PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11/00 C‘JCIRC/DatoDIanS-DJ4

COMPARING PATCH- AND SURFACE-BASED METRICS OF PATCHINESS
AND ABRUPTNESS ON SIMULATED ECOTONES

By

Mark Allen Bowersox

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Geography

1 999

ABSTRACT

Comparing patch- and surface-based metrics of patchiness and abruptness on
simulated ecotones

By

Mark Allen Bowersox

The use of landscape metrics to infer ecological process at alpine treeline
requires knowledge of metric sensitivity to specific ecotone characteristics. In
this study, a set of patch-based metrics was compared with a set of surface-
based metrics to assess their suitability as patchiness and/or abruptness
quantifiers of simulated ecotone landscapes. A simulation produced 25 groups
of ecotones that vary systematically in their degree of patchiness and
abruptness. Factorial ANOVA was used to evaluate metric sensitivity to the
known differences among the simulated groups. lndices of metric suitability
based on the ANOVA results were developed and used to rank the metrics in
order of their suitability. Metrics from each set were suitable as patchiness and
abruptness quantifiers, but the patch-based metrics were on average more
suitable for both characteristics than the surface-based metrics. Both sets of
metrics were more consistent as abruptness metrics than they were as
patchiness metrics. Specific recommendations of which metrics to use for
patchiness and abruptness were made. The results of this research are
meaningful to applications dealing with ecotones that rely on the interpretation of

patch- or surface-based metrics.

To my parents, Thomas and Karen Bowersox

ACKNOWLEDGMENTS

First and foremost, I would like to thank Dr. Daniel Brown for helping me
develop the topic of this research and for providing continual oversight and
support through its completion. I am glad I had the opportunity to work with him.
In the summer of 1999 Dr. Brown accepted a position in the School of Natural
Resources and Environment at the University of Michigan. Upon his move, Dr.
Judy Olson took over as the chair of my committee but Dr. Brown remained my
primary advisor. I wish to thank Dr. Olson for being so accommodating, for her
constant willingness to brainstorm ideas, and for her guidance in general. Dr.
Randall Schaetzl provided a helpful review of my thesis, but more importantly I
would like to thank him for his honest criticism and encouraging praise. Thank
you to Geoffrey Jacquez and Susan Maruca of BioMedware for the use of their
GEM software and their suggestions on my methodology. My sincere
appreciation goes to my lab mates Scott Drzyzga, Geoffrey Duh, and Brad
Shellito for taking the time to assist me with my day-to-day problems no matter
how busy they were. In addition to those mentioned above, I would also like to
recognize Julie Colby, Paul Rindfleisch, Colleen Garrity and Yushuang Zhou for
their support as friends and colleagues. I am grateful to my friend Seamus
Mooney for providing me with a place to stay and work. Last but by no means
least, I wish to thank Courtney Abeln for being my inspiration and for her ever-

present consolation and patience.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................... VII
LIST OF FIGURES ............................................................................................. IX
LIST OF ABBREVIATIONS ................................................................................ XI
INTRODUCTION .................................................................................................. 1
Alpine treeline ecotones ............................................................................. 1
Patch-based metrics .................................................................................. 3
Surface-based metrics ............................................................................... 6
Objectives and Research Questions .......................................................... 9
METHODOLOGY ................................................................................................ 1 1
Simulation ................................................................................................ 1 1
Calculating patch-based metrics .............................................................. 23
Calculating surface-based metrics ........................................................... 26
Choosing threshold values ............................................................ 28

Subgraph statistics and hypothesized function .............................. 29

Cumulative boundary elements ..................................................... 32

Data summary .......................................................................................... 34
Analysis of Variance ................................................................................. 36

Main effects ................................................................................... 38

Interaction Effects .......................................................................... 38

Simple effects ................................................................................ 4O

Metric suitability ranking ................................................................ 43

RESULTS ........................................................................................................... 45
Patch-based metric ANOVA results ......................................................... 45
Patch-based metrics ranked by simple effects .............................. 52

Patch-based metric suitability ranking ........................................... 54
Surface-based metrics ANOVA results .................................................... 56
Subgraph statistics ........................................................................ 56

Cumulative boundary elements ..................................................... 70
Surface-based metrics ranked by simple effects ........................... 72
Surface-based suitability ranking ................................................... 75

Patch-based suitability vs. Surface-based suitability ................................ 77
DISCUSSION AND CONCLUSIONS .................................................................. 81
Research Questions Revisited ................................................................. 81
Considerations ......................................................................................... 87
Summary .................................................................................................. 89
APPENDIX A ...................................................................................................... 91
Bonferroni Comparisons for patch-based metrics .................................... 91

APPENDIX B .................................................................................................... 100
Bonferroni comparisons for surface-based metrics ................................ 100
APPENDIX C .................................................................................................... 109
Bonferroni Comparisons for the CBE metric .......................................... 109
APPENDIX D .................................................................................................... 112
Source Code .......................................................................................... 1 12
Simdetennc ................................................................................ 1 12

Matrix.aml .................................................................................... 1 15

Gemit.aml .................................................................................... 1 18

Connect2.c .................................................................................. 122

Dispersionc ................................................................................. 129
BIBLIOGRAPHY ............................................................................................... 135

vi

LIST OF TABLES

TABLE 1. MATRIX OF SIMULATION GROUPS. ............................................... 16

TABLE 2. DESCRIPTIVE STATISTICS AND MORAN’S I SPATIAL
AUTOCORRELATION COEFFICIENT FOR THE SIMULATED ECOTONES,
SUMMARIZED BY ECOTONE GROUP ...................................................... 20

TABLE 3. PATCH-BASED METRICS AND HYPOTHESIZED FUNCTION.
ADAPTED FROM MCGARIAGAL AND MARKS (1993). ............................. 24

TABLE 4. SUBGRAPH STATISTICS AND HYPOTHESIZED FUNCTION. ....... 31

TABLE 5. MAIN AND INTERACTION EFFECTS FOR FACTORIAL ANOVA ON
PATCH—BASED METRICS. ......................................................................... 46

TABLE 6. SIMPLE EFFECTS RESULTS FOR ANOVA ON PATCH-BASED
METRICS. MAXIMUM OMEGA SQUARED VALUES ARE IN BOLD,
MINIMUM ARE UNDERLINED. AN “NS" INDICATES AN ANOVA THAT

WAS NOT SIGNIFICANT. ........................................................................... 47
TABLE 7. PATCH-BASED METRICS RANKED BY SINGLE-FACTOR OMEGA

SQUARED VALUES. ................................................................................... 53
TABLE 8. PATCH-BASED METRICS RANKED BY PATCHINESS AND

ABRUPTNESS SUITABILITY SCORES. ..................................................... 55

TABLE 9. MAIN AND INTERACTION EFFECTS RESULTS FOR ANOVA ON
LEVEL 1 SUBGRAPH STATISTICS. SIGNIFICANCE IS FOR or = 0.01. A
“NS" INDICATES THAT THE VARIATION IN THE SOURCE WAS NOT
SIGNIFICANT DIFFERENT. ........................................................................ 60

TABLE 10. MAIN AND INTERACTION EFFECTS RESULTS FOR ANOVA ON
LEVEL 9 SUBGRAPH STATISTICS. SIGNIFICANCE IS FOR or = 0.01. A
“NS” INDICATES THAT THE VARIATION IN THE SOURCE WAS NOT
SIGNIFICANT DIFFERENT. ........................................................................ 64

TABLE 11. OMEGA SQUARED VALUES FOR EACH SINGLE-FACTOR
ANOVA FOR EACH SUBGRAPH STATISTIC. MAXIMUM OMEGA
SQUARED VALUES ARE IN BOLD, MINIMUM ARE UNDERLINED. AN
“NS” INDICATES AN ANOVA THAT WAS NOT SIGNIFICANT. ................. 65

TABLE 12. MAIN AND INTERACTION EFFECTS FOR ANOVA ON THE
CUMULATIVE BOUNDARY ELEMENTS METRIC. AN “NS” INDICATES
THAT THE VARIATION IN THE SOURCE WAS NOT SIGNIFICANT
DIFFERENT ................................................................................................. 71

vii

TABLE 13. SIMPLE EFFECTS RESULTS FOR ANOVA ON THE CUMULATIVE
BOUNDARY ELEMENTS METRIC. THE MAXIMUM OMEGA SQUARED
VALUE IS IN BOLD; THE MINIMUM IS UNDERLINED. AN “NS”
INDICATES AN ANOVA THAT WAS NOT SIGNIFICANT ........................... 71

TABLE 14. SURFACE-BASED METRICS RANKED BY SINGLE-FACTOR
OMEGA SQUARED VALUES ...................................................................... 74

TABLE 15. SURFACE-BASED METRICS RANKED BY PATCHINESS AND
ABRUPTNESS SUITABILITY SCORES. ..................................................... 76

TABLE 16. PATCH- AND SURFACE-BASED METRICS RANKED
ACCORDING TO PATCHINESS SUITABILITY SCORES. PATCH-BASED
METRICS ARE UNDERLINED. ................................................................... 78

TABLE 17. PATCH- AND SURFACE-BASED METRICS RANKED ACCORDING

TO ABRUPTNESS SUITABILITY SCORES. PATCH-BASED METRICS
ARE UNDERLINED. .................................................................................... 79

viii

LIST OF FIGURES

FIGURE 1. FIVE DETERMINISTIC SURFACES USED TO SIMULATE
ABRUPTNESS. CONTROL OVER THE ABRUPTNESS OF EACH
SURFACE WAS ACCOMPLISHED BY ALTERING THE SLOPE
PARAMETER OF THE FUNCTION THE SURFACES WERE MODELED
AFTER. THE BRIGHTNESS OF THE SURFACE SHADING REPRESENTS
THE STRENGTH OF TREE PRESENCE. ................................................... 14

FIGURE 2. FIVE PERTURBATION SURFACES USED TO SIMULATE
PATCHINESS. PATCHINESS WAS CONTROLLED BY ALTERING THE
DEGREE OF SMOOTHING TO WHICH EACH PERTURBATION SURFACE
WAS SUBJECTED. THE BRIGHTNESS OF THE SURFACE SHADING
REPRESENTS THE STRENGTH OF TREE PRESENCE ........................... 15

FIGURE 3. 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. ................................................... 18

FIGURE 4. AN EXAMPLE OF THE SIMULATED ECOTONE SURFACES THAT
HAVE BEEN CLASSIFIED INTO TREE PRESENCE/ABSENCE FROM
EACH OF THE 25 GROUPS. BLACK REPRESENTS TREE PRESENCE,
WHITE REPRESENTS TREE ABSENCE (|.E., TUNDRA SPECIES, ROCK,
AND BARE SOIL). ....................................................................................... 19

FIGURE 5. MEAN MORAN’S I SPATIAL AUTOCORRELATION COEFFICIENT
FOR EACH GROUP OF SIMULATED ECOTONE SURFACES. THE
VARIATION IN SPATIAL AUTOCORRELATION BETWEEN GROUPS WAS
USED AS EVIDENCE THAT THE SIMULATION PRODUCED ECOTONE
SURFACES THAT WERE DIFFERENT WITH REGARD TO SPATIAL
PATTERN. ................................................................................................... 22

FIGURE 6. DEPICTION OF THE CUMULATIVE BOUNDARY ELEMENTS

METRIC FOR AN ECOTONE OF LOW ABRUPTNESS VS. AN ECOTONE
OF HIGH ABRUPTNESS. THE CHART IN THE CENTER ILLUSTRATES
THE DIFFERENCE IN THE BOUNDARY ELEMENT RESPONSE CURVES
OF THE TWO ECOTONES. MAPS OF THE BOUNDARY ELEMENTS FOR
EACH ECOTONE AT 20 ROC SLICE LEVELS ARE SHOWN ABOVE AND
BELOW THE CHART. BOUNDARY ELEMENTS ARE SHOWN IN BLACK.
..................................................................................................................... 35

FIGURE 7. MEAN COEFFICIENT OF VARIATION OF THE 8 SUBGRAPH
STATISTICS BY ROC SLICE LEVEL. THE COEFFICIENT OF VARIATION

ix

SUMMARIZED THE POTENTIAL AMOUNT OF INFORMATION
AVAILABLE THAT COULD BE USED TO DISTINGUISH DIFFERENCES
AMONG THE 1250 ECOTONES. ................................................................ 57

FIGURE 8. AN EXAMPLE FROM EACH OF THE 25 GROUPS OF THE
SUBGRAPH MAPS CREATED FROM THE LATTICE DELINEATION THAT
USED SLICE LEVEL 1 AS THE GRADIENT THRESHOLD. AREAS OF
HOMOGENEOUS COLOR REPRESENT INDIVIDUAL SUBGRAPHS OR
SINGLETONS .............................................................................................. 58

FIGURE 9. AN EXAMPLE FROM EACH OF THE 25 GROUPS OF THE
SUBGRAPH MAPS CREATED FROM THE LATTICE DELINEATION THAT
USED SLICE LEVEL 9 AS THE GRADIENT THRESHOLD. AREAS OF
HOMOGENEOUS COLOR REPRESENT INDIVIDUAL SUBGRAPHS OR
SINGLETONS .............................................................................................. 59

LIST OF ABBREVIATIONS
Simulation
p patchiness
a abruptness
t type (of alpine treeline ecotone; defined by levels of a and p)

Patch-based metrics
AWMPF D area-weighted mean patch fractal dimension

AWMSI area-weighted mean shape index
CONTAG contagion

LPl largest patch index

MPS mean patch size

NP number of patches

PSCV patch size coefficient of variation
TE total edge

Surface-based metn'cs

BEs number of boundary elements
Disp subgraph dispersion

L".ax maximum subgraph length

me mean subgraph length

Lmin minimum subgraph length

Lstdv standard deviation of subgraph length
N number of subgraphs

NS number of singletons

CBE cumulative boundary elements
Suitability score

SME main effect sensitivity

SSE simple effects sensitivity

CE external consistency

C. internal consistency
Miscellaneous

NDVI normalized difference vegetation index
LAI leaf area index

GNP Glacier National Park

TM Thematic Mapper

ROC rate of change

xi

INTRODUCTION

Ecotones are zones of transition between adjacent ecological systems.
They have a set of characteristics uniquely defined by space and time scales and
by the strength of the interactions between adjacent ecological systems (Holland,
1988). Conceptual consideration of ecotones began in the 1930’s. However, it
was not until the 1970’s that ecotones became recognized as dynamic landscape
entities that deserve separate attention (Risser, 1995). Ecotones under the
above definition include what are referred to as edges (Orloci and Orloci, 1990),
transitional areas (Kent et al., 1997) and boundaries (Wiens et al., 1985) but
unlike definitions of these three terms, the definition of an ecotone does not
specify that the transition must be distinct. An ecotone may be distinct or less
obvious and may be highly heterogeneous or less heterogeneous depending on

the ecological processes acting on it.

Alpine treeline ecotones

The alpine treeline ecotone occurs along an increasingly stressful
altitudinal gradient and forms the zone of transition between lower-elevation sub-
alpine forest and high-elevation alpine tundra. There is a threshold within this
ecotone above which the potential for tree species establishment and growth is
zero (Armand, 1985). As the threshold is approached, tree species’ growth form
and patterns of occurrence begin to change. Generally, the stature and
abundance of trees decreases as altitude and environmental stress increases.

At treeline, krummholz, shrub-like formations of tree species, become more

conspicuous and occupy the less environmentally stressful sites often creating a
patchy landscape. Other components of alpine treeline such as tundra species,
rock, and bare soil are found adjacent to krummholz patches in the matrix of the
alpine treeline landscape.

Variation in the spatial arrangement or pattern of the ecotone components
presumably occurs because of differences in ecological processes acting upon
them. In order to decipher which spatial patterns are representative of specific
ecological processes, reliable data on spatial pattern must first be collected.

This research focuses on the measurement of two characteristics of
spatial pattern thought to be representative of specific ecological processes at
treeline: patchiness and abruptness. The impetus for this investigation came
from the early stages of research conducted by Malanson and Brown (1997) at
alpine treeline in Glacier National Park, MT. They required a suitable
methodology to measure ecotone patchiness and abruptness in order to
establish a link between spatial pattern and ecological process. The goal here is
to investigate two methodologies and determine which is the most suitable for
quantifying ecotone patchiness and abruptness.

Ecotone patchiness is defined as spatial heterogeneity or unconformity; an
ecotone is patchy when neighboring areas are frequently dissimilar. Processes
that may be responsible for creating patchy patterns include natural and human
disturbance (Krummel et al., 1987) and differential species establishment due to
an uneven distribution of nutrient-rich sites (Stevens and Fox, 1991) or seed rain

(Malanson, 1997).

Abruptness is a measure of the rate at which one ecological system
changes to another across the ecotone; abrupt alpine treeline ecotones change
rapidly from trees to alpine tundra while gradual ecotones are characterized by
slower transitions. Abrupt ecotones that occur along a gradual environmental
gradient suggest the presence ecological processes such as species competition
(Armand, 1992; Malanson and Butler, 1994), and positive feedback mechanisms
(Wilson and Agnew 1992; Malanson, 1997). These processes prohibit the
species at treeline from exhibiting change consistent with the change in
environmental gradient.

Landscape ecology and the use of landscape metrics provides the
methods necessary to develop the link between pattern and process (Turner,
1989; Fomlan and Godron, 1986). The approaches to developing metrics that
describe landscape pattern have been divided into two categories; a categorical
or patch-based approach and a spatially continuous or surface-based approach
(Gustafson, 1998). This research evaluates each approach with the goal of

selecting the most suitable metrics for quantifying patchiness and abruptness.

Patch-based metrics

The patch-based approach typically involves classifying satellite imagery
or aerial photography to produce a map in which similar types of ecological
communities or vegetation types are grouped together. These groups, referred
to as classes, are represented within a raster map as contiguous clusters of cells

called patches. After classification, patches are considered internally

homogeneous and the boundary between patches of different classes is a
distinct one. Patches from similar and different classes agglomerate over an
area to form a landscape (Forman and Godron, 1986). The programs SPAN
(Turner, 1990), r. le (Baker and Cal, 1992) and FRAGSTATS (McGariagal and
Marks, 1993), which are compatible with geographic information systems,
generate a variety of patch-based metrics that mathematically define the spatial
pattern of a landscape. Patch-based metrics quantify patch density, size, shape
and variability, landscape edge, core area, landscape diversity, contagion and
interspersion (McGariagaI and Marks, 1993). Several of these metrics are
reviewed by Haines-Young and Chopping (1996) for application to forested
landscapes.

Baker and Weisberg (1995), in Rocky Mountain National Park, CO and
Allen and Walsh (1996), in Glacier National Park, MT applied the patch-based
approach to quantify landscape pattern at alpine treeline. Both were able to
discern 6 unique types of alpine treeline ecotone using cluster analysis of patch-
based metric data. Patch-based metrics measuring the number of patches,
patch interspersion, patch richness, and edge density were used by both studies.
Baker and Weisberg (1995) also included an evenness of patch types metric and
several variables describing relative amounts of different land cover types in
each ecotone. Unlike Baker and Weisberg (1995), Allen and Walsh (1996) used
mean landscape fractal dimension as a metric of landscape complexity. Fractal
dimension quantifies the departure of a shape from Euclidean geometry or, in

other words, how well a shape fills a plane (Haines-Young and Chopping, 1996).

Differences in the classification scheme used by each study make the
comparison of ecotone types between studies difficult. However, the studies did
find similar ecotone types. Of particular interest here is that each study found
ecotone types that were best described according to their patchiness and
abruptness characteristics. Baker and Weisberg (1995) found ecotones that
were “long” and “short” with variable amounts of patchiness, while Allen and
Walsh ( 1996) separated patchiness and abruptness into ecotone types that were
labeled “heterogeneous” or “highly heterogeneous” and “moderately zonal” or
“zonal”, respectively. Results from Allen and Walsh suggest that ecotones were
characterized by a certain degree of patchiness or a certain degree of
abruptness, but not both. On the other hand, Baker and Weisberg suggest that
patchiness and abruptness (length of ecotone) were not independent
characteristics of ecotone pattern. This difference between the two studies might
be attributed to regional differences in treeline at the two study areas; or it could
have been a result of misinterpretation due to a lack of information concerning
patch-based metric behavior in response to combinations of ecotone patchiness
and abruptness.

