this review is to show that categoricaland point-data analysis techniques each provideuseful and complementary information.

Description of MethodsThe fundamental assumption of classical inferentialstatistical analysis is the independence of observa-tions. However, because the spatial structures wefind in nature are commonly patches or gradients,this assumption is usually violated at specific (andusually unknown) scales of sampling. Therefore it isimperative to know something about the degreeand scale of spatial dependence in the system beingstudied. Geostatistics provide methods to both de-scribe spatial structure and to make statistical infer-ences that are robust in the presence of spatiallydependent relationships. Although point data canbe used to generate interpolated numerical maps(for example, digital elevation models (DEM), mapsof climate variables, and habitat suitability maps),they are typically analyzed to make inferencesabout some system property in a spatial context.

The method used to represent the spatial varia-tion in the system in digital maps is an arbitraryconstruct. Raster maps (constructed of grid cells)represent boundaries as the interface between cellsof different classes, so that boundaries must con-form to the underlying lattice structure. This canhave marked effects on the delineation of patches,as the lattice structure may affect the contiguity ofpixels. Maps constructed of vectors (digitized lines)also characterize boundaries as a sequence of straightlines, but these can be oriented in any direction, andeach segment can be very short.

Categorical maps. Quantification of spatial hetero-geneity is often focused on categorical (or thematic;

choropleth) maps, in which system properties arerepresented qualitatively and given arbitrary labels(for example, cover type, soil series name, or habitatquality), although sometimes they are quantitativeproperties that exhibit discrete spatial discontinui-ties (for example, forest stand age). Categoricalmaps tend to ignore the spatial variation withinspatial units and trends in system properties acrosslandscapes. A large number of indices have beendeveloped to quantify spatial heterogeneity on cat-egorical maps. These indices fall into two generalcategories: those that evaluate the composition ofthe map without reference to spatial attributes, andthose that evaluate the spatial configuration ofsystem properties, requiring spatial information fortheir calculation. Spatial configuration indices areeither patch oriented or neighborhood oriented.Patch-oriented indices are calculated consideringonly a single patch and its edges. Neighborhood-oriented indices are calculated using spatial neigh-borhoods and may consider complete patches withinthe neighborhood or only neighboring pixels. Anexhaustive list of all published indices is beyond thescope of this review. I present here a representativesample of published indices, with references tomore complete descriptions of their calculation.

Composition. Composition is readily quantified,and is typically described by (a) the number ofcategories or classes in the map, (b) the proportionof each class relative to the entire map, and (c)diversity. The number of classes and their propor-tions are generated by simple counting algorithms.Diversity measures typically combine two compo-nents of diversity: richness, which refers to thenumber of classes present, and evenness, whichrefers to the distribution of area among the classes.Examples are the Shannon’s (Shannon and Weaver1949), Simpson’s (Simpson 1949), and modifiedSimpson’s (Pielou 1975) diversity indices. The rich-ness and evenness components can also be mea-sured independently (Romme 1982). Dominance isthe complement of evenness (evenness 5 1-dominance), indicating the extent to which the mapis dominated by one or a few classes (O’Neill andothers 1988a) and has been used widely in land-scape ecology research. A less widely known diver-sity index is mosaic diversity as recently proposed byScheiner (1992). Some composition indices havebeen spatially referenced by calculating the indexwithin a moving window that is passed across themap. The calculated index value at each point isdisplayed (Riitters and Wickham 1995; Perera andBaldwin forthcoming), allowing visualization of thespatial variability of the index at a specific scale ofresolution (size of the window).

146 E. J. Gustafson

Spatial configuration. The spatial configuration ofsystem properties is much more difficult to quantify,and attempts have focused on describing spatialcharacteristics of individual patches (patch based)and the spatial relationships among multiple patches(neighborhood based). Other metrics evaluate neigh-borhood properties without reference to patches,using only the pixel representations of system prop-erties. Patch characteristics of an entire landscapeare sometimes reported as a statistical summary (forexample, mean, median, variance, and frequencydistribution) for all the patches of a class (Baskentand Jordan 1995). When the configuration of asingle patch type is of particular interest, theseanalyses are often conducted on simple binarymaps, where there are only two classes: the class ofinterest and all others combined.

