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thesis entitled

TOWARDS THE DEVELOPMENT OF SPATIALLY
UNBIASED LANDSCAPE FRAGMENTATION INDICES

presented by
Dante Gideon K. Vergara

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TOWARDS THE DEVELOPMENT OF SPATIALLY UNBIASED LANDSCAPE
FRAGMENTATION IN DICES

By

Dante Gideon K. Vergara

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS
Department of Geography

2003

ABSTRACT

TOWARDS THE DEVELOPMENT OF SPATIALLY UNBIASED LANDSCAPE
FRAGMENTATION INDICES

By

Dante Gideon K. Vergara

Most current landscape indiees suffer from the effects of scale, precluding
comparative studies between sites and/or with other continuous data over the landscape.
A new index to describe landscape fragmentation as a spatial process based on patch
density, patch shape complexity, and patch arrangement is proposed. The maximum
support size encompassing a single spatial process centered on a point is determined by
detecting changes in spatial structure as nested supports increases, through changes in
significance of Moran’s I and Geary’s C as well as a changes in the structure of the
variogram model. The support size over which to compute the fragmentation index now
has basis and need not be arbitrary. The vertices of the polygon and its minimum
spanning tree define the locations where the fragmentation index should be computed.
Spatial interpolation on the indices derived at these selected points would generate the
landscape fragmentation intensity surface, converting the initially discrete index into a
continuous variable. Doing so will allow for comparative studies between sites and/or
with other continuous data. Relating patterns to processes on the landscape would then be
feasible. The methodology was tested on co-incident. classified ETM+ and Ikonos

imagery of the Legal Amazon forest near Uruara. Para in Brazil.

Dedicated to my family

iii

ACKNOWLEDGEMENTS

To my mentors:

Percy, Michael, and Ben,

For starting me on this line of scholarship;

Bob, Ashton, and Qi,

For their instruction, encouragement, comments, and kind words;

And in memory of my Dad,

Who taught me to think of possibilities;

THANK YOU ALL.

iv

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES
INTRODUCTION
1. Patterns and processes
11. Landscape fragmentation indiees
III. Scale
IV. The bias of scale
V. Limitations of the current landscape fragmentation indiees
VI. Statement of problem
VII. Objectives
VIII. Fragmentation
A. Patch density, arrangement and shape complexity
B. Estimates of index parameters
C. Development of the proposed index
IX. Overcoming the bias of spatial support

A. Smoothing the surface
1. Low pass or mean filter
2. Pycnophylatic interpolations
3. Kriging

B. Spatial processes

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1. Areal approach
2. Point estimation approach

C. Generating the fragmentation intensity surface

MATERIALS AND METHODS

1. Satellite imagery

II. Nested supports

III. Fragmentation index

IV. Variograms and first lag gammas
V Autocorrelation

VI. Regressions

RESULTS AND DISCUSSIONS

I. Fragmentation index

11. Moran’s I and Geary's C

III. Variogram modelling

IV. Encompassing spatial processes
V. Estimating for null support
CONCLUSIONS

1. Areal approach

11. Point estimation approach
RECOMMENDATIONS
APPENDICES

APPENDIX A: Summaries for Fragmentation Parameters and Indices

APPENDIX B: Summaries for Increasing Support of Fragmentation Study

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APPENDIX C: Variograms for Increasing Supports of Two Cases Each of 62
Three Forest Scenarios

LITERATURE CITED 8]

REFERENCES 85

vii

LIST OF TABLES

Table 1. Non-redundant univariate landscape fragmentation indiees (Riitters et 5

al 1995).
Table 2. Summarized slope and intercept estimates with standard errors. 40
Table A.1.a. Forested 1 53
Table A.1.b. Forested 2 53
Table A.2.a. Non-Forested 1 54
Table A.2.b. Non-Forested 2 54
Table A.3.a. Intermediate 1 55

Table A.3.b. Intermediate 2 55

viii

LIST OF FIGURES
Figure 1.1. Increasing number and sizes of patches
Figure 1.2. Uniform, random, and clustered patches
Figure 1.3. Patch shape complexity
Figure 2. Pycnophylactic interpolation

Figure 3. Satellite imagery, Uruara in the Legal Amazon, showing forest
fragmentation at different resolutions.

Figure 4. Typical increasing support for fragmentation study

Figure 5. Typical variogram with exponential model showing nugget,
sill and range variances.

Figure 6.1. Non-forested l, 17 x 17 pixels
Figure 6.2. Non-forested 2, 17 x 17 pixels
Figure 7.1 First lag gamma and support size for six supports.

Figure 7.2 Ratio of first lag gamma and variance vs support size for six
supports.

Figure 8. Typical areal estimation.

Figure 9.1. Non-convex hull polygons with centroids and vertices.
Figure 9.2. Internal vertices determined for spanning tree

Figure 9.3. Typical spanning tree within the polygon.

Figure 9.4. Thiessen triangles for surface generation

Figure 10. Typical point estimation

Figure B. 1 .a. Summaries of increasing support for fragmentation study:
forested 1

Figure B. 1 .b. Summaries of increasing support for fragmentation study:
forested 2

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Figure B.2.a. Summaries of increasing support for fragmentation study:

non-forested 1

Figure B.2.b. Summaries of increasing support for fragmentation study:

non-forested 2

Figure B.3.a. Summaries of increasing support for fragmentation study:

intermediate 1

Figure B.3.b. Summaries of increasing support for fragmentation study:

intermediate 2

Figure C.1.1.a. Forested l, 17 x 17 cells
Figure C.1.1.b. Forested 1, 33 x 33 cells
Figure C.1.1.c. Forested 1, 50 x 50 cells
Figure C.1.l.d. Forested 1, 67 x 67 cells
Figure C.1.1.e. Forested l, 83 x 83 cells
Figure C.1.1.f. Forested 1, 100 x 100 cells
Figure C.1.2.a. Forested 2. 17 x 17 cells
Figure C.1.2.b. Forested 2, 33 x 33 cells
Figure C.1.2.c. Forested 3, 50 x 50 cells
Figure C.1.2.d. Forested 2, 63 x 63 cells
Figure C.1.2.c. Forested 2, 83 x 83 cells
Figure C.1.2.f. Forested 2, 100 x 100 cells
Figure C.2.1.a. Non-forested 1, 17 x 17 cells
Figure C.2.1.b. Non-forested 1, 33 x 33 cells
Figure C.2.1.e. Non-forested l, 50 x 50 cells
Figure C.2.1.d. Non-forested 1, 67 x 67 cells

Figure C.2.1.e. Non-forested 2, 83 x 83 cells

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Figure C.2.1.f. Non-forested 1, 100 x 100 cells
Figure C.2.2.a. Non-forested 2, 17 x 17 cells
Figure C.2.2.b. Non—forested 2, 33 x 33 cells
Figure C.2.2.e. Non-forested 2, 50 x 50 cells
Figure C.2.2.d. Non-forested 2, 67 x 67 cells
Figure C.2.2.e. Non-forested 2, 83 x 83 cells
Figure C.2.2.f. Non-forested 2, 100 x 100 cells
Figure C.3.1.a. Intermediate 1, 17 x 17 cells
Figure C.3.l.b. Intermediate 1, 33 x 33 cells
Figure C.3.1.c. Intermediate 1, 50 x 50 cells
Figure C.3.1.d. Intermediate 1, 67 x 67 cells
Figure C.3.1e. Intermediate 1, 83 x 83 cells
Figure C.3.1.f. Intermediate 1, 100 x 100 cells
Figure C.3.2.a. Intermediate 2, 17 x 17 cells
Figure C.3.2.b. Intermediate 2, 33 x 33 cells
Figure C.3.2.c. Intermediate 2, 50 x 50 cells
Figure C.3.2.d. Intermediate 2, 67 x 67 cells
Figure C.3.2e. Intermediate 2, 83 x 83 cells

Figure C.3.2.f. Intermediate 2, 100 x 100 cells

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M.

INTRODUCTION

Landscape fragmentation is an inevitable consequence of human alteration of the
environment (Forman and Godron 1986). Numerous studies have shown the adverse
effects of habitat fragmentation on species behavior (e.g. Wiens and Milne 1989),
population levels (e.g. Walker et a1. n.d.), biodiversity trends and endemicity (e.g.
Vergara 1997). Fragmentation threatens the viability of wildlife populations by reducing
and isolating habitats, limiting population dispersal and interaction, interfering with host-
predator relationships, and causing genetic drift (Walker et a1. n.d.).

Fragmentation studies are not limited to wildlife habitats. Young and Jarvis (2001)
explored methodology to characterize the physical disruption to the structure of urban
landscapes, focusing on the connectivity of areas with different qualities and emphasizing
the “adverse effects of urbanization on the connections between quality habitats”.

As long as there are human habitation utilizing resources on landscapes for
sustenance, their fragmentation will continue. Understanding the relation of
fragmentation patterns and ecological processes is imperative in guiding policy to
optimize resource utilization without jeopardizing their sustainability.

However, even after two decades of quantitative landscape ecology (Kronert et al.
2001), fragmentation patterns still cannot be related to ecological processes on the
landscape (Wu and Hobbs 2002). Fragmentation indices are necessarily measured over a
given area, certainly not at just one point. The indices are thus discrete and may vary with
changes to the area it is measured over. On the other hand, fluxes of matter and energy
that describe ecological process are of a continuous nature. I argue that this

incompatibility between the factors of patterns and processes is the greatest hindrance to

relating them on a landscape, precluding comparisons between sites and/or with other
continuous variables and seriously limiting the scope of research that can be conducted.

Aside from new and further development of landscape metrics, Kronert et a1. (2001)
advocate standardized model parameters for comparison of indicators from different
studies. I maintain that this may not be enough, as the fundamental problem, aside from
model standardization, is the effects of scale inherent in landscape metrics. Many have
attempted to unravel the scaling relations of current landscape indiees (e.g. Frohn 1998,
Wu et al. 2002) in an effort to resolve the effects of scale, as data may not always be in
the appropriate scale for the desired inferences. Knowledge of scaling relations may lend
confidence to inferences drawn from data that were scaled up or down. But this research
is still a long way from relating landscape patterns and processes. Using the same scale
for studies at different sites does not guarantee they can be compared, unless the areas on
which the indices are computed were standardized for all sites. But are standardized areas
for all sites feasible or even plausible?

I propose a different approach to the same problem. Spatial processes form the bases
for methodology in developing a new index and strategies in identifying the areas over
which they should be measured. By computing such an index at selected areas or points
within the extent of the study, the fragmentation intensity surface could then be generated
through spatial interpolation, converting the initially discrete index into a continuous
variable over the landscape. With the generated fragmentation intensity surfaces, results
between sites would be comparable in the language of surface topology. Relating
landscape patterns to continuous processes on the landscape would now be possible, and

not just temporal changes over the same site or qualitative studies within a site.

I.

Patterns and Processes

Patterns strongly influence ecological processes (Turner 1989). But processes
may also affect observed patterns. Walker et a1. (2003) describe how social and
economic processes influence deforestation patterns in the Amazonian frontier. Arima
et a1. (unpub.) link deforestation in the Amazon to the spatial process of road
building. Understanding the underlying processes is critical in explaining phenomena.
Explaining phenomena is the essence of science. The need for landscape pattern
methodology has been often voiced, realizing that therein lay the key for
understanding the relationships between patterns and processes in landscapes (e.g.,
O’Neill et al. 1988, Golley 1989, Turner 1990, McGarigal and Marks 1995, Opdam et
a1. 2002, Wu and Hobbs 2002). Indeed, a major focus of landscape ecology is
quantifying relationships between landscape patterns and ecological processes
(McGarigal, et a1 2002). The future of landscape ecology depends on whether the
geographical (pattern) and the ecological (process) approaches in landscape studies
can be integrated (Opdam et al. 2002).

Bastian (2001) identifies as the outstanding features of landscape ecology its
focus on structures, processes, and changes; spatial and hierarchical aspects; and the
complexity of different factors in a landscape. He relegates to geography the essential
task of spatial processing and structuring of aggregated and integrative parameters of
landscape ecology. Landscape functions are not just a matter of “fluxes of energy,
minerals, nutrients, and species between landscape elements” (Forman and Godron
1986), but also how these fluxes relate to human society (Bastian 2001). However, to

achieve this view, and thus unify the science of landscape ecology, methodology is

II.

needed to relate anthropogenic activity on a landscape to the consequent ecological
and biophysical changes it will induce. Future ecologically sustainable systems will
need spatial arrangements that support the required ecological processes on those

landscapes (Opdam et al 2002).

Landscape fragmentation indiees

A landscape is an area of spatial heterogeneity in at least one factor of interest
(Turner et al. 2001 , F orman 1995). The most basic element in a landscape is the patch
or fragment, a surface area that is different in nature or appearance from its
surroundings. Perceptions of landscapes and patches are relative to the observer
(Wiens and Milne 1989). Thus modelling landscapes and patches are scale dependent.

