The Gradient Concept of Landscape Structure: Or, Why are ...The Gradient Concept of Landscape...

The Gradient Concept of Landscape Structure...McGarigal and Cushman

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30 April, 2002Kevin McGarigalDepartment of Natural Resources ConservationUniversity of Massachusetts304 Holdsworth Natural Resources CenterBox 34210Amherst, MA 01003Fax: (413)545-4358Phone: (413)577-0655Email: [email protected]

The Gradient Concept of Landscape Structure: Or, Why are There so Many

Patches

Kevin McGarigal1, Samuel A. Cushman1

1Department of Natural Resources Conservation, University of Massachusetts, Amherst, MA

01003

Abstract

Understanding pattern-process relationships hinges on accurately characterizing heterogeneity in

a manner that is relevant to the organism or process under consideration. In this regard,

landscape ecologists have generally adopted a single paradigm–the patch mosaic model of

landscape structure–in which a landscape is represented as a collection of discrete patches. While

this paradigm has provided an essential operating framework for landscape ecologists, and has

facilitated rapid advances in quantitative landscape ecology, we believe that further advances in

landscape ecology are somewhat constrained by its limitations. We advocate the expansion of


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the paradigm to include a ‘gradient-based’ concept of landscape structure which subsumes the

patch mosaic model as a special case. The gradient approach we advocate allows for a more

realistic representation of landscape heterogeneity by not presupposing discrete structures,

facilitates multivariate representations of heterogeneity compatible with advanced statistical and

modeling techniques used in other disciplines, and provides a flexible framework for

accommodating organism-centered analyses.

Key Words: landscape patterns, landscape structure, categorical map patterns, surface patterns,

FRAGSTATS.

1. Introduction

Landscape ecology deals fundamentally with how, when and why patterns of environmental

factors influence the distribution of organisms or the actions of ecological processes, and

reciprocally, how the actions of organisms and ecological processes feedback to influence

ecological patterns (Urban et al. 1987, Turner 1989). The landscape ecologists’ purpose is more

than to simply describe interesting relationships. Rather, the goal is to determine where and

when spatial and temporal heterogeneity matter, and how they influence processes in action. A

fundamental issue in this effort revolves around the choices a researcher makes regarding how to

depict and measure heterogeneity, specifically, how these choices influence the “patterns” that

will be observed and what mechanisms may be implicated as potential causal factors (Turner

1989, Wiens 1989). Indeed, it is well known that observed patterns and their apparent

relationships with response variables often depend upon the scale that is chosen for observation

and the rules that are adopted for defining and mapping variables (Wiens 1989). Thus, success in


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understanding pattern-process relationships hinges on accurately characterizing heterogeneity in

a manner that is relevant to the organism or process under consideration.

In this regard, landscape ecologists have generally adopted a single paradigm–the patch

mosaic model of landscape structure (Forman 1995). Under the patch mosaic model, a landscape

is represented as a collection of discrete patches. Major discontinuities in underlying

environmental variation are depicted as discrete boundaries between patches. All other variation

is subsumed by the patches and either ignored or assumed to be irrelevant. This model has

proven to be quite effective. Specifically, it provides a simplifying organizational framework that

facilitates experimental design, analysis, and management consistent with well established tools

(e.g., FRAGSTATS) and methodologies (e.g., ANOVA). Indeed, the major axioms of

contemporary landscape ecology are built on this perspective (e.g., patch structure matters, patch

context matters, pattern varies with scale, etc.). However, even the most ardent supporters of the

patch mosaic paradigm recognize that categorical representation of environmental variables

often poorly represents the true heterogeneity of the system, which often consists of continuous

multi-dimensional gradients. Yet, alternative models of landscape structure based on continuous

environmental variation are poorly developed. We believe that further advances in landscape

ecology are constrained by the lack of methodology and analytical tools for effectively depicting

and analyzing continuously varying ecological phenomena at the landscape-level.

Before launching into our major thesis, a brief digression may be helpful to define some

concepts. Scientists nearly universally recognize the central role that scale plays in determining

the outcome of observations (Levin 1992, Schneider 1994, Peterson and Parker 1998).


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Observation scale is usually defined by two interrelated attributes, the grain and the extent of

observation (Forman and Godron 1986, Turner et al. 1989, Wiens 1989). Grain refers to the

minimum size of the observation units, often denoted as patches or as pixels. The grain defines

the smallest entities that can be distinguished. No conclusions can be made about processes

acting below the grain of the study. Extent is simply the spatial domain over which the system is

studied and for which data are available. The extent sets the bounds on the generality and scope

of inferences. Other processes may be operating in different regions, or at larger scales in the

same region, but conclusions must be bound within the extent of the study. However, in the

discussion of observation scale, little attention is typically given to what is measured, only the

spatial scale, in terms of grain and extent, it is measured at. Clearly, though, the rules that are

followed in defining what is measured, and the resolution at which it is measured, are just as

important as the spatial grain and extent of the measurement. This implies that there is a third

dimension to environmental scaling in addition to grain and extent. We will call it intensity. We

define intensity as the resolution in environmental variation discriminated by a given variable. It

is conceptually equivalent to radiometric resolution in remote sensing, which refers to how finely

the spectral differences among locations are measured. A single variable may be recorded at any

number of intensities. For example, soil temperature may be coarsely measured as either high or

low, or by 1 degree, or 0.01 degree increments. Also, a single variable at one intensity may be

recorded as several variables at a higher measurement intensity. For example, a low intensity

binary variable called “forest” may be subdivided into biomass in different strata, stem density

and basal area, or other continuous variables at a higher measurement intensity. The intensity of


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measurement has dramatic influences on the types of associations that can be made and on the

nature of the patterns that would be mapped from that variable.