A patch-based approach may not be the most appropriate one when
attempting to quantify ecologically meaningful characteristics of ecotones. The
central problem lies with the classification process necessary to compute patch-
based metrics. Classification reduces the amount of information available by
transforming continuous, locally variable, data into discrete classes. The patch-

based approach represents an ecotone as the boundary line between adjacent

 

patches, which is a line of zero thickness. This is particularly troublesome
because even though most ecotones occur along a continuum they are
represented as discrete elements resulting from a method of classification that is
based on arbitrary ecological differences (Wood and Foody, 1989; Brown, 1998).
At best, only the length of ecotone and the classes that it separates can be
directly quantified. The patch-based approach does not allow ecotone width or
ecological contrast to be calculated because this information is lost to
classification (Johnston and Bonde, 1989). Also important, there is no way to
represent a gradual transition between neighboring patches; all transitions

between patches are represented as abrupt.

Surface-based metrics

The surface-based approach attempts to avoid the classification step and
instead calculates metrics directly on a variable derived from satellite imagery,
aerial photography, or other continuous field sampling approach. It is called a
surface-based approach because the metrics are calculated from the distribution
of a continuous variable over a two-dimensional area. Surfaces possess the
mathematical property of continuity and have only one value at any point.
Surfaces of ecologically relevant variables are frequently generated from satellite
imagery; examples include percent vegetative cover, the Normalized Difference
Vegetation Index (NDVI) and Leaf Area Index (LAI). The ecological surface is

represented as a regularized grid where each cell of the grid contains a unique

data value. In essence, the regularized grid is only an approximation of the real
ecological surface.

There are several surface-based methods available to extract information
on landscape pattern. Legendre and Fortin (1989) provide a thorough review of
methods employing spatial autocorrelation coefficients, correlograms,
variograms, spectral analysis, and the Mantel test to measure spatial pattern.
Kent et al. (1997) reviews similar methods with respect to ecotone analysis.

Among the several surface-based methods available, one group focuses
directly on transitional areas and relies on the concepts of edge detection. The
goal of edge detection is to locate discontinuities along transects (Ludwig and
Cornelius, 1987) or within two-dimensional maps (Johnston et al., 1992) using
algorithms that accentuate areas with high rates of change within a given
variable or set of variables mapped over the landscape. Once detected, spatially
contiguous locations with high rates of change are referred to as edges or
boundaries. The location, width, shape, or distribution of these edges can be
used to characterize the transition.

One method of boundary detection within two-dimensional ecological data
was proposed by Womble (1951) and has since been rejuvenated by Barbujani
et al. (1989), Fortin (1994), and Jacquez and Maruca (1998). The method is
commonly called either Wombling or lattice delineation. The method first
computes a rate of change value for every point on a regularized surface (lattice).
The t0p k percent of the rate of change locations are then selected from the rate

0f change surface, where k is an arbitrary threshold set by the researcher. The

top k percent of rate of change locations are called boundary elements.
Boundary elements are connected to one another based on an aspect threshold,
also set by the researcher, to form subgraphs. Subgraphs represent locations
where the rate of change is high and the direction of change is similar. The
number of subgraphs and singletons (singletons consist of only one boundary
element), as well as subgraph length and diameter statistics are calculated and
used to measure the cohesiveness of the detected boundary.

The use of the subgraph statistics as a surface-based approach has
typically focused on delineating contiguous boundaries that are statistically
significant. Lattice delineation will always find locations whose rate of change is
higher than others which make significance tests necessary to determine if the
boundaries would occur on the basis of chance alone (Fortin, 1994). Subgraph
statistics were developed specifically to determine whether the boundaries
detected in the collected data differed significantly from null models based on
spatial randomness (Oden et al., 1993). While the subgraph statistics have been
found to be effective in determining the significance of detected boundaries
(Fortin 1994; Fortin and Drapeau, 1995), less is known about how the subgraph
statistics can be used to quantify specific boundary or landscape pattern. For
instance, a small number of long subgraphs may indicate that an ecotone is
abrupt, while a large number of shorter subgraphs may indicate the ecotone is
more gradual. The subgraph statistics may also prove to be good indicators of
ecotone patchiness. For example, a large number of singletons would indicate

that the boundaries are not contiguous which may indicate ecotone patchiness.

One concern with the lattice delineation approach is the arbitrary nature of
the rate of change threshold. Researchers have commonly used a threshold of
the top 5 or 10% of the rate of change values; however this may not be optimal
for all applications. A method that uses the lattice delineation approach to obtain
subgraph statistics at multiple threshold levels may avoid the effects due to an
ad-hoc analysis at one threshold level. For example, information on how the
statistics change from one threshold level to another could constitute a unique
metric. One such metric, called cumulative boundary elements, is developed and
presented here. The cumulative boundary elements metric was conceived to
measure ecotone abruptness alone, but it may measure patchiness as well, in
which case it will not be a good abruptness quantifier because its value could

indicate “patchy”, “abrupt”, or both.

Objectives and Research Questions

Applications of both patch- and surface-based metrics measuring the
patchiness and abruptness of alpine treeline ecotones are affected by
uncertainties. The goal of this research is to evaluate the two approaches in an
experiment where patchiness and abruptness characteristics are controlled
through a simulation of alpine treeline ecotones. This approach allows for an
objective comparison between patch- and surface-based methodologies and
should provide evidence of the information content and behavior of the metrics

when applied to alpine treeline and ecotones in general. The experiment also

allows for an examination of the interacting effects of patchiness and abruptness

on each set of metrics.

A series of specific research questions was addressed involving the

capability of the metrics to measure and distinguish differences in ecotone

patchiness and abruptness:

Do patch-based metrics only measure patchiness or are they capable of
measuring abruptness? Which of the patch-based metrics are most
suitable as patchiness metrics and which are most suitable as abruptness
metrics?

Are the surface-based metrics able to measure patchiness? Do the
surface-based metrics measure abruptness? Which of the surface-based
metrics are most suitable as patchiness metrics and which are most
suitable as abruptness metrics?

How does the degree of one ecotone characteristic affect the
measurement of another? Is it possible to measure the degree of
patchiness at multiple degrees of abruptness and visa versa? Is metric
performance dependent on the level of patchiness or abruptness? For
instance, does a patchiness metric lose its ability to measure patchiness
as abruptness increases?

10

METHODOLOGY

The research was conducted in three phases. First, ecotone surfaces with
known patchiness and abruptness characteristics were simulated, classified, and
organized into a matrix structure that would later allow a factorial Analysis of
Variance (ANOVA) experimental design to be used. In the second phase, patch-
and surface-based pattern metrics were calculated on the simulated data. The
third phase included ANOVA and the interpretation of planned comparisons

between groups of simulated ecotones.

Simulation

The simulation was designed to produce values of a hypothetical
continuous variable for each cell in a square grid that mimicked a real world
study area containing an ecotone. Simulated data were used rather than real
world data to control the patchiness and abruptness characteristics of each
ecotone by systematically altering the parameters of the simulation. Real world
data were less advantageous because the relative degree of patchiness or
abruptness between samples would not be known prior to testing the metrics on
them.

The first goal of the simulation was to create ecotone data similar to what
would be obtained from a LANDSAT Thematic Mapper (TM) satellite image.
Inspection of alpine treeline ecotones captured in a TM scene of Glacier National
Park (GNP), MT provided a visual model for the simulations. A majority of the

treelines examined in the TM image had transition lengths less than

11

approximately 600m. Transition length was defined as the distance between
closed canopy forest and open alpine tundra as measured along the profile of
vegetation change. The extent of the areas used in the simulations was set at
630m2 to accommodate the maximum transition lengths observed in the image.
The simulated data set used a cell size of 30m, which corresponds to the TM cell
size. The area for each simulated ecotone was approximately 40 ha which was
comparable to some of the smaller two-dimensional transects used by Baker and
Weisberg (1995). The GNP TM scene was consistently referred to throughout
the development of the simulation to assure at the very least that the simulated
ecotones visually resembled real ecotones.

The second and foremost goal of the simulation was to provide control
over the degree of patchiness and abruptness assigned to each ecotone. For
simulation purposes, abruptness was defined as the rate at which the surface
variable changed from its maximum to its minimum across the zone of transition.
High variable values represented a strong presence of trees while low variable
values represented a strong presence of tundra, bare soil or rock. Abruptness
was controlled using a deterministic function in the form of an “s-shaped” curve
where the slope term of the function was changed to produce variable levels of
abruptness. The deterministic function used was:

y = sign(x) . |x| ("I . 0.5
Where:
y = surface variable value

x = position along transition
n = slope parameter

12

This function was not chosen for any known ecological significance, but
rather for its ability to model a smooth transition from low values of a variable to
high. In a similar fashion, Churkina and Svirezhev (1995) and Timoney et al.
(1993) used sigmoid functions to mathematically model ecotones; however, their
work was done at the biome scale and not a local scale, as is the case here.
Other sigmoid functions were explored for use in this study but none of them
were able to model transitions of varying degrees over a constant x range while
holding the y range constant. The function used here is capable of modeling
transitions of variable abruptness without a change in the minimum and
maximum y values.

Four deterministic functions with slope parameters of 1, 0.5, 0.25 and
0.0625 were extended into two—dimensional maps resulting in four deterministic
surfaces varying in their degree of abruptness. This was accomplished by
mapping the value of the function over the interval x = -1 to x = 1 at 0.1 unit
increments to consecutive cells in each column of the two-dimensional map.
Each of the four deterministic surfaces had a surface value range of -0.5 to 0.5.
Lower magnitude slope terms produced surfaces where the transition from high
to low variable values was less abrupt. In fact, using a slope term of 1 produces
a planar transition. A fifth deterministic surface was created manually, so that the
transition between -0.5 and 0.5 occurred between two cells, which created the
most abrupt surface possible. In effect, it was the deterministic function with the

slope parameter set to zero. All of the surfaces lacked plan curvature while

13

profile curvature was dictated by the parameters of the deterministic function

used. Perspective views of the five deterministic surfaces are shown in Figure 1.

 

‘1 "fist-n.

 

low high

Abruptness

Figure 1. Five deterministic surfaces used to simulate abruptness. Control over
the abruptness of each surface was accomplished by altering the slope
parameter of the function the surfaces were modeled after. The brightness of the
surface shading represents the strength of tree presence.

Patchiness was introduced to the simulation by creating perturbation
surfaces that were later added to the five deterministic surfaces. First, a surface
containing normally distributed spatially random cell values was generated. The
spatial randomness of the cell values produced a surface that lacked spatial
autocorrelation. Positive spatial autocorrelation was then added by passing a
square averaging filter over the random surface. Changing the neighborhood
size of the filter effectively changed the amount of spatial autocorrelation added
to the perturbations. Larger neighborhoods had a larger smoothing effect, which
contributed more spatial autocorrelation. Perturbations with a high degree of
spatial autocorrelation were considered less patchy. The perturbation surface

without smoothing and the surfaces smoothed by neighborhoods of 2x2, 3x3, 4x4

14

and 5x5 cells represented five different levels of patchiness. To assure that
patchiness was the only variant, the perturbation surface means and standard
deviations were adjusted prior to smoothing so that they were approximately
equal for all levels of patchiness after smoothing. The normally distributed
spatially random surfaces were also adjusted so that cell values would range
from —O.5 to 0.5 after smoothing. Examples of the perturbation surfaces for each

of the five levels of patchiness are shown in Figure 2.

 

       

Patchiness

 

Figure 2. Five perturbation surfaces used to simulate patchiness. Patchiness
was controlled by altering the degree of smoothing to which each perturbation
surface was subjected. The brightness of the surface shading represents the

strength of tree presence.

The final surface representation of a simulated ecotone was produced
using simple map algebra. Each of the five perturbation surfaces was combined
with each of the five deterministic surfaces by adding the value of each cell in the
perturbation surface to the corresponding cell of the deterministic surface. In

doing so, 25 unique simulated ecotone surfaces were created. Since both the

15

perturbation and deterministic surfaces ranged from -0.5 to 0.5 the final
simulated ecotone surfaces ranged from approximately -1 to 1.

All of the simulated ecotone surfaces were then transformed into binary
representations of tree presence and absence to create the classified maps of
alpine treeline ecotone. The classification was accomplished by mapping all
surface variable values above the median surface value into a class of tree
presence and all values below the median into a class of tree absence. This
method assured that for each map approximately 50% of the ecotone was
classified as trees and 50% as not trees.

The simulation produced 2 sets of maps, a surface set and a classified
set, each representing 25 types, tap, of alpine treeline ecotones. The 25 types
resulted from unique combinations of 5 levels of patchiness and 5 levels of

abruptness. The simulated ecotone types are summarized in Table 1.

Table 1. Matrix of simulation groups.

 

 

 

 

 

 

 

Abruptness
A1 A2 A3 A4 A5
P1 111 I12 113 114 115
P2 121 122 I23 I24 I25
Patchiness P3 t3, t32 t33 t34 t35
P4 141 I42 I43 I44 145
P5 151 152 Isa 154 155

 

 

 

 

 

 

 

 

 

 

16

The subscript p denotes patchiness levels where increasing p corresponds to
increasing patchiness and subscript a denotes increasing abruptness levels. For
example, ecotones in tn were the least patchy and least abrupt while ecotones in
t55 were the most patchy and most abrupt. The simulation was repeated 50 times
for each type (tap) in order to obtain replicates for statistical analysis.

Accordingly, the 50 simulations produced 1250 randomized maps. For the
remainder of this text, the 50 ecotones that were simulated to be of the same
patchiness and abruptness level are said to belong to the same “group”.

Figure 3 contains one example of a simulated ecotone surface from each
of the 25 groups simulated. Figure 4 contains the classified maps derived from
the same surfaces. Although each ecotone is just one example of the patterns
created by the simulations, there are observable differences between types.
Changes in the degree of abruptness are most noticeable among the lower levels
of abruptness; abruptness levels 4 and 5 are not as separable. It appears that
differences in the degree of patchiness are easier to distinguish at lower levels
abruptness.

Table 2 contains descriptive statistics and Moran’s l spatial autocorrelation
coefficients for the simulated ecotones, summarized by ecotone group. As
planned, the mean range and surface mean values were similar between groups
and the group surface means were very close to zero. The group mean surface
standard deviation was similar within abruptness groups and variable between

abruptness groups. It was variable within patchiness groups. Adjusting the

17

 

Simulated Ecotone Surfaces

Abruptness level (A)

N

 

 

i 7 ~ ‘- -_ V l I, .
id 143-; :8 in be:

Patchiness level (P)

 

Note: The ecotone transition was simulated in the north-south direction.

 

Figure 3. 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.

18

Cla

Patchiness level (P)

Blacl

Note;

EQUIP r
3358315:

Psprefelq

 

 

Classified Simulated Ecotone Surfaces

Abruptness level (A)
1 2 3 4 5

1HH_
any i:
saw -H
4wuuuu
wuuu

Black = Tree presence White = tree absence

 

 

 

 

 

   

 

 

 

 

 

 

 

Patchiness level (P)

 

 

 

 

Note: The ecotone transition was simulated in the north-south direction.

 

Figure 4. An example of the simulated ecotone surfaces that have been
classified 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).

19

Table 2. Descriptive statistics and Moran’s l spatial autocorrelation coefficient for
the simulated ecotones, summarized by ecotone group

 

 

 

 

 

 

 

 

 

 

 

 

Range Mean Stdv Moran’s I
P1 1.828 0.009 0.369 0.916
P2 1.777 0.001 0.369 0.916
A1 P; 1.859 0.003 0.373 0.892
P4 1.816 0.000 0.365 0.877
P; 1.687 -0.001 0.361 0.811
P1 1.848 0.002 0.419 0.942
P2 1 .746 -0.004 0.408 0.938
A; P; 1.838 0.000 0.414 0.922
P4 1.774 0.001 0.406 0.907
P; 1 .678 0.000 0.403 0.858
P1 1.797 0.003 0.457 0.953
P; 1.787 0.000 0.449 0.949
A3 P3 1.862 0.000 0.456 0.934
P4 1.781 -0.001 0.447 0.923
P; 1.713 0.001 0.444 0.881
P1 1 .886 -0.006 0.496 0.953
P; 1.827 -0.009 0.496 0.953
A4 P3 1.929 -0.001 0.499 0.941
P. 1.848 0.000 0.493 0.933
P; 1.759 -0.003 0.490 0.898
P1 1.851 0.004 0.518 0.956
P; 1.840 -0.002 0.516 0.955
A; P; 1.935 0.003 0.519 0.942
P4 1.878 0.003 0.511 0.935
P; 1.779 0.001 0.508 0.902

 

20

 

standard deviations of the perturbation surfaces during the simulation effectively
minimized the difference in surface standard deviation between patchiness levels
in the same abruptness group. However, the difference in surface standard
deviation between abruptness groups was unavoidable. In order to create
different degrees of abruptness for the same value range, the deterministic
surfaces had to have variable standard deviations. As a result, the combination
of deterministic surfaces with variable surface standard deviations with the
perturbation surface with similar surface standard deviations produced ecotones
whose surface standard deviation increases slightly with abruptness level. The
simulation produced surfaces that were sufficiently similar in terms of the
numerical distribution of surface values that any observable differences among
surfaces from different groups should be attributable solely to the spatial pattern
of surface values.

The Moran’s l spatial autocorrelation coefficient was used as a preliminary
measure of spatial pattern to make sure the groups displayed differences in
spatial pattern before testing the metrics on them. Moran’s I was first thought of
as a baseline index of surface patchiness but it also seemed to be affected by
differences in abruptness. Figure 5 graphically illustrates the relationship of
Moran’s l with patchiness and abruptness. As was expected, Moran’s l
decreased with increasing patchiness. Moran’s I also decreased with increasing
abruptness. At each level of abruptness, differences in Moran’s l between
patchiness levels 1 and 2 were small as were differences between abruptness

levels 4 and 5 at each level of patchiness. The variability in Moran’s l reveals that

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J_-___ - ____.. __ __ __ __ _ __ u
096 9 8
0.94 ———————-LL ——————fl— —— __— §_.___ R , _____ II pl
0 92 ‘W— ‘6— __ —— h — ——————D— —_ __L_ ___4 L°V°|
- O 0 Cl 01
E 0.90 ~~—~ -—— fl — — ___ _ -3 -2 ., ___ _ g_-__ _ m
g 0.88 -__z--_-_ — — —— - e - _e._. .. _,- 6 z-.- _-A__..___ - A3
X4
0.86 ——-—-— ““4, , w —477_4A4_._ _--_ __._____2
U 05
0.84 A#
0.82 g-e — — __z __, A _
o
0.80 . ,
0 1 2 3 4 5 6

Patchiness level

 

Figure 5. Mean Moran’s I spatial autocorrelation coefficient for each group of
simulated ecotone surfaces. The variation in spatial autocorrelation between
groups was used as evidence that the simulation produced ecotone surfaces that
were different with regard to spatial pattern.

there were differences in the spatial pattern among groups. However, the
differences did not appear to be linear across all levels of patchiness or across all
levels of abruptness (i.e., differences between level 1 and 2 were not the same
as differences between levels 4 and 5). In fact, differences between patchiness
levels increased with patchiness level while differences between abruptness
levels decreased with abruptness level. This phenomenon is also apparent in

the examples shown in Figures 3 and 4. Non-linearity in the simulated
differences between patchiness and abruptness levels requires a careful
interpretation of metric sensitivity. Since the differences between abruptness

levels 4 and 5 are small, metrics capable of detecting the differences might be

22

considered more sensitive to abruptness than metrics that could only distinguish
differences between levels 3 and 4. The same applies to metric sensitivity to
patchiness since the differences between patchiness levels 1 and 2 appear to be
minute. The non-linearity in the simulation provides an additional test for the
metrics. Metrics truly measuring patchiness or abruptness should produce
difference magnitudes that are smallest between patchiness levels 1 and 2 and

abruptness levels 4 and 5.