Patch-based measures of pattern include size,number, and density of patches. These measures canbe calculated for all classes together or for a particu-lar class of interest. Useful edge information mayinclude perimeter of individual patches, total perim-eter of all patches of a particular class, the frequencyof specific patch adjacencies, and various edge met-rics that incorporate the contrast (degree of dissimi-larity) between the patch and its neighbors (Bakerand Cai 1992; McGarigal and Marks 1995). Forexample, there is less contrast between a matureforest stand and a young stand than there is be-tween the mature stand and a pasture. Patches cantake an infinite variety of shapes, making shape adifficult characteristic to quantify. Most shape indi-ces (including many fractal methods) use a perim-eter–area relationship (Forman and Godron 1986;Milne 1991; Riitters and others 1995). Other meth-ods have been proposed—average radius of gyration(Pickover 1990), contiguity (LaGro 1991), linearityindex (Gustafson and Parker 1992), and elongationand deformity indices (Baskent and Jordan 1995)—but these have not yet become widely used. Awidely used index related to both patch size andshape is core area (interior). Core area is the portionof a patch that is further than some specifieddistance from an edge and presumably not influ-enced by edge effects. Some edge effects may extenda relatively short distance into a patch [for example,microclimate (Chen and others 1992)], whereasothers may be more pervasive [for example, preda-tion rates on nesting birds (Robinson 1992)]. Thedistance used to define the patch core must bederived for a specific organism or process of interest.This distance is usually assumed to be fixed regard-less of the contrast and orientation of the edgeadjacent to the core. Baskent and Jordan (1995)describe a technique that uses a varying edge dis-

tance to determine core area. GISFrag is a relatedindex (Ripple and others 1991) that is calculated byfinding the average distance to the nearest edge ofall the pixels of the class of interest. It is used as anindex of patch fragmentation and can functionequally well as a landscape-level or patch-levelindex.

The interspersion and juxtaposition index devel-oped by McGarigal and Marks (1995) measures theextent to which patch types (patches of differentclasses) are interspersed. This patch-based index isconceptually similar to the pixel-based contagionindex (see below), but rather than evaluating pixeladjacencies, the interspersion index evaluates onlypatch adjacencies. Whereas contagion measures the‘‘clumpiness’’ of maps, this index measures thejuxtaposition of patch types.

Patch cohesion (PC) was recently proposed bySchumaker (1996) to quantify the connectivity ofhabitat as perceived by organisms dispersing inbinary landscapes. Because the behavior of PC hasnot been described over the full range of habitatproportions, I digress momentarily to illustrate itspotentially useful features. I examined PC on mapswhere habitat cells were assigned at random, calcu-lating PC by using all patches on the map by

PC 5 11 2Sp

S(pÎa)2 11 21

ÎN221

where p is patch perimeter, a is patch area, and N isthe number of pixels in the map. It is well knownthat, on random maps, patches gradually coalesce asthe proportion (p) of habitat cells increases, forminga large, highly connected patch that spans the latticeat a critical proportion (pc) (Stauffer 1985). The pc

varies with the rule used to connect cells to delin-eate a patch: when cells must share a common edgeto be considered connected, pc 5 0.59, and whenthey need only touch at a corner, pc 5 0.41 (Stauffer1985). The PC index has the interesting property ofincreasing monotonically until an asymptote isreached near pc (Figure 1). I also examined PC onmaps generated by randomly placed clumps ofhabitat [as in Gustafson and Parker (1992)]. Herethe slope of the increase was much less, but asymp-tote was still reached near pc (Figure 1).