Landscape indices were developed to quantify patterns in the landscape. They
abound in the literature, measuring everything from distribution, area, density, edge,
shape, and core area; isolation and proximity; contrast; contagion, interspersion,
connectivity, and diversity; etc., and at all levels, from a single patch, to classes of
patches, to the whole landscape (McGarigal et al. 2002). Literally hundreds of indices
have been proposed to characterize fragments (e.g. O’Neill et a1 1988, McGarigal and
Marks 1995, and Frohn 1998). Most are single parametered (McGarigal et a1. 2002),
but a few have multiple parameters (e. g. Hurd et al n.d, Vergara 1997).

Landscape indiees have so proliferated that McGarigal et a1. (2002) report
attempts by others to derive a minimum parsimonious set of independent indiees to
describe the landscape. No consensus has yet been reached and they conclude that a

single parsimonious set may not even exist. Factor analysis by Cain et a1. (1997) and

Riitters et a1. (1995) find many indiees redundant, but agree on factors interpreted as
texture, patch perimeter-area scaling (an indicator of patch complexity) and number
of attribute types. However, the former did not test for patch density-area scaling (an
indicator for patch density), which the later included in their contributory factors.

None tested for patch arrangement.

Table 1. Non-redundant univariate landscape fragmentation indices (Riitters et a1 1995).

 

 

 

 

 

 

 

 

 

 

Index Formulation Description
Average patch OER PA-l measures average patch compaction.
area perimeter- l/m * kazl --------- OE... is the number of outside edges

area ratio sqrt(Sk) enclosing patch k and St. is the area of patch
(PA-1) k in pixels.
Shannon Z‘Fl 23:1 vi]- ln(vU) SHCO measures image texture.
contagion 1 + ij = Aij/ E‘FIZ‘H Azjj for adjacency matrix
(SHCO) 2 ln(t) A (O’Neill et a1 1988, Li and Reynolds
1993)
Average St. NASQ measures average patch shape. OE],-
normalized l/m * ka=1 16 ---- and 8;, are as above.
area, square OEzk
model (N ASQ)

Patch [5; is the estimated slope of regression of
perimeter—area l/Bz ln(OCk) on ln(Sk) for all patches that do not
scaling (OCFT) touch the border of the map. OCR is the

number of outside pixels enclosing patch k
and St is as above.
Number of The number of different kinds of patches
attribute classes there are.
(N TYP)

 

 

Wu and Hobbs (2002) identify the following as some of the top research topics of
landscape ecology in the 21St century: ecological flows in landscape mosaics; causes,
processes, and consequences of land use and land cover change; integrating humans
and their activities into landscape ecology; and relating landscape metrics to

ecological processes. But for all the landscape metrics developed and used in the past

two decades, the technical and ecological understanding of these metrics are still
inadequate, and the relation of landscape metrics to ecological processes remains

unresolved.

III. Scale

Unfortunately, the geographic concept of scale has contradictory contexts for
geography and the other branches of science (Atkinson and Tate 2000). Cartography
treats scale strictly as a ratio of map distance to its corresponding distance on the
ground. Hence 1:20,000 is a larger scale than 1:5,000,000 in the cartographic sense.
However, scale is also equated with the scope of a phenomenon in most other
sciences. Analysis in a township, such as would be covered by a 120,000 map, is
considered of smaller scale in biology, physics, and sociology than would that in a
region described by a 1:5,000,000 map. To avoid ambiguity in this study, reference to
the direction of the change in scale will be minimized or expounded in reference to its
aspects.

Scale may also be taken to mean the level of detail observable (Tate and Atkinson
200]), corresponding to the geographic sense of the word. The higher the scale, the
more detail observable, and vise-versa. Moreover, a change in scale may also mean a
change in the size of the aggregation area used for computing attributes of the data.
Hence, geographically, increasing scale may increase the map-to-ground distance
ratio used, increase the amount of detail observable, and/or decrease the area over

which data is aggregated, and vice-versa.

IV.

Patterns detected in a landscape are a function of scale (F orman and Godron 1986,
Levin 1992). The landscape ecology literature (McGarigal et al. 2002, Turner et al
1989, O’Neill et al. 1996) lists only two aspects of scale that affect observations on a
landscape, i.e., extent and grain. Extent is the boundary of the study, corresponding to
the upper limit of spatial analysis. Grain is the smallest unit of observable space, akin
to spatial resolution in raster imagery.

A fundamental difference between how landscape ecologists and geographers
perceive and model space is the concept of spatial support. Akin to the kernel in
image processing and geographic information science (ESRI 1992), spatial support is
the discrete space or spatial unit of aggregation over which a spatial process is
considered (Cressie 1993), or the domain over which a geographical variable may be
measured (Tate and Atkinson 2001). It may be as small as the grain, as large as the
extent, or anywhere in between, and need not be restricted by size, shape, or
orientation. Spatial supports may not be mentioned in the landscape ecology
literature, but landscape indiees necessarily have to be computed over them. And
whenever the areas of interest are larger than the grain or smaller than the extent, the

support will have to be distinct from either.

The bias of scale

The Mirriam-Webster online dictionary defines bias, among others, as:

1) deviation of the expected value of a statistical estimate from the quantity it
estimates, or

2) systematic error introduced into sampling or testing by selecting or

encouraging one outcome or answer over others.

The use of “the effect of scale” seems to me as defining the problem by itself,
which I find ambiguous. Although a systematic error is the more prevalent definition
of bias in statistics, I propose rather to use “the bias of scale” to mean changes
induced in the quantification of a spatial index as the scale at which it is being
measured is changed, regardless of direction.

Grain and extent influences the numerical results of landscape metrics (Gergel
and Turner 2002). Changes in extent affect observations at the edges, and edge effect
biases may be controlled systematically or by simply using an extent larger than the
area of interest and then clipping off the edges of the resulting image to obtain a
smaller extent.

Changes in grain size have a more profound effect on the observed patterns
themselves. Details are lost as grain becomes coarser (Turner et al. 1989). Using
different grain or raster cell sizes in examining landscape metrics give rise to
quantitative differences in the indices (Kronert et a1. 2001). Wu et al. (2002) report on
the effect of changing grain and extent on landscape metrics, and classified landscape
indices into three categories according to their scaling relationships, i.e. predictable,
stepwise, or erratic. But, like others have done in the landscape ecology literature
(e.g. Frohn 1998, Turner et al. 1989), change in grain sizes were simulated by using a
majority filter, aggregating the original data. Doing this avoids the multi-dimensional
problems of using satellite imagery from different platforms, where differences in
observations may be attributed to differences not only in spatial but also in temporal,

radiometric, and spectral resolutions as well.

In the geographical information literature, Dungan (2001) analyzes the effect on
vegetation studies of varying remote sensing scales by using products of different
spatial resolution from two platforms. However the reference is to a change in
support, even if the study acknowledges work by others where remotely sensed data
are artificially degraded to study the effects of changing pixel sizes.

The apparent confusion as to which is grain and which is support lies in the fact
that grain can be as small as the support. Support size depends on what the area of
interest is. Landscape metrics are influenced by spatial support or the size of the area
over which they are computed. Observations are affected through changes in
aggregation, one aspect of the familiar modifiable areal unit problem or MAUP (Tate
and Atkinson 2001). MAUP is the dependence of spatial relations on supports
(Cressie 1993), and different approaches for its solution have been suggested. (e.g.

Hay et al. 2001).

Limitations of the current landscape indiees

Most landscape indices are defined discretely (McGarigal and Marks 1995) over
an arbitrary spatial support, or use a moving kernel of fixed dimension to derive the
supports (Hurd et al. n.d.). Changing the support or kernel size may change the
resulting index (Riiters et al. 2000). Hence landscape fragmentation studies are
usually limited to a single site, either at one point in time, as most applications in the
literature are, or involve changes over a temporal dimension (e.g. Southworth, et a1.
2002). The dependence of landscape indices on support or kernel size is precisely

what causes the bias, precluding studies across sites or with other continuous data.

Thus we need to “refine landscape metrics to reflect gradient values and allow a more
realistic representation of landscape pattern” (Theobald n.d.).

Moreover, all landscape indices in the literature give only an indication of
fragmentation. Hurd et al. (n.d.) measure forest fragmentation in six qualitative terms,
combinations of two forest continuity levels and three levels of proportion of forests.
O’Neill et al. (1996) used three landscape indiees (contagion, dominance, and shape
complexity) and proposed to measure fragmentation within the three dimensional data
space of these indices as the relative distances of a given landscape to landscapes that
were totally fragmented and totally intact. Riiters et a1. (1996) consider using single-
valued indices in describing landscapes the ideal case precisely to avoid multivariate
state spaces. Which begs the question: How do we relate fragmentation to any other
quantity, and thus provide insight on the relationship of patterns and processes in a

landscape, when fragmentation itself can not as yet be quantified?

VI. Statement of problem
There is currently no normalized, single-valued, multi-parameter landscape index
that is not biased by at least support size and will measure fragmentation as a
continuous spatial process on the landscape. Such limitation precludes comparative
landscape fragmentation studies across sites and/or with other biophysical attributes,
seriously hampering research and understanding of patterns and processes on the

landscape.

10

VII. Objectives
To relate fragmentation to processes in the landscape, I seek to explore methods
to develop an index that describes fragmentation as a continuous phenomenon over
the landscape and propose strategies to determine the location and size of the areas
over which they should be computed. Specifically, I will:

1. Propose a landscape fragmentation index that describes fragmentation as a
spatial process.

2. Derive nested supports of actual landscape scenarios from classified satellite
imagery over which the proposed index will be computed to evaluate its
ability to describe fragmentation.

3. Explore methods to confirm the bias of grain through regression of the
computed landscape indices from two co-registered images of different
resolutions against their corresponding supports.

4. Develop methodological frameworks to determine the location of points or

supports for generating the landscape fragmentation intensity surface.

VIII. Fragmentation
The landscape fragmentation index discussed here is an improvement of one I
proposed (Vergara 1997) to answer the question: “Given two landscapes defined on
equal supports, which is more fragmented, and by how much”? I consider patch
density, patch arrangement, and patch shape complexity as the essential attributes of

fragments to characterize their configuration within an area and thus describe

11

fragmentation as a spatial process, although I defer defining spatial processes until

later.

The landscapes to be analyzed were derived from satellite data classified into

binary images of forest-dominated landscapes (represented as 0’s) and non-forest

fragments (represented as 1’s). The raster data model is used throughout. In the

discussion, non-forest fragments are synonymous to patches and will be used

interchangeably. Index and metric will also be taken to mean the same thing.

A.

Patch density, arrangement and shape complexity

To illustrate how the above attributes affect fragmentation, consider three
hypothetical landscapes where one of the above parameters is varied while the
other two are held constant. First define landscape fragmentation as the breaking
up of the original cover, say forest. Fragmentation decreases the areas of
contiguous, convex-hull polygons of forested areas, i.e. polygons such that any
two points within can only be joined by a line that may touch but does not cross
an edge.

For the first case, obviously as the number and/or size of fragments in the
landscape increases, so will the proportion of patch pixels to non-patch pixels,
increasing fragmentation (Figure 1.1).

The arrangement of patches over a landscape also affects fragmentation.
Assume next a landscape with a fixed number of equally sized fragments that can
be moved about. Fragmentation decreases as the average distance between

patches decreases, i.e. as patch arrangement changes from uniform, to random, to

12

clustered, and the size of whole and contiguous non-patch areas increases without

changing the patch pixel to non-patch pixel ratio (Figure 1.2).

 

 

 

 

 

 

 

 

 

 

Fragmentation J

Figure 1.1. Increasing number and sizes of patches

 

 

 

 

 

 

 

 

 

 

 

 

I . - I I
I I
. I
I I
I I I I .
I I I
I . I
I I I ' I
4 Fragmentation

 

Figure 1.2. Uniform, random, and clustered patches

For the last case, consider a landscape with a single large but pliant patch.

Fragmentation increases as patch shape complexity increases, i.e. as the shape of

the patch goes from compact or regular to a complex, non-convex hull, and the

13

size of whole, contiguous, non-patch areas decreases despite constant patch pixel

to non-patch pixel ratio (Figure 1.3.).

 

 

 

 

 

 

 

 

 

Fragmentation >

Figure 1.3. Patch shape complexity

From the above illustrations, it should be clear that landscape fragmentation is
more intuitively related to the number and size of the patches, rather than to the
complexity of their shapes. Shape complexity can increase, increasing
fragmentation, without increasing the patch to non-patch pixel ratio. Increasing
patch density necessarily increases this ratio, and thus affects fragmentation

dramatically.

B. Estimates of index parameters
Patch density
Patch number and size can be estimated by patch density, which is simply the
proportion of patch pixels to the total number of pixels in the support. With all
else held constant, the higher the patch density, the more fragmentation there is.

