Statistics Data

Plant Ecology ***: ** ***, ****.

**** ****** ******** **********. ******* in the Netherlands.

Measuring the abruptness of patchy ecotones

A simulation-based comparison of landscape pattern statistics

Mark A. Bowersox1 & Daniel G. Brown2,

Department of Geography, Michigan State University, East Lansing, MI 48823-1115, USA ( current

address: Cayuga County Planning Department, 160 Genesee Street, Auburn, NY 13021, USA); 2 School

of Natural Resources and Environment, University of Michigan, Ann Arbor, MI 48109-1115, USA

(e-mail: ********@*****.***)

Key words: Boundaries, Ecotones, Landscape statistics, Simulation, Spatial analysis

Abstract

The use of statistics of landscape pattern to infer ecological process at ecotones requires knowledge of the speci c

sensitivities of statistics to ecotone characteristics. In this study, sets of patch-based and boundary-based statistics

were evaluated to assess their suitability as measures of abruptness on simulated ecotone landscapes. We gener-

ated 50 realizations each for 25 groups of ecotones that varied systematically in their degree of abruptness and

patchiness. Factorial ANOVA was used to evaluate the sensitivity of statistics to the known differences among

the simulated groups. Suitability of each index for measuring abruptness was evaluated using the ANOVA results.

The statistics were then ranked in order of their suitability as abruptness statistics based on their sensitivity to

abruptness, the consistency of the relationship, and their lack of sensitivity to patchiness. The two best statistics for

quantifying abruptness were those we developed based on lattice delineation methods, and are called cumulative

boundary elements and boundary element dispersion. The results of this research provide support for studies of

ecotone process that rely on the interpretation of patch or boundary statistics.

Introduction gradual treelines are characterized by gradual inter-

spersion of types and a decrease of tree cover until

Ecotones are zones of transition between adjacent trees are completely absent. Abrupt ecotones may re-

ecosystems. Their characteristics are uniquely de ned sult from abrupt environmental gradients. However,

by the interactions between the ecosystems (Holland abrupt ecotones may also occur along gradual environ-

1988). Ecotones under the above de nition have also mental gradients, suggesting the action of ecological

been referred to as edges (Orloci & Orloci 1990), tran- processes such as species competition (Armand 1992;

sitional areas (Kent et al. 1997) and boundaries (Wiens Malanson & Butler 1994) and positive feedback mech-

et al. 1985). Although these alternative terms often anisms (Wilson & Agnew 1992; Malanson 1997). One

imply abruptness, an ecotone may be abrupt or gradual approach to testing hypotheses about these processes

and may be highly heterogeneous or less heteroge- is to quantitatively compare the level of abruptness

neous depending on the ecological processes acting on that results from spatial process models that include

it. varied process assumptions and to compare model re-

This research focuses on the measurement of eco- sults with observations of real ecotones (e.g., through

tone abruptness, which is thought to be representative remote sensing). Quantitative comparisons, however,

of speci c ecological processes operating at the alpine require statistics of pattern that measure the property

treeline ecotone. Abruptness refers to the rate at which of interest (e.g., abruptness). The goal here is to inves-

one ecosystem spatially transitions to another across tigate multiple statistics and determine which are most

the ecotone; for example, abrupt alpine treeline eco-

tones change rapidly from trees to alpine tundra while

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suitable for quantifying ecotone abruptness for use in represented within a raster map and used to identify

model and observation comparison. contiguous clusters of cells, called patches. After clas-

Patchiness across an ecotone can confound at- si cation, patches are considered internally homoge-

tempts to detect differences in ecotone abruptness. For neous and the boundary between patches of different

this reason, the best statistics for measuring abruptness classes is a distinct one. The programs SPAN (Turner

will be those that are sensitive to abruptness, but rela- 1990), r.le (Baker & Cai 1992) and FRAGSTATS (Mc-

tively insensitive to patchiness. Ecotone patchiness is Garigal & Marks 1993), which are compatible with

de ned as spatial heterogeneity or the degree of spatial geographic information systems (GIS), generate a va-

randomness in the pattern; an ecotone is patchy when riety of patch statistics that mathematically de ne the

neighboring sites (e.g., grid cells) tend to be dissimilar. spatial pattern of a landscape. Among the charac-