Pattern indices that examine a spatial neighbor-hood have either a patch orientation or a pixelorientation. Patch-level neighborhood indices usu-ally consider patches on a binary map. The simplestof these is isolation, which is calculated as thedistance to the nearest neighboring patch of thesame class. Some indices identify a neighborhood oflimited extent around a focal patch (usually a buffer


some specific distance around the edge of eachpatch), and the index is calculated based on thecharacteristics of the patches found within thatbuffer (neighborhood). These indices were devel-oped primarily to predict relative connectivity ofhabitat islands. A simple example is the averageisolation of all patches within the neighborhood(Gustafson and Parker 1992). Some explicitly con-sider physical connections (for example, corridorsor hedgerows) and are supported by network theory(Lowe and Moryadas 1975; Lefkovitch and Fahrig1985). Others are based on island-biogeographytheory (MacArthur and Wilson 1967) and incorpo-rate both island (patch) size and isolation [isolationindex (Whitcomb and others 1981) and proximityindex (Gustafson and Parker 1992)]. Verboom andothers (1991) developed a connectivity index in thecontext of metapopulation theory (Levins 1970).

The most commonly used pixel-based neighbor-hood index is contagion (O’Neill and others 1988a),designed to quantify both composition and configu-ration. Li and Reynolds (1993) have corrected anerror in the original formula, and Riitters and others(1996) have clarified subtleties in the way adjacen-cies are tabulated. Contagion ignores patches per seand measures the extent to which cells of similarclass are aggregated. The index is calculated usingthe frequencies with which different pairs of classesoccur as adjacent pixels on the map. The indextypically does not distinguish differences in aggrega-tion that may exist for different classes, but summa-rizes the configuration of all classes. Contagion issometimes mistakenly used as a surrogate for aggre-

gation of a specific type (for example, forest), point-ing out the need for care in the interpretation of thecontagion index. [Alternative methods are availablefor calculating class-specific contagion measures(Pastor and Broshart 1990; Gardner and O’Neill1991).] Contagion has also been mapped by display-ing the contagion value calculated within a movingwindow (Riitters and Wickham 1995; Perera andBaldwin forthcoming). The use of the contagionindex is appealing because it appears to summarizethe overall clumpiness of maps effectively, but isproblematic because it is a single-valued index usedto represent complex interacting patterns.

Lacunarity analysis is a multiscale method used todetermine the heterogeneity (or texture) of a sys-tem property represented as a binary response inone, two, or three dimensions (Plotnick and others1993, 1996). The technique uses a gliding box(moving window) algorithm to describe the probabil-ity distribution of the class of interest as the box ispassed over the data (a map, a transect, or points).Valuable insight into the spatial heterogeneity of thesystem and the domains of the scale of variation inthat pattern can be achieved by using a number ofbox sizes and plotting lacunarity as a function of boxsize.

Point-data analysis. In contrast to techniques ap-plied to thematic maps of categorical data, point-data analysis is applied to data collected by samplingrather than to maps. The analysis assumes that thesystem property varies continuously in space, andapplies mathematical techniques for modelinggradual spatial change (Journel and Huijbregts 1978;Davis 1986; Rossi and others 1992; Burrough 1995).These techniques are not constrained by the need todelineate boundaries on the map and are verydifferent from the techniques applied to categoricalmaps. Only a superficial discussion of these tech-niques is possible here. The interested reader isreferred to excellent reviews by Legendre and For-tin (1989), Turner and others (1991), Rossi andothers (1992), and Burrough (1995).

Trend surface. One simple way to estimate thegradual, broad-scale spatial variation (trend) in asystem property is to fit a surface to the data usingregression techniques (trend-surface analysis). Theestimated model represents the systematic trend inthe data, and the deviations from the surface repre-sent the random component of the system. Forexample, biomass accumulation rates at a regionalscale might exhibit a spatial trend related to growingdegree-days. Although easy to calculate, this tech-nique must be used with care due to a number ofproblems (Burrough 1995). Regression techniques

Figure 1. Patch cohesion (Schumaker 1996) as a functionof proportion of habitat on randomly generated maps(100 3 100 cells). Habitat was assigned to cells indepen-dently (random) or in clumps (clumped). Patches weredelineated by one of two rules: (a) cells must share acommon edge to be considered connected (edge rule), or(b) cells need only touch at a corner (diagonal rule).

148 E. J. Gustafson

are susceptible to outliers when data points are few,edge effects can be severe with higher-order equa-tions, and the assumption of spatially independent,normal residuals is often violated. Also, finding atrend has little value unless it has a physical orecological explanation. Trend surface analysis isuseful to remove large-scale trends to enable inves-tigation of other, more-fine-scale structures (Legen-dre and Fortin 1989).