Patch density is bounded between 0 and 1.

l4

Patch arrangement

The arrangement of patch pixels can be estimated by the quadrat coefficient.
Quadrats are equal areal divisions of space. The QUADRAT module of IDRISI
assumes the input image contains integer counts in its pixels. It computes the
mean number of counts or density per pixel, and the variance in the totals of the
counts in the quadrats (Eastman 1996). The quadrat coefficient is the ratio of the
quadrat variance to the mean. In binary images, since a pixel can have at most a
value of 1 (for patch pixels), the quadrat mean is also the patch pixel density, the
proportion of patch pixels to the total number of pixels.

The quadrat coefficient ranges from 0 to 2. Values less than 1 indicate more
uniform arrangements, those closer to 1 indicate random arrangements, and those
greater than 1 indicate clustering. When the patch density is 1, every pixel in the
support is 1 as well, and the quadrat variance will be 0, so the quadrat coefficient

is 0. However, when the patch density is 0, the quadrat coefficient is undefined.

Patch shape complexity

Due to limitations in the indices used to measure it, patch shape complexity is
the parameter that is most difficult to model. The fractal dimension (de Cola
1989) is one of the earliest indiees developed to measure patch shape complexity.
Ranging from I for regular polygons to values just less than 2 for more complex
shapes, it is usually computed as the slope of the regression of the logarithm of
the patch perimeters against the logarithm of the square roots of the corresponding
patch areas. Issues concerning the fractal dimension include computational as

well as conceptual ones (Frohn 1998, Leduc et al. 1994). Firstly, it is impossible

15

to compute the fractal dimension of a single patch by regression. Secondly, most
patches are not true fractals, which should be self-similar at whatever scale they
are observed (Mandelbrot 1982) and hence violate an assumption on its
application.

Frohn (1998) developed an alternative to the fractal dimension that is more
appropriate to the spatial raster data model on which satellite imagery is based:

SqP = 1 — (4 * sqrt(Area) / perimeter).

Both area and perimeter are that of pixels and are thus unitless. SqP ranges
from 0 for a square, the most compact polygon in a raster model using square
pixels (since 4 * sqrt(area) = perimeter, thus SqP = O), to values closer to 1 for
more complex shapes (where 4 * sqrt(area) < perimeter, thus SqP > 0). The issues
related to the fractal dimension are thus resolved since SqP can now be computed
even for a single patch. Neither should there be concern on whether the patches
are true fractals since that was never an assumption of SqP.

However, SqP has its own limitations. One issue with SqP is when there is no
patch, and both area and perimeter are 0. Will SqP equal 1, corresponding to the
most complex shape even if there is none, will it be undefined, or does it go to
negative infinity? There are other issues regarding the use of SqP in classifying
satellite data (Messina et al n.d.). But the major concern with SqP is the inability
to distinguish between a single patch and a group of patches. A simple example
should illustrate the point.

Imagine a single, square patch pixel. SqP = l — (4 * sqrt(l)/4) = O in this case

for the most compact shape in the raster data model. Now imagine four separate

16

and distinct patch pixels. Logically, SqP should also be 0 since the number of
patches should not affect patch shape complexity if the patches have the same
shapes. But SqP = 1 — (4 * sqrt(4)/ 16) = 1 — (4 * 1/8) = 1 — ‘/2 = ‘/2 in this case,
which is the shape complexity of a patch polygon composed of four patch pixels
joined only at the corners.

But despite this major issue concerning SqP, rather than use two indices for
different instances, i.e. either a single or several patches, I will continue to model
patch shape complexity with SqP for the work to proceed, until a better alternative

is available.

Development of the proposed index

As a start, since fragmentation increases with patch density and shape
complexity, and decreases as clustering increases, i.e. as the quadrat coefficient
increases, the fragmentation index should be of the form of the explicit function
below:

g( d. SqP)

M?)

f

 

, where g and h are functions,

with the following assumptions:
f := fragmentation index
d := patch density, 0 <= d <= 1,

SqP := patch shape complexity, 0 <= SqP <= 1, such that SqP = 1 if d = 0, and
SqP = 0 if d = 1,

Q := quadrat coefficient, 0 <= Q < 2, such that Q = 0 ifd = 0 or d = 1.

l7

To keep f defined when Q = 0, let h(Q) = l + Q. Let g(d, SqP) = d * SqP,
thus:
d * SqP
f= ,suchthat0<=f<=l
l + Q

sinced <= 1, SqP<=1,andl+Q>=l.

 

But there is a computational issue involved with this definition. When d = 0,
i.e. when there is no patch, Q = O and SqP = 1 by the above assumptions, hence
f = 0, which is what we want. However, when d = 1, i.e. every pixel is a patch
pixel, then SqP = 0 and Q = 0 again from the above assumptions, and f = 0 as
well, which is contradictory.

To remedy this, we instead let g(d, SqP) = d * (1 + SqP), thus:

d*(1+SqP)

 

l + Q
so when d = 1, SqP = 0, Q = 0, and f= l, which is what we need.

But for d and SqP close to 1, Q could be close to 0 and it is possible that the
index will be greater than 1. Hence the numerator must be weighted for the
bounds to hold. Since the index is more intuitively related to patch density than to
SqP, we let g(d, SqP) = d * ( l + ( 0t * SqP)) to control the influence of SqP so as
to keep f bounded, thus:

d*(1+(a*SqP))

 

1 + Q
such that or is a constant to control the influence of patch shape complexity on the

index, and 0 <= f <= 1 holds.

18

IX.

As it is presently defined, the weight on SqP is the most questionable part of
this model. But until improved estimates of the parameters are available and
definite relationships between them are established, the above model will be used

for the purpose of continuity.

Overcoming the bias of spatial support

Assume fragmentation intensity to be continuous over the landscape. Computing

the fragmentation index at the critical locations to define the characteristic intensity

surface is an essential step in its generation. Although there are methods to derive the

surface directly from the binary data, the surface generated in this manner may not be

appropriate for analysis, as will be explained below.

A.

Smoothing the surface

Raster images of discrete, polygon features can be converted into a surface of
continuous values by a number of ways. Image processing and geographical
information systems rely on numerical techniques to smooth images of discrete
areas or points, such as those for deriving digital elevation models (Eastman
1996). Statistical methods exist for interpolating surfaces from zonal (areal) data
(Bailey and Gatrell 1995). Brief descriptions and discussions on their

inappropriateness follow.

1. Low pass or mean filter
Generally used in image processing, this technique involves a moving

kernel, usually square of odd dimension, over the whole input image,

19

averaging the pixels surrounding the center of the kernel, and assigning the
result to the corresponding pixel in the output image (Eastman 1996). The
result is an averaged image with smoothened boundaries between discrete
polygons. The process is repeated until the desired effect is achieved. This
technique is a quick and easy way to handle extreme values in image
processing or geographical information systems but would be an inappropriate
method for producing the fragmentation intensity surface. The primary issue
is the subjectivity in deciding when the desired surface smoothness has been

achieved.

Pycnophylatic interpolations

This technique smoothens discrete surfaces, such as populations in census
tracts, while maintaining the volume under the areas of the original surface
(Tobler 1979) and satisfying a non-negativity constraint (Figure 2.). What is
taken off one cell is added to another within the same area. Again the
technique is applied repeatedly until a number of iterations has been reached
or a smoothing requirement satisfied.

However, since the fragmentation data has only 0’5 and 1’s, smoothing the
surface in this manner would necessarily produce cells that would be greater
than 1 near the centers of the fragment polygons. Moreover, the non-
negativity constraint would have to be relaxed to smooth the non-patch areas,

and may produce cells that are less than 0. Aside from the arbitrariness of

20

deciding when to terminate the procedure, interpretation of values greater than

I or less than 0 would be ambiguous.

 

 

0 iteration S

  

5 iterations

 

10 iterations

 

  

 

15 iterations

 

 

20 iterations

 

  

26 its ratio ns

 

Pycnophylactic Interpolation

Figure 2. Pycnophylactic interpolation

 

Kriging

Kriging is a statistical method of spatial interpolation for point, lattice, or

zonal (areal) data (Cressie 1993). There are different approaches for handling

various data types, and all produce a measure of uncertainty of the predictions.

Block kriging is used for zonal or areal data, the closest to thematic images.

But the complex, non-convex hull shapes that non-forest fragments may take

are not amenable to use as zones in block kriging. Neither do the thematic

values of 1’s and 0’s reflect the true intensity of fragmentation.

21

 

B.

Spatial processes

To develop the approaches for determining the size and locations of the
supports over which to compute the indices to be used in generating the
fragmentation intensity surface, the concept of spatial processes must be
introduced. A spatial process is a stochastic process operating over a structure of
spatial objects. The random, stochastic process defines the relationships between
the deterministic spatial objects, as manifested by their current configuration.
Observed patterns in a landscape, i.e. the arrangements of spatial objects, are
assumed to be random manifestations of the many possible realizations of the
hypothesized underlying spatial process (Tiefelsdorf 2000). In our case, the
spatial objects are the patch and non-patch pixels, and the supports over which
they are observed. All other features (points, lines, and polygons) in a binary
raster model of space would have to be defined by the arrangements and states of
the pixels within the support. Hence detection of a spatial process is dependent
upon the support used as well as the arrangement and values of the pixels
contained therein. Spatial processes are not mutually exclusive and several may
be operating under any given area being observed, complicating analysis. At
present only the most fundamental manner of detecting a spatial process will be

explored.

1. Areal approach

By finding the support size when a change in spatial structure is detected,

the maximum support encompassing the underlying spatial process may be

22

determined. Fragmentation is assumed to be homogenous over the support for
which a single spatial process was detected, and computing the index over this
support assures us that we are free of the pitfalls of the modifiable areal unit
problem (Tiefelsdorf 2000). Determining the range of supports where the
spatial structure changes then is instrumental, and the search can be applied
iteratively to obtain the maximum support just before the change in spatial

structure occurs, or until the least non-trivial support size is reached.

Point estimation approach

To estimate the fragmentation index at a point or a pixel, a spatial
autoregression model over nested supports of increasing sizes centered on that
pixel is needed. Although conceptually easy to comprehend, the total
dependence of sequential observations would lead to autocorrelation in the
observations and/or the residuals. Autocorrelation in the residuals would lead
to inaccurate standard errors or invalid inferences (Poole and O’Farrel 1971,
Johnston 1978) for ordinary least squares regression To remedy this,
information on the nature of the autocorrelation function is needed. If an
autoregressive model could be specified to handle totally dependent
observations, the intercept and standard errors at the points already estimated
could be used to determine the search radius to the next point. Criteria for
determining the spatial arrangement of points to be estimated and used for

generating the surface would now exist.

23

C.

Generating the fragmentation intensity surface

Once the fragmentation index has been computed for the determined supports
or estimated for a sample of points or pixels, there are again numerical and
statistical techniques for generating the surface. By converting the areal data into
points through computation of the centroids, the surface could be generated
graphically as a digital elevation model or a gravity model (Eastman 1996).
However no estimate of accuracy, precision, or uncertainty of the generated
surface will be produced.

The surface can also be generated statistically from point data by ordinary
kriging on the centroids, or from zone (areal) data by block kriging on the
supports, with corresponding estimates of the uncertainty of the results (Bailey
and Gatrell 1995). These techniques will not be expounded here as they are
beyond the objectives of the study, although it is worth mentioning the
compromises in accuracy and precision when generating the fragmentation

intensity surface from point or areal data by numerical or statistical interpolation.

24

MATERIALS AND METHODS

To explore methods that may help confirm the bias of grain on the proposed
fragmentation index, two satellite images of the same area acquired as close as possible
in time but at different spatial resolutions will be analyzed. Since it has already been
established that more detail will be observable with finer grain, and that changes may be
due to more than just spatial resolution, it seems reasonable to expect the index to vary
for co-incident supports on images of different grain. However, it remains to be seen if
the estimates of the fragmentation index for null supports are also affected by changes in
grain. In the methodology for overcoming the bias of spatial support, only the ETM+

imagery will be used.

I. Satellite imagery

The satellite images used are ETM+ (30 x 30 m) acquired in 1999 and IKONOS
(4 x 4 m) acquired in 2001 of the Legal Amazon in Uruara, Para in Barzil (Figure 3).
The images were co-registered and corrected for atmospheric effects. Supervised
classification with ERDAS Imagine using all seven bands (including the thermal
Band 6) in the ETM+ image produced 14 classes, which were reclassed into five
themes: forest, deforested, cloud, shadow, and water. The final classification shows
only forest and non-forest classes. Unsupervised classification on the Ikonos image
produced at least 40 isodata classes, which were reclassed to the same five themes,

then generalized into forest and non-forest classes as well.

25

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26

      

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Forest
a Non-forest

Note:

Figure 4. Typical increasing support for fragmentation study

27

II.