Landscape statistics can be computed from spa- teristics that patch statistics represent are patch den-

tial representations of a landscape through a variety sity, patch size, patch shape and patch variability,

of approaches. We focus on two approaches that in- landscape edge, landscape core area, landscape diver-

volve identifying (a) homogeneous patches on the sity, contagion and interspersion (McGarigal & Marks

landscape or (b) locations of rapid change on the land- 1993).

scape (boundaries). This research evaluates patch and Baker & Weisberg (1995), in Rocky Mountain Na-

boundary statistics with the goal of selecting those tional Park, CO, and Allen & Walsh (1996), in Glacier

most suitable for quantifying abruptness. The goal is National Park, MT, used the patch approach to quan-

to evaluate the statistics in an experiment where patch- tify landscape pattern at the alpine treeline ecotone.

iness and abruptness characteristics are controlled Both were able to discern 6 unique types of alpine

through simulation of ecotones, modeled according treeline ecotone using a cluster analysis of patch sta-

to alpine treeline ecotones observed in a satellite im- tistics. Each study found ecotone types that were best

age. This approach allows for an objective comparison described according to their patchiness and abrupt-

among statistics and provides evidence of the informa- ness characteristics. Baker & Weisberg (1995) found

tion content and behavior of the statistics with respect ecotones that were long and short with variable

to abruptness when applied to ecotones with varying amounts of patchiness, while Allen & Walsh (1996)

degrees of patchiness. The experiment also allows for differentiated ecotone patchiness and abruptness by

an examination of the interacting effects of abruptness labeling types as heterogeneous or highly heteroge-

and patchiness on each set of statistics. neous and moderately zonal or zonal, respectively.

Following a general description and de nition of A central element of the patch approach is the clas-

terms associated with patch and boundary statistics, si cation and patch delineation necessary to compute

we describe our approach to simulating ecotones at patch statistics. The patch approach uses a line of zero

multiple levels of abruptness and patchiness and an- thickness to represent the transitions between adjacent

alyzing 11 different statistics for their suitability as patches. Because many ecotone-related questions ad-

abruptness measures. The results of an analysis of dress the contrast and spatial rate-of-change between

variance are presented for evaluating the suitability of adjacent ecosystems the classi cation procedure rep-

the statistics as abruptness measures based on their resents a loss of relevant information (Johnston &

sensitivity to abruptness, the consistency of the rela- Bonde 1989; Wood & Foody 1989; Brown 1998). At

tionship, and their lack of sensitivity to patchiness. best, only the length of the ecotone and the classes that

Finally, a discussion of the importance of understand- it separates can be directly quanti ed. There is no way

ing the interacting effects of patchiness and abruptness to represent a gradual transition between neighboring

and choosing the appropriate statistic is followed by patches using this approach.

our concluding remarks.

Boundary statistics

Patch statistics

Boundary statistics can avoid the classi cation step

The patch approach to describing landscapes typically by measuring information about boundaries that are

involves classifying satellite imagery or aerial photog- based on the spatial rate-of-change, i.e., slope, of

raphy to produce a map in which similar types of eco- a continuous variable surface. Many ecological sur-

logical communities or vegetation types are grouped faces possess the mathematical property of continu-

together. These groups, referred to as classes, are ity and have only one value at any point. Surfaces

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of ecologically relevant variables are frequently ap- Methods

proximated using satellite imagery; examples include

percent vegetative cover, the Normalized Difference The research was conducted in three phases. First, eco-

Vegetation Index (NDVI) and Leaf Area Index (LAI) tone surfaces with known patchiness and abruptness

(e.g., Brown, in press). The ecological surface is ap- characteristics were simulated, classi ed, and orga-

proximated as a regularized grid where each cell of nized into a matrix structure that facilitated a factorial

the grid contains a unique data value. analysis of variance (ANOVA) experimental design.