Autocorrelation. Often the variation in a systemproperty is such that points close together tend to bemore similar than points farther apart, but thepattern is not as simple as would be implied by atrend surface. The degree of similarity (or covari-ance) between observations separated by varyingdistances (lags) is computed. The autocovariance fora lag (t) is the covariance among all observationsthat are separated by a distance of t (Davis 1986).Plotting the autocorrelation (autocovariance normal-ized with respect to the variance of the observa-tions) against t gives the correlogram, which allowsvisualization of the spatial structure of the systemproperty (Legendre and Fortin 1989) and is usefulto determine the grain and extent of repeated spatialpattern (Turner and others 1991). Semivariogramsare conceptually related to correlograms, plottingsemivariance against lag distance. The semivarianceis the sum of squared differences between all pointsthat are separated by distance t. If the comparedpoints are increasingly different as t increases, thesemivariance increases. As t increases, the pointseventually become unrelated to each other so thatthe semivariance equals the average variance of allsamples, and the slope becomes zero (Figure 2). Thesemivariogram can be used to estimate the scale(s)of patchiness of a mosaic (Turner and others 1991).For example, if we imagine that the semivariogramin Figure 2A represents vegetation height sampledon a transect, a good estimate of the scale ofpatchiness of height might be the range (a) of thesemivariogram.

Anisotropy (autocorrelation that differs in differ-ent directions) is often found in ecological databecause spatial patterns are sometimes produced bydirectional geophysical phenomena. Anisotropy canbe detected by computing the autocorrelation amongpoints oriented to each other in specific directionsand comparing them (Legendre and Fortin 1989).As an example, Burrough (1995) used the semivar-iogram to analyze the spatial variation in soil tex-ture and found patchiness at scales between 600and 950 m in a north–south direction, but patcheswere longer than 1800 m in an east–west direction.

INTERPRETATION OF INDICES

The quantification of spatial pattern is not as easy asthe proliferation of indices might suggest. Measuresof spatial pattern often incorporate multiple aspectsof composition and pattern in their calculation,

Figure 2. Some forms of semivariograms right and thepossible forms of variation they describe left. In a typicalform of the semivariogram A, the range (a) is the distance(h) at which the semivariance [g(h)] levels out, and thislevel is known as the sill, which is theoretically equal tothe variance of the system. The nugget variance (C0)represents random error. B The same as A, but with nosingle clearly defined distance between abrupt changes. CA linear trend produces a semivariogram whose slopeincreases with h. D A periodic signal produces a cyclicsemivariogram. E Structureless variation or ‘‘noise’’ re-sults in a semivariogram that is 100% nugget variance.Adapted from Burrough (1995).


making interpretation fraught with pitfalls. There isseldom a one-to-one relationship between indexvalues and pattern (that is, several configurationsmay produce the same index value).

Domains of scale appear to exist in which arelationship established at a particular scale may bereliably extrapolated at similar scales, but may breakdown when applied at very different scales (Krum-mel and others 1987; Urban and others 1987; Wiensand others 1993). For example, least flycatchers andAmerican redstarts are negatively associated in for-ests of the northeastern United States, but arepositively associated at a regional scale (Sherry andHolmes 1988). Because the transition between do-mains of scale is not intuitively obvious, care mustbe exercised about the scale to which researchfindings are applied. Geostatistical techniques (Jour-nel and Huijbregts 1978; Legendre and Fortin 1989;Turner and others 1991; Rossi and others 1992) andfractal analysis (Mandelbrot 1977) have shown thepotential to identify transitions between landscapepattern scale domains (Palmer 1988; Milne 1991;Krummel and others 1987).