III.

Nested supports

The classified images were imported into IDRISI32 for subsetting. Two co-
incident cases each of forest-dominated (forested 1 and forested 2), non-forested
(non-forested 1 and non-forested 2), and intermediate (intermediate 1 and
intermediate 2) scenarios with the least variation were selected from the classified
ETM+ and Ikonos imagery. The maximum support common to all areas without
going beyond the extents was determined, and as an exploratory start, five other
nested supports were generated (for typical nested supports, see Figure 4). A total of

72 (2 images x 3 scenarios x 2 cases x 6 supports) images were subsetted.

Fragmentation index

The fragmentation index to be used is:

d*(l+(a*SqP))
f= ,where

1+Q

 

f := fragmentation index, 0 <= f <= 1, such that f = 0 if totally forested and f = 1 if
totally deforested,

d := patch density, 0 <= (1 <= 1, such that d = 0 if totally forested and d = 1 if
totally deforested,

SqP := patch shape complexity, 0 <= SqP <= 1, such that SqP = l ifd = 0, and
SqP = 1 if d = 0,

Q := quadrat coefficient, 0 <= Q < 2, such that Q = 0 ifd = O or d = l, and

or := constant, 0 <= 0t <= 1.

The index was computed on the determined supports using six values for or (0, 1,

0.25, 0.125, 0.01, and 0.001) and the effect of (1 on the index was noted. The total

28

patch area, total patch perimeter, quadrat coefficient, and patch density were derived
in IDRISI32 for all 72 supports. SqP and the fragmentation index were computed in

Windows Excel.

IV. Variograms and first lag gammas
Variograms are devices used to explore spatial dependence or the second order

properties in a spatial process (Bailey and Gatrell 1995). The subsetted, classified

 

ETM+ images were converted into ASCII vectors of 0’s and 1’s and turned into
spatial point process objects in S+. Variograms (formally called semi-variogram) for
the lagged pixel distances were computed, plotted and modelled using the exponential
variogram model with an emphasis on the front end of the variogram.

Theoretically, Bailey and Gatrell (1995) define the variogram as the difference in
the variance and covariance due to distance h as:

you) = 62 — C(h),

where h = s, - sj is the vector difference between s, and 31- for a spatial stochastic
process Y(s) with variance 02(5), mean u(s), and covariance C(h) due only to the
distance h, thus:

C(h) = C(Si - SJ) 2 C(St, 8}) = E((Y(Si) - H(Si))(Y(SJ) — 11(5))»-

A spatial process is stationary if the mean and variance are constant and
independent of location, i.e., p(s) = p and 02(5) = 62. The variogram model assumes a
stationary process. If first order or global trends do not affect a spatial process,
variance levels off as distance between point pairs increases. The maximum variance

that the variogram model reaches is called the sill. The distance between pixels where

gamma

the variogram model reaches its sill is defined as the range, the maximum distance up
to where spatial dependence manifests, beyond which the spatial objects are
independent, and the variogram model is flat in that region as it approaches an

asymptote (Figure 5).

 

 

 

 

 

 

 

L0 _" sill ——“‘— v .3
O
8 _ o o o o
' o
D 0 0 ° 0
L0 0
Q
Q ——1
Q
'd’
C)
Q ~—
C)
m
C)
D. —
(:3
N
C3
Q —
D
S
Q —t
C, range
_ objective = 0
D.
C) ‘l
I I I l
U 10 2D 3!]
distance

Figure 5. Typical variogram and exponential model showing nugget, sill and

range variances.

The variogram model starts at its nugget, which ideally should be 0. When it is
not, there are may be sampling errors or variability in the very shortest distances that
the model cannot account for. The behavior of the range, the first lagged gamma
values, and the overall variances over the changing supports were plotted,
summarized, and explored, with the aim of determining if these could be used to

indicate a change in spatial structure.

30

An absence of spatial dependence in a spatial process, as what would happen
within a totally forested or deforested support, would result inia flat variogram,
referred to as a pure nugget model. A change from a pure nugget model to one that
has a range and sill indicates a change in spatial structure, hence a change in spatial
process. A variogram that is not pure nugget but has no apparent sill or range is

unbounded, with spatial dependence increasing with distance between pixels.

V. Autocorrelation

The spatial point process objects of 0’s and 1’s from the classified ETM+ images
were converted into lattices, and Moran’s I and Geary’s C based on 1000
permutations (Kaluzny et al. 1997) were computed using the 8 nearest equally
weighted neighbors. The king’s case with equal weights includes normal as well as
diagonal joins and thus equally considers adjacent pixels in any direction. Moran’s I
and Geary’s C were tested over the number of permutations for statistical difference
from a random arrangement of the pixels in a support. The change in statistical
significance of the measures of spatial autocorrelation was used as an indicator of a

change in spatial structure as support size changed.

VI. Regressions
The fragmentation index at different 0t levels over the supports from the ETM+
and Ikonos images were evaluated, and an or was chosen for the regression. The

computed indices were regressed against the absolute areas of the corresponding

supports using ordinary least squares (OLS). As an exploratory start to confirm the

31

bias of grain size, the slope and intercept estimates of the two images for all cases
were compared. If the estimates of one image are within the bounds of the standard
errors of the corresponding case of the other image, this line of investigation may be

worth looking into in more depth.

32

1.

RESULTS AND DISCUSSION

Fragmentation index

At or = l, 0.25, and 0.125, the fragmentation indices were greater than 1 for some
of the non-forested and intermediate cases, notably for the smaller supports (Table
A.2.a, Table A.2.b, and Table A.3.a in Appendix A). The influence of patch shape
complexity to the index in these cases was excessive. At or = 0, patch shape
complexity is not considered, i.e., f = d / (1 + Q), and the resulting index intuitively
corresponds to patch density, such that there seems to be as much fragmentation as
patch density indicates (Tables A.1.a to Table A.3.b in Appendix A). However at this
or level the index can not distinguish between the smallest supports of the two non-
forested cases (Figures 6.1 and 6.2), which have one patch each, equal numbers of
patch and non-patch pixels (113 and 6, respectively) albeit arranged differently, and

equal fragmentation indiees (f = 0.959255 at or = 0).

Figure 6.1. Non-forested 1, 17x17 pixels Figure 6.2. Non-forested 2, 17x17 pixels

f= 0.959255 at or = 0 f= 0.959255 at or = 0
f=0.964148 ator=0.01 f=0.961831at0t=0.01
f= 0.959744 at or = 0.001 f= 0.959512 at or = 0.001
Note Forest
Non—forest

 

II.

At CL = 0.01, this deficiency is resolved and the index identifies the support with
more contiguous non-patch pixels as less fragmented (f = 0.961831) than the other
(f = 0.964148). Limiting the value of or to 0.01 affects only the second significant
figure of the index, thus 0 <= f <= 1 holds for the current cases and supports. At or =
0.001, at most only the third significant figure of the index is affected, and going by
patch density alone, 0 <= f <= 1 should hold for all cases and supports smaller than or
equal to 100 x 100 pixels. The index is again intuitively akin to patch density yet
differences in spatial arrangement of the fragments are discemable, while still being

bound between 0 and 1 (Table A. l .a to Table A.3.b in Appendix A).

Moran’s I and Geary’s C

Moran’s I and Geary’s C, computed over the closest 8 neighbors in a particular
support, should give a measure of how similar adjacent pixels are in that support.
However, what matters are not the values per se, but detecting changes in their
significance. Although the spatial structure and hence the spatial process may change
from one support to the next even if these statistics remain significant, a change in the
significance of these statistics from one support size to the next would indicate a
change in the spatial structures encompassed within these supports. The interest is in
the first such detected change, and the two supports at which this change was
detected. Performing this test iteratively, it should be possible to determine the
maximum support that encompasses a particular spatial process.

In the forested cases (Figure B.1.a and Figure B.1.b in Appendix B), both

Moran’s I and Geary’s C are not significant for the first two supports, but are

34

significant for the remaining four supports. The spatial structure thus changed
significantly between the second and third supports, and it is in that range of supports
that an iterative search should be performed. In the non-forested cases (Figure 8.2.3
and Figure B.2.b in Appendix B), Moran’s I and Geary’s C are not significant for the
first support, and significant for the rest. Hence the change in spatial structure
occurred between the first and second support. Moran’s I and Geary’s C are
significant for all supports in the intermediate cases (Figure B.3.a and Figure B.3.b in
Appendix B). The support should hence be further reduced until a change in the
significance of the autocorrelation measures is detected or the support reaches the

minimum, non-trivial case of 3 x 3 pixels where the search terminates.

III. Variogram modelling

Changes in the variogram model from one support to the next could verify
changes in the spatial structure encompassed by succeeding supports. A change from
a pure nugget model to one with definite structure indicates that a change in spatial
structure. In the forested cases, this occurs between the second and third supports
(Figure C.1.l.b and Figure C.1.1.c; Figure C.1.2.b, and Figure C.1.2.c in Appendix
C), which correspond with the results from the spatial autocorrelation measures. In
the non-forested cases, the change in the structure of the variogram model occurs
between the first and second supports (Figure C.2.1.a and C.2.1.b; Figure C.2.2.a and
Figure C.2.2.b in Appendix C), again corresponding with the results from the spatial
autocorrelation measures. For the intermediate cases, the variograms for all supports

show definite unbounded structure from the start (Figure C.3.1.a to Figure C.3.2.f in

35

Appendix C), and so a change in spatial structure cannot be concluded. So as in the

previous discussion, the supports should be further reduced, until a change is detected

or the minimum, non-trivial support size is reached.

The first lag gammas are the difference between the overall variance and the

covariance of adjacent pixels due to distance. If this value is low, then almost all the

variance in the spatial process can be explained by adjacent pixel covariance due to

distance. Plotting the first lag gammas for the six cases against support size (Figure

7.1) does not show any hint for use as an indicator of change in spatial structure as

support changes for all cases. Likewise, neither does the ratio of the first lag gammas

to overall variance (Figure 7.2).

Lag 1 GamavsStpportSizefa'SixSmports
0.1 ,
(lights?
0.08 j—a-fawed2 l
i—o—intemedatet l
0% da—immredaez 1
,+mfaeded1 4‘
0-04 57-: Massey?
0.02«
o

 

 

 

 

W (FIXES)

 

 

Figure 7.1 First lag gamma and support size for six supports.

36

IV.

Gamma 1 [Variance vs Support Size .

1.7 “
i \

\\

  

 

0.8 \
I, i 8 l \\ V1222 2,2222... .22j
. 5 ‘ \ .+forested1 ,
.l .: 1 \ tl+forested 2
. W 06 s . . . I.
‘ 2 ' \ “1+ intermediatet I
: ‘ \ ‘1», + intermediate 2 i,
. E l \ . . > ‘+ non-forested 1 I.
E 0.4 7 \\ . 4r non-forested 21
. (D ’ \ :- 2 2 .2 ..
- . l—
l .
0.2 .
0 .: 1' . . w ""1 ‘ ».
17 33 50 67 83 ‘ 100 ' " 1

Support (pixels)

Figure 7.2 Ratio of first lag gamma and variance vs support size for six supports.

Encompassing spatial processes

The methodology outlined above now forms basis for choosing the support size to
use when computing the fragmentation index. Support shape and orientation are
considered as well since only square supports were used. All that is left to consider is
the effect of locating the supports. Due to edge effects, centering the nested supports
at any other point or pixel within the determined support that encompasses a spatial
process could change the location and possibly the size of the resulting encompassing
support. Nesting the supports at a comer or an edge rather than centering them on a

pixel would also affect the location of the encompassing support.

37

V.

Estimating for null support

A spatial autoregressive model that can deal with totally dependent residuals
would be the ultimate method of rendering indiees computed over supports unbiased
by those supports. Tiefelsdorf (2000) describes methodology to detect spatial
structure by analyzing the spatial autocorrelation of the partial dependence of the
residuals of a spatial autoregressive model over non-overlapping zones or areas.

At present there is no autoregressive model to handle observations taken from
nested zones or areas. Such a model would be helpful in the remote sensing discipline
as well. As an example, to develop methodology for monitoring lake water clarity,
Wiangwang (2002) regressed Secchi disk depth measurements of Michigan lakes
greater than 25 acres against the trophic state index of those lakes calculated from
satellite imagery over standardized supports. A Secchi disk is usually a metal disk
painted black and white on alternating quadrants and attached to the end of a
graduated tape or cord used to measure turbidity in the water column. The Secchi disk
depth is the recorded water depth where the lowered disk just disappears from view.
For known sampling locations with global positioning system (GPS) coordinates, a
support of 15 pixels centered on the GPS fixes was used. Larger supports of up to 100
pixels were used for the deepest part of the lake where there were no GPS fixes. If a
nested spatial autoregression model can be specified, then there would be basis for
estimating the trophic state index at a point or pixel.