Legendre & Fortin (1989) provide a review of Second, patch and boundary statistics were calculated

methods for analyzing surfaces that employ spatial au- on the simulated data. Finally ANOVA was conducted

tocorrelation coef cients, correlograms, variograms, and interpreted for comparison between groups of

spectral analysis, and the Mantel test to measure spa- simulated ecotones and among statistics.

tial pattern. Kent et al. (1997) review similar methods

with respect to ecotone analysis. Among the sev- Simulation

eral surface-based methods available, one group, the

A goal of the simulation was to create ecotone data

boundary statistics, focuses on transitional areas and

similar to what would be obtained from a LANDSAT

relies on the methods of boundary detection. The goal

Thematic Mapper (TM) satellite image. Inspection of

of boundary detection is to locate discontinuities along

alpine treeline ecotones captured in a TM scene of

transects (Ludwig & Cornelius 1987) or within two-

Glacier National Park (GNP), MT provided a visual

dimensional maps (Johnston et al. 1992) using algo-

model for the simulations. A majority of the treelines

rithms that accentuate areas with high rates of change

examined in the TM image had transition lengths less

on a given variable or set of variables mapped over

than approximately 600 m. Transition length was de-

the landscape. Womble (1951) proposed a method of

ned as the distance between closed canopy forest and

boundary detection within two-dimensional maps that

open alpine tundra as measured along the pro le of

has been modi ed by Barbujani et al. (1989), Fortin

vegetation change. The extent of the areas used in

(1994), and Jacquez & Maruca (1998). The method

the simulations was set at 630 m 630 m to accom-

is commonly called either Wombling or lattice delin-

modate the maximum transition lengths observed in

eation and identi es boundaries as spatially contigu-

the image. The simulated data set used a cell size

ous locations with high rates of change. The location,

of 30 m, which corresponds to the cell size of TM.

width, shape, or distribution of these boundaries can

The size of each simulated ecotone was approximately

be used to characterize the transition.

40 ha, which was comparable to some of the smaller

While boundary statistics have been found to be

two-dimensional transects used by Baker & Weisberg

effective in determining the signi cance of detected

(1995). The GNP TM scene was consistently referred

boundaries (Fortin 1994; Fortin & Drapeau 1995),

to throughout the development of the simulation to

less is known about how the boundary statistics can

assure, at the very least, that the simulated ecotones

be used to quantify speci c boundary characteristics

visually resembled real ecotones (Figure 1).

or landscape patterns. For instance, a small number

Simulated data were used to control the patchiness

of long boundaries may indicate that an ecotone is

and abruptness characteristics of each ecotone by sys-

abrupt, while a large number of shorter boundaries

tematically altering the parameters of the simulation.

may indicate that an ecotone is more gradual.

The simulation produced values of a hypothetical con-

A limitation of the lattice delineation approach

tinuous variable for each cell in a square grid that

is the arbitrary nature of the gradient threshold used

mimicked a real world study area containing an eco-

to de ne boundaries. A method that uses the lattice

tone. High variable values were intended to represent

delineation approach to obtain boundary statistics at

more tree cover while low variable values represented

multiple gradient threshold levels may avoid the limi-

lower tree cover and more tundra, bare soil or rock.

tations of an analysis at only one threshold level. We

We used simulation to produce 2 sets of maps, a

developed the cumulative boundary elements (CBE)

continuous set (Figure 2) and a classi ed set (Fig-

statistic to take advantage of this approach and to

ure 3), each representing 25 different types of eco-

measure ecotone abruptness. CBE is tested for its suit-

tones. The 25 types resulted from unique combinations

ability as an abruptness statistic along with other patch

of 5 levels of patchiness and 5 levels of abruptness.

and boundary statistics.