Much of the thinking about spatial heterogeneityin ecology and conservation biology has been shapedby island-biogeography theory (MacArthur and Wil-son 1967). This has resulted in essentially a binarymodel of habitat: suitable habitat and unsuitablehabitat (Wiens 1994). We have learned that forbinary maps the predominant determinant of spa-tial pattern is proportion of the class of interest(O’Neill and others 1988b; Gustafson and Parker1992). This compositional characteristic determinesthe probable range of many patch configurationcharacteristics. Often, knowing the proportion of atype of interest tells you almost as much as knowingmany other measures of heterogeneity. If propor-tion is low, generally the patches are small andisolated, and do not have enough area to formconvoluted shapes. For example, the most signifi-cant explanatory variable used to predict the qualityof a landscape as wild turkey habitat in Indiana wasproportion of forest (Gustafson and others 1994).Cowbird parasitism and nest predation rates onforest birds in the midwestern United States werenegatively correlated with proportion of forest coveron nine landscapes (Robinson and others 1995),and forest fragmentation indices were correlatedwith proportion of forest. Andren (1994) reviewedstudies of birds and small mammals and discoveredthat when the proportion of suitable habitat was lessthan 10%–30%, the effects of patch area andisolation became greater than effects expected fromhabitat loss alone.

We have also learned that many indices used toquantify spatial heterogeneity are correlated (O’Neilland others 1988a; Riitters and others 1995) andexhibit statistical interactions with each other (Liand Reynolds 1994). Furthermore, many indicesare confounded, that is, they appear to measuremultiple components of spatial pattern (Li andReynolds 1994; Riitters 1995). Some have arguedthe desirability of indices that combine multiplecomponents of pattern into a single value to reducethe number of variables carried in a multivariateanalysis (Riitters and others 1996; Scheiner 1992).Others have argued that it is difficult enough tointerpret indices that measure a single componentof spatial pattern, and that any method to quantifyspatial pattern must be examined theoretically andtested under controlled conditions (Li and Reynolds1994). A proposed solution is to describe fundamen-tal components of spatial pattern that are indepen-dent and develop a suite of metrics to measure thosecomponents (Li and Reynolds 1994; Riitters andothers 1995). A first approximation of those factorshas recently appeared in the literature. Li andReynolds (1995) divided spatial heterogeneity intofive components using theoretical considerations:compositional components are (a) number of typesand (b) proportion of each type; and spatial compo-nents are (c) spatial arrangement of patches, (d)patch shape, and (e) contrast between neighboringpatches. Riitters and others (1995) conducted afactor analysis of 55 indices of spatial pattern calcu-lated for 85 maps of land cover and identified fiveindependent factors that they have interpreted as(a) average patch compaction, (b) overall imagetexture (pixel distributions and adjacencies), (c)average patch shape, (d) patch-perimeter scaling(fractal measures), and (e) number of types. McGari-gal and McComb (1995) conducted a principlecomponent analysis of 30 indices calculated forlate-seral forest landscapes in the northwesternUnited States and identified three principal compo-nents: (a) patch shape and edge contrast, (b) patchdensity, and (c) patch size.

It is often useful to explore the response andsensitivity of indices to variation in heterogeneity(Haines-Young and Chopping 1996). Neutral mod-els (Gardner and others 1987; Gardner and O’Neill1991) have been used extensively for this purpose(Li and Reynolds 1993; Gustafson and Parker 1992;Milne 1992) because they control the process gener-ating the heterogeneity, allowing unconfoundedlinks between variation in heterogeneity and thebehavior of the index. Additionally, they are used togenerate patterns expected in the absence of ahypothesized process that can be compared with

150 E. J. Gustafson

those produced by the process under study (Withand King 1997). An understanding of the responseof an index to a neutral model of a process canquickly illuminate what might otherwise be intrac-table.

Critical to any study of the link between spatialpattern and ecological process is to remember thetemporal dynamics of pattern. Land-use mosaicschange at various temporal and spatial scales. Cropsare rotated on an annual basis in agricultural land-scapes, and silvicultural practices and regenerationsignificantly change forested landscapes over de-cades. Human development and land abandonmenthave dramatically changed many landscapes over aperiod of decades and centuries. Fire and winddisturbances reshape vegetation mosaics over aperiod of decades and centuries. In seeking toestablish the spatial pattern needed to support anecological process, it is imperative to understandthat a range of pattern conditions must be identi-fied. Very few heterogeneity indices include a mea-sure of temporal variation. This is a serious defi-ciency because it encourages a static view of natureand an unrealistic concept of change in naturalsystems (Huston 1994). Because change and variabil-ity are part of ecological systems, an understandingof what constitutes a significant change in ecologicalheterogeneity is critical to understanding the conse-quences for ecological processes. Statistical tech-niques can detect significant changes in the mean ofa pattern index with known variation, but ecologi-cally significant change is much more difficult toassess. For example, a number of pattern/processrelationships appear to be threshold phenomena(Birney and others 1976; O’Neill and others 1989;With and Crist 1995; Wiens and others 1997),where only changes in pattern near the thresholdare ecologically important. Also, the ecological sig-nificance of pattern measured at one point in time isdifficult to assess without an understanding of thehistorical variability of that pattern.