Now let us return to our major thesis. Our premise is that the truncation of landscape-level

environmental variability into categorical maps collapses the intensity of continuously varying

attributes, resulting in a substantial loss of information, and troublesome issues of subjectivity

and error propagation. We suggest that the traditional focus on categorical map analysis, to the

exclusion of other perspectives, limits the flexibility and efficiency of quantitative analysis of

spatially structured phenomena, and contributes to the persistent disjunction between the

methods and ideas of community and landscape ecology, as well as slowing the integration of

powerful geostatistical and multivariate methods into the landscape ecologist’s toolbox. We

suggest that adopting a perspective that explicitly considers variable intensity as a third attribute

of scale and conducting investigations over appropriate ranges of intensity will facilitate the

resolution of some of these difficulties, and lead to a more robust and flexible analytical science

of scale.

In the paragraphs that follow we outline our argument. We begin by reviewing the traditional

approach to landscape analysis, and identify its limitations. We argue that many of these

difficulties can be overcome by adopting a gradient approach in landscape analysis with explicit

consideration to grain, extent and intensity of measurements. Then we discuss some of the

specific benefits of gradient approaches, and illustrate them with several simple examples using

real data. Note, it is not our intent to argue that the patch mosaic model is no longer appropriate.

On the contrary, we recognize the great heuristic value of the patch mosaic model, and believe


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that it is the appropriate model to use under many circumstances, e.g., when anthropogenic

activities have created sharp discontinuities in the environment. However, we believe that a

landscape ecology paradigm built solely on this perspective is limiting, and propose that the

paradigm be expanded to include both the categorical and continuous perspectives in a single

conceptual and analytical framework.

2. Our Categorical History

Landscape ecology owes much of its origin to the traditions of geography and cartography.

Many of its most laudable perspectives and most effective techniques have their origins in

classical cartographic analysis (Forman and Godron 1986). However, we feel that one of the

most deeply rooted perspectives to be assimilated into landscape ecology from cartography has

limiting consequences as to how landscapes are mapped, and thus how patterns and processes

are perceived to interact. The first step in any landscape ecology analysis is to map the system. It

has become traditional in geography to abstract the world into non-overlapping regions, or

polygons. This perspective was adopted in the infancy of landscape ecology. Now it is so deeply

ingrained that many have difficulty thinking of the world as other than a mosaic of patches.

Indeed, the most commonly adopted definitions of landscape are as a mosaic of non-overlapping

patches of different land-cover types (Forman and Godron 1986, Forman 1995). It is, without

question, the dominant paradigm in landscape ecology. In terms of observational scale, this kind

of mapping truncates the intensity of measured variables into categories. Quantitative

information about how variables vary through space and time is lost, leaving rigid, internally

homogeneous patches. While this perspective has been tremendously useful for many


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applications, we believe it is important to consider how it influences our measurements and

analyses.

2.1. Implications of Categorical Mapping of Landscapes

Maps are abstractions derived from reality. They are models which emphasize certain aspects

of the system, while ignoring others. Any real-world system contains variability in a multitude of

factors across an almost limitless scale range. Traditional choropleth mapping begins with

decisions about what to map, and then moves directly to the process of defining discontinuities

in those factors. Importantly, discontinuities are presupposed; the world is assumed to be

inherently discrete. When the entities to be mapped are actually categorical, such as political

units for example, there is no difficulty. However, when mapping continuously scaled

phenomena, the cartographer must decide where boundaries are to be placed. This is usually

based on the areas of sharpest change in the measured variables. These ‘discontinuities’ then

serve as the basis for defining boundaries, and thus mapping patches.

There are three major losses of information in reducing quantitative landscape variation to

categories. First, patch boundaries are established based on some criteria defined by the

observer. This is no guarantee that these boundaries are meaningful or even perceived by the

organism or process that is being investigated. Subjectivity is often required in deciding what

attributes to categorize and how to set the boundaries. The decisions of what to map and how to

define the boundaries absolutely defines what patterns will be seen and what relationships are

inferred. Second, as mentioned above, the intensity of the variable is truncated into classes or

patch types. Once patches are created, all internal variability within and among patches of the


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same class is eliminated. All locations of the same class are treated as identical, all patches of the

same type are treated as internally homogeneous, and all interclass differences are reduced to

categorical differences. This is the second source of information loss, and it can be a dear one.

The third is even more expensive. Categorical patches define the regions of assumed

homogeneity in a single or composite attribute. Once defined, all variability in that attribute not

used to define the patch is discarded. The remaining information in the variable, or that in other

variables, can of course be used to define other layers of patches based on their own boundary

definitions. The boundaries of these two or more layers of patches can be overlaid in GIS and

analyzed using traditional map algebra. This is the standard approach to analyzing multi-level

categorical map patterns. However, the boundaries of patches in different layers are often poorly

related, as they reflect slices through the distributions of independently varying environmental

attributes based on different classification rules. Thus, not only is information lost within

individual map layers, but information is lost between layers in terms of the covariation of

environmental factors.

This information loss increases multiplicatively with additional choropleth layers. In the

traditional patch-based model, analyzing many layers of patches results in intractably vast

numbers of unique combinations of map categories. When a researcher attempts to predict a

response variable, such as the habitat suitability for a particular species, as a function of a

number of landscape-level attributes across several categorical data layers, prediction can only

be based on combinations of categories. In contrast, if the same response variable is predicted on

the basis of several layers of quantitative predictor variables, the prediction can be based on how


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the quantitative landscape-level variables covary along dimensions that are related to the habitat

suitability for the species in question (McGarigal et al. 2000). With each combination of

categorical map layers the information loss multiplies, as does the potential bias and subjectivity.

There is no such penalty for combining quantitatively scaled variables. In addition, subjectivity

is lessened by preserving quantitative ecological factors. The subjectivity of boundary definition

is replaced by the subjectivity of the intensity of measurement, e.g. the length of the increment,

which is often a less restrictive assumption than category width and boundary definitions.

Retaining the quantitative scale of ecological variables also allows the scientist to analyze many

response variables simultaneously, with each responding individualistically to multiple

landscape structure gradients.