Calculating patch-based metrics

FRAGSTATS (McGariagal and Marks, 1993) was used to calculate the
patch-based metrics used in the analysis. The program calculates metrics at the
landscape, class, and patch levels of aggregation. Landscape level metrics
quantify characteristics of the entire area of interest, class level metrics quantify
characteristics of each class in the landscape, and patch metrics are generated
for every patch of each class. Each of the classified ecotone maps was
submitted to FRAGSTATS and all possible landscape, class, and patch metrics
were calculated. Later, the patch and class level metrics were excluded and
metrics for final analysis were chosen from the landscape level only.
Table 3 contains a list and brief description of the patch-based metrics evaluated.
This particular set of metrics was selected because each was thought to have
some utility for measuring either patchiness or abruptness. As a patch-based
approach, all metrics calculated by F RAGSTATS relate to some degree to the

patchiness or, as the name implies, fragmentation of the landscape. It might

23

Table 3. Patch-based metrics and hypothesized function. Adapted from
McGariagal and Marks (1993).

 

 

 

 

 

 

 

 

 

 

 

 

Metric Description Hypothesized function
Area- Average fractal dimension over Patchiness and
weighted all patches weighted by area. abruptness metric
mean patch Fractal dimension is a measure
fractal of the degree of complexity of
dimension planar shapes. A shape with a
(AWMPFD) high fractal dimension is more
plane filling than a shape with a
low fractal dimension.
Area- Average perimeter to area ratio Patchiness and
weighted for all patches weighted by area. abruptness metric
mean shape
index
(AWMSI)
Contagion Measures both patch Patchiness and
(CONTAG) interspersion (the interrnixing of Abruptness metric
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.
Largest patch Percentage of landscape area Patchiness metric
index (LPI) comprised by the largest patch.
Patch size Measures the relative variability Patchiness metric
coefficient of of patch size about the mean
variation patch size; variability as a
(PSC\Q percentage of the mean.
Mean patch Average area of all patches in Patchiness metric
size (MPS) the ecotone.
Number of A count of all patches in the Patchiness metric
atches @P) ecotone.
Total Edge Absolute measure of total edge Patchiness and
(TE) between all patches. Abruptness metric

 

24

 

be difficult to find patch-based metrics that exclusively measure patchiness or
abruptness. Unfortunately, the suite of available patch-based metrics lacks a
metric specifically designed to measure abruptness but surrogates may exist.
Four of the metrics chosen for use, LPI, MPS, NP and PSCV, were designated
as the primary patchiness metrics and were thought to have little utility for
measuring abruptness. These metrics were suspected to be the best indicators
of patchiness because they rely on the relationship between the number of
patches determined by classification and the fixed size of the ecotone landscape.
As patchiness increases, the number of patches (NP) should increase and mean
patch size (MPS) along with the largest patch index (LPI) should decrease.
PSCV should increase with patchiness as the chance for patches of a variety of
sizes should also increase with the number of patches.

The remaining patch-based metrics, the area weighted shape metrics
(AWMPFD and AWMSI), contagion (CONTAG) and total edge (TE), were
expected to be equally useful for measuring patchiness and abruptness. Values
of AWMPFD and AWMSI should decrease with abruptness. Patches in gradual
ecotones should form more complex shapes because they are not confined by a
steep gradient and are able to spread out on the landscape. As abruptness
increases the complexity of patch shape should decrease as patches are
confined to smaller areas of transition. As ecotone patchiness increases, the
potential for patches to form convoluted shapes increases. Therefore, values of
AWMPFD and AWMSI were expected to increase with increasing patchiness.

Since CONTAG is supposed to measure the degree to which patches of different

25

classes are intermixed, or patch interspersion, it should be sensitive to
patchiness. Patch interspersion increases as patchiness increases especially
when only two classes are present. CONTAG also measures patch dispersion,
in other words, the degree to which patches of the same class are separated.
Patch dispersion would be low for an abrupt ecotone because the zone of
transition is short causing patches to be clumped together. Low values of
CONTAG were expected as patchiness increased, while high values were
expected as abruptness increased. Total Edge (TE) measures the length of the
boundaries between all patches of different classes. TE was expected to
increase with increasing patchiness and decrease with increasing abruptness.
As patchiness increases, the number of patches increases and patch size will
decrease leading to an increase in the amount of edge between patches. This is
akin to the surface area of fine grain material being greater than that of a coarse
grain material. TE was expected to decrease with increases in abruptness. An
abrupt transition should form patches without complex shapes and therefore

generally less edge.

Calculating surface-based metrics

Lattice delineation and the collection of subgraph statistics were
performed using the capabilities of ARC/INFO geographic information system
and two supplemental programs written in C. The work of Fortin (1994) and the
alpha version of the program GEM: Geographic Boundary Analysis Software

(Jacquez and Maruca, 1998), were used as a template for the ARC/INFO and C

26

routines in this study. Although GEM is fully functional and adaptable to the file
structure of the simulated ecotones, it was not used due to the lack of an efficient
method for batch processing. To accommodate processing in ARC/INFO the
methods used here differ slightly from those used in GEM and by others
(Barbujani et al., 1989; Fortin, 1994).

Lattice delineation of each ecotone required that a rate-of—change surface
be computed for all variables under investigation. Only one variable was
investigated so only one rate-of-change surface was generated for each ecotone.
A rate-of-change (ROC) or slope surface is a vector surface (as opposed to a
scalar surface) consisting of two components: gradient and aspect (Chrisman,
1997). Gradient defines the magnitude of change while aspect equals the
direction of change. Each of these components was used in the lattice
delineation. All locations on the ROC surface with gradient values greater than a
given threshold were selected (discussed in detail below). These locations,
termed boundary elements, became candidates to form subgraphs, contiguous
areas of high ROC. Subgraphs are formed through the application of a threshold
on the aspect component of the boundary elements. The boundary elements
whose aspects differed less than the threshold were connected to form
subgraphs. This rule was imposed in order to prevent areas of rapid, unrelated

variation from being classified as boundaries (Jacquez et al., 1999).

27

Choosing threshold values

When performing lattice delineation, threshold value selection is
subjective. However, conventional values have been used. The rules most
commonly used in lattice delineation result in the selection of the top 5 or 10% of
gradient values as boundary elements (Barbujani et al., 1989; Fortin 1994; Fortin
and Drapeau, 1995) and use 30 degrees as the aspect threshold for boundary
element connection (Barbujani et al., 1989; Jacquez et al., 1998). Under these
rules, the number of boundary elements selected is roughly equal to the
threshold percentage (i.e., 5%) of the total number of ROC locations. The 30-
degree aspect threshold was used here but a different approach to selecting
boundary elements was chosen. Instead of selecting locations of high rate-of-
change until an area threshold is met, each ROC surface was divided into 20
equal intervals based on the range of ROC values. Each interval could contain a
different number of ROC locations, but the value of each ROC location was
within the same range of ROC values. After forming the intervals, the ROC
locations in each interval were combined to form “slices”. A slice was produced
by selecting all ROC locations whose value is greater than or equal to the lower
limit of a specific interval. For example, the first slice would contain all locations
whose ROC values were greater than or equal to the maximum ROC minus the
range of ROC values divided by 20 (e.g., gradient thresholdsnoe number = ROCmax —
[slice number * (ROCrange * .05)]). Subgraphs were formed for each of the 20

slices. This approach was taken in order to develop the cumulative boundary

28

elements metric (discussed below) and to explore a less arbitrary approach to
selecting a gradient threshold at which to calculate subgraph statistics.

Avoiding the use of an arbitrary gradient threshold level was accomplished
by selecting the ROC slice that provided the most information content among all
of the subgraph statistics. Information content was judged by the variation
present within the subgraph statistics at each of the slice levels; a large variation
indicated a potentially large amount of information. Here it was assumed that
variability was the result of the unique response of each subgraph statistic to the
simulated differences among ecotone types and not experimental error. The
coefficient of variation (CV = standard deviation * 100 I mean ) was used as
relative measure of variation within the subgraph statistics. The CV of each
subgraph statistic was calculated over the 1250 realizations that make up each
ROC slice. Then the mean CV of the 8 subgraph statistics was computed for
each ROC slice. The ROC slice that produced the maximum mean CV was

chosen for use in the calculation of subgraph statistics.

Subgraph statistics and hypothesized function

Statistics were generated for each subgraph formed during lattice
delineation. The number of subgraphs and singletons, descriptive statistics on
subgraph length and a subgraph dispersion metric were calculated for each
ecotone. The length of a subgraph is equal to the number of boundary elements

it contains. The dispersion metric summarized the mean distance of each

29

boundary element from the geographic center of all boundary elements. A list
and description of the subgraph statistics is provided in Table 4.

The surface-based metrics were generally thought to be more capable of
measuring abruptness than patchiness. However, certain surface-based metrics
were expected to be at least somewhat sensitive to ecotone patchiness. The
extent to which these surface-based metrics are capable of measuring ecotone
patchiness depends in part on how many boundary elements were included in
the calculation of the subgraph statistics. Subgraph statistics calculated on too
few boundary elements may not adequately represent the patchiness
characteristics of the entire ecotone landscape but rather just the patchiness of
the local edges the boundary elements represent. The number of singletons
(NS) and the standard deviation of subgraph length (Lstdv) were hypothesized as
patchiness metrics among the surface-based set. NS was expected to increase
as patchiness increased because patchiness would tend to promote locations of
rapid, unrelated change that would not be connected during lattice delineation.
Lstdv was expected to decrease as patchiness increased. As patchiness
increases, the probability of forming long subgraphs decreases and short
subgraphs of similar lengths should form.

The number of boundary elements (BEs), the number of subgraphs (N)
and the boundary element dispersion metric were hypothesized primarily as
abruptness metrics. As an ecotone becomes more abrupt, the area of transition
decreases, which means the number of high ROC locations also decreases. For

this reason, the number of boundary elements (BEs) along with the number of

30

Table 4. Subgraph statistics and hypothesized function.

 

Submph statistic

Description

Hypothesized function

 

Number of
Boundary Elements
(BEs)

Count of boundary elements
(ROC locations) selected
duringlattice delineation

Abruptness metric

 

 

boundary elements

Number of Count of subgraphs, two or Abruptness metric
subgraphs (N) more connected boundary

elements
Number of Count of boundary elements Patchiness metric
singletons (NS) not connected to any other

 

Minimum length

(Lmln)

Minimum number of
boundary elements in any
one subgraph

Patchiness and
abruptness metric

 

Maximum length

(Lmax)

Maximum number of
boundary elements in any
one subgraph

Patchiness and
abruptness metric

 

Mean length (Lmean)

Average number of
boundary elements per
subgraph

Patchiness and
abruptness metric

 

Standard deviation
of length (Lstdv)

Standard deviation of
sugqraph length

Patchiness metric

 

Subgraph
dispersion (Disp)

 

 

The average distance of
each BE from the centroid
of all BEs combined. The
distance used is the y
distance from the centroid
of all BEs to the centroid of
each BE.

 

Abruptness metric

 

31

 

subgraphs (N) was expected to decrease as abruptness increased. Subgraphs
form by connecting boundary elements of similar aspects. Abrupt ecotones
should have distinct transition areas that result in boundary elements with similar
aspects. Therefore, boundary elements of abrupt ecotones should have a higher
connectivity, which translates into fewer subgraphs (N) per ecotone. The
boundary element dispersion metric (Disp) was developed to measure the
degree of boundary element clumping in the direction of the ecotone transition.
The more contiguous boundaries produced by abrupt ecotones were expected to
form boundary element patterns that were clumped together resulting in a low
Disp value. Gradual ecotones were hypothesized to have higher values of Disp
since boundary elements would tend to be more dispersed throughout the
ecotone.

The remaining subgraph statistics, Lmax, Lmean and Lmin were proposed to
be useful for measuring both patchiness and abruptness. As patchiness
increases, Lmax, Lmean and Lmin are expected to decrease because of the negative
effect patchiness has on the connection of boundary elements into long
subgraphs. Lmax, Lmean and Lmin are expected to increase as abruptness
increases because abrupt transitions should have similar aspect values and thus

be more connectable.
Cumulative boundary elements

Cumulative boundary elements, an alternative surface-based metric, was

developed and calculated for each ecotone surface. Cumulative boundary

32

elements uses data from each of the 20 slice levels avoiding the need to choose
one specific gradient threshold at which to conduct the analysis. This metric was
called cumulative boundary elements (CBE) because it was calculated as the
total sum of boundary elements over all threshold levels. CBE roughly
approximates the integral of the response curve produced by plotting the number
of BEs verses ROC slice level. It was thought that ecotones with different
abruptness characteristics would produce noticeably different response curves
and that the integral of these curves would provide a numerical means to
differentiate them.

It is easier to conceptualize the differences in response curves for
progressively more abrupt ecotones than for progressively patchy ecotones. The
area under the response curve for an abrupt ecotone should be less than the
area under the curve depicting a more gradual ecotone. When the transition is
confined to a small geographic range, as is the case with an abrupt ecotone, the
rate-of-change values are very high within the area of transition. When the rate-
of-change surface for an abrupt ecotone is sliced into equal intervals, each
successive slice will contain a few more boundary elements until all of the area of
transition is sliced through. The transition of a more gradual ecotone occurs over
a broader geographic range and produces a rate-of-change surface with a lower,
more equally distributed, range of values. When a gradual rate-of-change
surface is sliced into equal intervals, each successive slice contains
proportionately more boundary elements than would each successive slice of an

abrupt rate-of-change surface. It is in this way that the response curves for

33

gradual and abrupt ecotones differ. CBE was the metric used to quantify the
differences in response curves and was hypothesized to be a prime metric of
abruptness.

Figure 6 shows the boundary elements of a gradual (i.e., less abrupt) and
an abrupt ecotone for each of 20 ROC slice levels. Also shown is the boundary
element response curve for both ecotones. Notice that the area under the abrupt
ecotone’s response curve is noticeably smaller than the area under the less
abrupt ecotone’s response curve. The less abrupt ecotone had a CBE value of

3049 verses a CBE of 1835 for the abrupt ecotone.

Data summary

The calculation of the patch- and surface-based metrics produced a
substantial amount of data. The entire set of data can be summarized using the
idea of a data cube. The dimensions of the patch-based data cube were: 8
patch-based metrics x 25 ecotone groups x 50 replicates per group. The data for
the surface-based metrics includes those for the 8 subgraph statistics calculated
at a particular ROC slice and those for the cumulative boundary elements metric
The subgraph statistics form a cube with the dimensions: 8 surface-based
metrics x 25 ecotone groups x 50 replicates per group. The CBE data were

collected for each of the 50 replicates of the 25 ecotone groups.

34

200.: :. 850:0 00 08080.0 E0ccaom

80:0 05 30.0: 0:0 0>on0 850:0 0.0 0.05. 02.0 00m cm 00 0:280 :000 .00 08080.0 200:8: 05 Co 0005. 00:280
02: 0:: Co 0028 00:800. 8080.0 E0055: 0:“ :_ 00:00.20 0:. 02080:... .0800 05 :_ t0:o 0: ._. 00050350

:9: .0 0:980 :0 .0> 00050200 26. B 0:980 :0 :8 0508 08080.0 E0050: 020.388 0:: Co 8.8.000 .0 0.50.“.

 

9.

cm

NF mw m.. 3. Mr N.. I. or m w h w

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

002:0 .0600. 0:280 00050350 30..

0.6.8.0 m e 0
000.920.9320.:S 0 0 n 0 0 v n N.
4 i. . . + ..-. 1......ITT. 1.1.- +IIHII.1\LIIL o
.. i l.\I\lI\I\. E l l I 1
$6.00. 0:280 00050350 :9... \lm\ _
8
V N _.
00050350 :9: +_ 8.
. ‘ I I
.. om. #
\ m...
.\ _ W
_ s... n
\ u
\ 00050350 26. . may
8 2 \ .80
W
I l _ a
.. 08 w
a
. w
. 80 s
mw t. e. .
4 09.
.
H

 

 

00:800. 8080.0 E0058
8.. _m>m_ Qo=m
N. F F 2 0 0 n 0 0 v m N F
I I I 1 I E E U I .
II . . I II .1 I. .- I a a .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

35

Analysis of Variance

The metric data were organized to facilitate a pair of two-way factorial
ANOVA experiments (see Bhattacharyya and Johnson, 1977); one for the patch-
based approach and another for the surface-based approach. The two-way
ANOVA design was chosen because it simultaneously considers the effects of
two factors (independent variables), including the effects due to their interaction,
on a dependent variable. The ecotone characteristics, patchiness and
abruptness, were considered as factors, while each of the patch- and surface-
based metrics were treated as dependent variables.

Since a simulation was used to produce definite differences between the
ecotone groups, the ANOVA results reflect the degree to which each metric was
capable of detecting the simulated differences. In other words, the ANOVA
statistically determined whether the metrics were sensitive to differences
resulting from simulated combinations of unique patchiness and abruptness
levels. The unique combinations of patchiness and abruptness are called
treatments and the quantitative differences in patchiness and abruptness
themselves are known as treatment effects. Since all of the metrics were
subjected to the same simulated differences in patchiness and abruptness, those
that produced the largest treatment effects were said to be the most sensitive.

The ANOVA determines the presence and magnitude of treatment effects
through the comparison of two independent estimates of population variance.
One estimate is obtained from the variance between subjects given the same

treatments; it is called the within group variance. The other estimate, known as

36

the between group variance, is obtained from the variance measured among
subjects that were treated differently. Since it is calculated on subjects that were
treated alike, the within groups variance is attributed to unsystematic errors
beyond the control of the experimenter. The variation between groups on the
other hand, is the result of unsystematic error along with an added component
due to treatment effects (Sokal and Rohlf, 1995). The ratio of the two estimates
(between group variance / within group variance), referred to as the F-ratio, is
used to test whether the treatment effects are larger than what would be
expected by chance alone.

Keppel (1991) summarizes the three important pieces of information that

can be gleaned from a factorial ANOVA regarding treatment effects:

First, we have the simple effects, which refer to the results of the
component single-factor experiments making up the factorial design.
These effects reflect treatment effects associated with one of the
independent variables, with the other one held constant. Second, we
have interaction effects, which reflect a comparison of the simple effects.
Interaction is present when the component single-factor experiments
produce different results; interaction is absent when the results are the
same. Finally, we have the main effects, which essentially transform the
factorial design into two single-factor experiments. Main effects are of
primary interest in the absence of interaction, when it is safe to combine

the data from the component experiments. (p. 191-192)

37

Main effects

The main effects of the factorial design were examined first to determine
whether the metrics were sensitive to one or both of the factors. The main
effects tested the null hypothesis that the mean metric values observed at each
level of one factor are equal when the effects of the second factor are
disregarded. Rejection of the null hypothesis based on the F-ratio provided by
the ANOVA indicated that the mean metric value under study was significantly
different for at least two of the factor levels. For each metric being tested there

was a null hypothesis for each main effect:

Ho: there are no differences between the mean values of the metric at
each level of patchiness.
Ho: there are no differences between the mean values of the metric at

each level of abruptness.

Failure to reject the null hypothesis indicated that the metric being analyzed was
not sensitive to the treatments effects produced by that factor. When the null
hypothesis was rejected, the metric was determined capable of detecting

differences in either patchiness or abruptness.