APPLICATION OF INDICES

The proper application of landscape pattern analysisis not trivial. Perhaps the most critical step is toidentify properly the scale of the heterogeneity(patchiness) of the landscape, so that subsequentanalyses will be conducted at an appropriate scale.This may require a geostatistical and/or fractalanalysis, although the scale of patchiness may beobvious in some systems. Although critical, this stepis almost always omitted because it usually involvescollecting extra (point) data and conducting spatialstatistical analyses with unfamiliar (to many) tech-

niques. If maps are not available at the scale identi-fied as appropriate, compilation of a new landscapemap may be required. Considerable thought mustalso be given to the scale at which the ecologicalprocess being studied operates, and/or the scale atwhich the organism(s) being studied perceives (orresponds to) the heterogeneity of the landscape(Wiens 1989).

The concept of ecological neighborhoods (Addi-cott and others 1987) has not been widely applied,but could be useful. The concept is centered on anecological process and/or organism, and an ecologi-cal neighborhood is defined by the time period andthe spatial area in which the organism or processinteracts with the environment. Patchiness is thenmeasured relative to the size of the neighborhood.For example, relative patch size is the ratio of patchsize to neighborhood size, and relative patch dura-tion is the ratio of patch duration to the time scale ofthe neighborhood. This conceptual framework alsoserves as a reminder that the pattern and process wemeasure is but a snapshot of a dynamic system.

In studies linking pattern and process, it is criticalto measure pattern and process at the same spatialscale (that is, both grain and extent). Many mea-sures of heterogeneity are made using a graindetermined by data collection or storage methods(for example, resolution of satellite imagery orminimum mapping unit of a GIS layer). Because thecollection of landscape data is expensive, thesesources are often used out of necessity. Data areroutinely resampled to change the grain, althoughthis should be done with caution because resam-pling introduces spatial errors (Turner and others1989) and produces nonspatial statistical effects inindices that depend on pixel adjacencies or patchcounts (Haines-Young and Chopping 1996). Mapboundaries usually truncate patches, so map extentcan affect patch measures. Considerable thoughtshould be given to the appropriate grain and extentto represent accurately the ecological neighborhoodbeing studied. This task is somewhat easier forsimulation studies, as the spatial information usedby the model (process) can be used to measure thepattern. However, this precaution is sometimesoverlooked. For example, Schumaker (1996) calcu-lated indices of habitat connectivity by using Land-sat multispectral scanner data (0.32 ha, squarepixels) and related them to the results of a dispersalmodel that operated on a hexagonal grid with cellsizes ranging from 12 to 331 ha. The grain withwhich system properties are represented has aprofound effect on many indices, particularly pixel-based measures and those that incorporate patchedges (Haines-Young and Chopping 1996).


It also crucial to understand what an indexactually measures relative to the process underconsideration. This cannot be overemphasized. Someindices appear intuitive, but their behavior displayssubtleties (for example, contagion) that can misleadinterpretation. Many indices measure multiple as-pects of pattern (for example, diversity indices,proximity index, patch cohesion, and many edgemeasures). These present difficult interpretationproblems that can only be overcome by carefulconsideration of how the metric is calculated, thecharacteristics of the map representing the pattern,and an understanding of the behavior of the indexas the pattern of interest changes. Simultaneousconsideration of several indices that measure spe-cific components of heterogeneity may be instruc-tive. As Li and Reynolds (1994) warn, it is importantto perceive correctly the spatial and nonspatial(composition) elements being measured.