Of course, categorical patch mosaics are often derived specifically to correspond

meaningfully to the scale and sensitivities of a particular organism or ecological process. In such

a case, the patch mosaic may meaningfully represent how the landscape is experienced by the

organism or process in question. However, in most cases there is considerable imprecision about

the scale and resolution of landscape variability that is pertinent to a particular question, and

patterns at several scales may simultaneously influence an organism or process. Reducing a

continuous ecological surface to a patch mosaic, even if based on the best information available,

eliminates information as a result of imprecision in boundary placement and class divisions, or

because ecological variation is important across several scale ranges. In addition, even if a patch

mosaic is constructed that ideally represents an organism’s ecological landscape, this mosaic is

very unlikely to also optimally fit a second or third organism, making comparisons between them


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based on a single landscape map questionable. Standard choropleth mapping can simplify

complex information, facilitating analysis, but we believe that scientists should carefully

consider the fee that is paid in terms of information loss, subjectivity and compounding errors,

and the considerable advantages that may arise from preserving the continuously scaled nature of

quantitative landscape attributes.

In practice, ecologists don’t simply use combinations of categories as predictor variables in

landscape-level modeling and analysis. In fact, they generally convert the categorical map into

quantitative variables reflecting, for example, the amount of mature forest within 1 km, the patch

richness within a certain radius, the interspersion of forest and meadow at a 3-km scale, etc.,

each of which can be represented as continuous gradient. Indeed, nearly all grid-based modeling

already uses a gradient approach. However, the tools commonly used to quantify landscape

structure are not gradient based (e.g. FRAGSTATS, McGarigal and Marks 1995).

2.2. The Hierarchical Model of Patch Structure

Over the past 15 years, landscape ecologists and hierarchy theorists have developed methods

to quantify patchiness simultaneously across a range of scales based on any number of

environmental attributes (Allen and Star 1982, O’Neill et al. 1986, Schneider 1994) . In

landscape ecology, a great interest has arisen in approaches based on defining multiple scales of

patchiness. John Wiens (1989) introduced the concept of ‘scale domains’ to describe regions of

the scale continuum that exhibit constancy in pattern-process relationships. He suggested that

there may exist multiple scale domains for the same phenomena; that is, that there may be

several discrete regions of the scale continuum within which pattern-process relationships are


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constant, implying a hierarchical organization to pattern-process relationships. Building on these

concepts, Kotliar and Wiens (1990) proposed a model of heterogeneity that effectively merges

scaling and hierarchy theory with the patch mosaic model of landscape structure. They proposed

that landscape heterogeneity is better characterized as a hierarchy of patch mosaics than a single

patch mosaic, recognizing that patches exist at many scales. This multi-scale characterization of

landscape structure provides a broad conceptual model that facilitates analysis of multi-level,

multi-scale phenomena in a theoretical context. In particular, it generates hierarchical conceptual

models of system organization and control which can be explicitly tested with empirical data

(Cushman and McGarigal, in review). However, the strict hierarchical patch perspective contains

several key assumptions which may not be justified in many analyses of continuously scaled

landscape attributes.

First, it assumes that the attributes being described are inherently discrete, at each level. This

is a justifiable model in many systems for some questions, but other systems may be better

characterized by combinations of continuously varying factors. Second, at every level, criteria

must be accepted to determine how the boundaries are drawn. The validity of a patch hierarchy

depends on the validity of the patches defined in each level and the researcher must decide on

what basis to define patches, not only in terms of the categories to split the factor into, but about

the threshold of change sufficient to define a boundary between entities. Third, the choice of

boundary definition is often problematic, because it is usually the case that one of the goals of

the research is to determine what constitutes ‘patchiness’ to the study organism or process in

terms of scale and intensity. But in order to study the response of the organism or process to


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environmental structure, a model of environmental structure must be provided. Thus, the choice

of how the boundaries are drawn will strongly influence how the organisms are perceived to

respond, and may yield erroneous conclusions. The problem is not readily solved by testing

organism or process response to several realizations of patch structure based on several boundary

definition rules, as the model to which the organism or process responds best does not

necessarily reflect the way the organism actually views and responds to the environment.

Finally, when dealing with multivariate structure in a hierarchy, these patch structure models are

overlaid. In this process, their inaccuracy regarding organism perception or process response

compounds. Even if errors in each layer are relatively small, multiplication across several levels

can result in large biases. Thus, while hierarchical models based on multiple-scales of patchiness

have emerged as putative solutions to the problems of how to accommodate complex,

multivariate, and multi-scale data in map analysis, they can give rise to troubling issues of

compounding subjectivity and compounding error.

Despite these limitations, hierarchical approaches which decompose the effects of multiple

factors at multiple organizational levels, and across temporal and spatial scales, are essential

tools for landscape ecologists (Allen and Star 1982, O’Neill et al. 1986). We feel that combining

the hierarchical patch-mosaic model proposed by Kotliar and Wiens (1990) with a gradient-

based characterization of environmental variability will make hierarchical approaches more

flexible and realistic. In particular, categorical mapping of patch-based organizational levels can

be readily coupled with continuous characterization of how environmental factors vary within

levels. In addition, levels, as defined in a hierarchical conceptual model, need not be


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characterized as patches. They can also be modeled as pattern gradients with scale and resolution

selected to match the hierarchical conceptual model being evaluated. Furthermore, the reality of

hierarchical levels can be assessed using gradient approaches to test the nestedness and

discreteness of particular hierarchical conceptual models.

3. The Gradient Concept of Landscape Structure

We believe that choosing an appropriate intensity measure for each variable is just as

important as choosing a pertinent grain and extent. A priori we see no reason to assume that

environmental variability is usually categorical or that organisms or ecological processes

respond categorically to it. Indeed, it seems less tenuous to assume that most environmental

factors are inherently continuous and that many of them are perceived and responded to as such

by organisms and ecological processes. Accordingly, we propose a conceptual shift in landscape

ecology akin to that which occurred in community ecology in the decades following Gleason’s

(1926) seminal statements on the individualistic response of species in a community and their

refinement by Whittaker (1967). Thus, to supplement the current patch mosaic paradigm, we

believe it will be useful for landscape ecologists to adopt a gradient perspective, along with a

new suite of tools for analyzing landscape structure and the linkages of patterns and processes

under a gradient framework. This framework will include, where appropriate, categorically

mapped variables as a special case, and can readily incorporate hierarchical and multi-scaled

conceptual models of system organization and control. In the sections that follow we outline how

a gradient perspective can be of use in several areas of landscape ecological research.

