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REPLICATING NATURAL TREE STAND PATTERNS IN A
NORTHERN MICHIGAN ROCK OUTCROP LANDSCAPE: A
FRACTAL BASED METHOD AND APPLICATION FOR
REFORESTING A RECLAIMED MICHIGAN SURFACE MINE
presented by
Wade J Lehmann
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REPLICATING NATURAL TREE STAND PATTERNS IN A NORTHERN
MICHIGAN ROCK OUTCROP LANDSCAPE: A FRACTAL BASED METHOD AND
APPLICATION FOR REFORESTING A RECLAIMED MICHIGAN SURFACE
MINE
BY
Wade J Lehmann
A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of
MASTER OF ARTS
Environmental Design
2009
Abstract
REPLICATING NATURAL TREE STAND PATTERNS IN A NORTHERN
MICHIGAN ROCK OUTCROP LANDSCAPE: A FRACTAL BASED METHOD AND
APPLICATION FOR REFORESTING A RECLAIMED MICHIGAN SURFACE
MINE
BY
Wade J Lehmann
Landscape planners and designers are interested in
replicating natural landscape patterns to reclaim degraded
landscapes to match existing conditions. One approach that
shows promise is the use of fractal geometry to create
natural landscape patterns. While the measurement of the
actual fractal dimension of an object is difficult, the
box-counting method (developed at Agrocampus Ouest, Angers,
France) approximates the fractal dimension of an object.
This process is illustrated by measuring and replicating a
stand of trees in the Upper Peninsula of Michigan and
applying the method for a planting plan on a Northern
Michigan surface mine. The estimated fractal dimension of
each tree is; 0.329 for TSuga canadensis carriers, 0.674
for Thuja occidentalis L., 0.607 for Acer rubrum L, 0.345
for Acer saccharum Marshall, 0.442 for Pinus strobus L.,
and 0.359 for Picea glauca (MOench) VOss.
Dedicated to my parents Bill Lehmann and Holly Weller
ACKNOWLEDGEMENTS
I would like thank my graduate advisor Dr. Jon B.
Burley for his guidance and advice during both my graduate
and undergraduate careers. I would also like to thank my
two other graduate committee members Dr. Patricia L.
Machemer and Dr. Robert E. Schutzki. Special thanks is
extended to Cyril Fleurant of Agrocampus Ouest, Angers,
France for his contribution of the inverse box-counting
method, utilized in this research. Lastly, I would like to
thank Katherine R. Latocki for her contributions as
proofreader and editor.
LIST OF TABLES
TABLE OF CONTENTS
vi
LIST OF FIGURES Vii
INTRODUCTION 1
1.1 ORIGIN OF FRACTALS 3
1.2 FURTHER DESCRIPTIONS ILLUSTRATING FRACTALS .............. 4
1.3 GEOMETRIC PROPERTIES OF FRACTALS 7
1.4 FRACTAL DIMENSIONS 8
1.5 INVERSE BOX-COUNTING METHOD: A TOOL FOR
REPLICATING LANDSCAPES 12
1.6 PLANNING AND DESIGN APPLICATIONS 14
METHODOLOGY 16
RESULTS 72
APPLICATION AND DISCUSSION 31
APPENDIX 45
LITERATURE CITED 56
LIST OF TABLES
3.1 Dependent and independent variables for
regression analysis
3.2 Mean and standard deviations for each species
trial. EH-Eastern Hemlock, NWC-Northern White
Cedar, RM-Red Maple, SM-Sugar Maple, WP-White
Pine, WS-White Spruce
vi
24
27
Cantor’s
Equation
Equation
Equation
objects
Equation
Location
Dust fractal object
for the
for the
for the
LIST OF FIGURES
fractal dimension
fractal dimension
fractal dimension
of a line
10
’
of an area
of self-similar
for the
of the study area in Michigan
fractal dimension
of Cantor's Dust ......
Forest stand at the study site (notice the rocky
terrain and exposed bedrock)
Pre-settlement vegetation map (study area is outlined
10
11
11
17
17
by the rectangle in the upper right of image) ............... 18
Fractal dimension equation for all species 72
Fractal dimension equation for Eastern hemlock 27
Fractal dimension equation for Northern white cedar ...... 28
Fractal dimension equation for Red maple 98
Fractal dimension equation for Sugar maple 99
Fractal dimension equation for White pine 79
Fractal dimension equation for White spruce 3O
Fractal based planting plan for all species 33
vii
.2 Fractal
.3 Fractal
.4 Fractal
.5 Fractal
.6 Fractal
.7 Fractal
based planting plan
based planting plan
based planting plan
based planting plan
based planting plan
based planting plan
for
for
for
for
for
for
.8 Example of a waste rock pile in
Peninsu
. 1 Eastern
.2 Eastern
.3 Eastern
.4 Eastern
.5 Eastern
.6 Northern
.7 Northern
.8 Northern
.9 Northern
.11 Red map
Eastern Hemlock .............. 34
Northern White Cedarm35
Red Maple 36
Sugar Maple 37
White Pine 38
White Spruce 39
Michigan's Upper _
40
la
hemlock trial 1 45
hemlock trial 2 45
hemlock trial 3 46
hemlock trial 4 .46
hemlock trial 5 46
white cedar trial 1 47
white cedar trial 2 47
white cedar trial 3 47
white cedar trial 4 48
.10 Northern white cedar trial 5 48
1e trial 1 48
vm
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
Red maple trial 2
Red maple trial 3
Red maple trial 4
Red maple trial 5
Sugar
Sugar
Sugar
Sugar
Sugar
White
White
White
White
White
White
White
White
White
maple trial
maple trial
maple trial
maple trial
maple trial
pine trial 1
pine trial 2
pine trial 3
pine trial 4
pine trial 5
spruce trial
spruce trial
spruce trial
spruce trial
49
49
49
50
50
50
51
51
52
52
52
53
53
53
54
54
54
A.30 White spruce trial 5
55
INTRODUCTION
Surface mine reclamation is an important subject which
involves land planning, ecology, landscape design, and site
engineering. Reclaiming surface mines is the process of
successfully converting a material resource exhausted
environment into one that can accomplish a new land use
(Burley, 2001). Mine reclamation has become an area of
interest in the past half decade possibly because of
increased environmental awareness. The Surface Mining
Control and Reclamation Act of 1977 mandated that all
abandoned surface mines be reclaimed. The western United
States alone houses over 500,000 abandoned and active
mines, spanning millions of acres (Berger, 2008). The
amount of surface mines in the United States and the
harmful effect of abandoned mines require attention from
landscape planners and designers.