Interaction Effects

Interaction occurs when the affect of one factor on the dependent variable

changes at different levels of the other independent variable (Keppel, 1991). The

38

factorial ANOVA provided an F-ratio to test for significance of interaction. The

null hypothesis is:

Ho: interaction is not present; the mean value observed at each level of

one factor does not depend on the level of the other factor

Rejection of the null indicated significant interaction. The presence of interaction
did not allow for an easy interpretation of the main effects. Significant interaction
meant that the metric being tested did not perform consistently across all levels
of one or both of the independent variables. For example, a metric may have
been more sensitive to abruptness at lower levels of patchiness than at higher
patchiness levels. lnforrnation elucidating such patterns was not available in the
main effects. However, examination of the simple effects (discussed below)
provided a means to uncover the patterns of interaction.

In the two-way factorial ANOVAs used here there were four component
sources of variance contributing to the experiment. They were: the variance due
to patchiness treatments (02p), abruptness treatments (0%), the interaction of
patchiness and abruptness (02,, x a) and experimental error (02%,). Estimates of
the four component variances were calculated using the average variance
estimates from the ANOVA and were used to compute a relative index of
treatment magnitude for each component. The index used is called Omega
Squared and was chosen because it provided a way to standardize the

measures of treatment effects so they could be compared within and between

39

experiments (Keppel, 1991 ). For example, the treatment effect observed for a
patch-based metric could be compared with the treatment effect observed for a
surface-based metric. Omega squared was calculated as the ratio of the
treatnent effect variance estimate to the combined variance estimates of all
sources. For example, the Omega squared calculation for the patchiness main

effects was:

2

00 p = czp/(ozp + 02

2 2
a+ 0' pxa T 0' error)

Omega squared ranges from 0 to 1 with higher values representing larger
treatment effects (Keppel, 1991). Metrics that produced high Omega squared
values were interpreted as being more sensitive to differences produced by the

simulation than those with lower Omega squared values.

Simple effects

If one of the metrics produced significant interaction effects in the factorial
ANOVA, the simple effects of the metric were analyzed. This was accomplished
by decomposing the factorial ANOVA into 10 single-factor ANOVA experiments.
Each single-factor ANOVA was equivalent to holding the level of one factor
constant while studying the effects of the other factor. The simple effects of
patchiness were evaluated at each level of abruptness as well as the simple
effects of abruptness at each level of patchiness. In the single factor ANOVAs

there were only two variance components, variance due to the independent

40

variable (i.e., either patchiness or abruptness) and variance due to experimental
error. Therefore, the Omega squared for the single factor ANOVAs is equal to
the treatment effect variance estimate divided by the treatment effect variance
estimate plus the error variance estimate (Keppel, 1991). For an ANOVA set up

to examine patchiness simple effects the Omega squared calculation was:

2 _. 2 2 2
wp-Gp/(Up+Oerror)

Again, metrics exhibiting higher Omega squared values were considered more
sensitive to the differences in the factor being examined.

For each of the single factor ANOVAs, pain/vise multiple comparisons
between factor levels were conducted using the Bonferroni method. The
Bonferroni comparisons determine when the mean metric value between two
levels of the same factor is significantly different. There were 10 pairwise
comparisons for each of the simple effects, equal to the number of unique pairs
that can be taken from 5 levels of one factor. The calculation of pairwise
comparisons provided the magnitude and sign of the mean difference between
each pair of factor levels. The sign of the difference provided evidence of
whether the metric was performing the way it was suspected to perform. For
example, the number of patches metric (NP) was expected to increase with
patchiness level which means the subtraction of NP at patchiness level 5 from
NP at patchiness level 4, P4 — P5, would be negative. A magnitude difference

between a comparison made at one factor level and the same comparison made

41

at a different factor level also indicates the presence of interaction. The
magnitude differences were interpreted to determine at which levels of one factor
was the metric most capable of detecting differences in the other factor. This
information was examined at each level of patchiness and abruptness to resolve
the patterns of interaction.

The Bonferroni comparisons were also used to evaluate whether the
metrics were behaving consistently with the known differences in patchiness and
abruptness. Two different aspects of consistency were considered. The first,
termed external consistency, summarized the degree to which the hypothesized
response (sign of the mean difference) of the metric changed as factor level
changed. The hypothesized response of a metric with a high external
consistency changed little with factor level. A low external consistency indicated
that the interaction was such that the response (sign) of the metric changed as
factor level varied. External consistency was calculated as the ratio of the
number of significant comparisons that produced the hypothesized sign (which
indicated the metric was responding as expected) to the total number of
significant comparisons. This ratio was referred to as the external consistency
index.

The second aspect of consistency, termed internal consistency,
summarized the degree to which significant comparisons were observed in a
logical sequence. For example, a metric is said to be highly sensitive to
patchiness if one observes high Omega squared values for the patchiness simple

effects. Assume that further examination determines that the metric is capable of

42

detecting the difference between patchiness levels 1 and 2 but not between
patchiness levels 1 and 5. If a metric is internally consistent and it detected a
small difference in patchiness, it should also detect a larger difference in
patchiness. Of course, the fact that a metric detects large differences in an
ecotone characteristic does not guarantee it will detect small ones. Internal
consistency was calculated as the ratio between the number of significant
comparisons and the number of significant comparisons that would have been
detected if the metric was responding in a 100 percent logical manner. This ratio

was referred to as the internal consistency index.

Metric suitability ranking

Four properties were used to rank the patch- and surface-based metrics
according to their ability to quantify patchiness and abruptness. The properties
were: 1.) main effect sensitivity (SME); 2.) simple effect sensitivity (SSE); 3.)
external consistency (CE); and 4.) internal consistency (0.). Each metric received
a standardized score for each property and the sum of the four scores was used

for the final ranking. The score for main effect sensitivity was calculated as:

_ 2 2
SME-O) Immax

where 02......x was the Omega squared value of the metric with the highest main

effect sensitivity and 02 was the Omega squared value for the metric for which

43

the score was being calculated. The score for simple effect sensitivity was

calculated as:

SSE: Z (032i /(02i max)
.lfl

I'I

 

where n equals the number of simple effects (5) and 02. and 02. max were the
Omega squared value for a particular metric and the maximum Omega squared
value for each simple effect, respectively. The external consistency score was

calculated as:

CE = external consistency / maximum external consistency

and the internal consistency score was calculated as:

C. = internal consistency / maximum internal consistency

The scores for each property ranged from 0 to 1 with 1 representing the highest
performance for a property. Accordingly, the metrics’ total scores ranged from 0
to 4 with 4 representing the best overall suitability. This method was used to
produce a ranking for each set of metrics as well as a ranking for the two sets

combined.

44

RESULTS

Patch-based metric ANOVA results

The factorial ANOVAs for the patch-based metrics are summarized in
Table 5. The F-ratios, Omega squared values, and p-values at the 0.01
significance level are given for each metric for the patchiness and abruptness
main effects and the interaction effect. It is immediately apparent that each
patch-based metric was at least somewhat sensitive to the simulated differences
in patchiness and abruptness. Also, as expected, certain metrics were more
sensitive to the simulated differences than others. However, significant main
effects in Table 5 are not entirely conclusive of metric behavior because
interaction effects were significant for each metric. The simple effects of each
metric were evaluated to provide a better description of metric sensitivity at
different factor levels. The Omega squared values for each of the component
single-factor ANOVAs (i.e., simple effects) are summarized in Table 6. The
Bonferroni comparisons for each of the single factor ANOVAs are provided in

Appendix P.

AWMPFD and AWMSI
Not surprisingly, the two patch shape metrics produced very similar
results. Both AWMPF D and AWMSI were proposed as useful metrics for
patchiness or abruptness but each performed better as an abruptness metric. In
fact, they were the top two metrics most sensitive to abruptness (Table 5). Their

sensitivity to patchiness was among the worst of all the patch-based metrics.

45

Table 5. Main and interaction effects for factorial ANOVA on patch-based

 

 

 

 

metrics.

Source Metric F 0’ P

Patchiness LPI 84.023 0.170 0.000
NP 378.237 0.128 0.000
PSCV 114.323 0.067 0.000
TE 202.863 0.063 0.000
CONTAG 197.974 0.061 0.000
AWMPFD 112.687 0.046 0.000
AWMSI 114.468 0.045 0.000
MPS 47.757 0.037 0.000

Abruptness AWMPFD 1983.536 0.815 0.000
AWMSI 2023.419 0.806 0.000
CONTAG 2537.143 0.790 0.000
TE 2427.515 0.758 0.000
MPS 846.495 0.671 0.000
PSCV 1099.728 0.647 0.000
NP 1443.752 0.490 0.000
LPI 9.504 0.017 0.000

Interaction NP 203.467 0.275 0.000
LPI 22.047 0.172 0.000
PSCV 44.151 0.102 0.000
TE 65.916 0.081 0.000
CONTAG 41.778 0.051 0.000
MPS 14.684 0.043 0.000
AWMSI 16.505 0.025 0.000
AWMPFD 7.604 0.011 0.000

 

 

 

 

 

 

46

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

000.0 005.0 005.0 500.0 050.0 500.0 005.0 _. ..5.0 an.
000.0 000.0 500.0 000.0 000.0 505.0 000.0 000.0 a.
050.0 550.0 050.0 0.0.0 0:0de 000.0 550.0 000.0 an.
050.0 000.0 ..00.0 000.0 500.0 000.0 050.0 000.0 «a
.003 .4010 two .300 000.0 500.0 00.010 500.0 .0
wk >000 02 005. .n... 9:200 .0522 9.0.2.540
0000.00 0.08.0 000500340
050.0 0:500.0 000.0 000.0 000.0 050.0 00 F .0 000.0 0<
£40 8% 0:0000 8% 0:M0Io.|0. 000.0 000.0 000.0 J
050.0 000.0 000.0 500.0 000.0 050.0 550.0 050.0 3
500.0 005.0 00.20 005.0 000.0 000.0 @310 500.0 N<
000.0 000.0 000.0 93.0 0? ...0 550.0 00 ...0 0.. ...0 P.0
m... >000 02 00.... .n... 0.0.0200 5.2.5.0 CHEES<

 

 

 

 

 

 

 

 

 

 

£00.00 0.08.0 0008:0000

0:005:90 .0: 00.5 00:. <>Oz< :0 00.00.05 .0:.. :0. 00:80.8: 0.0 8088.8
.20: :. 0.0 00:.0> 000000 00080 838.05. 00.008 0000000000 :0 <>Oz< .0. 00.000. 0.0000 0.08.0 .0 0.00...

47

The interaction effects for AWMPF D and AWMSI were the weakest.
AWMPFD and AWMSI also exhibited the same interaction patterns. Table 6
indicates that the patchiness sensitivity of AWMPFD and AWMSI was highest at
abruptness level 1, reached a low at abruptness level 2 then increased with
subsequent patchiness levels. The abruptness sensitivity of AWMPFD and
AWMSI steadily increased with patchiness.

The relationships of AWMPFD and AWMSI to patchiness and abruptness
were externally consistent with hypotheses. Each was directly related to
patchiness and inversely related to abruptness. Also, AWMPF D and AWMSI
appear to be internally consistent. The Bonferroni comparisons (Appendix P)
indicate AWMPF D and AWMSI were not significantly different when patchiness
was one level apart. At intermediate abruptness levels, AWMPFD and AWMSI
did not differentiate high levels of patchiness. AWMPFD and AWMSI were not
sensitive to differences between intermediate and high levels of abruptness at

low levels of patchiness but this problem improved at higher levels of patchiness.

CONTAG and TE
CONTAG and TE also produced comparable results. Like AWMPF D and
AWMSI, CONTAG and TE were proposed as potential patchiness and
abruptness metrics and each was sensitive to changes in both characteristics.
CONTAG performed better as an abruptness metric while TE performed equally
well as a metric of both characteristics. CONTAG ranked third and TE ranked

fourth among the patch-based metrics in terms of abruptness main effects

48

sensitivity. T8 was the fourth most sensitive metric to patchiness main effects
and CONTAG the fifth.

The interaction effects for CONTAG and T8 were moderate and both
exhibited the same pattern of changing patchiness and abruptness sensitivity
over factor levels. Their patchiness sensitivity decreased as abruptness level
increased while abruptness sensitivity increased as patchiness increased.

As hypothesized, CONTAG was inversely related to patchiness and
directly related to abruptness. TE’s response was also consistent with
hypotheses as it was directly related to patchiness and inversely related to
abruptness. CONTAG and T8 were internally consistent as patchiness metrics
except at abruptness level 3. CONTAG was not internally consistent because it
was unable to detect differences between patchiness levels 2 and 4, and
between patchiness levels 3 and 5, when it successfully found differences
between levels 2 and 3 or 3 and 4. TE failed to detect patchiness differences 2
levels apart when it was successful at detecting patchiness levels 1 level apart.
The response of both metrics to abruptness was internally consistent at all levels
of patchiness. CONTAG and TE did not distinguish small and intermediate
differences In patchiness as abruptness increased. CONTAG did not detect
differences between intermediate and high levels of abruptness at low levels of
patchiness but did improve at higher levels of patchiness. TE did not detect
differences among intermediate and high level abruptness at all patchiness levels

except for patchiness level 3.

49

LPI, NP and PSCV

LPI, NP, and PSCV were hypothesized to perform best as patchiness
metrics. They were the top three metrics in regard to patchiness main effects
sensitivity but were also the patch-based metrics with the strongest interaction
effects.

The effects of interaction on LPI, NP and PSCV were very obvious; they
were highly sensitive to patchiness at low abruptness levels but sensitivity
decreased as abruptness increased. In fact, patchiness sensitivity decreases so
dramatically that LPI, NP and PSCV were not significantly related to patchiness
levels at abruptness level 4 and were nearly so at abruptness level 5 (Table 6).
These metrics perform very well as patchiness metrics but only on ecotones with
a low degree of abruptness.

LPI, NP and PSCV responded as hypothesized to changes in patchiness.
LPI decreased as patchiness increased while NP and PSCV increased as
patchiness increased. LPI responded internally consistently to differences in
patchiness at all levels of abruptness. LPI failed to detect one and two level
patchiness differences at both low and high levels of abruptness. NP behaved
internally consistently to patchiness at abruptness levels 1, 2 and 5 but not at
level 3. PSCV behaved internally consistently to patchiness at abruptness levels
1 and 2 but not at level 3. At abruptness level 3, both NP and PSCV were not
internally consistent because they were not sensitive to differences between
patchiness levels 1 and 4, 2 and 4 or 4 and 5. However, they were sensitive to

differences between patchiness levels 1 and 3, 2 and 3 and 3 and 4. PSCV and

50

NP were capable of detecting 1, 2, 3 and 4 level patchiness differences at low

levels of abruptness but failed to do so as abruptness increased.

MPS

MPS, although similar in many respects to LPI, NP, and PSCV, performed
poorly as a patchiness metric but had weaker interaction effects than LPI, NP
and PSCV. MPS was sensitive to differences in both patchiness and abruptness.
It was the least sensitive to the patchiness main effects and was moderately
sensitive to the abruptness main effects (Table 5). The poor performance of
MPS as a patchiness metric was surprising, especially since the other patch size
metrics (LPI and PSCV) performed well (at least at low abruptness levels). The
poor performance of MPS as a patchiness metric could be attributed to the
vagueness of information provided by a mean. The mean patch size is a good
representative of ecotone characteristics only if patch sizes are normally
distributed. Othenrvise, outliers, extremely large or small patch sizes, may bias
the value of MPS. A bias of this type may prevent MPS from distinguishing
patchiness differences.

MPS had the third weakest interaction effect. AWMSI and AWMPF D were
the only patch-based metrics with weaker interaction effects. The interaction was
such that the patchiness sensitivity of MPS decreased as abruptness increased
(Table 6). At abruptness level 4, MPS did not detect patchiness differences.

This is the same pattern exhibited by LPI, NP and PSCV but the difference in

patchiness sensitivities between abruptness levels is smaller for MPS.

51

As hypothesized, MPS decreased as patchiness increased. MPS
responded sensibly to patchiness differences at abruptness levels 1, 2 and 5 but
not at abruptness level 3. Just like NP and PSCV, MPS failed to find differences
between patchiness levels 1 and 4, 2 and 4 and 3 and 5 when differences
between 1 and 3, 2 and 3 and 3 and 4 were significantly different. MPS was able
to differentiate patchiness 1, 2, 3 and 4 levels apart at low abruptness levels but

failed to do so as abruptness increased.

Patch-based metrics ranked by simple effects

Table 7 shows the patch-based metrics ranked by Omega squared for
each of the patchiness and abruptness simple effects. The ranking clearly
indicates that relative metric performance was highly dependent on factor level.
For example, NP was the superior patchiness metric at low levels of abruptness
but it was outperformed at intermediate and high levels of abruptness. The same
is true for PSCV. Oppositely, performance of AWMPFD and AWMSI as
patchiness metrics improved as abruptness level increased. CONTAG, AWMSI,
AWMPF D, and TE are the superior abruptness metrics across all levels of
patchiness. MPS and LPI consistently perform as poor abruptness metrics
regardless of patchiness level.

There were larger differences in the metric rank order for the patchiness

simple effects than for the abruptness simple effects. This suggests that the

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

050.0 5.. 000.0 5.. 000.0 5.. 500.0 5.. 000.0 5..
500.0 05.2 000.0 05.2 050.0 05.2 000.0 05.2 500.0 05.2
005.0 >005 000.0 >005 5 50.0 >005 00.0 52 2.0.0 >005
35.0 Q55.2>>< 500.0 52 050.0 52 000.0 >005 50.0 52
005.0 52 000.0 .w.2>>< 000.0 n.55.2>>< 050.0 m... 000.0 05
005.0 .0230. 00.0 055.234. 050.0 0... 050.0 .0234. 000.0 .0230.
000.0 m... 000.0 0... 550.0 .w.2>>< 000.0 o..5.2>>< 500 .0 05.200
F000 0.05200 505.0 0.0.8200 000.0 05.200 000.0 9.9200 500.0 0.55—2.50.
«8 0.302 Nwe 0.302 NS 0.302 «6 0.302 we 0.302
0Q 00..— .1
800.50 0.58.0 000.35 :.n<
500.0 >005 000.0 5.. 000.0 >005 500.0 Q55.2>>< 0.. 50 055.230.
000.0 05.2 000.0 05.2 500.0 05.2 000.0 .w.2>>< 05 5.0 5..
000.0 52 000.0 >005 000.0 52 000.0 5.. 005.0 .0230.
000.0 5.. 000.0 52 000.0 5.. 500.0 0... 505.0 05.2
050.0 0.. 000.0 0... 050.0 0... 000.0 9:200 550.0 00.8200
050.0 05.200 000.0 05.200 050.0 0552>>< 00 v .0 05.2 000.0 0...
000.0 0552>>< 000.0 055.22? 050.0 O<5200 00 .. .0 >005 000.0 >005
00 P .0 .0230. 000.0 .0.2>>< 550.0 .w2>>< 00 F .0 52 000.0 52
as 0.302 «a 0.302 «a 0.302 «a 0.302 «e 0.302
.< z 2 .< .<
030030 0.58.0 0005:0305

.00:_0> 00.0000 00080 530000.050 >0 002:0. 00.308 000098305 .5 .030...

53

decision to use a particular metric to measure patchiness at a given abruptness
level is more consequential than the decision to use a particular metric for

abruptness at a given patchiness level.

Patch-based metn'c suitability ranking

Table 8 contains the scores used to calculate the metric suitability ranking
for the patch-based metrics. Standardized scores for main effects sensitivity,
simple effects sensitivity, external and internal consistency were totaled and
provided the basis to judge the metrics’ overall suitability as patchiness and/or
abruptness metrics. The size of a score is relative to the highest score recorded
for the property that was measured. A metric that performed the best for each of
the properties would receive a score of 4. A score of less than 4 indicated that at
least one property of the metric was inferior to the others.