3.1. Gradient Attributes of Categorical Patterns


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Even when categorical mapping is appropriate, conventional analytical methods often fail to

produce unbiased assessments of organism responses. Organisms often experience categorical

environments as pattern gradients. For example, take a species that responds to landscape

structure, as measured by the density of edges in the landscape weighted by their structural

contrast. Traditional landscape pattern analysis would measure the total contrast-weighted edge

density for the entire landscape. This would be a global measure of the average property of that

landscape. In other words, the analysis implicitly assumes that the landscape is homogeneous,

with the contrast-weighted density of edges at every point in the landscape equal to that of the

landscape as a whole. However, landscape patterns are rarely stationary; there may be no place

in the landscape with a contrast-weighted edge density equivalent to that calculated for the

landscape as a whole. Supposing that the landscape is relatively large with respect to the

organism’s home range, the organism is unlikely to even experience the global average structure

of the landscape. The organism responds to the local structure within its immediate perception,

within its daily foraging bout area, and within its home range. Thus, a more useful description of

landscape pattern would be a location-specific measure at a scale relevant to the organism or

process of interest. Studies often attempt this by analyzing neighborhoods of a certain size

assumed to correspond to the process (e.g., dispersal) under investigation around each point

location of interest (e.g., nest site) and then calculating the desired parameter within that area.

One problem with this approach is that it gives measures of landscape structure only in the

vicinity of known organism locations and no measure of how that structure varies across the

study area, which limits our understanding of how organism response may vary through space.


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We propose that organisms experience landscape structure, even in categorical landscapes, as

pattern gradients that vary through space according to the perception and influence distance of

the particular organism. Thus, instead of analyzing global landscape patterns, we would be better

served by quantifying the local landscape pattern across space as it may be experienced by the

organisms of interest, given their perceptual abilities. Until recently, no tools were readily

available to accomplish this. However, FRAGSTATS (McGarigal et al. 2002) now contains a

moving window option which allows the user to set a circular or square window size for

analyzing selected class- or landscape-level metrics. The window size should be selected such

that it reflects the scale at which the organism or process perceives or responds to pattern. If this

is unknown, the user can vary the size of the window over several runs and empirically

determine which scale the organism is most responsive to. The window moves over the

landscape one cell at a time, calculating the selected metric within the window and returning that

value to the center cell. The result is a continuous surface which reflects how an organism of that

perceptual ability would perceive the structure of the landscape as measured by that metric. The

surface then would be available for combination with other such surfaces in multivariate models

to predict, for example, the distribution and abundance of organism continuously across the

landscape.

3.2. Gradient Analysis of Continuous Field Variables

When patch mosaics are not clearly appropriate as models of the variability of particular

environmental factors, there are a number of advantages to modeling environmental variation as

individually varying gradients. First, it preserves the underlying heterogeneity in the values of


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variables through space and across scales. The subjectivity of deciding on what basis to define

boundaries is eliminated. This enables the researcher to preserve many independently varying

variables in the analysis rather than reducing the set to a categorical description of boundaries

defined on the basis of one or a few attributes. In addition, the subjectivity of defining cut points

is eliminated. Imprecision in scale and boundary sensitivity is not an issue, as the quantitative

representation of environmental variables preserves the entire scale range and the complete

gradients to test against the response variables. The only real subjectivity is the increment or

intensity at which to measure variability. By tailoring the grain, extent, and intensity of the

measurements to the hypotheses and system under investigation, researchers can capture a less

equivocal picture of how the system is organized and what mechanisms may be at work. An

important benefit is that one can directly associate continuously-scaled patterns in the

environment, space, and time with continuous response variables such as organism abundance. A

specific advantage is that by not truncating the patterns of variation in the landscape variables to

a particular scale and set of categories, a scientist can use a single set of predictor variables to

simultaneously analyze a number of response variables, be they species responding

individualistically along complex landscape gradients, or ecological processes acting at different

scales. Comparison between organisms or processes is not compromised, because each can be

optimally predicted by the surface or combinations of surfaces without altering the data set in

ways that limit its utility for predicting other response variables. Importantly, this facilitates

efficient multivariate associations involving many response and many predictor variables

simultaneously, and allows the use of powerful analytical methods such as path analysis (Wright


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1921, 1960), partial canonical ordination (ter Braak 1988) and gradient interpolation methods

(Ohmann and Gregory, in press) to test hypotheses about the nature and strength of system

control.

3.3. Gradient-based Measures of Landscape Structure

Landscape ecologists often use comparative mensurative studies to explain pattern-process

relationships (McGarigal and Cushman 2002). Such studies involve comparing the structure of

different landscapes or of the same landscape over time and relating the observed differences to

some process of interest. When categorical maps are appropriate, conventional landscape metrics

based on the patch mosaic model are quite effective, and scientists have developed a plethora of

metrics for this purpose (e.g., Baker and Cai 1992, McGarigal et al. 2002). However, when

environmental variation is better represented as continuous gradients, it is not as simple to

summarize the structure of each landscape in a metric. In this case, each landscape is represented

as a continuous surface, or several surfaces corresponding to different environmental attributes.

The challenge lies in summarizing the structure of a surface in a metric.

The two fundamental attributes of a surface are its height and slope. The patterns in a

landscape surface that are of interest to landscape ecologists are emergent properties of particular

combinations of surface heights and slopes across the study area. The challenge is to develop

metrics that describe meaningful attributes of surface height and slope that can be used to

characterize surface patterns and to derive variables that are effective predictors of organismic

and ecological processes.