The results of the OLS regressions over the supports from the two images of
different resolutions show that none of the forested intercept estimates are significant

(Table 2). The reason may be that the regression was conducted for all six supports,

38

even if the preceding discussions show that the maximum support size should be
between the second and the third supports. The results may have been different if the
analysis were conducted over supports with the maximum somewhere between the
second and the third supports. In other words, the results from the autocorrelation and
variogram analysis would have been useful in identifying the supports over which to
conduct the regressions.

Of the remaining intercept estimates only one (non-forested 1) falls within the
bounds of their corresponding standard errors. All had equal corresponding signs for
their slope estimates. Except for one forested and one intermediate case, the slope and
intercept estimates and standard errors aren’t that far off. But it is inconclusive
whether they are significantly different or not. A two-way (resolution x case)
ANOVA test with two sub-samples each for six supports and four statistics (intercept
estimate and standard error, and slope estimate and standard error) will suffer from
very low degrees of freedom since only 12 observations can be used for each test.
Even if a three-way (resolution x case x support) ANOVA test for statistical
significance using all 72 observations shows that support as well as grain matters in
the cases presented, it will not be enough to conclude that the proposed index is or is
not affected by changes in grain. Hence the bias of the index due to grain cannot be

confirmed as yet.

39

 

7T able 2. Summarized slope and intercept estimates with standard

2 2.12 .272 ‘w

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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L j ETM+ Ikonos i
[forested 1 estimate std error estimate std error 1
l intercept 0.000931 0.000824 0.0001162 8.258E-05 I
L SL096 4.11E-07 1.69E-06 *6.08E-07 *1 .69E-07

L l ETM+ Ikonos _l
forested 2 [ estimate std error estimate std error j
12 interceptl-0.024683 0.015486 -0.021611 0.0138089 1.
L # #_§lgpej*0.000165 *0.000031 *0.000139 *0.000028 j
I ‘ l
f ETM+ Ikonos l
{Intermediate 1 estimate std error estimate std error I
l: intercept *0.518843 *0.064561 *0.256548 *0.020722

L slopel*-0.00043 *0.000132 *-0.00014 *0.000042

l l

I l ETM+ Ikonos

[intermediate 2 estimate std error estimate std error

I intercept *0.252256 *0.012867 *0.213569 *0.011228

f slope -5.096E-05 2.643E-05 -4.762E-05 2.3066E-05
l

P ETM+ Ikonos ,
[gen-forested 1 estimate 1 std error estimate std error i
i intercept *0.755530 *0.099767 *0.733656 *0.079684

E slope *-0.00069 *0.000204 *-0.00066 *0.000163

l .-
l l ETM+ Ikonos l
lnon-forested 2f estimate 1’ std error estimate std error 1
L intercept *0.978665 *0.015969 *1 .018715 *0.016883

1 slope *-0.00056 *0.000032 *-0.00059 *0.000034

 

 

 

 

 

 

40

CONCLUSIONS

The proposed index as defined so far was developed to provide a normalized,
intuitive measure of landscape fragmentation capable of discerning differences due to
patch density, patch arrangement, and patch shape complexity as the necessary conditions
to define fragmentation as a spatial process. To maintain the bounds, the estimate for
patch shape complexity was weighted to control its influence on the index as the
parameter in the numerator with supposedly lesser contribution to fragmentation. This
approach does seem arbitrary and since it was tested only for one site with a particular
fragmentation pattern, it arguably may not be applicable for other sites with different
fragmentation patterns.

The emphasis here is on the methodology to deveIOp the index and render it unbiased
by at least support size, not on the index itself. Further research on both natural and
synthetic landscapes preferably with improved estimates for patch shape complexity
would still be needed to verify the applicability of the weights or uncover the actual
relations between the estimates of the parameters of fragmentation. Be that as it may, at
least for this site an index to directly quantify fragmentation of binary landscapes is now
available.

The methodology to make the fragmentation index unbiased by the size of the spatial
support is independent of the index itself. This means that if such methodology could be
developed, any index computed over a spatial support that intends to measure an attribute
of the encompassed spatial process may be made unbiased by support size using the
methodology outlined above. The application of such methodology would be a

tremendous help in landscape ecology. Firstly, the generated fragmentation intensity

41

surfaces of two different sites would now be comparable in the language of surface
topology of cusps, manifolds, attractors, singularities, bifurcations, etc., as long as the
scale of analysis are roughly equal. Secondly, since biophysical processes on the
landscape are assumed to be continuous, their surfaces may also be interpolated from
sample data and compared to the generated fragmentation intensity surface, thus
providing a framework for relating patterns and processes on the landscape.

The preceding discussions outline methodology to detect a change in spatial structure
from one nested support to the next to determine the maximum support that encompasses
a spatial process, over which the fragmentation index is computed. The detection of
changes in spatial structure from the numerical results of the spatial autocorrelation
statistics corresponds to the graphical results from variogram modeling, and though the
methodology as developed so far may be elementary, it is nonetheless fundamental. It is
however possible that the spatial process may change between two supports but still be
indiscemible by the methodology presented. Further research into other indicators of
changes in spatial processes for such cases is thus needed. But despite this shortcoming,
the choice of support over which to compute the fragmentation index now has basis and
need not be arbitrary. The next concern then would be generating the fragmentation

intensity surface.

I. Areal approach
Methodology to determine the maximum support encompassing a spatial process
(over which to compute the fragmentation index) has been outlined above. But

defining a tessellation of the extent, i.e. non-overlapping supports to cover the entire

42

extent (Tiefelsdorf 2000), that is not prone to the complications of the modifiable
areal unit problem due to the bias of location of supports that the methodology as so
far described introduces, is another matter. The size and arrangement of the supports
for the same image may vary for every solution, as would the generated
fragmentation intensity surfaces. The generated surfaces will agree in general
structure but may have local variations. Thus the areal approach as outlined above
will at best be only heuristic.

A better strategy would be to predeterrnine the points or pixels at which the
supports should be located. Most GIS or image processing applications are capable of
distinguishing polygons greater than eight pixels without diagonal joins. The
centroids can then be found, even if they lie outside the bounds of the polygon, at
which point the search for the appropriate support begins, using center-nested
supports. Then the search shifts to a comer of the determined support over the
centroid, and using edge or corner nested supports, the process continues along all
edges and corners of the first determined support. The search continues to the external
supports until the polygon is covered fully by non-overlapping supports. Smaller
supports would thus cover the areas closer to the edges of the polygons, with larger
supports elsewhere (Figure 8). Doing this for all patches should cover the points of
interest within the extent. All that would be left are the areas with no patches or those
with patches less than nine pixels. Again the centroids are found for these areas and
appropriate supports determined. Even if all the supports do not cover the whole

extent edge to edge, all areas of interest would still be considered.

43

 

Figure 8. Typical areal estimation.

A better strategy for this approach is to pre-determine not only the areas of
interest, i.e. the patch polygons, but also the points at which to generate the surface. A
brief digression into elementary graph theory is needed here before the next
procedure can be expounded.

A graph is a set of vertices (or points) and edges (or lines) connecting any two
vertices, regardless of length, shape or manner of join (Matousek and Nesetril 1998).
A path is a trace of edges joining any two vertices. A graph is connected if a path

exists for every pair of vertices in the graph. A cycle is a path traversing all vertices

44

II.

once and returning to the starting vertex. A tree is a connected graph with no cycles.
There exists exactly one path for any two vertices on a tree. A spanning tree of a
graph is a sub-graph that contains all the vertices of the original graph. A minimum
spanning tree is a spanning tree with weighted edges such that the sum of the edges is
a minimum.

Hence all polygons are cyclic graphs. The vertices that define the polygon are
searched for and a minimum spanning tree with distance-weighted edges wholly
circumscribed within the polygon is then derived (Figures 9.1 to 9.4). The difference
in the definition above is the liberty of locating the vertices of the minimum spanning
tree. The solution should be unique if derived in this manner, although this is
postulated and stated here without proof. The vertices of the polygon and the
minimum spanning tree are precisely the points to use in generating the surface, for
which the appropriate supports would be determined, over which the corresponding

indiees computed.

Point estimation approach

Conceptually the ultimate approach to remedy the bias of support, it is
computational imprecise at this point until a spatial autoregression model is specified
that could handle observations taken from nested zones or areas. But if such a model
could be specified, there should be no ambiguity in determining the location of points
to be estimated, given each estimate has a measure of uncertainty. Using the standard
error of the intercept and the corresponding slope estimate of the regression line, a

pixel radius could be computed for every estimated point. By starting at a corner and

45

 

 

 

 

 

 

 

 

Figure 9.1. Non-convex hull polygons Figure 9.2. Internal vertices determined
with centroids and vertices. for spanning tree.

 

 

 

 

 

 

 

 

 

Figure 9.3. Typical spanning tree within Figure 9.4. Thiessen triangles for
the polygon. surface generation.

using the computed radius to locate the next point along an edge, selected points
along all edges could be iteratively located. And just as a jigsaw puzzle is solved after

the edges are determined, the internal points are determined next until the whole

46

extent is covered from the edges to the center. Following this procedure will result in
smaller search disks near the edges of the polygon and hence more points to generate
the surface, and larger search disks and hence less points elsewhere (Figure 10).

With a lattice of points now determined, the surface can be generated as a digital
elevation model (DEM) or by kriging. The arrangement of points may be over-
specified for the DEM, and thus be non-optimal, but each point would have an
estimate of uncertainty from which the integrity of the generated surface may be
gleaned. Kriging would also generate the surface with corresponding estimates of
uncertainty as well. Either result might not be distinguishable from the other, given
the extensiveness of the sample points from which to generate the surface.

However, since the point samples will be over-specified anyway, there really is no
compunction in using the OLS estimates for the points. The true estimates won’t be
that far off, just that the standard errors may be too optimistic. But an over-specified
sample of points could compensate for the loss of precision in the standard errors.

Still the best solution would be to pre-determine the points for generating the
surface, as in the previous discussion of finding the minimum spanning tree, and then
estimate the fragmentation index at those points using nested spatial autoregression.
Done this way, the configuration of the points for surface generation should be

optimal, although again this is postulated and stated here without proof.

47

 

Figure 10. Typical point estimation

48

RECOMMENDATIONS

A multi-parametered, single-valued, normalized landscape fragmentation index is
proposed that can discern differences in patch density, patch shape complexity and patch
arrangement and describe fragmentation intuitively.

The underlying spatial processes is now the basis for determining the size of the
supports to use in calculating landscape fragmentation. Doing so avoids the
complications due to the modifiable areal unit problem. Generating the continuous
fragmentation intensity surface from the computed indices encompassing single spatial
processes on the determined supports will result in a fragmentation intensity value
anywhere in the extent.

Although computationally intensive and requiring at least thrice the amount of
computer memory to store the original image, the frameworks outlined here for
generating the fragmentation intensity surfaces are tractable. The methodology was
developed for forest fragmentation analysis, but it should be applicable to any binary
landscape, such as urban and non-urban areas, agricultural and non-agricultural sites, etc.
Four major issues remaining are:

1. The availability of better estimates of patch shape complexity to improve the

model and determination of their relationships,

2. Specifying a nested spatial autoregression model that can handle totally dependent

observations,

3. Confirming the uniqueness of the solution for the minimum spanning tree given

any polygon, and

49

4. Optimizing the search for the vertices of the minimum spanning tree at which to

compute the indices and generate the fragmentation intensity surface.

There are other fundamental issues as well. Three parameters necessary to
characterize landscape fragmentation were identified, i.e. patch density, patch shape
complexity, and patch arrangement. Are they the sufficient conditions as well? If so, then
the model may be treated as a mathematical function, and if continuous over a range, the
inverse may exist. Identifying the necessary and sufficient conditions of a functional
relationship lends confidence to the inferences that may be drawn by using them.
Moreover, verifying that the inverse function exists paves way for fiirther research into its
applicability. Is 1 — f the fragmentation index for the non-patch pixels?

For now, these are the only parameters that come to mind, although that gives no
assurance that they are the only ones. Future research may help to identify other factors
that characterize fragmentation in the landscape, or confirm that these indeed are
necessary as well as sufficient.

The appropriateness of the estimates of the model parameters may also be questioned.
Does the quadrat coefficient really measure patch arrangement? ls SqP the best estimator
for patch shape complexity? For now these are what are available, and the model would
definitely benefit from improved parameter estimates subject to their availability.

Lastly the assignment of the weight for SqP may also be scrutinized as arbitrary. The
parameters to characterize landscape fragmentation are inter-related, and more work is
required to determine their relationships. The assignment of 01 = 0.00] may work for the

current configuration, as only one fragmentation pattern was analyzed. Will it hold for

50

other regions as well, even if the anthropogenic view of the world with its corresponding
scale is maintained?

As an exploratory first step, this research elicits more questions than answers. But
such is the nature of exploration, hence the title of this research. I hope I have stirred

enough interest to warrant more research along these lines of scholarship.