Very few pattern indices produce values that areuseful by themselves. Their most instructive use isin comparing alternative landscape configurations,either the same landscape at different times orunder alternative scenarios, or different landscapesrepresented by using the same mapping scheme andat the same scale. Pattern indices are also commonlyused as independent variables in models attemptingto establish a link between spatial pattern andecological process. The relationship between ecologi-cal processes and absolute values of indices is rarelyknown, although sometimes the relative responseof ecological systems to the direction of change inindex values is understood.

Many patch-based indices are commonly summa-rized for an entire landscape by calculating themean and variance of the index for all patches ofeach class. This may be misleading when the distri-bution of patch sizes is greatly skewed towardsmaller patch sizes, as is typically the case. Area-weighted means or medians may provide betterestimates of central tendency. Another commonapproach to landscape analysis involves identifyinga subset of the landscape around a patch or studysite of particular interest, and indices are calculatedfor that subset to quantify the landscape context ofthe site. Composition indices are most commonlyused, but some patch-based measures may be calcu-lated. Patch-based measures will be affected byedges that truncate patches, and this problem willbe severe for subsets that are small relative to thesize of patches.

When investigating the relationship between spa-tial pattern and ecological process, there is always atemptation to calculate as many indices of pattern aspossible and look for relationships. The availability

of software to automate the quantification of spatialpattern makes this temptation difficult to resist. It isvery rare that some ecological mechanism for arelationship between pattern and process can not beproposed, and explanatory studies should be fo-cused by these putative relationships. Only the mostpreliminary exploratory studies should embark on‘‘fishing’’ expeditions and then only with a clearunderstanding of potential mechanisms driving theprocess and explicit knowledge of what the patternindices are measuring.

RESEARCH NEEDS

In 1989, John Wiens challenged researchers tomake scaling issues a primary focus of researchefforts. Little progress has been made due to thedifficulty of scaling problems. Nevertheless, thesuccessful application of research findings dependscritically on both identifying the appropriate scalefor the application and the ability to extrapolatefindings across scales.

Much of the need for spatial pattern indices isdriven by the desire to predict the response of someecological entity (for example, fire, organism, ornutrient flux) to the spatial heterogeneity of amanaged landscape. These responses are often com-plex, caused by interactions between various compo-nents of heterogeneity and scale. Many of theindices that have been developed are an attempt tocapture the elements of pattern that are importantto a specific ecological entity [for instance, connec-tivity (Verboom and others 1991) or proximity(Gustafson and Parker 1994)]. For example, Ver-boom and others (1991) developed a connectivityindex to quantify patch isolation that was relevantto a metapopulation of European nuthatches. Theindex significantly improved predictions of patchoccupancy by nuthatches. However, indices devel-oped for specific purposes or systems are by defini-tion not very general, and their usefulness may belimited to narrow applications. Nevertheless, withcareful application, linking specific ecological pro-cesses with special indices may provide the ability topredict those processes by measurement of spatialheterogeneity.

There also appears to be some impetus to developindices that capture all relevant aspects of heteroge-neity in a single value (Scheiner 1992; Riitters1996), or a suite of indices where each indexmeasures exactly one component of spatial hetero-geneity (Li and Reynolds 1994). This may be desir-able, but perhaps is not sufficient. It is unlikely thatall ecological process/pattern relationships can beadequately described by using a small suite of

152 E. J. Gustafson

metrics. Also, it must be verified that a combinationof indices can adequately capture the heterogeneityto which specific species or ecological processes arerelated. It should also be mentioned that it may notbe possible to know all the important pattern/process relationships, and that general indices maybe useful even when there is not an explicit link tospecific species or ecological processes. Such a viewis consistent with the coarse-filter strategy to main-tain biodiversity proposed by Hunter (1990).