Geostatistical techniques have been developed that allow us to summarize the spatial


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autocorrelation of such a surface (Webster and Oliver 2001). Measures such as Moran’s I and

semi-variance, for example, indicate the degree of spatial correlation in the quantitative variable

(i.e., the height of the surface) at a specific lag distance (i.e., distance between points). They are

plotted against a range of lag distances to summarize the spatial autocorrelation structure of the

landscape. The correlogram and semi-variogram can provide useful indices to quantitatively

compare the intensity and extent of autocorrelation in quantitative variables among landscapes.

However, while they can provide information on the distance at which the measured variable

becomes statistically independent, and reveal the scales of repeated patterns in the variable, if

they exist, they do little to describe other interesting aspects of the surface. For example, the

degree of relief, density of troughs or ridges, and steepness of slopes are not measured.

Fortunately, a number of gradient-based metrics that summarize these and other interesting

properties of continuous surfaces have been developed in the physical sciences for analyzing

three-dimensional surface structures (Barbato et al. 1996, Sout et al. 1994, Villarrubia 1997). In

the past ten years, researchers involved in microscopy and molecular physics have made

tremendous progress in this area, creating the field of surface metrology (Barbato et al. 1996).

Surface Metrology.--In surface metrology, several families of surface pattern metrics have

become widely utilized (Table 1). These have been implemented in the software package SPIP

(SPIP 2001). One so-called family of metrics quantify intuitive measures of surface amplitude in

terms of its overall roughness, skewness and kurtosis, and total and relative amplitude. Another

family records attributes of surfaces that combine amplitude and spatial characteristics such as

the curvature of local peaks. Together these metrics quantify important aspects of the texture and


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complexity of a surface. A third family measures certain spatial attributes of the surface

associated with the orientation of the dominant texture. The final family of metrics are based on

the surface bearing area ratio curve, also called the Abbott curve (SPIP 2001). The Abbott curve

is computed by inversion of the cumulative height distribution histogram. The curve describes

the distribution of mass in the surface across the height profile. Generally, the curve is divided

into three zones, called the peak zone, corresponding to the top 5% of the surface height range,

core zone, corresponding to the 5% - 80% height range, and valley zone, which corresponds to

the bottom 20% of the height range of the surface. A number of indices have been developed

from the proportions of this cumulative height-volume curve which describe structural attributes

of the surface (SPIP 2001).

Many of the classic landscape metrics for analyzing categorical landscape structure have

ready analogs in surface metrology. For example, the major compositional metrics such as patch

density, percent of landscape, and largest patch index are matched with peak density, surface

volume, and maximum peak height. Major configuration metrics such as edge density, nearest

neighbor index, and fractal dimension index are matched with mean slope, mean nearest

maximum index and surface fractal dimension. Many of the surface metrology metrics, however,

measure attributes that are conceptually quite foreign to conventional landscape pattern analysis.

Landscape ecologists have not yet explored the behavior and meaning of these new metrics; it

remains for them to demonstrate the utility of these metrics, or develop new surface metrics

better suited for landscape ecological questions.

Fractal Analysis.–Fractal analysis has been well developed for the analysis of two-


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dimensional surface patterns, but is just as suited for analyzing continuous variables as three- or

higher-dimensional surfaces. Fractal analysis provides a vast set of tools to quantify the shape

complexity of surfaces. There are many algorithms in existence that can measure the fractal

dimension of any surface profile, surface or volume (Mandelbrot 1982, Pentland 1984, Barnsley

et al. 1988). One such index that is implemented in SPIP calculates the fractal dimension along

profiles of the surface from 0 to 180 degrees. This profile fractal index is calculated for the

different angles by analyzing the Fourier amplitude spectrum. For each angle, the Fourier profile

is extracted and the logarithm of the frequency and amplitude coordinates calculated. The fractal

dimension for each direction is calculated as 2.0 minus the slope of the log-log curves. The

fractal dimension can also be evaluated from 2D Fourier spectra by application of the log-log

function. If the surface is fractal, the log-log graph should be highly linear (SPIP 2001). A

number of other fractal algorithms are available for calculating the overall fractal dimension of

the surface, rather than for particular profile directions (Ewe et al. 1994). Variations on these

approaches will certainly yield metrics that quantify important attributes of surface structure for

comparison between landscapes, between regions within a landscape, and for use as independent

variables in modeling and prediction of ecological processes. In addition, there are surface

equivalents to lacunarity analysis of categorical fractal patterns. Lacunarity measures the

gapiness of a fractal pattern (Plotnick et al. 1993). Several structures with a given fractal

dimension can look very different because of differences in their lacunarities. The calculation of

measures of surface lacunarity is a topic which deserves considerable attention. It seems to us

that surface lacunarity will be a useful index of surface structure, which measures the ‘gapiness’


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in the distribution of peaks and valleys in a surface, rather than holes in the distribution of a

categorical patch type.

Spectral and Wavelet Analysis.--Spectral analysis and wavelet analysis are ideally suited for

analyzing surface patterns. The spectral analysis technique of Fourier decomposition of surfaces

could find a number of interesting applications in landscape surface analysis. Fourier spectral

decomposition breaks up the overall surface patterns into sets of high, medium and low

frequency patterns (Kahane and Lemarie 1995). The strength of patterns at different frequencies,

and the overall success of such spectral decompositions can tell us a great deal about the nature

of the surface patterns and what kinds of processes may be acting and interacting to create those

patterns. They also provide potential indices for comparing among landscapes and for deriving

variables describing surface structure at different frequency scales that could be used for

prediction and modeling (Kahane and Lemarie 1995). Similarly, wavelet analysis is a family of

techniques which has vast potential applications in landscape surface analysis (Bradshaw and