The simulation was repeated 50 times for each type in

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Figure 1. Two sample alpine treeline ecotone sites from a Landsat image of Glacier National Park, Montana, USA. The images represent

estimates of the average leaf area index (LAI) of trees within each 30 m pixel, following Brown (in press).

Figure 2. An example of the simulated ecotone surfaces from each of the 25 groups. Each surface is the result of adding a perturbation surface

of a given patchiness level to a deterministic surface of a given abruptness level. The brightness of the surface shading represents the strength

of tree presence.

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Figure 3. An example of the simulated ecotone surfaces that have been classi ed into tree presence/absence from each of the 25 groups. Black

represents tree presence, white represents tree absence (i.e., tundra species, rock, and bare soil).

order to obtain replicates for statistical analysis. Ac- & Svirezhev (1995) and Timoney et al. (1993) used

cordingly, 1250 simulated maps of each type (contin- sigmoid functions to mathematically model ecotones;

uous and classi ed) were produced. For the remainder however, their work was done at the biome scale and

of this text, the 50 ecotones that were simulated to be not a local scale, as is the case here. The function

of the same patchiness and abruptness level are said to used here is capable of modeling transitions of vari-

belong to the same group . able abruptness without a change in the minimum and

Abruptness was controlled using a deterministic maximum z values.

function, which is in the form of a sigmoid curve: Four values for the slope parameter (n) were used

to generate two-dimensional maps of varying degrees

z = sign(y) y n 0.5, (1)

of abruptness: 1, 0.5, 0.25 and 0.0625. This was ac-

where z is the surface variable, sign is an operator complished by mapping the value of the function on

that returns the sign (i.e., + or ) of y, y is the y over the interval [ 1, 1] at 0.1 unit increments to

position along the transition, and n is the slope (or consecutive cells in each row of the two-dimensional

abruptness) parameter. This function was chosen for map. Each of the four resulting deterministic surfaces

had a value range [ 0.5, 0.5]. Lower magnitude slope

its ability to model a smooth transition from low val-

ues of a variable to high. In a similar fashion, Churkina terms produced surfaces on which the transition from

94

high to low variable values was more abrupt. Using a value into a class of tree presence and all values below

slope term of 1 produced a planar transition. A fth the median into a class of tree absence. This method

deterministic surface was created manually, so that assured that for each map approximately 50% of the

the transition between 0.5 and 0.5 occurred over one ecotone was classi ed as trees and 50% as not trees.

incremental unit. All of the surfaces lacked planform

curvature (i.e., change in slope in the direction paral- Patch statistics

lel to elevation contours) while pro le curvature (i.e.,

FRAGSTATS (McGarigal & Marks 1993) was used

change in slope perpendicular to elevation contours)

to calculate the patch statistics used in the analysis.

was dictated by the parameters of the deterministic

Each of the classi ed ecotone maps was submitted to

function used.

FRAGSTATS and the landscape-level statistics listed in

Patchiness was introduced to the simulation by cre-

Table 1 were calculated. The small size of sample sites

ating perturbation surfaces that were later added to

limited the value patch statistics for characterizing

each of the ve deterministic surfaces through map

pattern because:

algebra. The level of spatial autocorrelation of the

large patches at the site borders result in patch

perturbation surfaces was adjusted to represent vari-

edges that are artifacts and not real landscape

ations in landscape patchiness. First, surfaces contain-

edges,

ing normally distributed spatially random cell values

at 50% trees in each simulated site, the number of

were generated to represent the most patchy condition.

distinct patches is too small, and

Next, positive spatial autocorrelation was added to the

the small number of grid cells (21 by 21) intro-

random surfaces by using a variable-size square aver-

duces cell-based artifacts (e.g., stair-step edges)

aging lter kernel. Larger kernels had a larger smooth-

into the patch-delineation process.

ing effect, and the resulting surface had a greater

degree of spatial autocorrelation and less patchiness.