51

APPENDICES

APPENDIX A

Summaries for Fragmentation Parameters and Indices

Table A.1.a. Forested 1

etm+

ikonos

1089
2500
4489
6889
10000

n
15625
62500

140625
250000
390625
562500

(1
0.00000
0.00000
0.00600
0.00334
0.00218
0.00150

d
0.00000
0.00032
0.00067
0.00108
0.00087
0.00122

Table A. l .b. Forested 2

etm+

ikonos

1089
2500
4422
6972
10000

n
15625
62500

140625
250000
390625
562500

d

0.00000
0.00000
0.00920

0.02058

0.12134
0.25710

d

0.00000

0.00034

0.00399
0.01536

0.09863

Q
0.00000

0.00000
0.99440
0.99688
0.99797
0.99860

Q
0.00000

0.99970
0.99934
0.99892
0.99914
0.99878

Q
0.00000

0.00000

0.99120

0.97964
0.87878
0.74297

Q

0.00000

0.99968
0.99602
0.98464
0.90137

0.22280 0.77720

SqP
0.00000
0.00000
0.29582
0.29582
0.29582
0.295 82

SqP
0.00000
0.52925
0.80989
0.88071
0.89786
0.92764

SqP
0.00000
0.00000
0.49518
0.65931
0.78923
0.82019

SqP
0.00000
0.54174
0.85985
0.91569
0.93002
0.94452

a = 0
to
0.00000
0.00000
0.00301
0.00167
0.00109
0.00075

a = 0
10

0.00000
0.00016
0.00033
0.00054
0.00043
0.00061

a = 0
10
0.00000
0.00000
0.00462
0.01040
0.06459
0.14751

a = 0
10
0.00000
0.00017
0.00200
0.00774
0.05188
0.12537

53

r= (d * (l + (a * SqP)))/(I+Q)

a = 1
fl
0.00000
0.00000
0.00390
0.00217
0.00141
0.00097

a = 1
fl

0.00000
0.00024
0.00060
0.00102
0.00082
0.001 18

a = 0.25
12
0.00000
0.00000
0.00323
0.00180
0.001 17
0.00081

a = 0.25 a = .125 a =0.01

12
0.00000
0.00018
0.00040
0.00066
0.00053
0.00075

a = .125
13
0.00000
0.00000
0.00312
0.00174
0.00] 13
0.00078

13
0.00000
0.00017
0.00037
0.00060
0.00048
0.00068

a :00]
f4
0.00000
0.00000
0.00302
0.00168
0.00109
0.00075

14
0.00000
0.00016
0.00034
0.00055
0.00044
0.00062

r= (d * (I + (a * SqP)))/(|+Q)

a = 1
f 1
0.00000
0.00000
0.00691
0.01725
0.1 1556
0.26849

a = 1
f1

0.00000
0.00026
0.00372
0.01483
0.10012
0.24378

3 = 0.25 a = .125 a “—001

12
0.00000
0.00000
0.00519
0.0121 1
0.07733
0.17775

a=025
12
0.00000
0.00019
0.00243
0.00951
0.06394
0.15497

13
0.00000
0.00000
0.00491
0.01 125
0.07096
0.16263

3 = .125
f3
0.00000
0.00018
0.00221
0.00863
0.05791
0.14017

14
0.00000
0.00000
0.00464
0.01046
0.06510
0.14872

a :0.01
f4

0.00000
0.00017
0.00202
0.00781
0.05236
0.12655

a = .001
f5
0.00000
0.00000
0.00301
0.00167
0.00109
0.00075

a = .001
f5
0.00000
0.00016
0.00033
0.00054
0.00043
0.00061

a = .001
f5
0.00000
0.00000
0.00462
0.01040
0.06464
0.14763

a = .001
f5

0.00000
0.00017
0.00200
0.00775
0.05192
0.12549

Table A.2.a. Non-forested l

etm+

ikonos

n
289
1089
2500
4489
6889
10000

n
15625
62500

140625
250000
390625
562500

d
0.97924
0.77502
0.64640
0.52439
0.44491
0.38190

(1
0.93280
0.80130
0.64004
0.52830
0.43512
0.373 72

Table A.2.b. Non-forested 2

etm+

ikonos

n
289
1089
2500
4489
6889
10000

n
15625
62500

140625
250000
390625
562500

(1
0.97924
0.97062
0.92520
0.84161
0.74917
0.65970

(1
0.99046
0.97330
0.96084
0.88537
0.77346
0.65304

Q
0.02083

0.22518
0.35374
0.47571
0.55517
0.61816

Q
0.06720

0.19871
0.35996
0.47170
0.56489
0.62628

0
0.02083

0.02941
0.07483
0.15842
0.25087
0.34033

Q
0.00954

0.02670
0.03916
0.1 1463
0.22655
0.34696

SqP
0.51010
0.75350
0.78377
0.80380
0.80298
0.82343

SqP
0.79468
0.91573
0.94542
0.95564
0.94266
0.92755

SqP
0.26858
0.40346
0.53079
0.66320
0.75481
0.80475

SqP
0.43963
0.76170
0.87219
0.92376
0.93290
0.92601

10
0.95925
0.63258
0.47749
0.35535
0.28609
0.23601

a = 0
10
0.87406
0.66847
0.47063
0.35898
0.27805
0.22980

a = 0
10
0.95925
0.94288
0.86079
0.72652
0.59891
0.49219

a = 0
10
0.981 1 1
0.94798
0.92463
0.79432
0.63060
0.48482

54

f1
1.44857
1.10923
0.85173
0.64098
0.51581
0.43035

a = 1
f1
1.56866
1.28060
0.91558
0.70203
0.54015
0.44295

a = 1
f1
1.21689
1.32330
1.31769
1.20834
1.05098
0.88828

a = 1
fl
1.41243
1.67006
1.73108
1.52808
1.21888
0.93377

a=025
f2
1.08158
0.75174
0.57105
0.42676
0.34352
0.28459

a = 0.25
12
1.04771
0.82150
0.58187
0.44474
0.34358
0.28309

a = .125
13
1.02042
0.69216
0.52427
0.39105
0.31480
0.26030

3 = .125
13
0.96088
0.74498
0.52625
0.40186
0.31081
0.25645

f= (d * (l + (a * SqP)))/(1+Q)

a =0.01
f4
0.96415
0.63734
0.48123
0.35821
0.28838
0.23795

a =0.01
f4
0.88101
0.67459
0.47508
0.36241
0.28067
0.23193

r= (d * (l + (a * SqP)))/(I+Q)

a=0.25 a= .125 a =0.01

12
1.02366
1.03799
0.97501
0.84697
0.71193
0.59121

a = 0.25
12
1.08894
1.12850
1.12624
0.97776
0.77767
0.59706

13
0.99146
0.99043
0.91790
0.78675
0.65542
0.54170

a = .125
13
1.03502
1.03824
1.02544
0.88604
0.70413
0.54094

f4
0.96183
0.94669
0.86536
0.73134
0.60344
0.49615

a =0.01
f4
0.98542
0.95520
0.93269
0.80166
0.63648
0.48931

a = .001
f5

0.95974
0.63305
0.47787
0.35563
0.28632
0.23620

a = .001
f5
0.87475
0.66908
0.47108
0.35932
0.27831
0.23002

a = .001
15
0.95951
0.94326
0.86124
0.72700
0.59937
0.49259

a = .001
15

0.98154
0.94870
0.92544
0.79505
0.631 18
0.48527

Table A.3.a. Intermediate 1

etm+

ikonos

289
1089
2500
4489
6889

10000

15750
62500
140250
250000
391250
562500

(1
0.76817
0.66208
0.50360
0.39497
0.34925

0.35260

(1
0.40502
0.44094
0.33895
0.28427
0.26073
0.26325

Table A.3.b. Intermediate 2

etm+

ikonos

n
272
1 122
2500
4422
6972
10000

n
15625
62500

140625
250000
390625
562500

(1
0.43382
0.39483
0.36560
0.34713
0.36446
0.35760

(1
0.38099
0.34173
0.31937
0.29954
0.30728
0.30922

Q
0.23264

0.33824
0.49660
0.60517
0.65084
0.64747

0
0.59502

0.55907
0.66105
0.71574
0.73928
0.73675

Q
0.56827

0.60571
0.63465
0.65302
0.63563
0.64246

Q
0.61905

0.65828
0.68064
0.70046
0.69273
0.69078

SqP
0.63659
0.75700
0.80923
0.84744
0.86394
0.87719

SqP
0.90975
0.93892
0.95755
0.96686
0.96101
0.95302

SqP
0.32108
0.56603
0.65645
0.75589
0.80833
0.83639

SqP
0.86923
0.91851
0.93891
0.95142
0.95771
0.94909

a = 0
10
0.62319
0.49474
0.33650
0.24606
0.21 156
0.21403

a = 0
10
0.25393
0.28283
0.20406
0.16568
0.14990
0.15158

a = 0
10
0.27663
0.24589
0.22366
0.21000
0.22282
0.21772

a = 0
10
0.23532
0.20607
0.19003
0.17615
0.18153
0.18288

55

a = 1

f1
1.01991
0.86925
0.60880
0.45458
0.39433
0.40177

a = 1

f1
0.48493
0.54838
0.39945
0.32588
0.29397
0.29604

3 = 1
f1
0.36544
0.38507
0.37047
0.36873
0.40294
0.39982

a = 1
f1
0.43986
0.39535
0.36845
0.34375
0.35538
0.35646

r= (d * (1 + (a * SqP)))/(I+Q)

a = 0.25 a = .125 a =0.01

12
0.72237
0.58837
0.40457
0.29819
0.25725
0.26096

13
0.67278
0.54155
0.37053
0.27212
0.23441
0.23749

f4
0.62716
0.49848
0.33922
0.24814
0.21339
0.21590

a = 0.25 a = .125 a=0.01

12
0.31 168
0.34921
0.25291
0.20573
0.18592
0.18769

a=025
12

0.29883
0.28069
0.26036
0.24968
0.26785
0.26325

a=025
12

0.28645
0.25339
0.23463
0.21805
0.22499
0.22628

13
0.28280
0.31602
0.22848
0.18571
0.16791
0.16964

a = .125
13
0.28773
0.26329
0.24201
0.22984
0.24534
0.24048

a = .125
13
0.26089
0.22973
0.21233
0.19710
0.20326
0.20458

f4
0.25624
0.28548
0.20601
0.16728
0.15135
0.15302

f= (d * (1 + (a * SqP)))/(1+Q)

a =0.01
f4
0.27751
0.24728
0.22512
0.21 158
0.22462
0.21954

a =0.01
f4
0.23736
0.20797
0.19181
0.17783
0.18327
0.18462

a = .001
f5

0.62358
0.49511
0.33677
0.24627
0.21 174
0.21421

a = .001
f5
0.25416
0.28309
0.20425
0.16584
0.15005
0.15172

a = .001
f5

0.27672
0.24603
0.22380
0.21015
0.22300
0.21790

a = .001
f5

0.23552
0.20626
0.19021
0.17632
0.18170
0.18306

APPENDIX B

Summaries for Increasing Support of Fragmentation Study

Pa; -.-12

.....
u.» .. ~.;.-.