Spies 1992, Cuhi 1992, Kaiser 1994, Cohen 1995). Traditional wavelet analysis is conducted on

transect data, but the principle is easily extended to two-dimensional surface data. There have

been great advances in wavelet applications in the past few years, with many software packages

now available for 1- and 2-dimensional wavelet analysis. For example, comprehensive wavelet

toolboxes are available for S-Plus, MATLAB and MathCad. Wavelet analysis has the advantage

that it preserves hierarchical information about the structure of a surface pattern while allowing

for pattern decomposition (Bradshaw and Spies 1992). It is ideally suited to decomposing and

modeling signals and images, and is useful in capturing, identifying, and analyzing local, multi-


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scale, and nonstationary processes. It can be used to identify trends, break points, discontinuities,

and self-similarity (Cuhi 1992, Kaiser 1994). In addition, the calculation of the wavelet variance

enables comparison of the dominant scales of pattern among landscape surfaces or between

different parts of a single surface (Bradshaw and Spies 1992). Thus, wavelet decomposition and

wavelet variance have a great potential as sources of new surface-pattern landscape metrics and

novel approaches to analyzing landscape surfaces.

3.4. Synthesis of Community Ecology and Landscape Ecology

Just as a gradient approach has proven most useful in studying the structure of biological

communities and the factors that create and maintain structure, we believe it will also prove so in

landscape ecology. Indeed, one of the great challenges we perceive facing landscape ecology is

the integration of landscape- and community-level patterns and processes into a unified analysis.

A major impediment to this integration is the incomparability of categorical map-based

landscape methods with gradient-based community analysis approaches. We believe that by

integrating the approaches this synthesis will occur readily. Indeed, modern community analysis

techniques such as discriminant analysis, multiple regression, TWINSPAN, ordination and

constrained ordination are ideally suited for analyzing spatial, temporal and structural gradients

in landscape data. Once landscape- and community-level analyses use a common library of

analytical techniques, it will be more feasible to conduct robust analyses of the important

questions related to how variability in landscape patterns influences the structure and stability of

biological communities.

We by no means suggest eliminating the patch concept. Clearly there are many cases where


23

categorical patches are the most meaningful model for the question at hand. The approaches we

advocate can readily incorporate both categorical patch-based data and continuously scaled data.

It is conceptually akin to a transition from an analysis of variance framework to a regression

framework. In a gradient-based perspective of landscape analysis, categorical variables and

categorical patterns are readily incorporated as special cases of binary gradients. The regression,

logistic regression and ordination approaches we envision as the analytical workhorses of this

perspective can readily include categorical variables, coded as dummy variables, where

appropriate, along with continuous field data. The great advantage of the approach is that it can

incorporate the many powerful and dearly won techniques from categorical pattern landscape

ecology seamlessly into a broader and more inclusive gradient-based framework.

If there is one criticism of adopting a generalized gradient framework for landscape ecology

it is certainly that it demands much more computational power. More sophisticated statistical

techniques and longer computer processor time are required to analyze layers of surfaces rather

than a single layer of non-overlapping patches. In the past, limitations of computing power and

software availability limited the development of gradient methods in landscape ecology.

However, these are no longer an impediment. With multiple gigahertz processor computers

containing multiple gigabytes of memory available presently, computing power is becoming less

an issue. Also, multivariate and spatial analysis software packages are widely available. In the

past 20 years there has been an explosion of gradient-based multivariate analysis techniques.

Many of these approaches are particularly well suited to analyzing continuous field landscape-

level data. There are also a number of developing statistical approaches, such as gradient


24

interpolation techniques, fractal analysis, spectral analysis and wavelet analysis, we believe will

catalyze rapid advancement in applications of the gradient perspective to analysis of landscape

pattern and process.

4. Examples of the Gradient Concept

The following two examples illustrate the gradient concept of landscape structure, but by no

means represent the breadth of situations in which gradient-based approaches are appropriate.

The examples are intended to show that the gradient concept is not just a theoretical concept, but

can be implemented in practice in a variety of situations.

4.1. Gradient Modeling of Categorical Map Patterns

Even when categorical maps make sense, it may be more appropriate to map the categorical

environment as a density gradient. In this example, let’s assume that we are interested in the

connectedness of forest across the landscape shown in figure 1a, as it might be perceived by a

forest-obligate organism with a maximum dispersal ability of 500m. The landscape is

represented as a simple binary patch mosaic consisting of forest patches surrounded by a

nonforest matrix. The aggregation index is a measure of connectedness based on the aggregation

(i.e., clumping) of like-valued cells, in this case, forest, and is computed as a percentage based

on the ratio of the observed number of like adjacencies (i.e., cells of forest next to forest) to the

maximum possible number of like adjacencies given the total area of forest (He et al. 2000). The

maximum number of like adjacencies is achieved when the class is clumped into a single

compact patch, which does not have to be a square. The aggregation index for the forest class is

easily computed for the entire landscape using FRAGSTATS (McGarigal et al. 2002) and equals


25

86.77. While this value can be interpreted, it is unlikely that an organism in this landscape would

ever experience this level of habitat aggregation, unless their movement range encompassed the

entire landscape. It may be more meaningful to express aggregation at the local scale based on

the vagility the organism. If we assume that the vagility of the organism extends to a

neighborhood of 500 m radius, we can compute this index as a continuously varying quantity

using a moving window analysis (Fig. 1b). The result is a continuous surface in which the height

of the surface at any point represents the aggregation of forest (i.e., connectedness) as might be

perceived by an organism at that location.

The local calculation of landscape metrics using a moving window is also useful for

characterizing the structure of a multi-class patch mosaic. For this example, we analyzed a 25

km2 landscape in western Massachusetts. The landscape was classified into nine vegetation

classes based on natural breaks in a vegetation index derived from Landsat TM imagery (see Fig.