Nonetheless, we present the analysis of patch-based

Five different levels of patchiness were simulated us-

statistics, bearing these limitations in mind, because

ing the unsmoothed random surfaces, plus surfaces

smoothed with four kernel sizes: 2 2, 3 3, 4 4 they are commonly used in landscape studies and

and 5 5 cells. The random surfaces were adjusted their value in studies of treeline ecotone pattern has

been demonstrated (Baker & Weisberg 1995; Allen &

prior to smoothing, through multiplication by a scalar

Walsh 1996).

value, so that their means and standard deviations after

The area weighted shape statistics (AWMPFD and

smoothing were approximately equal across all pertur-

AWMSI), contagion (CONTAG) and total edge (TE)

bation surfaces. The adjusted surface values were all

in the range [ 0.5, 0.5] after smoothing. Since both were expected to be useful for measuring abrupt-

ness, but not without some sensitivity to patchiness.

the perturbation and deterministic surfaces were in the

range [ 0.5, 0.5] the simulated ecotone surfaces, after Values of AWMPFD and AWMSI should decrease

with abruptness. Patches in gradual ecotones were

adding the surfaces together, had values in the range

[ 1, 1]. expected to form shapes that were more complex be-

cause they are not con ned by a steep gradient and

We used Moran s I, a coef cient of spatial autocor-

tend to spread out on the landscape. As abruptness in-

relation, as an indication of spatial pattern that resulted

creases the complexity of patch shape should decrease

from the simulations and to verify that the groups

as patches are con ned to smaller areas of transition.

displayed differences in spatial pattern before testing

However, as ecotone patchiness increases, the poten-

the landscape statistics on them. Moran s I is positive

tial for patches to form convoluted shapes increases

when positive spatial autocorrelation (i.e., clustering)

also. Therefore, values of AWMPFD and AWMSI

is present in a pattern and negative when negative spa-

were expected to increase with increasing patchiness.

tial autocorrelation (i.e., regularity) is present. It varies

from 1 to +1. A value close to zero suggests that the Higher values of CONTAG were expected as abrupt-

ness increased, while lower values were expected as

pattern is spatially random.

patchiness increased. Since CONTAG measures the

All of the simulated ecotone surfaces were then

degree to which patches of different classes are in-

transformed into binary representations of tree pres-

termixed, or patch interspersion, it should also be

ence and absence to create classi ed maps of the eco-

affected by patchiness. Patch interspersion increases,

tone. The classi cation was accomplished by mapping

and CONTAG decreases, as patchiness increases es-

all surface variable values above the median surface

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Table 1. Patch and boundary statistics evaluated for suitability as abruptness measures. All statistics were hypothesized

to be sensitive to ecotone abruptness.

Patch statistics Abbr. Description

Area-weighted mean patch AWMPFD Average fractal dimension over all patches weighted by area.

fractal dimension Fractal dimension is a measure of the degree of complexity of

planar shapes. A shape with a high fractal dimension is more

plane lling than a shape with a low fractal dimension.

Area-weighted mean shape index AWMSI Average perimeter to area ratio for all patches weighted by area.

Contagion CONTAG Measures both patch interspersion (the intermixing of different

patch types) and patch dispersion (the spatial distribution of a

patch type). Low values of CONTAG are equated with a high

degree of patch interspersion and/or dispersion.

Total edge TE Absolute measure of total edge between all patches.

Boundary Statistics

Number of Boundary BE Count of boundary elements (ROC locations) selected during

Elements lattice delineation

Number of subgraphs N Count of subgraphs, two or more connected boundary elements

Minimum length LMIN Minimum number of boundary elements in any one subgraph

Maximum length LMAX Maximum number of boundary elements in any one subgraph

Mean length LMEAN Average number of boundary elements per subgraph

Subgraph dispersion DISP The average distance of each BE from the centroid of all BE

combined. The distance used is the y distance from the centroid

of all BE to the centroid of each BE.