The patch-based metrics in order of their suitability as patchiness metrics
were: LPI, NP, AWMSI, CONTAG, TE, AWMPFD, PSCV and MPS. Of the
metrics that were predicted to be most useful as patchiness metrics, only LPI and
TE were in the top half of the suitability ranking. PSCV and MPS received scores
that placed them in the bottom half of the ranking. MPS had the lowest score for
each of the four properties contributing to the final score. PSCV, while not the
least sensitive to patchiness, ranked low because of its low consistency scores.
The suitability scores of CONTAG, TE and AWMPFD were very close and they

ranked as mediocre patchiness metrics. It should be noted that only three of the

54

Table 8. Patch-based metrics ranked by patchiness and abruptness suitability

 

 

 

 

 

 

 

 

 

 

 

 

scores.
Patchiness suitability scores
Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency
LPI 1.00 0.29 1.00 1.00 3.29
NP 0.75 0.51 0.91 0.85 3.01
AWMSI 0.27 0.68 1.00 1.00 2.94
CONTAG 0.36 0.63 0.97 0.94 2.90
TE 0.37 0.61 0.97 0.94 2.89
AWMPFD 0.27 0.61 1.00 1.00 2.88
PSCV 0.39 0.39 0.95 0.84 2.58
MPS 0.22 0.26 0.90 0.84 2.23
Abruptness suitability scores
Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency

CONTAG 0.97 0.99 1.00 1.00 3.96
AWMPFD 1.00 0.94 1.00 1.00 3.94
AWMSI 0.99 0.93 1.00 0.98 3.90
TE 0.93 0.96 1.00 1.00 3.89
PSCV 0.79 0.62 1.00 1.00 3.42
MPS 0.82 0.58 1.00 1.00 3.40
NP 0.60 0.65 1.00 1.00 3.25
LPI 0.02 0.15 0.43 0.95 1.55

 

 

 

 

 

 

55

 

 

patch-based metrics, LPI, AWMSI, and AWMPF D had perfect external and
internal consistency scores.

The patch-based metrics in order of their suitability as abruptness metrics
were: CONTAG, AWMPFD, AWMSI, TE, PSCV, MPS, NP and LPI. This order
was convincing because the metrics that were hypothesized as poor abruptness
metrics (LPI, MPS, NP and PSCV) ranked lower in terms of their abruptness
metric suitability. The patch-based metrics were far more consistent as
abruptness metrics than as patchiness metrics. LPI was an exception due to its

very low external consistency.

Surface-based metrics ANOVA results

Subgraph statistics

The results from two ANOVA experiments, one that used the subgraph
statistics produced at slice level 1 and another that used subgraph statistics from
slice level 9, were evaluated. The decision to use slice level 1 came from the
interpretation of the mean metric coefficient of variation at each of the 20 slice
levels (Figure 7). Slice 1 had the highest mean metric coefficient of variation and
was chosen because it was thought have produced the most information to
distinguish differences among ecotone groups. The decision to use slice 9 was
based on the traditional method of selecting boundary elements discussed
previously. Slice level 9 was chosen because the mean number of boundary

elements (43.5) for the 1250 ecotones was the closest of all the slices to 10

56

 

160 ----—--——— _, ___z — ~ —~* ——-—--——— ————— —— *1

 

14o - -— _- — -_ —~——~~wrw — -- 3* --__.,_,.. _-- , 0—0

120

 

 

100 -,

so * - ., .__----
00.-....0 \
.0...-_______-_

20 L-.- - ._ __-.._-_--, -- - ..--- - - ---__¥0

 

 

CV

 

 

 

f—T ffi f
v- C") LO N O) 1- 0') IO N O)
‘- ‘— 1- ‘- ‘-
Sficelevel

 

Figure 7. Mean coefficient of variation of the 8 subgraph statistics by ROC slice
level. The coefficient of variation summarized the potential amount of information
available that could be used to distinguish differences among the 1250 ecotones.

percent of the total ecotone area. The 10 percent value is a typical value used
during lattice delineation.

Figure 8 shows a subgraph map from each of the 25 simulated ecotone
groups when slice level 1 was used for lattice delineation. Figure 9 shows a
subgraph map from each of the 25 simulated ecotone groups when slice level 9

was used for lattice delineation.

SLICE LEVEL 1
The main effects of the slice level 1 subgraph statistics are summarized in

Table 9. The most noticeable property of the subgraph statistics produced by

57

 

Sllce 1 Subgraph maps

Abruptness level (A)

 

 

 

 

 

 

 

 

 

1 2 3 4 5
I

1 . II III- saw-rut
A 2 I I mar» 00
$
5
>
2
§ 3 '- - ' J ' "
.E
5
.9.
m t
O.

4 I n a.

5 . I h I II n

 

 

 

 

 

 

 

 

 

 

 

Note: The ecotone transition was simulated in the north-south direction.

 

Figure 8. An example from each of the 25 groups of the subgraph maps created
from the lattice delineation that used slice level 1 as the gradient threshold.
Areas of homogeneous color represent individual subgraphs or singletons.

58

 

Slice 9 Subgraph Maps

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Abruptness level (A)
3 4 5
1 J‘r—tfi: hz_rum.u.. . ‘ : a IIMtMfinnmfi-u 2.1
I

2 I ;. ‘ + Ila-1.55:! fightmzuuuq "".'.-.:.;a='.::-=.:.'-j
A I
[L
v
'5
> I -
.92
g _
0 3 “a"- “3...... [I *
.8
.C
2
(U
D.

4 rim_fi:fl m man-mm

5 r- 1.. .1... v...-- -—-

 

 

 

 

 

 

 

 

 

 

 

Note: The ecotone transition was simulated in the north-south direction.

 

Figure 9. An example from each of the 25 groups of the subgraph maps created
from the lattice delineation that used slice level 9 as the gradient threshold.
Areas of homogeneous color represent individual subgraphs or singletons.

59

Table 9. Main and interaction effects results for ANOVA on level 1 subgraph
statistics. Significance is for a = 0.01. A “ns” indicates that the variation in the

source was not significant different.

 

 

 

 

 

 

Source Metric F (.32 P
Patchiness Lmean 12.242 0.067 0.000
Lmax 1 1.837 0.065 0.000
Lmin 1 1.291 0.062 0.000
NS 23.704 0.062 0.000
N 20.832 0.051 0.000
BEs 17.107 0.046 0.000
Disp 0.522 -0.001 0.719 ns
L5,...V 0.794 -0.001 0.530 ns
Abruptness N 66.582 0.165 0.000
Disp 62.110 0.164 0.000
NS 40.766 0.106 0.000
883 32.382 0.087 0.000
Lmax 2.876 0.011 0.022 ns
Lmean 2.224 0.007 0.065 ns
Lstdv 1.531 0.003 0.191 ns
Lmjin 1.438 0.003 0.220 ns
Interaction Disp 1.129 0.001 0.322 ns
BEs 1.112 0.001 0.338 ns
NS 0.988 0.000 0.467 ns
N 0.971 0.000 0.487 ns
Lmin 0.944 -0.001 0.518 ns
Lstdv 0.934 -0.002 0.530 ns
Lam.n 0.810 -0.004 0.675 ns
Lmax 0.698 -0.007 0.797 ns

 

 

 

 

60

 

slice level 1 is the lack of significant interaction effects. The response of each
slice 1 metric in regard to one factor was not dependent on the level of the other
factor. This is important because it means that the metric response to ecotone
characteristics, patchiness or abruptness, was consistent over all degrees of
each characteristic. The lack of interaction also allows for a less extensive
investigation of the metrics; the simple effects do not have to be evaluated to
uncover patterns of interaction. Instead, the main effects are wholly
representative of the component ANOVAs that make up the simple effects.
available for detecting differences between the 1250 simulated ecotone surfaces
in each slice level.

Disp and Ls...V did not produce significantly different values among
patchiness groups while Lmax, Lmean, Lstdv, and Lm... were not significantly different
among abruptness groups. Insignificant differences in the Disp metric within the
patchiness main effects was consistent with hypotheses because it was not
designed to measure patchiness, however insignificance in Lam was surprising.
The insignificance of Lm... within the abruptness main effects was predictable
because every set of subgraphs produced by slice 1 had a minimum subgraph
length (Lmin) of 2. Therefore the mean differences between the ecotone groups
would be zero and insignificant. L...ax and Lmean, were not hypothesized to be
insignificant within the abruptness main effects. It is possible that the surprising
insignificance of Lam in the patchiness main effects and Lmax and Lmean in the
abruptness main effects was tied to the fact that slice level 1 produced metric

values that were based on a very small proportion of the ecotone area. The

61

average number of BEs for slice level 1 was 2.32 or 0.5% of the total ecotone
area. The small area used by slice level 1 did not allow the metrics to differ
enough to produce significant differences in the ANOVA when in fact the
ecotones were different in terms of their characteristics. It was for this reason
that metrics from slice level 9 were chosen and further evaluation of slice level 1

was abandoned.

SLICE LEVEL 9

There are four notable differences between the subgraph statistics
produced using slice 1 and those produced using slice 9. First, the average
percentage of ecotone area selected as boundary elements is much greater for
slice 9: 9.9 vs. 0.5 for slice 1. Second, all of the slice 9 metrics showed
significant differences in the patchiness and abruptness main effects (Table 10).
Third, the slice 9 metrics were much more sensitive (higher omega squared
values) to abruptness differences than were the metrics from slice 1. Note
however that patchiness sensitivity of the two slices was quite similar. The fourth
difference is that each slice 9 metric had significant interaction effects whereas
the slice 1 metrics did not. The first three differences were advantages of using
slice 9; using a larger area should make the metrics more accountable for the
simulated differences and finding the metrics with a high sensitivity was a primary
goal. Unfortunately, the significant interaction effects with slice 9 are a
disadvantage since another goal was to find patchiness and abruptness metrics

that operate consistently across factor levels. Despite the presence of

62

interaction, the slice 9 metrics were chosen over the metrics calculated at slice
level 1 for comparison against the patch-based metrics.

The factorial ANOVAs for the slice 9 subgraph statistics indicate that all of
the metrics were at least somewhat sensitive to both patchiness and abruptness.
Table 10 shows the F-ratios, Omega squared values, and p-values of the
factorial ANOVA for each metric and each factor. While the sensitivity of the
metrics was variable among patchiness and abruptness main effects, sensitivity
to patchiness was more variable than sensitivity to abruptness. Interaction of
factors was significant for each of the subgraph statistics, requiring interpretation
of the simple effects to reveal specific patterns of metric behavior at different
factor levels. Table 11 contains the results of the surface-based simple effects.
NS exhibited the strongest interaction, the interaction effects of Lstdv and Lmam
were the next strongest while Lmin, Lmean, Disp, 888, and N exhibited weaker
interaction. Appendix S contains the results of the Bonferroni comparisons that

were used to further examine the interaction effects.

NS and L9,...V
NS and Lstdv were introduced as patchiness metrics. NS was the
metric most sensitive among the surface-based metrics to the patchiness main
effects, Lstdv was the fourth most sensitive (Table 10). Both metrics were also
sensitive to abruptness but ranked among the lower half of the metrics in that
regard. Lstdv was the least sensitive of all the surface-based metrics to

abruptness. The interaction effects for NS and L9,...v were very strong and the

63

Table 10. Main and interaction effects results for ANOVA on level 9 subgraph
statistics. Significance is for a = 0.01. A “ns” indicates that the variation in the
source was not significant different.

 

 

 

 

 

Source Metric F (92 P
Patchiness NS 166.576 0.103 0.000
L"...Ix 70.049 0.101 0.000
Lmean 37.256 0.035 0.000
Lstdv 1 1.159 0.024 0.000
Lmin 19.344 0.022 0.000 W
N 15.503 0.013 0.000 t
BEs 7.867 0.010 0.000
Disp 28.801 0.009 0.000
Abruptness Disp 2772.977 0.880 0.000
N 826.595 0.710 0.000 _
Lmean 708.935 0.661 0.000
Lmin 522.168 0.602 0.000 J
NS 891.857 0.544 0.000
BEs 380.321 0.538 0.000
Lmax 293.554 0.420 0.000
[.5th 73.251 0.170 0.000
Interaction NS 66.955 0.161 0.000
Lstdv 8.644 0.072 0.000
Lmax 6.069 0.029 0.000
Lmin 4.216 0.015 0.000
Lmean 4.239 0.012 0.000
Disp 10.096 0.012 0.000
BEs 2.712 0.010 0.000
N 3.642 0.009 0.000

 

 

 

 

 

 

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.00 000.0 20.0 E~0 50.0 000.0 «55.0 3.0 m.
.000 00040 .000 03.0 00. N. 0.. 000.0 0500 00.0 .5
«00 0.5.0 0000 $00 0000 00.0 0000 0&0 an.
0000 050.0 000.0 ~80 03.0 000.0 .0010. 5000 «5
000.0 0.000 000.0 0000 0000 000.0 0000 00~0 .5
s} z 02 5a... 5...... x2... 00.0 000
0300.00 0.58.0 00055332

300 0000 «.0 N00 :00 0810 m: 0.010 m: 0000 m: «.000... ...<
0:03 500 R00 000 000 50.0 m: 0000 0.. 00.0 2
«00 000.0 000.0 0.00 0000 0.0.0 500 m: 0000 m0
300 000.0 8.0 m: .0000 0. .0 0. .0 0000 :00 N0..
300 m: 0.000 500.0 m: 0000 3.0.0 3.00 w: 0000 m: 0000 F<
,2... z 02 5..... $2.... 5..... 00.0 000

 

 

 

 

 

 

 

 

 

 

0:005:00 .o: 003 .05 <>02< :0 00.00.08 0:. :< .00:__.00:: 0.0 8388.8 .060 :_ 0.0 00:_0>
00.0300 00080 82800.2 00000.0 8.0.0030 :000 .o. <>02< 830000.080 :000 .o. 0029 000300 00080 ..... 0.00...

0000.00 0.58.0 00082005

65

interaction effects of NS were twice as strong as the interaction effects of Lstdv.

As abruptness increased, the patchiness sensitivity of NS and Lstdv
decreased (T able 11). Interaction was so strong that NS failed to detect
differences in patchiness at abruptness level 5 even though it was the most
sensitive to patchiness at abruptness levels 1, 2 and 3 and the second most
sensitive at abruptness level 4. Lstdv found significant differences in patchiness at
all levels of abruptness.

NS was hypothesized to increase as patchiness increased. This
relationship was confirmed by examining the Bonferroni comparisons for NS
(Appendix B). The difference was negative when the mean NS of a higher
patchiness level was subtracted from the mean NS of a lower patchiness level.
There were three comparisons where this did not occur. The comparisons
between patchiness levels 3 and 5 and 4 and 5 at abruptness level 3 were
positive when 5 was subtracted from 3 and 4 respectively. This indicates that NS
was lower at patchiness level 5 when it was expected to be higher. Since this did
not occur at every level of abruptness, it must be attributed either to an
interaction effect or an unusual error. Also, other comparisons between
patchiness level 5 and lower patchiness levels were not significant at higher
levels of abruptness. In other words, this metric cannot make distinctions at
higher levels of abruptness. NS was externally consistent with regard to
patchiness differences at abruptness levels 1 and 2 but not at abruptness levels
3, 4 and 5. NS detected differences between patchiness levels 1, 2, 3 and 4

levels apart at abruptness levels 1 and 2 but not at 4 and 5. At abruptness level

66

3, NS differentiated patchiness differences of 1, 2 and 3 levels but did not
differentiate a 4 level difference in patchiness.

The response of Lstdv exhibited interaction. Lstdv was hypothesized to
decrease as patchiness increased. Lstdv met hypotheses at low levels of
abruptness but at high levels of abruptness Lsmv increased with patchiness. Also,
L9,...v responded internally consistently to patchiness at low levels of abruptness
but not at high abruptness levels. L3...v detected 1, 2, 3 and 4 level differences in

patchiness at abruptness levels 1 and 2 but not at abruptness levels 3, 4 and 5.

88s, Disp and N

BEs, Disp and N were hypothesized to be most useful as abruptness
metrics. Each was sensitive to abruptness as well as patchiness, but they
excelled as abruptness metrics. Disp and N ranked as the top two subgraph
statistics in terms of sensitivity to the abruptness main effects while 883 ranked
sixth (Table 10). N, BEs and Disp were the worst of the subgraph statistics in
terms of patchiness sensitivity. The interaction effects for BEs, N and Disp were
the weakest.

For 88s and N, the presence of interaction produced similar patterns of
abruptness sensitivity within the abruptness simple effects (Table 11).
Abruptness sensitivity for both 883 and N was fairly constant at patchiness levels
1, 2 and 3, dropped to a minimum at patchiness level 4, then increased slightly at

patchiness level 5. Abruptness sensitivity of Disp was the opposite, exhibiting a

67

trend of increasing abruptness sensitivity as patchiness increased. Disp had
maximum abruptness sensitivity at patchiness level 5.

885, N and Disp each responded to abruptness differences in a
predictable and internally consistent manner (Appendix B). As hypothesized,
883, N and Disp decreased as abruptness increased. Disp was significantly
different for comparisons between all abruptness levels except the comparisons
between abruptness levels 4 and 5. Similarly, 383 and N did not detect a
difference between abruptness levels 4 and 5. N and 885 did not distinguish the
difference between intermediate and high levels of abruptness at certain
patchiness levels. In addition, both N and B83 failed to detect a difference
between abruptness levels 1 and 2 at patchiness level 3. The difference
between 1 and 2 should have been easily distinguished since the surface with
abruptness level 1 was planar while the surface with abruptness level 2 was

curvilinear. The fact that N and BEs did not detect this difference was curious.

Lmax. Lmean and Lmin
Lmax, L"...an and Lm... were hypothesized as functional patchiness and
abruptness metrics. Each metric had significant patchiness and abruptness main
effects as well as significant interaction effects (Table 10). With regard to the
patchiness main effects, Ln...x ranked second, Lmean ranked third and Lmin ranked
fifth. As abruptness metrics, Lmean ranked third, Lm... fourth and Lmax seventh.

Lmean had the weakest interaction effects of the three followed by Lmin then Lmax.

68

The interaction effects on Lmax were nearly twice as strong as the interaction
effects on Lmean and Lmin.

For Lmax and Lmean, the presence of interaction resulted in a decrease in
patchiness sensitivity as abruptness increased (Table 11). in fact, Lmax did not
distinguish any difference in patchiness at abruptness level 5. Lmean found
significant differences at all levels of abruptness. Lmin was unable to detect
patchiness differences at abruptness levels 1 and 2 but its patchiness sensitivity
increased as abruptness increased.

The interaction effect for Lmam was such that abruptness sensitivity
increased steadily as patchiness increased (Table 11). L"...an and Lmin exhibited
patterns of interaction more similar to N and L9,...V than Lmax. Maximum
abruptness sensitivity for Lmean and me occurred at patchiness level 2 while the
minimum occurred at patchiness level 4 and it fluctuated at the remaining
patchiness levels.

Lmax, Lmean and L"...1 were less predictable and responded with less internal
consistency as patchiness metrics than they did as abruptness metrics (Appendix
B). Lmax was the only metric to respond as hypothesized for every patchiness
comparison; L...ax decreased as patchiness increased. Lmean and Lmin also had
this relationship with patchiness except in comparisons involving patchiness level
5. The same pattern was observed for NS and Lady; both deviated from expected
responses in comparisons involving patchiness level 5. Lmax did not respond to
patchiness in an internally consistent manner at abruptness levels 3 and 4. For

example, Lmax found the comparison between patchiness levels 2 and 4

69

significantly different but not the comparison between level 2 and 5. Lmean and
Lmin had similar problems at abruptness levels 3, 4 and 5.

As abruptness metrics, Lmax, Lmean and Lmin performed as hypothesized for
every abruptness comparison made (Appendix 8). Values of Lmax, Lmean and Lm...
increased as abruptness increased. Lmean and Lm... responded internally
consistently to abruptness differences at each patchiness level while Lmax had an
internally inconsistent comparison at patchiness level 1 and 2. None of the three
metrics distinguished a difference between abruptness levels 4 and 5 and 1 and

2 but did distinguish large and small degree differences in abruptness.