3a). For the example, it isn’t important what the patch mosaic represents, only how the local

structure of the mosaic can be measured and mapped. We illustrate this for three metrics at two

window sizes (Fig. 2). The metrics are total edge, which is the linear distance of all edges

between patches in the landscape, mean shape index (shape_mn), which records the mean shape

complexity of all patches in the landscape, and Simpson’s diversity index (SIDI), which

measures the landscape diversity of patch types. These were implemented at 500 m and 250 m

radii window sizes. The output maps are shown in figure 2. These scale-specific

characterizations of neighborhood landscape structure provide critical information about how

patterns vary within individual landscapes, and provide the input data needed for modeling and


26

prediction of organism and process responses to landscape structure across heterogeneous

landscapes.

4.2. Gradient Modeling of Continuous Environmental Variation

In this example, we will explore the structure of two continuous field variables in two 25-km2

landscapes in western Massachusetts. The two variables are the normalized difference vegetation

index (NDVI) and the topographic wetness index (TWI). NDVI provides a continuous measure

of the intensity of green vegetation at a site based on Landsat TM imagery, while TWI measures

the predicted wetness of a site based on a combination of its local slope and the area of the

watershed that drains into it (Moore et al. 1991). NDVI is derived from 30 m cell size imagery,

and resampled to 15 m, while the TWI is derived at 15 m resolution. There are two main goals of

this example. First, the goal is to provide an intuitive picture, through 3-D visualization, of the

differences between categorical and gradient representations of these variables in one of the

landscapes. This will illustrate the truncation and loss of information across scales when

continuous field variables are categorized. Second, we will apply a select number of surface

metrics developed in surface metrology to quantify the structural attributes of the surfaces.

Finally, we will discuss how these surface pattern analyses could be incorporated in analyses of

particular hypotheses regarding the relationships between landscape structure and ecological

processes.

We classified the NDVI image into nine categories using natural breaks to define the class

limits, as in the previous example (Fig. 3a). This resulted in a highly heterogeneous patch

mosaic, with patch types representing slices of the gradient from low vegetation intensity (red),


27

to very high intensity of green vegetation (dark green). The mosaic looks comfortingly familiar,

as we are all accustomed to seeing maps dividing landscapes into patches. However, when we

visualize the actual NDVI surface, the boundaries of these patches disappear and are replaced by

a complex continuous surface (Fig. 3b). Categorizing this surface into patches truncates the

variability of the vegetation index, collapsing it into a few categories, resulting in the loss of any

information about the internal heterogeneity within classes and quantitative information about

how the classes differ. In addition, it is not clear that the classification criterion has found

boundaries that are meaningful given the questions we may be investigating. While the surface

pattern depicted in figure 3b may look intimidating at first, it preserves the actual variability in

the index without truncation or division. As we zoom in on smaller portions of the images, the

loss of internal variability and questions about the reality of patch boundaries become even more

dramatic (Figs. 2c-f). At the finest resolution we see that areas that are homogeneous patches in

the categorical map do indeed possess substantial internal heterogeneity, and that the boundaries

of the patches do not necessarily correspond with major discontinuities in the vegetation index

(Figs. 2e,f).

Similarly, we classified the TWI into nine categories using natural breaks, resulting in a

mosaic of patch types representing a range of wetness values from very dry (red) to extremely

wet (dark blue). This mosaic looks very different than the vegetation map, as it emphasizes the

linear nature of the hydrological network (Fig. 3a). However, just as in the case of the vegetation

index, it is not clear that the mapped patches reflect ecologically meaningful subdivisions of the

wetness gradient, or that the boundaries reflect actual discontinuities in the index (Fig. 3b).


28

These issues are clearly visible at all resolutions (Figs. 3c-f).

For the second part of the example, we calculated a selection of nine surface structure

metrics (see Table 1 for a brief description of each metric) for both the NDVI and TWI indices in

the two landscapes (Fig. 4). These metrics quantify attributes of the surface as a whole, rather

than the structure of a patch mosaic. These and other surface metrics provide a quantitative basis

for comparing the structure of surfaces, just as traditional metrics do for categorical patterns.

They are ready alternatives to traditional landscape metrics for quantitative variables, and are

suited to all the uses landscape metrics are employed for, such as description, comparison,

modeling and prediction. In addition, as is the case for categorical landscapes, one can use a

windowing function to calculate these surface structure metrics for focal areas of specified size

creating maps of the neighborhood value of each surface metric. This would facilitate scale-

dependent modeling of organism or process response to neighborhood surface structure, across

any number of quantitative predictor variables.

5. Conclusions

Landscape ecology has emerged over the past decade as the study of spatial and temporal

heterogeneity, and under what circumstances it matters to organisms, communities, and

ecological processes (Turner et al. 2001). The patch mosaic model of landscape structure has

become the operating paradigm of the discipline. While this paradigm has provided an essential

operating framework for landscape ecologists, and has facilitated rapid advances in quantitative

landscape ecology, we believe that further advances in landscape ecology are somewhat

constrained by its limitations. We advocate the expansion of the paradigm to include a gradient-


29

based concept of landscape structure which subsumes the patch mosaic model as a special case.

The gradient approach we advocate allows for a more realistic representation of landscape

heterogeneity by not presupposing discrete structures, facilitates multivariate representations of

heterogeneity compatible with advanced statistical and modeling techniques used in other

disciplines, and provides a flexible framework for accommodating organism-centered analyses.

Perhaps the greatest obstacle to the adoption of gradient approach is the lack of familiarity

with tools for conducting gradient-based landscape analyses and inexperience in the application

of surface metrics to landscape ecological questions. While familiar tools now exist for

conducting gradient analyses of categorical map patterns (e.g., moving-window analysis in

FRAGSTATS), landscape ecologists have not yet fully taken advantage of these. In addition,

while numerous surface metrics have been developed for characterizing continuous landscape

surfaces, and the software tools for computing them are now available, it remains for landscape

ecologists to investigate how these metrics behave and what information they provide in

landscape surface analysis, and to develop additional metrics that quantify specific surface

attributes of importance in landscape ecology. This is an interesting and important challenge, and

until such measures are understood in the context of landscape analysis, and until additional

metrics are tailored to the specific needs of landscape ecologists, the full potential of gradient-

based methods will not be realized. We believe that landscape ecology, as a discipline, is poised

on the verge of tremendous advances; the gradient concept is an organizational and

methodological construct that we believe will facilitate these advances.