 

gamma 1 = 0
variance = 0
gaml/var = 0

Moran’s I = NaN

 

gamma 1 = 0
variance = 0
gaml/var = 0

Moran’s I = NaN

 

17 x 17 cells 33 x 33 cells 50 x 50 cells
f=0,0t=0 f=0,0r=0 f=0.003008427,0t=0
f=0,0t=0.001 f=0,0t=0.001

f= 0.003009, or = 0.001
gamma 1 = 0.003618
variance = 0.005964
gaml/var = 0.606595

 

 

 

Moran‘s I = 0564*
Min = NaN Min = NaN Min = -0.00639

Max = NaN Max = NaN Max = 0.04437

Geary’s C = NaN Geary’s C = NaN Geary’s C = 044*
Min = NaN Min = NaN Min = 0.955
Max = NaN Max = NaN Max = 1.027

67 x 67 cells 83 x 83 cells

f= 0.00167361, 0. = 0

 

f= 0.001674, or = 0.001
gamma 1 = 0.001244
variance = 0.00333
gaml/var = 0.37347
Moran‘s I = 0.5652*

 

 

f= 0.001089608, O. = 0
f= 0.00109, or = 0.001
gamma 1 = 0.000961
variance = 0.002173
gaml/var = 0.442239

Moran’s I = 0.5657* Moran’s I = 0566*
Min = -0.00349 Min = —0.00227 Min = -0.00156
Max = 0.04681 Max = 0.03130 Max = 0.03191
Geary’s C = 0.4347* Geary’s C = 0.4342* Geary’s C = 0.4339*
Min = 0.949 Min = 0.960 Min = 0.968
Max = 1.024 Max = 1.023 Max = 1.022

 

100 x 100 cells
f= 0.000750525, or = 0
f= 0.000751, or = 0.001
gamma 1 = 0.00066
variance = 0.001498
gaml/var = 0.440571

 

 

Figure B. 1 .a. Summaries of increasing support for fragmentation study: forested 1

56

 

 

 

 

 

 

 

 

 

17 x 17 cells 33 x 33 cells 50 x 50 cells
f=0,0t=0 f=0,0t=0 f=0.0046203,0t=0
f= 0, or = 0.001 f= 0, or = 0.001 f= 0.004623, or = 0.001
gamma 1 = 0 gamma 1 = 0 gamma 1 = 0.00501882
variance = 0 variance = 0 variance = 0.00911536
gaml/var = O gaml/var = 0 gaml/var = 0.55058944
Moran’s I = NaN Moran’s I = NaN Moran’s I = 0.5119*
Min = NaN Min = NaN Min = -0.00969
Max = NaN Max = NaN Max = 0.05649
Geary’s C = NaN Geary’s C = NaN Geary’s C = 0.4825*
Min = NaN Min = NaN Min = 0.946
Max = NaN Max = NaN Max = 1.025
67 x 67 cells 83 x 83 cells 100 x 100 cells

f= 0.0103953, or = 0
f= 0.010402, or = 0.001
gamma 1 = 0.00516588
variance = 0.02015543
gaml/var = 0.25630225

Moran’s I = 0.6721*

Min = -0.01821
Max = 0.02393
Geary’s C = 0.3204*
Min = 0.972
Max = 1.026

 

f= 0.0645859, or = 0
f= 0.064637, or = 0.001
gamma 1 = 0.01717843

variance = 0.1066185
gaml/var = 0.16112053

Moran’s 1 = 0.8328*

Min = -0.02067
Max = 0.02071
Geary’s C = 0.1663*
Min = 0.982
Max = 1.020

 

f= 0.1475065, or = 0
f= 0.147627, at = 0.001
gamma 1 = 0.02568398

variance = 0.1909996
gamI/var = 0.13447135

Moran’s I = 08628"

Min = -0.01370
Max = 0.01429
Geary’s C = 0137*
Min = 0.985
Max = 1.016

 

Figure B. 1 .b. Summaries of increasing support for fragmentation study: forested 2

57

 

 

 

 

17 x 17 cells
f= 0.959255, or = 0
f= 0.959744, or = 0.001
gamma 1 = 0.01420455
variance = 0.02033022
gaml/var = 0.69869141
Moran’s I = 0.06655

 

33 x 33 cells
f= 0.632577, or = 0
f= 0.633053, 01 = 0.001
gamma 1 = 0.05625
variance = 0.1743624
gaml/var = 0.32260396
Moran’s I = 0.6715*

 

50 x 50 cells
f= 0.477492, or = 0
f= 0.477866, a = 0.001
gamma 1 = 0.07007528
variance = 0.228567
gaml/var = 0.30658529
Moran’s I = 0763*

 

 

 

 

Min = -0.0243 Min = -0.04809 Min = -0.03599
Max = 0.1273 Max = 0.04478 Max = 0.02948
Geary’s C = 0.8693 Geary’s C = 0.3298* Geary’s C = 0.2367*
Min = 0.837 Min = 0.957 Min = 0.972
Max = 1.092 Max = 1.059 Max = 1.035

67 x 67 cells 83 x 83 cells 100 x 100 cells

f= 0.355349, or = 0
f= 0.355635, or = 0.001
gamma 1 = 0.0425147

variance = 0.249405
gaml/var = 0.17046451

Moran’s I = 0.7995*

 

f1 = 0.286086, or = 0
f2 = 0.286316, or = 0.001
gamma 1 = 0.04364375
variance = 0.2469653
gaml/var = 0.17672017
Moran’s I = 0823*

 

f1 = 0.236009, or = 0
f2 = 0.236203, or = 0.001
gamma 1 = 0.0383483]
variance = 0.2360524
gaml/var = 0.16245677
Moran’s I = 0.8373*

Min = -0.02300 Min = -0.02068 Min = -0.01864
Max = 0.02591 Max = 0.01512 Max = 0.01728
Geary’s C = 0.2004* Geary’s C = 0.1769* Geary’s C = 0.1626*
Min = 0.971 Min = 0.984 Min = 0.985
Max =1.024 Max = 1.018 Max = 1.014

 

Figure B.2.a. Summaries of increasing support for fragmentation study: non-forested l

58

 

 

 

 

17 x 17 cells
f= 0.959255, or = 0
f2 = 0.959512, or = 0.001
gamma 1 = 0.02083333
variance = 0.02033022
gam 1/var = 1.024746904
Moran’s I = 0.06302

 

33 x 33 cells
f= 0.942883, or = 0
f= 0.943263, or = 0.001
gamma 1 = 0.02223558
variance = 0.02852129
gam l/var = 0.779613405
Moran’s I = 0.1872*

 

50 x 50 cells
f= 0.860787, or = 0
f= 0.861244, or = 0.001
gamma 1 = 0.03086575
variance = 0.06920496
gam l/var = 0.446004882
Moran’s I = 0.6122*

 

 

 

 

Min = -0.0239 Min = -0.03087 Min = -0.02971
Max = 0.1486 Max = 0.05850 Max = 0.02785
Geary’s C = 0.9541 Geary’s C = 0.8102* Geary’s C = 0.3867*
Min = 0.859 Min = 0.941 Min = 0.967
Max = 1.092 Max = 1.037 Max = 1.027

67 x 67 cells 83 x 83 cells 100 x 100 cells

f= 0.726517, or = 0
f= 0.726999, or = 0.001
gamma 1 = 0.0313772
variance = 0.1333007
gam l/var = 0.235386611
Moran’s I = 0.7312*

 

f= 0.598915, or = 0
f= 0.599367, or = 0.001
gamma 1 = 0.04066888

variance = 0.1879166
gam l/var = 0.216419837
Moran’s l = 0.7818

 

f= 0.492191, or = 0
f= 0.492587, or = 0.001
gamma 1 = 0.04374143

variance = 0.2244959
garn l/var = 0.19484289
Moran’s I = 08034“

Min = -0.02465 Min = -0.02129* Min = -0.01360
Max = 0.02622 Max = 0.02100 Max = 0.01741
Geary’s C = 0.2688* Geary’s C = 02186” Geary’s C = 0.1966*
Min = 0.976 Min = 0.979 Min = 0.986
Max=1.019 Max=1.019 Max=l.016

 

Figure B.2.b. Summaries of increasing support for fragmentation study: non-forested 2

59

 

 

 

 

17 x 17 cells
f= 0.623188, or = 0
f= 0.627155, or = 0.001
gamma 1 = 0.10795455
variance = 0.1780869
gam l/var = 0.606190292
Moran’s I = 0.3893*

 

33 x 33 cells
f= 0.494737, or = 0
f= 0.495112, or = 0.001
gamma 1 = 0.10336538
variance = 0.2237316
gam l/var = 0.462006172
Moran’s I = 0.5405*

 

50 x 50 cells
f= 0.336496, or = 0
f= 0.336769, or = 0.001
gamma 1 = 0.10002091
variance = 0.249987
gam l/var = 0.400104445
Moran’s I = 0.6775*

 

 

 

 

Min = -0.08526 Min = -0.04096 Min = -0.02945
Max = 0.09070 Max = 0.04882 Max = 0.03546
Geary’s C = 0.6099* Geary’s C = 0459* Geary’s C = 0.3224*
Min = 0.887 Min = 0.952 Min = 0.971
Max = 1.073 Max = 1.051 Max = 1.031
67 x 67 cells 83 x 83 cells 100 x 100 cells

f= 0.246058, or = 0
f= 0.246267, or = 0.001
gamma 1 = 0.05857078

variance = 0.2389677
gam 1/var = 0.245099149
Moran’s I = 0.7164*
Min = -0.02965
Max = 0.02164
Geary’s C = 0.2836*
Min = 0.976
Max = 1.024

 

f= 0.21156, or = 0
f= 0.211743, (1 = 0.001
gamma 1 = 0.056966
variance = 0.2272752
garn 1/var = 0.250647673
Moran’s I = 0.7487*
Min = -0.02459
Max = 0.01865
Geary’s C = 0.2511*
Min = 0.980
Max = 1.019

 

f= 0.214026, or = 0
f= 0.214214, or = 0.001
gamma 1 = 0.05148216

variance = 0.2282732
gam l/var = 0.225528709
Moran’s I = 0.7736*
Min = -0.01605
Max = 0.01472
Geary’s C = 0.2263*
Min = 0.984
Max = 1.015

 

Figure B.3.a. Summaries of increasing support for fragmentation study: intermediate 1

60

 

 

l,’

1.3.: v. -.
1.“). .. '

 

 

17 x 17 cells
f= 0.276627, or = 0
f= 0.276715, or = 0.001
gamma 1 = 0.03033268
variance = 0.2456207
garn l/var = 0.123493989
Moran’s I = 0.8445*

 

33 x 33 cells
f= 0.245892, or = 0
f= 0.246031, or = 0.001
gamma 1 = 0.0420844
variance = 0.2389394
gam l/var = 0.176130015
Moran’s I = 0.8209*

 

50 x 50 cells
f= 0.223656, or = 0
f= 0.223803, or = 0.001
gamma 1 = 0.04337097
variance = 0.2319366
gam l/var = 0.186994937
Moran’s I = 0.8572*

 

 

 

 

Min = -0.0820 Min = —0.05094 Min = -0.03363
Max = 0.1281 Max = 0.04571 Max = 0.03485
Geary’s C = 0.1547* Geary’s C = 0.1794* Geary’s C = 0.1428*
Min = 0.888 Min = 0.948 Min = 0.971
Max = 1.090 Max = 1.041 Max = 1.030
67 x 67 cells 83 x 83 cells 100 x 100 cells

f= 0.209996, or = 0
f= 0.210155, or = 0.001
gamma 1 = 0.03231546

variance = 0.2266302
gam l/var = 0.142591146
Moran’s I = 0.8327*
Min = -0.02314
Max = 0.02837
Geary’s C = 0.1672*
Min = 0.973
Max = 1.024

 

f= 0.222824, or = 0
f= 0.223004, or = 0.001
gamma 1 = 0.03888422

variance = 0.2316283
gam l/var = 0.167873356
Moran’s I = 0832*
Min = -0.01826
Max = 0.01987
Geary’s C = 0.1679*
Min = 0.981
Max = 1.017

 

f= 0.217722, or = 0
f= 0.217904, 01 = 0.001
gamma 1 = 0.03850058

variance = 0.2297222
garn I/var = 0.167596253
Moran’s I = 0831*
Min = -0.01576
Max = 0.01543
Geary’s C = 0.1689*
Min = 0.986
Max = 1.015

 

Figure B.3.b. Summaries of increasing support for fragmentation study: intermediate 2

61

 

 

APPENDIX C

Variograms for Increasing Supports of Two Cases Each of Three Forest Scenarios

62

Variogram for etm1_1, n=0

 

0.005

 

 

 

 

 

 

 

 

3 o
E o" I I
8 l
l
l
1 l
8 1 j
o 4
c.>‘ I l
L—l—T—T *_I T— —— ——i_—— i _F l \J
0 2 4 6 a 10
distance
Figure C.1.1.a. Forested 1, 17 x 17 cells
Variogram for etm1_2, n=0
1
m l
l
‘1
m l l
E s j |
°’ 1
1 l
a l l
:31 l
l j
l 2 j
0 5 10 15 20

distance

Figure C.1.1.b. Forested 1,33 x 33 cells

63

Variogram for etm1_3, r=1.95, s=0.0065, n=0

 

(D 2 ”42.
O l /
81 /“ ..
.6 l /
o
0.1 /
o l
l /
3- /
O. l /
O .
m I [t
2 .. . /
a 8H 2
o 1 fl
j {i
N
8.J l
o 1
l
2 l 1
o l l
0% l
O 1 {I '1» ~21
. l
O l 1 . .
o’ objective=0
0 10 20

distance

Figure C.1.1.c. Forested 1, 50 x 50 cells

Variogram for etm1_4, r=3, s=0.00395, n=.0003

 

i
l
__l

._.2 ‘ __ ___.
I

30

 

 

 

 

 

f"* 7 —" * —— _*' ’W
3 ‘ ,K/‘T’T
0. l
o / f)
l
l / .
l / ,
8 l / ‘
O. “ ./
O ‘ {I
_/
l 1/
a
i /
E 8 f
(U o it ’1
°’ 0' '. l
2 l l '
o .’
0. 'j I
O . .‘
l l
‘ .
o z . .
o"? objective=0 J
22 2L2 _ ____‘ _ , 7 ,_
0 10 20 30 40
distance

Figure C.1.l.d. Forested 1, 67 x 67 cells

64

Variogram for etm1_5, r=2.5, s=0.00245, n=.00015

 