Cumulative boundary elements CBE Area under curve of BE versus slope threshold level.

pecially when only two classes are present. Total Edge aspect surface represents the compass direction of

(TE) measures the length of the boundaries between change. All locations with gradient values greater

all patches of different classes. TE was expected to than a given threshold were identi ed. These loca-

decrease with increasing abruptness. An abrupt transi- tions, termed boundary elements, became candidates

tion should form patches without complex shapes and to form subgraphs. Subgraphs are created to repre-

therefore generally less edge. As patchiness increases, sent the boundaries on the landscape. Subgraphs were

the number of patches increases and patch size will formed by joining neighboring boundary elements

decrease leading to an increase in the amount of edge (i.e., cells with gradient values above the threshold)

between patches. that were within a minimum angular divergence of

aspect (Jacquez et al. 2000).

Boundary statistics Threshold value selection is subjective. Conven-

tional rules-of-thumb commonly suggest the selection

Lattice delineation was performed using the capabil- of the top 5 or 10% of locations with the highest gra-

ities of ARC/INFO geographic information system dient values as boundary elements (Barbujani et al.

(GIS) and two supplemental programs written in C. 1989; Fortin 1994; Fortin & Drapeau 1995) and the

The work of Fortin (1994) and the alpha version of use of 30 deg as the aspect threshold for boundary

the program BoundarySeer (Jacquez & Maruca 1998), element connection (Barbujani et al. 1989; Jacquez

were used as a template for the ARC/INFO and C et al. 1998). Here, instead of selecting locations of

routines in this study. high rate-of-change until an area threshold is met, the

First, two surfaces related to the spatial change in range of values in each gradient surface was divided

the variable surface were computed for each simulated into 20 equal intervals and the threshold is based on

surface: gradient and aspect. The gradient surface the lower limit of one of the intervals. This method of

represents the magnitude of spatial change while an

96

threshold selection produces what we call slices of the different abruptness characteristics should produce no-

gradient surface, where each slice represents all loca- ticeably different response curves and the integral of

tions whose value is equal to or greater than the lower these curves would provide a numerical means to

limit of a particular interval. Lattice delineation and differentiate them (Figure 4).

subgraph formation was performed for each of the 20

slices. We selected slice level 9, about mid-range, for Analysis of variance

calculating most of our statistics because the average

The calculated statistic values were organized to facil-

number of boundary elements at or above that level

itate a two-way factorial ANOVA experiment (Bhat-

was close to 10%.

tacharyya & Johnson 1977). The ANOVA was used

The numbers of boundary elements (BE) and

to determine whether the statistics were sensitive to

subgraphs (N) and descriptive statistics on subgraph

differences resulting from simulated combinations of

length (LMEAN, LMIN, and LMAX) were calculated

unique patchiness and abruptness levels. The unique

for each ecotone (Table 1). These statistics were all

combinations of abruptness and patchiness are called

thought to have some sensitivity to the level of ecotone

treatments and the quantitative differences in abrupt-

abruptness. As an ecotone becomes more abrupt, the

ness and patchiness themselves are known as treat-

area of transition decreases, which means the num-

ment effects.

ber of locations of high gradient value should also

The main effects of the factorial design were ex-

decrease. For this reason, BE and N at or above any

amined rst to determine whether the statistics were

given level of steepness were expected to decrease

sensitive to abruptness, patchiness, or both. The main

as abruptness increased. Abrupt ecotones should have

effects tested the null hypothesis that the mean statistic

distinct transition areas that result in boundary el-

values observed across factor levels are equal when the

ements with similar aspects. Therefore, boundary

effects of the second factor are disregarded. Rejection

elements of abrupt ecotones should have a higher con-

of the null hypothesis indicates that the statistic was

nectivity, which also translates into fewer subgraphs

capable of detecting differences in either abruptness

(N) per ecotone.

or patchiness.