6. Literature Cited


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Table 1. Brief description of some common surface metrology metrics that might be applied in

landscape ecological investigations. Metrics are grouped into “families” based on the surface

properties measured; names and descriptions follows that given in the SPIP software program

(SPIP 2001). See the program documentation for formulas and full descriptions, as well as other

surface metrics.

Acronym Metric Name Description

Amplitude Metrics: give information about the statistical average properties, the shape of the

height distribution histogram, and about extreme properties

Sa roughness average average deviation of the surface height from

the global mean

Sq root mean-square roughness variance in the height of the surface

Ssk surface skewness asymmetry of the surface height distribution

histogram

Sku surface kurtosis peaked-ness of the surface topography

Sy peak-peak height height difference between the highest and

lowest pixel in the image

Sz ten point height average height fo the five highest local

maximums plus the average height of the five

lowest local minimums


35

Hybrid Metrics: reflect slope gradients of the local surface

Ssc mean summit curvature average of the principal curvature of the local

maximums on the surface

Sdq root mean square slope variance in the local slope across the surface

Sdr surface area ratio ratio between the surface area to the area of

the flat plane with the same x-y dimensions

Spatial Attributes: describe the density of summits and the orientation (direction) of the

surface texture

Sds summit density number of local maximums per area

Std dominant texture direction angle of the dominating texture in the image

calculated from the Fourier spectrum

Stdi texture direction index relative dominance of Std over other directions

of texture

Surface Bearing Metrics: based on the surface bearing area ratio curve, also called the Abbott

curve, computed by inversion of the cumulative height distribution histogram. The curve

describes the distribution of mass in the surface across the height profile. Generally, the curve

is divided into three zones, called the peak zone, corresponding to the top 5% of the surface

height range, core zone, corresponding to the 5% - 80% height range, and valley zone, which

corresponds to the bottom 20% of the height range of the surface.


36

Sbi surface bearing index ratio of the root mean square roughness (Sq) to

the distance from the top of the surface to the

height at 5% bearing area

Sci core fluid retention index void volume (area above the Abbott curve) in

the core zone

Svi valley fluid retention index void volume (area above the Abbott curve) in

the valley zone


37

Figure Captions

Figure 1. Comparison of global and neighborhood-based calculation of a landscape metric for a

categorical map. The Aggregation Index (AI) was calculated for the “forest” class (grey) in the

binary map on the left for the landscape overall, and within 500 m radius circular windows

centered on each pixel. The moving window calculation, shown on the right, produces a surface

whose height is equal to the neighborhood AI value. AI measures the connectedness of a class

based on the aggregation of like-valued cells and is computed as a percentage based on the ratio

of the observed number of like adjacencies to the maximum possible number of like adjacencies

given the total area of forest. There is a border classified as “no data” around the edge of the

landscape to a depth of the selected neighborhood radius. Higher AI values are light, lower

values are dark.

Figure 2. An example moving window calculation of three landscape-level metrics at two

neighborhood sizes (500 m and 250 m). The input map is depicted in figure 3a, and is a natural

breaks classification of the normalized difference vegetation index value across a 25-km2

landscape in western Massachusetts into nine classes. The three metrics calculated here are the

total edge length between different patch types, mean shape index (Shape_mn), which quantifies

the average shape complexity of patches, and Simpson’s patch diversity index (SIDI). The figure

shows how the internal patterns of these metrics differ across the landscape, and how changing

neighborhood size alters the local output values, with larger window sizes yielding smoother

grids. Higher values are light, lower values are dark.


38

Figure 3. Comparison of categorical and gradient mapping of the normalized difference

vegetation index (NDVI) for a 25-km2 landscape in western Massachusetts. (A) The landscape

classified into nine discrete classes using a natural breaks classification criterion. (B) The same

landscape depicted as a 3-dimensional surface whose height is proportional to the NDVI value at

each pixel (15m cell size). (C) and (D) show the same landscape at a higher resolution,

corresponding to the yellow box in A and B, respectively, while E and F show it at even higher

resolution corresponding to the yellow box in C and D, respectively. The figure shows

considerable internal variation within “patches” and uncertainty about the meaningfulness of

patch boundaries for this classification. For this variable in this landscape, mapping as a

continuous variable may be more appropriate than truncating its variability into categories, given

its highly variable, gradient nature.

Figure 4. Comparison of categorical and gradient mapping of the topographic wetness index

(TWI) for the same landscape as in figure 3. (A) The landscape classified into nine discrete

classes using a natural breaks classification criterion. (B) The same landscape depicted as a 3-

dimensional surface whose height is proportional to the TWI value at each pixel (15m cell size).

(C) and (D) show the same landscape at a higher resolution, corresponding to the black box in A

and B, respectively, while E and F show it at an even higher resolution corresponding to the

black box in C and D, respectively. The figure shows the inherently “smooth” gradient nature of

the wetness index across all resolutions, and the artificial truncation of this variability to

form“patches”. For this variable in this landscape, mapping as a continuous variable may be


39

more appropriate than truncating its variability into categories, given its smooth, gradient nature.

Figure 5. Calculation of nine surface pattern metrics for two continuous variables in two

landscapes. The landscapes are approximately 25-km2 and are located in western Massachusetts.

The two variables are the normalized difference vegetation index, whose value is proportional to

the intensity of green vegetation at a site, in A and B, and the topographic wetness index, whose

value is proportional to the expected wetness of the site based on a combination of its water-

gathering and water-holding capacity, in C and D. The nine surface pattern metrics include:

Mfract - mean profile fractal dimension, which is the mean fractal dimension of 180 profiles

taken at 1 degree increments across the surface, and several of the surface roughness metrics

described in table 1.


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Figure 1


41

Figure 2


42

Figure 3


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Figure 4


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Figure 5

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