Cumulative boundary elements

The factorial ANOVA results for CBE are found in Table 12. CBE was
sensitive to both the patchiness and abruptness main effects. Its Omega
squared value for the patchiness main effects (0.018) ranked CBE as the sixth
most sensitive metric of all the surface-based metrics. With an Omega squared
value of 0.806, CBE ranked as the second most sensitive metric to abruptness.
Although it was hoped the response of CBE would avoid significant interaction
effects, interaction was significant but very low (02 = 0.006).

The single factor ANOVA results for CBE are found in Table 13. CBE
detected significant differences in patchiness at every level of abruptness except
abruptness level 1. C88 also detected significant differences in abruptness at
every level of patchiness. CBE interaction effects exhibited a pattern of

increased CBE patchiness sensitivity as abruptness increased. Abruptness

7O

sensitivity decreased with successive patchiness level until patchiness level 5

where it greatly increased.

Table 12. Main and interaction effects for ANOVA on the cumulative boundary
elements metric. An “ns” indicates that the variation in the source was not
significant different.

 

 

 

 

 

 

Source F 02 P

Patchiness 33.260 0.018 0.000
Abruptness 1478.364 0.806 0.000
Interaction 3.880 0.006 0.000

 

Table 13. Simple effects results for ANOVA on the cumulative boundary
elements metric. The maximum Omega squared value is in bold; the minimum is
underlined. An “ns” indicates an ANOVA that was not significant.

 

 

Patchiness simple effects Abruptness simple effects
CBE CBE
A1 0.003 ns P1 0.501
A2 0.018 P2 0.479
A; 0.036 P3 0.458
A 0.070 P4 - 0.424
A; 0.122 P5 0.571

 

 

 

 

 

 

The Bonferroni comparisons of each single factor ANOVA computed for
CBE are included in Appendix C. The comparisons among the patchiness
simple effects are not externally consistent, the sign of the mean differences
between patchiness levels changes depending on the comparison being made.

Also, at abruptness levels 3 and 4, the significant comparisons of CBE did not

71

 

form an internally consistent pattern. No hypotheses were made as to how CBE
would respond to patchiness differences, but the sign changes and inconsistent
behavior suggest CBE is not a very good patchiness metric. On the other hand,
the comparisons among the abruptness simple effects were externally and
internally consistent. CBE, hypothesized to be inversely related to abruptness,
did decrease as abruptness increased at every comparison made. CBE did not
detect a difference between abruptness levels 4 and 5 at any patchiness level
nor between abruptness level 1 and 2 at patchiness levels 3 and 4. CBE found
significant differences for all other comparisons. CBE's pattern of significantly
different comparisons among abruptness is nearly identical to the pattern

exhibited by N.

Surface-based metrics ranked by simple effects

The surface-based metrics ranked in order of simple effects patchiness
and abruptness sensitivity are found in Table 14. Across the 5 abruptness levels,
NS and Lmean consistently ranked as high patchiness detectors, while B85 and
Ln...x consistently ranked low. The CBE, N and Lm... metrics improved in rank as
abruptness increased. Lstdv ranked low as a patchiness metric at abruptness
levels 3 and 4 but was among the best at abruptness levels 1, 2, and 5. At
abruptness level 3, Disp was the second ranked patchiness metric but was less
impressive at all other abruptness levels. Disp, CBE and N consistently ranked
as superior abruptness metrics, while Lstdv, Lmax, and B83 repeatedly ranked

among the lower half. At lower levels of patchiness, Lmin and NS were mediocre

72

abruptness metrics. Lm... decreased in rank as patchiness increased but NS
improved to be the second ranked abruptness metric at patchiness levels 4 and

5. Ln“...n was an average abruptness metric at all patchiness levels.

73

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

00.0 ,2... .~00 >2.... 00.0 .2... 000.0 >0...._ ~000 as.
3.0 000 03.0 .__.... 00.0 a} 3.0.0 as. 0000 ,2...
40~0 E. .0.0 000 0.~0 000 .80 000 00~0 000
000.0 55.. 00~0 .20 $00 520 000.0 02 00~0 .__....
.000 c0...... 00~.0 5.5 0~00 02 ~80 .__.... .000 02
0000 z 0.00 z 0~00 52... 0.0.0 52.. 0000 is.
0.0.0 02 .000 02 050.0 2 0500 2 .000 z
.500 000 «~40 000 000.0 000 054.0 000 .000 000
~550 00.0 050.0 00.0 0~00 00.0 ~00 00.0 0~00 00.0

Ne 080: Ne 0805. «a 0.022 as 0.00.2 we 080:

0.. .0 .5 m. .5
. 0300.00 0.58.0 000553.22

000. 000 .000 000 000.0 000 0000 .__3 000.0 00.0
000 02 000.0 00.0 0.00 E. 5.00 000 000.0 000
000.0 .2... 0.0.0 .23 -00 .23 0.0.0 000 0000 000
000.0 00.0 5.00 .82.. 000.0 000 0~00 2 000.0 E.
0000 as... 0~00 5.5 0.0.0 .2; 000.0 00.0 000.0 2
..0.0 E. 0~00 .__.... 000.0 52.0 .000 ,2... 3.0 as.
«.00 5.. 50.0 02 000.0 2 2.0 52.. 00.0 a}
000 z 5~00 z .000 00.0 0. .0 as. 3~0 52..
-..0 000 050.0 000 0000 02 0~..0 02 500.0 02

Ne 0.00: «a 0822 “3 0.00: «a 080: Ne 0.00:

00 .< 2 N0. .<

 

 

 

 

 

 

0300.00 0.58.0 0008:0305

.00:.0> 00.0300 00080 .o.00.-0_0:.0 .5 00.80. 00.308 00000000030 .3. 0.00...

74

Surface-based suitability ranking

The suitability scores for each of the eight subgraph statistics and CBE
are presented in Table 15. In descending order, the most suitable surface-based
metrics for quantifying patchiness were: NS, Lmax, Lmean, Lmin, N, Disp, CBE and
BEs. The two surface-based metrics thought to be exclusively patchiness
metrics, NS and Lstdv, did rank among the best. Not surprisingly, the
hypothesized abruptness metrics, BE, CBE, Disp and N, performed poorly as
patchiness metrics. Lmax was a better patchiness metric than an abruptness
metric while me did not perform particularly well as either. Lmam was the only
surface-based metric that was completely externally consistent with regard to
patchiness. BEs was the only surface-based metric that was completely
internally consistent with regard to patchiness.

In descending order, the most suitable surface-based metrics for
quantifying abruptness were: Disp, CBE, N, Lmean, NS, Lmin, BEs, Lmax, Lstdv. This
order conforms to the expected function of the metrics. CBE and Disp were
designed specifically to measure abruptness and ranked as the most suitable
abruptness metrics. Lmean was the only metric to rank in the top half as both a
patchiness metric and abruptness metric. All of the surface-based metrics,
except for Lmax and Lstdv, were completely externally and internally consistent with

regard to abruptness.

75

Table 15. Surface-based metrics ranked by patchiness and abruptness

suitability scores.

Patchiness suitability scores

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency
NS 1.00 0.68 0.88 0.86 3.41
Lmam 0.98 0.42 1.00 0.87 3.27
Lmean 0.33 0.51 0.86 0.81 2.52
Lady 0.24 0.35 0.95 0.87 2.41
Lmin 0.21 0.14 0.75 0.62 1.71
N 0.12 0.26 0.69 0.59 1.67
Disp 0.09 0.24 0.42 0.86 1.60
CBE 0.17 0.50 0.00 0.75 1.43
BEs 0.10 0.04 0.00 1.00 1.14
Abruptness suitability scores
Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency

Disp 1.00 1.00 1.00 1.00 4.00
CBE 0.92 0.75 1.00 1.00 3.67
N 0.81 0.55 1.00 1.00 3.36
Lmean 0.75 0.49 1.00 1.00 3.24
NS 0.62 0.56 1.00 1.00 3.18
Lmin 0.68 0.40 1.00 1.00 3.08
BEs 0.61 0.31 1.00 1.00 2.92
Lmax 0.48 0.27 1.00 0.95 2.69
E339, 0.19 0.09 0.58 0.68 1.54

 

76

 

 

Patch-based suitability vs. Surface-based suitability

The primary goal of this research has been to determine which approach,
the patch- or surface-based, would be most suitable for quantifying ecotone
patchiness and abruptness. Both approaches were sensitive to patchiness and
abruptness, but the patch-based metrics were on average more sensitive to both
characteristics than the surface-based metrics. On average, the surface-based
metrics had weaker interaction effects. Both sets of metrics were more externally
and internally consistent as abruptness metrics than as patchiness metrics. In
order to determine which metrics performed the best relative to one another, they
were ranked according to their patchiness and abruptness suitability. The
patchiness suitability ranking is shown in Table 16 while the abruptness suitability
ranking is shown in Table 17. Mean suitability scores for each set of metrics are
also provided.

According to the patchiness suitability ranking of all the metrics (Table 16),
the patch-based set performed better as patchiness metrics than the surface-
based set. The patch-based mean patchiness suitability score (2.81) was
greater than the surface-based mean patchiness suitability score (1.92). On
average, the patch-based metrics received higher scores for main effects
sensitivity, simple effects sensitivity, external and internal consistency than the
surface-based metrics. Patch-based metrics occupied the 5 highest ranks for

patchiness detection.

77

Table 16. Patch- and surface-based metrics ranked according to patchiness
suitability scores. Patch-based metrics are underlined.

Patchiness suitability scores

 

 

 

 

 

 

 

 

 

 

 

 

Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency

Lfl 1.00 0.27 1.00 1.00 3.27
NP 0.75 0.49 0.91 0.85 3.00
AWMSJ 0.27 0.66 1.00 1.00 2.92
CONTAG 0.36 0.59 0.97 0.94 2.86
1!; 0.37 0.57 0.97 0.94 2.86
NS 0.61 0.52 0.88 0.86 2.86
AWMPFD 0.27 0.55 1.00 1.00 2.82
Lmax 0.60 0.31 1.00 0.87 2.77
PSCV 0.39 0.38 0.95 0.84 2.56
Lmean 0.20 0.38 0.86 0.81 2.26
Lstdv 0.14 0.25 0.95 0.87 2.22
.fl 0.22 0.25 0.90 0.84 2.21
Lmin 0.13 0.12 0.75 0.62 1.61
N 0.07 0.24 0.69 0.59 1.59
Disp 0.05 0.21 0.42 0.86 1.53
CBE 0.11 0.46 0.00 0.75 1.31
BEs 0.06 0.03 0.00 1.00 1.09
Patch-based
mean 0.45 0.47 0.96 0.93 2.81
Surface-
based mean 0.22 0.28 0.62 0.80 1.92

 

78

 

Table 17. Patch- and surface-based metrics ranked according to abruptness
suitability scores. Patch-based metrics are underlined.

Abruptness suitability scores

 

 

 

 

 

 

 

 

 

 

Metric Main effects Simple effects External Internal Total
sensitivity sensitivity Consistency Consistency

Disp 1.00 0.98 1.00 1.00 3.98
CONTAG 0.90 0.89 1.00 1.00 3.79
AWMPFD 0.93 0.84 1.00 1.00 3.77
AWMSI 0.92 0.83 1.00 0.98 3.73
IE 0.86 0.86 1.00 1.00 3.72
CBE 0.92 0.74 1.00 1.00 3.65
N 0.81 0.54 1.00 1.00 3.35
PSCV 0.74 0.57 1.00 1.00 3.30
m 0.76 0.53 1.00 1.00 3.29
me 0.75 0.48 1.00 1.00 3.23
NS 0.62 0.54 1.00 1.00 3.16
LLB 0.56 0.59 1.00 1.00 3.15
Lmin 0.68 0.39 1.00 1.00 3.08
BEs 0.61 0.30 1.00 1.00 2.91
me 0.48 0.26 1.00 0.95 2.68
E1 0.02 0.14 0.43 0.95 1.55
LN, 0.19 0.09 0.58 0.68 1.54
Patch-based
mean 0.71 0.66 0.93 0.99 3.29
Surface-
based mean 0.67 0.48 0.95 0.96 3.07

 

79

 

The abruptness suitability ranking (Table 17) indicated that the patch-
based set also performed better than the surface-based set as abruptness
metrics. The patch-based set had a mean suitability score of 3.29 verses a
mean suitability score of 3.07 for the surface-based set. The patch-based
metrics had, on average, higher main effects sensitivity, simple effects sensitivity,
and internal consistency scores. The surface-based metrics had, on average,
slightly better external consistency. Disp, the most suitable surface-based
abruptness metric, outperformed the most suitable patch-based abruptness
metric CONTAG. Disp had a nearly perfect suitability score (3.98) that was much
higher than the second ranked CONTAG (3.79).

It was surprising for the patch-based metrics to outperform most of the
surface-based metrics as both patchiness and abruptness metrics. The surface-
based approach was hypothesized as the superior approach for quantifying
abruptness but the suitability scores suggest otherwise. However, when applying
one approach or another to quantify patchiness or abruptness an entire set of
metrics need not be used. Only the metrics that were the most suitable for
quantifying the characteristic should be selected. In that case the patch-based
approach remains the superior approach for quantifying patchiness, and since
Disp ranked as the most suitable abruptness metric, the surface-based approach

might be the more attractive method for quantifying abruptness.

80

DISCUSSION AND CONCLUSIONS

Eight patch-based metrics and nine surface-based metrics were evaluated
to determine which were the most suitable for quantifying ecotone patchiness
and abruptness. Suitability was based on a ranking of scores totaled from 4
categories. The score for each category was based on the metrics’ response to
simulated patchiness and abruptness differences captured in a factorial analysis
of variance. As a whole, the patch-based set of metrics was more suitable than
the surface-based set as both patchiness and abruptness metrics. However,
individual surface-based metrics were modestly suitable as patchiness metrics

and highly suitable as abruptness metrics.

Research Questions Revisited

Do patch-based metrics only measure patchiness or are they capable of
measuring abruptness? Which of the patch-based metrics are most suitable as
patchiness metrics and which are most suitable as abruptness metrics?

All of patch-based metrics measured differences in both patchiness and
abruptness. LPI, NP, and AWMSI ranked as the most suitable patchiness
metrics while CONTAG, AWMPFD, and AWMSI ranked as the most suitable
abruptness metrics.

The fact that AWMSI excelled at measuring both characteristics could be
problematic. When AWMSI is applied to an ecotone of unknown patchiness and

abruptness characteristics it would be difficult to determine whether the metric is

81

measuring patchiness or abruptness. For instance, consider two ecotones A and
B; ecotone A has an AWMSI of 1.0 and ecotone B has an AWMSI of 1.5. Based
on the results of this study, the smaller AWMSI of ecotone A suggests that it is
more abrupt than ecotone B. However, the smaller AWMSI also suggests that it
is less patchy than ecotone B. If it is assumed that patchiness and abruptness
are independent of one another, AWMSI is difficult to interpret; its value could
either represent a degree of patchiness or a degree of abruptness. This
interpretation problem is avoided by using a metric that was only effective at
measuring one characteristic; in other words, one that had a high polarity
between patchiness suitability and abruptness suitability. Under this criterion,
LPI and NP stand out as a superior patchiness metrics. However, none of the
patch-based metrics that were highly suitable abruptness metrics were
particularly poor patchiness quantifiers. AWMPFD ranked low as a patchiness
metric but its perfect consistency scores suggest it was a fairly good patchiness
metric. CONTAG, although it showed high sensitivity to patchiness was not as
consistent as the other patch-based metrics. For this reason, CONTAG was
considered a poor patchiness metric and was considered most useful as an

abruptness metric.

82

Are the surface-based metrics able to measure patchiness? Do the surface-
based metrics measure abruptness? Which of the surface-based metrics are
most suitable as patchiness metrics and which are most suitable as abruptness

metrics?

The surface-based metrics also measured both patchiness and
abruptness. NS, Lmax and Lmean ranked as the most suitable surface-based
patchiness metrics and Disp, CBE, and N ranked as the most suitable surface-
based abruptness metrics. However, none of the surface-based metrics were
completely consistent with regard to patchiness. This is a big concern if they are
to be applied as patchiness metrics.

Of the most suitable surface-based patchiness metrics only Lmax had a
high polarity between patchiness suitability and abruptness suitability. Lmax was
the only surface-based patchiness metric unlikely to confuse patchiness and
abruptness. NS and Lme,am both showed relatively high suitability as abruptness
metrics which could lead to interpretation problems. On the other hand, the most
suitable surface-based abruptness metrics (Disp, CBE, and N) had a very high
polarity between their abruptness suitability and patchiness suitability. The risk
of interpretation problems between patchiness and abruptness when using Disp,
CBE, and N as abruptness quantifiers should be minimal.

The CBE metric was a very impressive abruptness metric. Several
properties make this metric the superior choice for quantifying abruptness. First,
the metric ranked as a highly suitable abruptness quantifier with a high polarity to

patchiness suitability. Second, the response of CBE exhibited a very weak

83

interaction effect. It had a completely consistent response, both externally and
internally, to abruptness. Lastly, unlike the subgraph statistics, the method by
which CBE is calculated avoids the use of an arbitrary ROC threshold.

Future research should be directed at improving the CBE metric. The
weak interaction effects it displayed might be further minimized or eliminated by
investigating the effect of the kernel size of the moving window used to compute
the ROC surface. CBE’s already weak interaction effects may prove to be
reduced further if a larger kernel size is used. A larger kernel size may continue
to capture the variation in the ecotone surface due to the transition present

without being as affected by the finer-grain variation due to patchiness.

How does the degree of one ecotone characteristic affect the measurement of
anothefl? Is it possible to measure the degree of patchiness at multiple degrees
of abruptness and visa versa. ls metric performance dependent on the level of
patchiness or abruptness? For instance, does a patchiness metric lose its ability

to measure patchiness as abruptness increases.

Each metric exhibited at least a weak interaction effect. Significant
interaction effects indicated that the metrics’ ability to measure one ecotone
characteristic was dependent on the strength of the other characteristic. In some
cases the interaction effect was so strong that the metric failed to detect
characteristic differences at one level when it was successful at detecting

differences at another. In some cases, interaction was such that metric response

84

was inconsistent. In other cases, the presence of interaction did not always
result in an inconsistent response.

The interaction effects of the patch-based metrics were stronger than the
interaction effects of surface—based metrics. On average the patch-based set
exhibited an interaction effect that was more than twice as strong as the
interaction effect of the surface-based set. The mean Omega squared for the
interaction effects on the patch-based metrics was 0.095 as compared to 0.036
for the surface-based metrics. The patch-based metric with the weakest
interaction effect, AWMPFD (02 = 0.109), had an interaction effect that was much
stronger (18x) than the interaction effect of the surface-based metric with the
weakest interaction effect, CBE (0)2 = 0.006). Despite the large difference in
interaction effects, the two sets of metrics were comparable in terms of external
and internal consistency, especially with regard to consistency as abruptness
metrics. This indicates that the presence of interaction did not always translate
into adverse consistency effects. Even so, interaction should still be considered
a poor property of these metrics.