LMAX, LMEAN and LMIN were expected to be

Interaction occurs when the affect of one factor

useful for measuring abruptness, but also affected

on the dependent variable changes at different levels

by patchiness. They were all expected to increase as

of the other independent variable (Keppel 1991). The

abruptness increased because abrupt transitions should

null hypothesis is that interaction is not present. Rejec-

have similar aspect values and thus be more connected.

tion of the null indicated signi cant interaction. The

As patchiness increases, LMAX, LMEAN and LMIN

presence of interaction did not allow for a simple in-

were expected to decrease because of the negative

terpretation of the main effects. Signi cant interaction

effect patchiness has on the connection of boundary

meant that the statistic being tested did not perform

elements into long subgraphs.

consistently across all levels of one or both of the in-

We developed two new statistics based on the lat-

dependent variables. For example, a statistic may have

tice delineation method for the express purpose of

been more sensitive to abruptness at lower levels of

measuring ecotone abruptness from images. The rst

patchiness than at higher levels of patchiness.

of these, called dispersion (DISP), was developed to

Four component sources of variance contributed to

measure the degree of boundary element clumping in

the factorial experiment: the variance due to abrupt-

the direction of the ecotone transition. DISP measures

2 2

ness treatments ( a ), patchiness treatments ( p ), the

the average distance of each BE from the centroid

2 ), and

interaction of patchiness and abruptness ( p a

of all BE combined. The more clumped boundaries

2

experimental error ( error ). An index, called Omega

produced by abrupt ecotones were expected to form

Squared, was used to compare the relative strength of

boundary element patterns that resulted in low DISP

the relationships between statistic values and abrupt-

values.

ness levels (Keppel 1991):

The second new statistic, the cumulative bound-

ary elements (CBE), uses data from each of the 20

a = a /( a + p + p a + error ).

2 2 2 2 2 2

(2)

slice levels, avoiding the need to choose one speci c

To evaluate the in uence of patchiness on the sta-

gradient threshold. CBE approximates the integral of

2

tistics, we used this same equation, with p in the

the response curve produced by plotting the num-

numerator. Omega squared ranges from 0 to 1, with

ber of BE versus gradient slice level. Ecotones with

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Figure 4. Illustration of the cumulative boundary elements statistic. (A) A surface variable with an ecotone exhibiting low abruptness (level 1);

(B) A surface variable with an ecotone exhibiting high abruptness (level 5); (C) and (D) The slope surfaces corresponding to the surfaces in A

and B. (E) The cumulative boundary element curves for each ecotone.

2

higher values representing stronger statistic sensitivity. tal error ( error ). Therefore, the Omega squared for the

Sensitivity to the differences produced by the simula- single factor ANOVAs is:

tion is directly related to the Omega squared values a = a /( a + error ).

2 2 2 2

they produce.

For each of the single factor ANOVAs, pairwise multi-

By decomposing the factorial ANOVA into 5

ple comparisons between factor levels were conducted

single-factor ANOVA experiments, we analyzed the

using the Bonferroni method (Keppel 1991). Bonfer-

abruptness simple effects of each statistic. Each

roni comparisons tested for signi cant mean differ-

single-factor ANOVA was equivalent to holding the

ence between each pair of abruptness levels. The sign

level of patchiness constant while studying the effects

of the mean differences was used to determine whether

of abruptness. In each of the single factor ANOVAs

the statistic responded to abruptness as expected (i.e.,

there were only two variance components, variance

it was externally consistent) and whether its response

2

due to abruptness ( a ) and variance due to experimen-

was consistent across patchiness levels (i.e., it was

internally consistent). As an example of external con-

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Table 2. Descriptive statistics for the simulated ecotones,

sistency, the number of boundary elements (BE) was averaged by ecotone group. The ecotone values were in the

expected to decrease with increasing abruptness level. range [ 1,1].