The effects of the methods we use to characterizespatial heterogeneity and relate it to ecologicalfunction have been inadequately explored. Forexample, we have become comfortable with charac-terizing landscapes as thematic (categorical) maps,and most of the commonly used pattern quantifica-tion methods are designed for use with categoricaldata. It is well established, however, that manyecological and environmental conditions are bestcharacterized as gradients (Chen and others 1992).Conditions within ‘‘patches’’ are never completelyhomogeneous, and even relatively high-contrastboundaries (edges) between habitat types are com-monly conceived as ecotones (that is, a gradient)(Fortin 1994; Fortin and others 1996). More re-search is needed to develop our ability to character-ize and quantify spatial point data to make themcompatible with current theory couched in a patchframework and to interpret the results of theseanalyses reliably.

CONCLUSION

This review has highlighted the pervasiveness of theisland paradigm in studies of spatial pattern (andecological process for that matter). With the excep-tion of edge and adjacency measures, all indicescalculated for categorical maps are based on a binarypatch model. However, habitat patches usually existin a complex landscape mosaic, and dynamics withina patch are affected by external factors related to thestructure of the mosaic (Wiens 1994; Gustafson andGardner 1996). Although the patch model is bestsuited to many landscape studies, some landscapemosaics are better represented as gradients (orcontinuously varying surfaces) of system properties.The island model has sometimes been forcibly im-posed on the study of such systems (Wiens 1994),and this has been a conceptual impediment toprogress in our ability to understand and model thelink between spatial pattern and ecological processin systems without discrete boundaries.

The available techniques for the study of continu-ously varying systems (point-data analysis) havenot been widely embraced. This may be partly

because the methods are more complex, and theirinterpretation is less intuitive than patch models. Itis also not widely appreciated that these methodslend themselves to powerful tests of spatial hetero-geneity hypotheses (Legendre and Fortin 1989;Turner and others 1991). Many of these methodsprovide information on the ‘‘patchiness’’ of thesystem or can be used to delineate ‘‘patch’’ bound-aries [for example, constrained clustering (Legendreand Fortin 1989)], pointing the investigator backtoward a less arbitrary patch model [see Fortin andDrapeau (1995)]. Spatial autocorrelation should beassumed for most ecological data. Investigatorsshould test for spatial autocorrelation and describethe spatial structure by using maps and spatialstructure functions (for example, correlograms andvariograms). Such analyses seem necessary to choosethe appropriate scale for landscape analysis (patch-based or continuous) and to avoid violating assump-tions of the analysis methods.

Several guidelines for sound analysis of spatialheterogeneity have emerged from this review: (a)Get the scale right. This includes determining thescale of patchiness and the hierarchical structure byusing geostatistical or fractal techniques. Equallyimportant is to understand the scale of the ecologi-cal process(es) of interest. This information deter-mines how the system should be represented (forexample, point data or map, categorical or continu-ous, minimum mapping unit, resolution, and ex-tent). (b) Choose the analysis method based on theobjectives of the analysis and the spatial characteris-tics of the system property of interest. (c) Choosemetrics by considering the heterogeneity that isrelevant to the ecological process of interest. Be-come familiar with the behavior of indices as hetero-geneity changes, building neutral models if neces-sary. When possible, calculate multiple indices thatmeasure the same component of heterogeneity toincrease the reliability of the measure of heterogene-ity. (d) Formulate a theoretical relationship betweenan index and the ecological process so that empiricalevidence can be rationally related to the results ofthe analysis. Remember that heterogeneity indicesfunction as surrogate measures of certain ecologicalconditions, and empirical or theoretical links be-tween them are necessary.

In summary, spatial heterogeneity indices repre-sent a link between pattern and process. Patch-based landscape analyses have been the primaryfocus of landscape ecological analysis, and are in-deed appropriate for a majority of ecological ques-tions. However, the appropriate spatial and tempo-ral scales at which these analyses are applied shouldbe determined by geostatistical analysis. Geostatisti-


cal alternatives to a patch-based approach should beconsidered when the system property varies gradu-ally in space and/or time. These techniques are mostuseful when applied and interpreted in the contextof the organism(s) and ecological processes of inter-est, and at appropriate scales, although some may beuseful as coarse-filter indicators of ecosystem func-tion [sensu Hunter (1990)].

ACKNOWLEDGMENTS

I thank Marie-Josee Fortin, Curt Flather, and twoanonymous reviewers for critical reviews that greatlyimproved the manuscript.

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