The presence of interaction could lead to misinterpretation of metric
values. Interaction indicated that the response of a metric to a difference in one
factor changes with the degree of another factor. For example, the mean
difference between NP of patchiness level 1 and NP of patchiness level 5 was
16.680, 5.240 and 0.860 at abruptness levels 1, 2 and 3 respectively. At
abruptness levels 4 and 5 the difference was not significant. Although the way in

which the difference between patchiness levels 1 and 5 was simulated did not

85

change over abruptness levels, the value of the NP metric did. The difference in
NP between abruptness levels was attributed to the interaction of patchiness and
abruptness. Therefore, when interaction was present, an NP difference of a
particular magnitude at a low level of abruptness did not represent the same
difference in patchiness as the same magnitude difference in NP at a higher
abruptness level. Interpretation of a metric in the presence of interaction must
take into account the way in which the combination of factors contributes to the
metric’s response. Without a priori knowledge of the patchiness and abruptness
characteristics of the ecotones being quantified (i.e., using real world data)
interpretation of the metric interaction effects would be difficult, if not impossible.
When the use of the metric suitability ranking was first considered, a score
based on the interaction Omega squared was included in the total score used for
the ranking. After further consideration, it was decided that the inclusion of this
score would overly weight interaction effects in the suitability total since the
simple effect sensitivity and the external and internal consistency scores
provided evidence of the effects of interaction. Inclusion of the interaction score
to the suitability rankings did change the results of ranking. With the interaction
score included, AWMPFD and AWMSI rank as both the best patch-based
patchiness and abruptness metrics. Lmax, NS, and Lmean are the top ranked
surface-based patchiness metrics and CBE, Disp, and N remain the top ranked
surface-based abruptness metrics when the interaction score is included. When

interaction is included in the suitability ranking of both sets combined, AWMPFD,

86

LPI, and AWMSI rank as the top patchiness metrics and CBE, Disp, and

AWMPFD rank as the top abruptness metrics.

Considerations

The simulated ecotones used to evaluate the metrics were oriented so
that the transition of the ecotone was parallel to the vertical dimensions of the
surface cells. When these methods are used on real world data, the transition
will most likely not be oriented in that manner (i.e., the transition as viewed by the
satellite image will not always be aligned with the grid). This orientation might
affect how the subgraphs are calculated and how the surface-based statistics are
interpreted. Calculation of the Disp metric is the most affected by the orientation
of the transition. Disp was designed to calculate the dispersion of subgraphs in
the direction perpendicular to the transition. This calculation is easily made when
the transition is parallel to the surface cells. When the transition is oriented
otherwise, the way in which Disp is calculated would have to account for the
difference between the orientation of the transition and the orientation of the
surface cells.

The surface data can be transformed so that the transition is parallel to the
vertical dimensions of the surface cells through a process called resampling.
However, resampling could result in two problems. First, how do you determine
the orientation of the transition? For alpine treeline ecotones, the orientation of
the transition is typically similar to the slope aspect of the terrain the ecotone is

situated on, but not always. Where terrain, disturbance regime or ecological

87

processes are especially complex, the transition may not be aligned with the
slope aspect. In that case, the orientation would have to be determined another
way. One way would be to compute an aspect surface for the ecological variable
surface (i.e., LAI or NDVI) and then determine the majority aspect value (the
aspect value that occurs the most frequently). The majority aspect value of the
variable surface should provide an acceptable estimate of the transition
orientation. The transition orientation may end up being a compromise between
the terrain aspect and the ecological aspect.

The other problem is the effect the resampling might have on the raw data
values. If the method of resampling is weighted, like that of bilinear interpolation
or cubic convolution, the data values prior to resampling are changed. Bilinear
interpolation is an interpolation method where the value of a location is obtained
by linear interpolation on two axes using four neighbors (Chrisman, 1997). Cubic
convolution is an interpolation method where the value is obtained by fitting a
third-order equation to the 16 neighbors surrounding the desired location
(Chrisman, 1997). The changes resulting from the resampling may be large
enough so that the integrity of the original data is sacrificed. For this reason, it
would be wise to employ the nearest neighbor method of resampling if
resampling is necessary. The nearest neighbor method uses the value of the

location nearest to the desired location as the interpolated value.

88

Summary

The goal of this research was to determine which approach was better
suited to measure the patchiness and abruptness patterns of alpine treeline
ecotones. The suitability scores indicated that the patch-based approach was
superior to the surface-based approach. However, the patch-based metrics as
abruptness quantifiers were thought to be easily misinterpreted as patchiness
quantifiers due to their low degree of polarity between patchiness and abruptness
suitability. Conversely, the surface-based metrics were considered good
abruptness quantifiers because they had a high patchiness/abruptness suitability
polarity. Therefore, this research suggests that the patch-based metrics were
optimal for quantifying patchiness while the surface-based metrics were optimal
for quantifying abruptness.

In application, it would be most efficient to use only one of the
approaches. It would be burdensome in terms of both time and effort to use one
approach for patchiness and one approach for abruptness. Each approach could
be used to quantify both patterns but the limitations of the approach being used
need to be recognized. Use of the patch-based approach runs the risk of
misinterpreting abruptness metrics for patchiness. Use of the surface-based
approach must consider that the surface-based metrics were not very consistent

patchiness quantifiers.

89

APPENDICES

90

APPENDIX A

Bonferroni Comparisons for patch-based metrics

The following tables contain the results of the Bonferroni pairwise multiple
comparisons that were calculated from the single-factor ANOVA for each metric.
The leftmost column indicates the factor level that was held constant in the
single-factor ANOVA. The second column from the left and the second row
index each of the comparisons made. The number in a cell in the table is the
mean difference in metric value for the comparison being made (column minus

row). Cells containing a . indicate that the comparison was not significantly

different at the 0.01 significance level.

91

 

 

Bonferonni comparisons for patchiness simple effects

   

Cont...

92

Bonferonni comparisons for patchiness simple effects

 

Cont...

93

Bonferonni comparisons for patchiness simple effects

 

Cont...

94

Bonferonni comparisons for patchiness simple effects

 

95

Bonferroni comparisons for abruptness simple effects

 

Cont...

96

Bonferroni comparisons for abruptness simple effects

 

Cont...

97

Bonferroni comparisons for abruptness simple effects

 

Cont...

98

Bonferroni comparisons for abruptness simple effects

 

99

APPENDIX B

Bonferroni comparisons for surface-based metrics

The following tables contain the results of the Bonferroni pairwise multiple
comparisons that were calculated from the single-factor ANOVA for each metric.
The leftmost column indicates the factor level that was held constant in the
single-factor ANOVA. The second column from the left and the second row
index each of the comparisons made. The number in a cell in the table is the
mean difference in metric value for the comparison being made (column minus
row). Cells containing a indicate that the comparison was not significantly

different at the 0.01 significance level.

100

Bonferroni comparisons for patchiness simple effects

 

Cont...

101

Bonferroni comparisons for patchiness simple effects

 

Cont...

102

Bonferroni comparisons for patchiness simple effects

 

Cont...

103

Bonferroni comparisons for patchiness simple effects

 

104

Bonferroni comparisons for abruptness simple effects

 

Cont...

105

Bonferroni comparisons for abruptness simple effects

   

Cont...

106

Bonferroni comparisons for abruptness simple effects

 

Cont...

107

Bonferroni comparisons for abruptness simple effects

 

108

APPENDIX C

Bonferroni Comparisons for the CBE metric

The following tables contain the results of the Bonferroni pairwise multiple
comparisons that were calculated from the single-factor ANOVA for the CBE
metric. The leftmost column indicates the factor level that was held constant in
the single-factor ANOVA. The second column from the left and the second row
index each of the comparisons made. The number in a cell in the table is the
mean difference in metric value for the comparison being made (column minus
row). Cells containing a indicate that the comparison was not significantly

different at the 0.01 significance level.

109

Bonferroni comparisons for patchiness simple effects

 

Cont...

110

Bonferroni comparisons for abruptness simple effects

 

111

APPENDIX D

Source Code

This appendix contains the AML and C programs used to for the
simulation and the calculation of the surface-based metrics. The first several

lines of each program provide a brief explanation of the program’s purpose.

Simdetenn. c

l*simdeterm.c was used to generate the five deterministic surfaces. The
deterministic surfaces where based on the equation y = sign(x)(abs(x"1ln)). The
(n) term controls the abruptness of the surface.*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

char ascii[20];

double x, y, lower, upper, increment, sum, mean;
double r = 1;

int i;

int n,a,q,c;

int nrows = 23;

int ncols = 23;

intcount

double column[50];

double grid[50][50];

long xIlcorner = 307630;
long yllcorner = 5395930;
long xIlcornerO = 307630;
long yllcornerO = 5395930;
int cellsize = 30;

FILE *fp;

FILE *fs;

int main()

if ((fs = fopen("simstats.txt", "a")) == NULL)
{

fprintf(stderr, " error opening stats file.\n\n");
exit(1);
}

112

fprintf(fs, "surface r sum mean lower upper\n");

for(n=1; n < 101; 0H)

{

sum = 0;
mean = 0;

if(n < 10)

sprintf(ascii, "det0°/od.asc", n);
else

sprintf(ascii, "det°/od.asc", n);

if ((fp = fopen(ascii, "a")) == NULL)

fprintf(stderr, "error opening input file.\n\n");
exit(1);

}

fprintf(fp,"ncols %d\n", ncols);
fprintf(fp,"nrows %d\n", nrows);
fprintf(fp,"xllcorner °/old\n", xIlcorner);
fprintf(fp,"yllcorner °/old\n", yllcorner);
fprintf(fp,"cel|size %d\n", cellsize);
fprintf(fp,"NODATA_value -9999\n");

x = -1.1;

for(a=0; a < nrows; a++)
{

if(x < 0)

{
column[a] = (-1*(pow(fabs(x),1lr)) * .5);
}

else

column[a] = (pow(x,1/r) * .5);

}
x=x+4;
Ma==1)

lower = column[a];

if(a == 21)

' upper = column[a];

113

if(3 <= a <= 23)
sum = sum + column[a];

}

mean = sum l21;
for(q = 0; q < nrows; q++)

for(c = 0; c < ncols; c++)

{
grid[qllc] = column[a];
fprintf(fp."%f gridIQIlcl);

}
}
fclose(fp);
fprintf(fs, "%s %f %f %f %f °/of\n", ascii, r, sum, mean, lower, upper);
xIlcorner = xIlcorner + 840;
if(n == 10)
yllcorner = yllcorner - 840;
xIlcorner = xIlcornerO;
}
if(n == 20)
yllcorner = yllcorner - 840;
xIlcorner = xIlcornerO;
}
if( == 30)
yllcorner = yllcorner - 840;
xIlcorner = xllcorner0;
}
if(n == 40)
yllcorner = yllcorner - 840;
xIlcorner = xllcorner0;
}
column[O] = '\0';
gridIOIIOI = ‘0';
r = r+.25;
}
fclose(fs);

}/*end of main*/

114

 

Matrix. aml

/" matrix.aml was used to combine the deterministic surfaces created by
simdeterm.c with /*the perturbation surfaces to create the simulated ecotone
surfaces. The surfaces are /*perturbed according to their row and column in the
matrix. Patchiness increases with l* row number, abruptness increases with
column number. Matrix.aml performs the /*simulation 50 times for each simulate
ecotone type (25), so 1250 surfaces are created.

f" open file to hold stats
&s fileunit := [open stats4matrix.txt openstat -w]
&if %openstat% ne 0 &then

&return &inform Could not open stats ouput file.

83 writestat := [write %fileunit% 'group, filename, normal mean, normal std,
normSM l, normSM mean, normSM std, surface I, surface min, surface max,
surface mean, surface std']

&svi=1
&doi=1&t050

l* loop for rows in matrix
&sv row = 1
&do row = 1 &to 5

&if %row% eq 1 &then
&do

&sv smooth = 5

&sv stdev = .8

&end

&if %row% eq 2 &then
&do

&sv smooth = 4

&sv stdev = .6

&end

&if %row% eq 3 &then
&do

&sv smooth = 3

&sv stdev = .49

&end

&if %row% eq 4 &then
&do

115

&sv smooth = 2
&sv stdev = .29
&end

&if %row% eq 5 &then
&do

&sv smooth = 1

&sv stdev = .14

&end

I'loop for columns in matrix
&sv col = 1

&do col = 1 &to 5

&if %col% eq 1 &then
&do

&sv r = 01

&sv base = aee

&end

&if %col% eq 2 &then
&do

&sv r = 05

&sv base = bee

&end

&if %col% eq 3 &then
&do

&sv r = 13

&sv base = cee

&end

&if %col% eq 4 &then
&do

&sv r = 61

&sv base = dee

&end

&if %col% eq 5 &then
&do

&sv r = l5

&sv base = eee

&end

I“ create zone grids for zonalstats
setcell det%r%

116

setwindow det%r%
mask%r°/o = 1

/* describe deterministic function surface to collect stdv
describe det%r%

l*create perturb surface and smooth accordingly

norm%col%%row% = normal() * %stdev%

smooth%col%%row% = focalmean(norm°/ocol°/o°/orow%, rectangle, %smooth%,
%smooth%)

%base%%row%-%i% = smooth%col%%row% + det%r%

l'classify surface into trees not trees
junk = int(°/obase°/o°/orow°/o-°/oi°/o * 10000000)
junkstat = zonalstats(mask°/or°/o, junk, median)
kill junk all

cursor cur declare junkstat INFO ro
cursor our open

cursor our 1

&Iistvar :cur.median

&sv med = %zcur.median% / 10000000
cursor cur remove

q

tables

kill junkstat

q stop

grid

setcell det%r%

setwindow det%r%

if (%base%%row%-%i% > %med%)

if (%base%%row%-%i% > 0)
%base%%row°/ocl-°/oi°/o = 1

end“

/* setup normal() surface for stat export
describe norm%col%%row%

&sv normalstd = %grd$stdv%

&sv nonnalm = %grd$mean%

/* setup smoothed normal for stat export
describe smooth%col%%row%

&sv normSMstd = %grd$stdv%

&sv normSMm = %grd$mean%

moran smooth%col%%row%

&sv normSMl = %.moran_out%

117

I* setup final surface for stat export
describe %base% °/orow°/o-°/oi°/o

&sv quadstd = °/ogrd$stdv%

&sv quadmean = %grd$mean%

&sv quadmin = %grd$zmin%

&sv quadmax = %grd$zmax%

moran %base°/o%row°/o-%i%

&sv quadl = %.moran_out%

&sv filename = %base°/o°/orow°/o-°/oi%

&s writestat := [write %fileunit% [quote
%base%%row%,%filename%,°/ononnalm°/o,%normalstd%,%normSMI%,%normS
Mm%,%normSMstd°/o,%quad|%,%quadmin°/o,%quadmax%,%quadmean%,%qua
dstd°/o]]

1* write to stats file F".

kill norm%col%%row% all
kill smooth%col%%row% all --_.
kill mask%r°/o all

l’kill %base%%row%-%i% all

l’kill °/oD8$6°/o°/of0W°/oCI-%I°/o all

 

&end l*end of column loop
&end I* end of row loop

&end /* end of 50 times

&s closestat := [close %fileunit%]
&retum

Gemit.aml

l" gemit.aml was used to do the lattice delineation and calculate the surface-
l‘based metrics. Gemit.aml was modeled after the capabilities of the GEM
/*Boundary Analysis software from Biomedware. The #be's, subgraph,
l’singletons, minlength, maxlength, meanlength, stdvlength, and dispersion
I‘metrics are calculated. These metrics are calculated for a user specified
l‘number of ROC thresholds by using the slice command in GRID.

l‘ The C programs from connect2.c and dispersion.c are called from this aml.

/* prompt user for parameters

&sv basename = [response 'Enter basename (i.e., aee* or bee*)' aee*]

&sv numintervals = [response 'Slice ROC surface into how many intervals' 20]
&sv aspectthreshold = [response 'Enter aspect threshold for BE connection' 30]
&s count := [filelist %basename% outfile -file]

&if %count% <= 0 &then

118

&return Error generating outfile file
&s unit := [open outfile ok -read]

I* open output file for statistics dump
&s fileunit := [open gemstatsatxt openstat -a]
&if %openstat% ne 0 &then

&retum &error Error opening gemstatstxt file

&s writestat := [write %fileunit°/o
'group,filename,i,#be's,subgraph,singletons,minlength,maxlength,meanlength,std
vlength,disp1 ,disp2,disp3,disp4']

I* open temporary file
85 tempunit := [open temptxt openstat -a]
&if %openstat% ne 0 &then

&retum &error Error opening tempory file

I* write aspect threshold value to temp file for connect2.c to read
&s writestat := [write %tempunit% [quote %aspectthreshold%]]
&s ok := [close %tempunit%]

/* Process each file in outfile list
&do] := 1 &to %count%

&s ezero := [read °/ounit% ok]
&s cover = [before %ezero% .]

I*import ezerozero file
import grid %ezero% %cover%
grid

l‘calculate slope and aspect

&ty Deriving slope and aspect for %cover%.
slope1 = slope(%cover%)

aspect1 = aspect(%cover%)

I*gridclip to study area, the surfaces were made with a one cell border to
l’eliminate edge effects when calculating the slope and aspect
&ty Removing outer edge of slope and aspect grids.

describe slope1

&sv imin = %grd$xmin% + %grd$dx%

&sv jmin = %grd$ymin% + %grd$dy%

&sv imax = %grd$xmax% - %grd$dx%

&sv jmax = %grd$ymax% - %grd$dy%

gridclip slope1 slope BOX %imin% %jmin% %imax% %jmax%
gridclip aspect1 aspect BOX %imin% %jmin% %imax% %jmax%
kill slope1 all

119

 

kill aspect1 all

i‘ slice the slope grid into x number of intervals
&ty Slicing rate of change surface into °/onumintervals% intervals.
sliced = slice(slope, eqinterval, %numintervals°/o)

/* process interval by interval

&sv i = 1

&do &until °/onumintervals°/o It 1

&ty Processing interval %i% for o/ocover°/o.

%cover%m_%i% = select(sliced, "value >= °/onumintervals%")

if (%cover%m_%i% > 0) then
%cover°/om2_°/oi°/o = 0
endfi

%cover°/oasp%i°/o = °/ocover%m2_°/oi°/o + aspect
besasp.asc = gridascii(°/ocover°/oasp°/oi%)

kill °/ocover°/oasp°/oi°/o all

kill %cover%m_%i% all

l*run connect2.c to connect be's based on aspect threshold
&ty Calling C program "connectum".

&sys .lconnectum

°/ocover°/obes°/oi°/o = asciigrid(beconnect.asc)

&sys rm besasp.asc

/*quit to arc, additem to separate sings and subs for stats

q
additem %cover%bes%i%.vat %cover%bes%i%.vat subgraph 4 4 b
grid

/* calculate subgraph item
reselect %cover°/obes°/oi°/o.vat INFO count gt 1
calculate °A3cover%bes°/oi°/o.vat INFO subgraph = 1

I*collect #boundary elements, min, mean, and max length
&ty Collecting subboundary statistics.

I*first, subgraph records

statistics %cover%bes%i°/o.vat INFO
min count

max count

mean count

standarddeviation count

end

120

 

&sv minlength = [show statistic 1 1]
&sv maxlength = [show statistic 2 1]
&sv meanlength = [show statistic 3 1]
&sv stdvlength = [show statistic 4 1]

f'select singleton records

aselect %cover°/obes%i%.vat INFO

reselect %cover%bes%i%.vat INFO subgraph eq 0
statistics %cover%bes°/oi%.vat INFO

sum count

end

&sv singleton = [show statistic 1 1]

I*select all

aselect %cover°/obes%i%.vat INFO
statistics °Aicover°/obes%i°/o.vat INFO
sum count

sum subgraph

end

&sv numbes = [show statistic 1 1]
&sv subgraph = [show statistic 2 1]

I* find centroids and call dispersion.c to calc distances from "mean treeline"
&ty Calculating centroids

meancent = zonalcentroid(°/ocover%m2_°/oi°/o)
meancentasc = gridascii(meancent)

kill meancent all

becents.asc = gridascii(°/ocover°/om2_°/oi%)
kill %cover%m2_%i% all

cents = zonalcentroid(%cover%bes%i%)
cents.asc = gridascii(cents)

kill cents all

&sys .ldispersum

&sys rm becents.asc

&sys rm cents.asc

&sys rm meancentasc

/* open output from dispersion.c and read into variables
&s dispunit := [open dispersion.txt openstats -r]
&if %openstats% ne 0 &then

&return &error Error opening dispersion.txt file

&do d := 1 &to 4
&s disp%d% := [read °/odispunit% ok]
&end

&s ok := [close %dispunit%]

121

 

I' Prepare output, write output.