This means that subtracting BE at abruptness level

Range Mean Stdv Moran s I

5 from BE at abruptness level 4, A4 A5, should

yield a positive value. External consistency was mea- A1 P1 1.828 0.009 0.369 0.916

sured as the proportion of signi cant comparisons that P2 1.777 0.001 0.369 0.916

produced the hypothesized sign. The hypothesized re- P3 1.859 0.003 0.373 0.892

sponse of a metric with a high external consistency P4 1.816 0.000 0.365 0.877

changed little with factor level. A low external consis- 0.001

P5 1.687 0.361 0.811

tency indicated that the interaction was such that the

A2 P1 1.848 0.002 0.419 0.942

response (sign) of the metric changed as factor level

0.004

P2 1.746 0.408 0.938

varied. Internal consistency summarized the degree

P3 1.838 0.000 0.414 0.922

to which signi cant comparisons were observed in a P4 1.774 0.001 0.406 0.907

logical sequence. A metric is internally consistent if, P5 1.678 0.000 0.403 0.858

when sensitive to a small difference in abruptness, it

A3 P1 1.797 0.003 0.457 0.953

is also sensitive to larger differences in abruptness. In-

P2 1.787 0.000 0.449 0.949

ternal consistency was calculated as the ratio between

P3 1.862 0.000 0.456 0.934

the number of signi cant comparisons and the number

0.001

P4 1.781 0.447 0.923

of signi cant comparisons that would have been de-

P5 1.713 0.001 0.444 0.881

tected if the statistic was responding in a 100 percent

0.006

A4 P1 1.886 0.496 0.953

consistent manner.

0.009

P2 1.827 0.496 0.953

0.001

P3 1.929 0.499 0.941

Statistic suitability ranking

P4 1.848 0.000 0.493 0.933

0.003

P5 1.759 0.490 0.898

Five properties were used to rank the patch and bound-

ary statistics according to their ability to quantify A5 P1 1.851 0.004 0.518 0.956

abruptness. The properties were: 0.002

P2 1.840 0.516 0.955

main effect sensitivity to abruptness (SMEa ); P3 1.935 0.003 0.519 0.942

simple effect sensitivity to abruptness (SSE ); P4 1.878 0.003 0.511 0.935

external consistency (CE ); P5 1.779 0.001 0.508 0.902

internal consistency (CI ); and

main effect sensitivity to patchiness (SMEp ).

All statistic values were standardized using the max-

imum observed values across all statistics. The stan- groups. Figure 3 illustrates the classi ed maps de-

dardized scores for each property ranged from 0 to rived from the same surfaces. Table 2 lists descriptive

1, with 1 representing the highest performance for a statistics for the simulated ecotones, summarized by

property. SMEp was standardized inversely to the high- ecotone group. The average range and mean of values

est score, such that the highest value was assigned a on the surfaces were similar between groups, and the

zero and the lowest a one. For the others, the maxi- surface mean values were very close to zero for all

mum statistic value was assigned a standardized score groups. The average standard deviations were similar

of one. Accordingly, the summary suitability scores within abruptness groups and variable between abrupt-

had a possible range of 0 to 5 with 5 representing the ness groups and within patchiness groups. Because

best overall suitability. we adjusted the standard deviations of the perturba-

tion surfaces during the simulation, the differences in

surface standard deviation between patchiness levels

Results and discussion in the same abruptness group were effectively mini-

mized. However, the differences in surface standard

Simulations deviation between abruptness groups were unavoid-

able. The resulting surfaces had standard deviations

Figure 2 shows one example of a simulated ecotone

that increased slightly with abruptness level. The sur-

surface from each of the 25 abruptness-patchiness

99

faces were suf ciently similar in terms of the numer-

ical distribution of surface values that any observ-

able differences among surfaces from different groups

Table 3. Main effect relationships between statistic values,

should be attributable solely to the spatial pattern of factor levels (abruptness and patchiness), and factor inter-

surface values. actions. 2 is standardized to allow comparison between

Moran s I was sensitive to differences in both statistics for a given factor (i.e., abruptness or patchiness).

All F values are signi cant at p

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