 

 

 

 

 

 

1 ° 0 C 0 O
O 0 (J
{J
I")
N
8 2.
O
_. ’J O .J O
E 2
5 8. ‘
m o
.6 I;
8 .. L}:
C). 131
o 1
g -l objective = 0
l l 1 l l f
0 10 20 30 40 50 60
distance

Figure C.1.1.c. Forested 1, 83 x 83 cells

Variogram for etm1_6, r=2.5, s=0.00175. n=0

 

0.0020
1
C

 

0.0015
L

gamma
0.0010
4

0.0005
1

 

0.0

 

 

 

1 T 'I
0 20 40
distance

Figure C.1.1.f. Forested 1, 100 x 100 cells

65

objective = 0
l

60

Variogram for etm6_1, n=0

r_r _' 2_-__2__,_. .__.__2___2__._..2__7~.__2—%

gamma
0.0

-0.005

 

 

 

 

0 2 4 6 8 10

distance

Figure C. l .2.a. Forested 2, 17 x 17 cells

Variogram for etm6_2, n=0

 

 

‘222 ' 1
I l
l .
t l
to l
8 4 I
o l
1 l
l l
E 1 l
O. .1
E o l 1
ca : l
l .
l l
l
.0 l
o - 2,, _.,,
9 ,1 §:_ _«
j t . 1
l
,1 l
l
"# j_‘ —'__fi *7" " —‘—__"l k—i 7* _- ‘—‘ 1 ’_ —*_J
0 5 10 15 20

distance

Figure C.1.2.b. Forested 2, 33 x 33 cells

66

 

 

Variogram for etm6_3, r=1.5, s=..0063, n=.001

 

 

 

in
O (
Q —<
O
a Q
o
to
O
Q ——4
O
l/
to 4'.
E /
E g /
O) Q 21
O /
/l
I 2 2 2;;

0.002

 

 

g j objective = 0
22.— —_" "——_ 22..2 l I
0

10 20 30

distance

Figure C. 1 .2.e. Forested 2, 50 x 50 cells

Variogram for etm6_4, r=1.75, s=.015, n=0

 

 

0.020

 

gamma
0.010 0.015
'4» |

0.005
212 2 2 2

0.0
r2222 2

 

‘ objecti e = 0.0001
222222 . , 2 -. ;22- - -2 .-

22—.

 

0 10 20 30 40

distance

Figure C.1.2.d. Forested 2, 63 x 63 cells

67

gamma

Variogram for etm6_5, r=8, s=.0675. n=.01

 

 

 

 

 

t: u
0' 7
‘2
cs 7 ..
0
0
0 O
O 2 0
‘5 /224_————
a _
o . 5. ‘H-
5..
3 _ '1 -
o‘ 3' At
N .‘T 3“
Q _j IX... ,- maul;
O
O . .
c," ‘ objective = 0.0057
1 I I l I l
10 20 30 40 50 60

distance

Figure C.1.2.e. Forested 2, 83 x 83 cells

gamma

Variogram for etm6_6, r=14, s=..16, n=.01

 

 

 

 

 

76’" 5’ ’
objective = 0.0127
1 l
40 60
distance

Figure C.1.2.f. Forested 2, 100 x 100 cells

68

 

Variogram for etm2_1, r=1.6, s=0.0063, n=0.011

 

0.020
2

0015
1

gamma
0 010

0.005
1

 

0.0

;_12
‘1
l
1
l
l
l
l
l
l
l

 

objective = 0.0001
1

 

distance

Figure C.2.1.a. Non-forested 1, 17 x 17 cells

10

Variogram for etm2_2, r=1.8, s=0.1, n=0.01

 

0.20

0.15

 

gamma
0 10

0.05 .
x

C?
o .

 

_._T— " 222. T 1
0 5 1 0

distance

Figure C.2.1.b. Non-forested 1, 33 x 33 cells

69

objective = 0.054

 

 

 

 

r
20

gamma

0.25

0.15

0.20

Variogram for etm2_3, r=6.6, s=0.214, n=0.0225

 

 

 

 

 

 

 

l /
/1/i
///

1 :1

|
4 objective = 0.0012

0 10 20 30
distance

Figure C.2.1.e. Non-forested l, 50 x 50 cells

gamma

0.05 0.10 0.15 0.20 0.25 0.30

0.0

Variogram for etm2_4, r=8.8, s=0.238, n=0.02

 

 

 

 

 

2. 22 .2122..- _ 2 ._2 l

0 10 20 30

distance

Figure C.2.1 .d. Non—forested 1, 67 x 67 cells

70

Variogram for etm2_5, r=10.5, s=0.226, n=0.02

 

0.30
l

0.25
1

gamma
0 15
l

0.10
J

 

9... “3.2.4.7..”

‘r
" 1
$2.21.: ‘eac..-'_..:.1

 

0.05
222 22

0.0

 

 

0 1 0 20 30 40 50

distance

Figure C.2.1.e. Non-forested 1, 83 x 83 cells

Variogram for etm2_6, r=7.7, s=0.178, n=0.015

objective = 0.0214
1

60

 

0.30

0.25

 

0.15 020
22- _ 1 2 2 2t

gamma

0.10

  

0.05

 

0.0

0 20 40 60

distance

Figure C.2.1.f. Non-forested 1, 100 x 100 cells

71

 

objective = 0.076
1

Variogram for etm4_1, n=.019, r=2, s=.003

 

 

 

o - : .. ”If ’ .
N _j 2/ i
I
l
to l
5.1
o !
.. l
e l
8 e j
0') o -
.. . l
‘ l
to l 1
O
O. 1
o l
|
} O22j
0 j
o' ; objective
——T’—_—i .— 222_222 _T——_ _—
0 2 4 6 8
distance

Figure C.2.2.a. Non-forested 2, 17 x 17 cells

Variogram for etm4_2, r=1.47, s=0.0106, n=0.0165

 

gamma
0.010 0 015 0.020

0.005

objective = 0

0 5 1O 15 20

0.0

 

 

 

distance

Figure C.2.2.b. Non-forested 2, 33 x 33 cells

72

 

Variogram for etm4_3, n=.015, r=3.5, s=.041

 

 

 

 

 

 

i
¢>|
Q _.
0 l g
‘ // if," U L I
i « ‘
s /
o __ _.
3 °' i /
E .
CU .
m 1 i,
8 l 2
o‘ 3
l
3 q objective = 0.0005
f—fi '(_—_ l T T
0 1o 20 30
distance

Figure C.2.2.e. Non-forested 2, 50 x 50 cells

Variogram for etm4_4, r=4.2, s=0.0805, n=0.015

 

,— ,.._..._ _____.,,

0.15

0.10

 

gamma

0.05

,,,,,

0.0

 

objective = 0.0122

1 7 ' _‘_"'T——‘—"— T *_ #

20 30 40

 

 

 

distance

Figure C.2.2.d. Non-forested 2, 67 x 67 cells

73

gamma

0.05

Variogram for etm4_5, r=6.1, s=0.137, n=0.017

 

 

‘; objective = 0.0137
‘ _l" V m l 7. " l
O 10 2O 30 40 50 60
distance

Figure C.2.2.e. Non-forested 2, 83 x 83 cells

gamma

Variogram for etm4_6, r=8, s=0.19, n=0.02

 

 

 

m ff ,,
(V a s
O l
l
l
O 1 _’,sdi«~‘f‘fii I _—
N. ./« a
o l / '
l /‘
,0 I /
‘—. ‘1 /
O //
/ j
/
O I
o' ,/

0.05

 

i
J objective = 00077 i

0.0

____.____._J

7»,,,_,_—_. _ .. a. _._ _ _

O 20 40 60

distance

Figure C.2.2.f. Non-forested 2, 100 x 100 cells

74

Variogram for etm3__1, n=.07, r=3.5, s=.135
____-_ T

 

020

015

 

005

gamma
. 0.10 . .

 

0.0

 

* objective = 0.0006
l

 

 

0 2 4 6 8 1O

distance

Figure C.3.1.a. Intermediate 1, 17 x 17 cells

Variogram for etm3_2, r=1.5, s=0.173, n=0.01

 

0.20

 

0.15
i

gamma
0.10

005
\
4:.

 

0.0

objective = 00248 J

fl

0 5 10 15 20

distance

Figure C.3.1.b. Intermediate 1, 33 x 33 cells

75

Variogram for etm3_3, r=4.5, s=0.176, n=0.045

 

0.25
_I

 

 

0.20

0.15

 

 

gamma

0.10

0.05

 

i -'~
‘1!!! .w‘ A

0.0

 

objective = 0.0279

— T —( W 7, 7 7 -— T—-———— r~»——— A A— * ~—~ ,,#_#, # ~‘ A IA ————-

0 10 20 30

distance

Figure C.3.1.c. Intermediate 1, 50 x 50 cells

Variogram for etm3_4, r=4.4, s=0.176, n=0.025

 

 

 

 

 

 

 

 

m 1 ~ I H J 0 it (I .- r. 1
N . g ,.
o. I u
l
1
8. -1 I ,._§_. - __
o 1 W
| /
1 /
ID 1 //
g 0'1 //
1%, ~ ,
O) 1 '
,9
c 1 I M4.
“0’. J /
o 1 i
1 3.. e...
O . .
0' ' objective=00441 A
' _ # i ” ' 7 '7‘ ’ I . _ i "
0 10 20 30 40
distance

Figure C.3.1.d. Intermediate 1, 67 x 67 cells

76

Variogram for etm3_5, r=5. s=0.152. n=0.025

 

 

 

 

 

 

LD
(‘44
O
8 o
OT 0
l0
vj—i
N O
E
E
N
U)
D
sf-
0
IO
0'!
O
Q-
o
I I

O 10 20

Figure C.3.1.e. Intermediate 1, 83 x 83

T
30

distance

40

objective = 0.0573
|

50

Variogram for etm3_6, r=5.8, s=0.148, n=0.025

60

 

0.0

gamma
0.05 0.10
_L___L___I____L

 

 

objective = 00631
I

 

 

0 20

Figure C.3.1.fIntermediate 1, 100 x 100

distance

77

60

Variogram for etm5_1, n=0, r=40, s=1.3

 

 

r___ .____i__‘_. _. _ _.___ ,_.__,\___ _.‘_____ 7.7

J (I

0.25 0.30

0.20

 

gamma
0 15

 

0.10

 

1

/'/ fl 1

 

0.05

0.0

 

’ objective = 0.0023 J
‘ 1' ‘_""*_‘ "““‘—1‘_ I I I

O 2 4 6 8 10

distance

Figure C.3.2.a. Intermediate 2, 17 x 17 cells

Variogram for etm5_2, n=.015, r=15, s=.36

 

0.20 0.25 0.30
1
|
1

gamma
0.15

 

0.10

0.05
_ 4_i__ __ _ l__ _

 

objective = 0.0002 1

L,- _.A_-_;_ I __-,_ _,fiiai?i~ _, _ -cAW ,, , __ _. _. ___,__,_ "c_c..

O 5 10 15 20

distance

Figure C.3.2.b. Intermediate 2, 33 x 33 cells

78

Variogram for etm5_3, r=15, s=0.265, n=0.015

 

0.25

0.20

 

 

 

E ‘2.
E o
(D
O)
53
o'
to
0
d
O . .
o‘ objective = 0.0098
M'TW’“' “’"fi *—W a
0 10 20 30

distance

Figure C.3.2.c. Intermediate 2, 50 x 50 cells

Variogram for etm5_4, r=6.7, s=0.138, n=0.015

 

 

 

 

 

 

 

 

[7‘7- 0
1 0 (x Q ‘
°1
1
O 1
(V. it
o 1
1
l0 1
(U F—j ME
E o"; ‘3 ,e I'"
E I /
N i
U) i ,
o I 3 ,
". j /
o i
1 /
to 1 ’/
81' /
1 /
oi - - -
o L 1 1____ I”# 1 _ __ Y obje_cti[ve-0.1238_1
O 10 20 30 4O

distance

Figure C.3.2.d. Intermediate 2, 67 x 67

79

gamma

0.05 0.10 0.15 0.20 025

0.0

Variogram for etm5_5. r=8.5, s=0.16, n=0.02

 

   

 

 

 

u 0 o a r
o J i) 0
D
[Pf/*—
1‘ -_ 1111
objective = 0.0642
T I i I l I I
0 1 0 20 30 40 50 60
distance

Figure C.3.2e. Intermediate 2, 83 x 83 cells

gamma

Variogram for etm5_6, r=10, s=0.17, n=0.02

 

0.10 0.15 0.20
1 I A I

0.05
I

 

0,0
I

 

   

 

 

objective = 0,0426
‘1—- y—j— ' ' | 1 fl
0 20 40 60

distance

Figure C.3.2.f. Intermediate 2, 100 x 100 cells

80

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