&ty Writing output to gemstatstxt.

&sv group = [substr %cover% 1 4]

&s writestat := [write °/ofileunit°/o [quote
%group%,%cover%,°/oi%,°/onumbes°/o,°/osubgraph°/o,%singleton%,%minlength%,
%maxlength%,°/omeanlength°/o,%stdvlength°/o,%disp1 %,%disp2%,%disp3%,%di
sp4°/oil

I* clean out grids and ascii files, increment counter
q

export grid °/ocover%bes%i°/o °/ocover°/obes°/oi°A).e00
grid

&sys mv %cover%bes%i%.eOO subgraphs

kill %cover°/obes°/oi% all

&s ok := [delete beconnectasc -file]

83 ok := [delete dispersion.txt -file]

&sv i = °/oi°/o + 1

&sv numintervals = %numintervals% - 1

&end I*belongs to process interval by interval
kill %cover% all

kill slope all

kill aspect all

kill sliced all

q

&end l‘process next surface

&s closestat := [close %fileunit%]
83 ok 2- [close %unit%]

&s ok : [delete outfile -file]

&s ok : [delete temp.txt -file]
&return

Connect2. c

/’ connect2.c was designed to be called from gemit.aml. connect2.c was used to
connect boundary elements that were within the aspect difference threshold.*l

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

int ncols, nrows, rowextent, colextent, cellsize, i, j, x, y, beid, l, d, max, opal;
long xIlcorner, yllcorner;

float grid[100][100]; I*holds aspect values read from gridascii*l

int connect[100][100];

122

 

int where[100][100];

float n, ne, e, se, 3, sw, w, nw;
float threshold = 30;

FILE *fp;

FILE *fpg;

FILE *ft;

int main()

printf("\n Successful call to connect2.c\n");

I‘ open temp.txt to get the aspect threshold entered at start of gemit.aml */
if ((ft = {fopen("temptxt", "r")) == NULL)

fprintf(stderr, "error opening temp.txt file.\n");
exit(1 );

 

}
fscanf(ft, "%f“, &threshold);

fclose(ft);

printf("\n Connecting all boundary elements within %f degree aspect threshold\n",
threshold);

if ((fp = fopen("besasp.asc", "r")) == NULL)
{

fprintf(stderr, "error opening besasp.asc file.\n");
exit(1);
}

fscanf(fp, "%*s %d\n%*s %d\n%*s %ld\n%*s %ld\n%*s %d\n%*s °/o*s\n",
&ncols, &nrows, &xllcorner, &yllcorner, &cellsize);
rowextent = nrows + 5;

colextent = ncols + 5;

I*fill matrix full of values that are not within the thresh of each other*/
opal = -10000 - threshold;

for (j=0; i < rowextent; i++)
{

for (i=0; i < colextent; i++)

grid[i][j] = opal;

123

opal = opal - threshold - 1;
}
for (j=1; j <= nrows; I“)

for (i=1; i <= ncols; i++)

{
fscanf(fp, "%f", &grid[i][j]);
if(grid[i][j] l= -9999)
{
/*convert to radians*/
grid[i][j] = grid[i][j] * .01745329252;
}

}
fclose(fp);

I*convert to radians*/
threshold = cos(threshold * .01745329252);
l*connect*/
for(j=1;j <= nrows; j++)
{
for(i=1; i <= ncols; i++)
{
I*first check to see if current cell is connected*/
if(connect[i][j] > 0)
{

beid = connect[ilfll;

if(grid[i][i-1] != -9999)

n = cos(grid[i][j] - grid[i][j-1j);
else

n = 0;

if(grid[i+1][j-1] != -9999)

ne = cos(grid[i][j] - grid[i+1][j-1]);
else

ne = 0;

if(grid[i+1][j] I: -9999)

e = 008(9ridliilil - grid[i+1llii);
else

e = 0;

if(grid[i+1][j+1] != -9999)

124

 

se = cos(grid[i][j] - grid[i+1][j+1]);
else
se = 0;

if(grid[i][j+1] != -9999)
s = cos(grid[i][j] - grid[illi+1]);
else

3 = 0;

if(grid[i-1][j+1] != -9999)

sw = cos(grid[i][j] - grid[i-1][j+1]);
else

sw = 0;

if(grid[i-1][j] != -9999)

w = 008(9ridlilli] - grid[i-lllil);
else
w = 0;

if(grid[i-1][j-1] != -9999)

nw = cos(grid[i][j] - grid[i-1][j-1]);
else

nw = 0;

if(n > threshold)

{
connect[i][i-1] = beid;
connect[i][i] = beid;

}

if(ne > threshold)

{

connect[i+1][j-1] = beid;
connect[i][j] = beid;

}
if(e > threshold)

{
connect[i+1][j] = beid;
connect[i][i] = beid;

I

if(se > threshold)

{
connect[i+1][j+1] = beid;
connect[i][i] = beid;

}

if(s > threshold)

{

125

 

connect[i][i+1] = beid;
connect[im] = beid;

}
if(sw > threshold)
{
connect[i-1][j+1] = beid;
connect[i][j] = beid;
}
if(w > threshold)
{
connect[i-1][j] = beid;
connect[i][j] = beid;
}
if(nw > threshold)
{
connect[i-1][j-1] = beid;
connect[i][j] = beid;
}
I*end of if‘/ }
else I*else #1 if current cell is not connected*/
{

if(grid[i][j] == -9999) I*deal with nodata*/
{
connect[i][j] = -9999;

else /*if not connected and not nodata*/

{

l“ calcuate the subgraph id by finding the highest value
in connect grid and add one *I

max = connect[1][1];

for (d=1; d <= nrows; d++)

{
for (l=1; I <= ncols; l++)
{
if(connect[l][d] > max)
max = connect[l][d];
}
}

beid = max +1;

126

 

if(grid[i][j-1] I: -9999)

n = cos(grid[i][j] - grid[illi-ll);
else

n = 0;

if(grid[i+1][j-1] != -9999)

ne = cos(grid[i][j] - grid[i+1][i-1l);
else

ne = 0;

if(grid[i+1][j] != -9999)

e = cos(grid[i][j] - grid[i+1][jj);
else

e = 0;

if(grid[i+1][j+1] l= -9999)
se = cos(grid[i][j] - grid[i+1][j+1]);
else

se = 0;

if(grid[i][j+1] I= -9999)

s = 008(9ridlillil - gridlilli+11);
else

s = 0;

if(grid[i-1][j+1] != -9999)

sw = cos(grid[i][j] - grid[i-1][j+1]);
else

sw = 0;

if(grid[i-1][j] != -9999)

w = c08(9ridlil[il - grid[i-lllil);
else
w = 0;

if(grid[i-1][j-1] != -9999)

nw = cos(grid[i][j] - grid[i-1][j-1]);
else

nw = 0;

if(n > threshold)

{
connect[i][j-1] = beid;
connect[i][i] = beid;

}
if(ne > threshold)

127

 

connect[i+1][j-1] = beid;
connect[i][j] = beid;

}
if(e > threshold)
{

connect[i+1][j] = beid;
connect[i][j] = beid;

 

}

if(se > threshold)

‘ ”I
connect[i+1][j+1] = beid; .. _
connect[i][j] = beid;

}

if(s > threshold)

{ f"-
connect[i][j+1] = beid; g
connect[i][j] = beid;

}

if(sw > threshold)

{
connect[i-1][j+1] = beid;
connect[i][j] = beid;

}

if(w > threshold)

{
connect[i-1][j] = beid;
connect[i][j] = beid;

}

if(nw > threshold)

{
connect[i-1][j-1] = beid;
connect[i][j] = beid;

}

/*deal with singletons, if no connections just assign
value to current connect cell*/

if(connect[l][j] == 0)
{

connect[i][j] = beid;
}l*end of inner else*/

]l‘end of else #1*/
}/"'end of i for*/

128

 

)l‘end ofj for*/

I*open the output file *I
if ((fpg = fopen("beconnect.asc", "w")) == NULL)
{

fprintf(stderr, "error opening connect.asc file.");
exit(1);
}

fprintf(fpg,"ncols %d\n", ncols);
fprintf(fpg,"nrows %d\n", nrows);
fprintf(fpg,"xllcorner %ld\n", xIlcorner);
fprintf(fpg,"yllcomer %ld\n", yllcorner);
fprintf(fpg,"cellsize %d\n", cellsize);
fprintf(fpg,"NODATA_value -9999\n");

for (j=1; j <= nrows: i++)
for (i=1; i <= ncols; i++)
fprintf(fpg, "%d ", connect[i][i]);
fprintf(f}pg, "\n");

}
fclose(fpg); I*close sub grid output file*/
}/*end of main*/

Dispersionc

/* dispersion.c was used to calculate the subgraph dispersion metric (Disp). Disp
was calculated as the mean distance from the centroid of all boundary elements
to each individual boundary element. Dispersionc was designed to be called
from gemit.aml and uses gridascii ouput created by gemit.aml. */

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

int ncols, nrows, cellsize, i, j, n, m;
long xIlcorner, yllcorner;

float cents[100][100];

float meancent[100][100];

129

 

float becents[100][100];

float dist, meandist, dist2, meandist2, bedist, bedist2, bemeandist, bemeandist2
subscent, subscent2, x, y, xcent, ycent, xcent2, ycent2; ’
FILE *fm;

FILE *fcs;

FILE *fbcs;

FILE *fo;

int main()
{
printf("\n Successful call to dispersion.c \n"); m

if ((fcs = fopen("cents.asc", "r")) == NULL)

 

fprintf(stderr, "error opening centroid file.\n");
exit(1);

}

fscanf(fcs, "%*s %d\n%*s %d\n%*s %ld\n%*s %ld\n%*s %d\n%*s %*d\n"
&ncols, &nrows, &xllcorner, &yllcorner, &cellsize); '

 

H-
‘L- _

for (i=0; j < nrows; j++)
for (i=0; i < ncols; i++)
fscanf(fcs, "%f", &cents[i][j]);
}
fclose(fcs);
if ((fm = fopen("meancent.asc", "r")) == NULL)
fprintf(stderr, "error opening centroid file.\n");

exit(1 );
}

fscanf(fm, "%*s %d\n%*s %d\n%*s %ld\n%*s %ld\n%*s %d\n%*s %*d\n",
&ncols, &nrows, &xllcorner, &yllcorner, &cellsize);

for (i=0; j < nrows; j++)
for (i=0; i < ncols; i++)

fscanf(fm, "%f", &meancent[i][j]);

130

}

fclose(fm);
if ((fbcs = fopen("becents.asc", "r")) == NULL)
fprintf(stderr, "error opening centroid file.\n");

exit(1 );
}

fscanf(fbcs, "%*s %d\n%*s %d\n%*s %ld\n%*s %ld\n%*s %d\n%*s
%*d\n", &ncols, &nrows, &xllcorner, &yllcorner, &cellsize);

for (j=0; j < nrows; j++)
for (i=0; i < ncols; i++)
fscanf(fbcs, "%f", &becents[i][j]);
}
fclose(fbcs);
l’find what row the centroid of all bes combined is*/
for (j=0; j < nrows; j++)

for (i=0; i < ncols; i++)

{
if(meancent[i][j] != -9999)
{
subscent = j;
xcent = i;
ycent = i;
}
}

}

I*find distance between centroid of each subgraph and the row of the centroid of
all subgraphs */

n=0;
dist=0;
dist2=0;

for (j=0; j < nrows; i++)

131

 

for (i=0; i < ncols; i++)

{

if(cents[i][j] I= -9999)
{

x=i;
y=i;

dist = dist + abs(subscent - j);
dist2 = dist2 + pow(pow(x - xcent, 2) + pow(y - ycent,2), .5);

n++;

}
}

I*average distance in meters*l

meandist = (dist / n) * cellsize;
meandist2 = (dist2 / n) * cellsize;

m=0;
bedist=0;
bedist2=0;

for (j=0;j < nrows; i++)

for (i=0; i < ncols; i++)

bedist = bedist + abs(subscent - j);
bedist2 = bedist2 + pow(pow(x - xcent, 2) + pow(y - ycent,2),

{
if(becents[i][j] l= -9999)
{
x = i;
Y =1;
-5);
m++;
}
}

}

I*average distance in meters*/

bemeandist = (bedist / m) * cellsize;
bemeandist2 = (bedist2 / m) * cellsize;

l*write output*/

132

 

 

if ((fo = fopen("dispersion.txt", "w")) == NULL)
{

fprintf(stderr, "error opening ouput file.\n");
exit(1);
}

fprintf(fo, "%f\n°/of\n°/of\n°/of\n", bemeandist, bemeandist2, meandist, meandist2);
fclose(fo);
printf("\n dispersion.c finished. \n");

}/*end of main*/

133

BIBLIOGRAPHY

134

BIBLIOGRAPHY

Allen, TR, and SJ. Walsh, 1996. Spatial and compositional pattern of alpine
treeline, Glacier National Park, Montana, Photogrammetric Engineering &

Remote Sensing, 62(1 1 ):1261-1268.

Armand, AD, 1992. Sharp and gradual mountain timberlines as result of
species interaction, Landscape Boundaries: Consequences for Biotic
Diversity and Ecological Flows, (A.J. Hansen and F. di Castri, editors),

Springer-Verlag, New York, pp. 361-378.

Armand, AD, 1985. The change of vegetation near the timber-line. lnter-Nord,
17:59-62

Baker, W.L. and Y. Cai, 1992. The r. Ie programs for multiscale analysis of
landscape structure using the GRASS geographical information system,

Landscape Ecology, 7:291-302.

Baker, W.L., and P.J. Weisberg, 1995. Landscape analysis of the forest-tundra
ecotone in Rocky Mountain National Park, Colorado, Professional

Geographer, 47(4):361-375.

Barbujani, G., N.L. Oden, and RR. Sokal, 1989. Detecting regions of abrupt
change in maps of biological variables, Sytematic Zoology, 38(4):376-389.

Bhattacharyya, GK, and RA. Johnson, 1977. Statistical concepts and
methods, John Wiley & Sons, New York.

Brown, D.G. 1998. Classification and boundary vagueness in mapping
presettlement forest types. Intemational Journal of Geographical

lnfonnation Science, 12(2):105-129.

Chrisman, N., 1997. Exploring Geographic lnfonnation Systems, John Wiley &
Sons, New York.

Churkina, G., and Y. Svirezhev, 1995. Dynamics and forms of ecotone of under
the impact of climatic change: mathematical approach, Joumal of

Biogeography, 22:565-569.

Fortin, M.J., 1994. Edge detection algorithms for two-dimensional ecological
data, Ecology, 75(4):956-965.

Fortin, M.J., and P. Drapeau, 1995. Delineation of ecological boundaries:
comparison of approaches and significance tests, Oikos, 72:323-332.

135

Fonnan, R.T.T., and M. Godron, 1986. Landscape Ecology, John Wiley & Sons,
New York.

Gustafson, E.J., 1998. Quantifying landscape spatial pattern: what is the state
of the art?, Ecosystems, 1:143-156.

Haines-Young, R., and M. Chopping, 1996. Quantifying landscape structure: a
review of landscape indices and their application to forested landscapes,

Progress in Physical Geography, 20(4):418:445.

Holland, MM, 1988. SCOPE/MAB Technical consultations on landscape
boundaries: report of a SCOPE/MAB workshop on ecotones, Biology

lntemational, Special Issue, 17:47-106.

Jacquez, GM, and S.L. Maruca, 1998. Geographic boundary detection,
Proceedings of the 8‘" lntemational Symposium on Spatial Data Handling
(T.K. Poiker and N. Chrisman, editors), International Geographical Union,

pp. 496-509.

Jacquez, G.M., S.L. Maruca, and M.J. Fortin, 1999. From fields to objects: a
review of geographic boundary analysis. Joumal of Geographic Systems,
in press.

Johnston, CA, and J. Bonde, 1989. Quantitative analysis of ecotones using a
geographic lnfonnation system, Photogrammetric Engineering and

Remote Sensing, 55(1 1 ):1643-1647.

Johnston, CA, J. Pastor, and G. Pinay, 1992. Quantitative methods for
studying landscape boundaries. Landscape Boundaries: Consequences

for Biotic Diversity and Ecological Flows, (A.J. Hansen and F. di Castri,
editors), Springer-Verlag, New York, pp. 107-125.

Keppel, G., 1991. Design and Analysis: A researcher's handbook, Prentice Hall,
Englewood Cliffs, New Jersey.

Kent, M., W.J. Gill, R.E. Weaver, and RP. Arrnitage, 1997. Landscape and
plant community boundaries in biogeography, Progress in Physical
Geography, 21 (3)231 5-353.

Krummel. J.R., R.H. Gardner, G. Sugihara, R.V. O’Neill, and PR. Coleman,
1987. Landscape patterns in a disturbed environment, Oikos, 48(3):321-

324.

Legendre, P., and M.J. Fortin, 1989. Spatial pattern and ecological analysis,
Vegetatio, 80: 1 07-1 38.

136

Ludwig, J.A., and J.M. Cornelius, 1987. Locating discontinuities along
ecological gradients, Ecology, 68(2):448-450.

Malanson, GP, 1997. Effects of feedbacks and seed rain on ecotone patterns,
Landscape Ecology, 12(1):27-38.

Malanson, GP, and D.G. Brown, 1996. Collaborative research on process and
pattern at alpine treeline, Project Proposal funded by National Science

Foundation (#97-14347).

Malanson, GP, and DR. Butler, 1994. Tree-tundra competitive hierarchies, soil
fertility gradients, and the elevation of treeline in Glacier National Park,

Montana, Physical Geography, 1 5: 166-1 80.

McGarigal, K., and B.J. Marks, 1993. FRAGSTATS: Spatial Pattem Analysis for
Quantifying Landscape Structure, Version 1.0, Forest Science
Department, Oregon State University, pp. 1-62 (software manual).

Oden, N.L., R.R. Sokal, M. J. Fortin, and H. Goebl, 1993. Categorical wombling:
detecting regions of significant change in spatially located categorical
variables, Geographical Analysis, 25:315-336.

Orloci, L., and M. Orloci, 1990. Edge detection in vegetation: Jornada revisited,
Journal of Vegetation Science, 1:311-324.

Risser, PG, 1995. The status of the science examing ecotones, BioScience,
45(5):318-326.

Sokal, RR, and F.J. Rohlf, 1995. Biometry: The principles and practice of
statistics in biological research, W.H. Freeman and Company, New York.

Stevens, G.C., and J.F. Fox, 1991. The causes of Treeline, Annual Review of
Ecological Systems, 22: 1 77-91 .

Timoney, K.P., G.H. LaRoi, and M.R.T. Dale, 1993. Subarctic forest-tundra
vegetation gradients: The sigmoid wave hypothesis, Journal of Vegetation

Science, 42387-394.

Turner, MG, 1989. Landscape ecology: the effect of pattern on process.
Annual Review of Ecology and Systematics, 20: 171-1 97.

TUrner, MG, 1990. Spatial and temporal analysis of landscape patterns,
Landscape Ecology, 4:21-30.

137

Wiens, J.A., Crawford, CS, and JR. Gosz, 1985. Boundary dynamics: a
conceptual framework for studying landscape ecosystems, Oikos, 45:421-
427.

Womble, W.H., 1951. Differential Systematics, Science, 114:315-322.
Wood, T.F., and Foody, GM, 1989. Analysis and representation of vegetation
continua from Landsat Thematic Mapper data for lowland heaths,

lntemational Joumal of Remote Sensing, 10(1):181-1 91 .

Wilson, J.B., and A.D.Q. Agnew, 1992. Positive-feedback switches in plant
communities, Advances in Ecological Research, 23:263-336.

138