Surface mine reclamation can utilize many different
end results. According to Burley (2001), a successful
reclamation process includes; recognizing the traditional
land use of the pre mining environment, and attempting to
return the post mining landscape to this condition or
another acceptable land use. Typical post mining land uses
include but are not limited to; agriculture, housing
development, parks and recreation, pasture, wildlife
habitat, and forested land. According to Berger (2008),
“mine sites enable designers to speculate over a landscape
that is not bound by, nor indebted to, historical filters,
aesthetic tradition, or strict contextualitym reclamation
can act as a laboratory for experimentation.”
The process of reforesting reclaimed landscapes is
typically achieved by mass plantings of the most
commercially viable trees for a particular site. This study
investigates a new fractal based procedure for replicating
natural patterns found in the landscape.
Landscape planners, designers, and environmental
specialists are concerned in evaluating the spatial
composition of landscape features such as composition of
vegetation, forms of water bodies, and shape of terrain to
unify disturbed landscapes with natural ones. However,
natural looking assemblies were difficult to mathematically
duplicate. Typical techniques used to replicate natural
systems include the gestalt methods and ecological field
methods (Fleurant, et al., 2009). The gestalt method was
heuristic in nature where one would creatively merge and
combine patterns together, until a desired condition was
achieved. The ecological field laboratory method used
scientific measures such as frequency, density, and size to
construct patterns. A new approach has evolved which
utilizes fractals to calculate spatial patterns in the
landscape (Fleurant et al., 2009). A fractal designates an
irregular or fragmented shape that can be divided into
parts, each of which is approximately a smaller copy of the
entire shape (Foroutan-pour et al., 1999).
1.1 ORIGIN OF FRACTALS
Fractals were originally noticed at the end of the 19th
century. However, the term “fractal” was coined later, the
Peano curves appear to be the first example of fractal
objects, first explained by Guiseppe Peano. The Peano
curves could fill a void through a series of iterations
utilizing only a few simple rules (Mandelbrot, 1982).
Fractals have been explored more thoroughly in the
latter half of the 20”‘century most notably by the French
mathematician Benoit Mandelbrot. Mandelbrot, while
researching “econometry” (mathematics applied to the
economy), found that there were no difference in the slopes
of curves predicting short-term and long-term market
prices. He compiled an extensive description of the curves
and created the term fractal (from the Latin word fractus,
meaning broken) to describe the objects where irregularity
separates them from typical Euclidian geometry curves. Upon
the discovery of fractals, their use and application has
broadened. Mandelbrot (1982) expresses the applications for
fractals as follows, “Nature exhibits a high level of
complexity in which typical Euclidian geometry classifies
as formless, these irregular and fragmented patterns around
us can be found using fractal geometry”. This is an
explanation of why fractals are used today in such sciences
as biology, ecology, and geology.
1.2 FURTHER DESCRIPTIONS ILLUSTRATING FRACTALS
To demonstrate the concept of fractals, picture the
rugged and rocky French coastline of Brittany. What is the
real length of the coastline? To determine the length one
could examine two forms of resolution.
1”.An aerial image from 10,000 meters high and calculating
the visible length of the coast.
:2.Another aerial image from 500 meters high and measuring
the details of the coast one meter at a time.
When measuring the length, one will determine that the
coastline is longer in the second case, and also more
accurate. If one were to examine the coastline at an even
finer resolution, the overall length would increase again.
The more defined the system of measurement, the greater the
length of the coastline will increase. The complexity of
the Brittany coastline (unable to be described with
Euclidian geometry) makes it a fractal object (Mandelbrot,
1982). As expressed by Mandelbrot previously, fractals are
everywhere in nature.
Fleurant et al.(2009) give a practical definition of
the concept of fractals as a “geometrical shape resulting
from infinite regular fragmentation of a given form”. It is
also proper to describe a fractal as a recurrence of the
same form on each part of the curve. If one looked closely
at any one part of a curve, it would resemble the entire
curve itself (Fleurant et al., 2009). Cantor's Dust
illustrates this property. Cantor’s Dust is an image which
results from Cantor's set, “a collection into a whole, of
definite, well distinguished objects of our perception or
thought." (Kamke, 1950) Cantor’s dust has the geometric
property where as the construction iteration process
increases towards infinity, the total length L increases
towards infinity. Imagine a straight line, then the same
line with the middle l/3rd removed. This process is
continued to infinity and eventually the divisions become
so small they are unobservable by the human eye (Figure
1.1). (Barnsley, 1988) The rings of Saturn are a real world
example of this phenomenon. Saturn’s ring was originally
thought to be one solid entity, upon closer examination
with higher powered telescopes it became clear that the
ring was actually comprised of many small rings.
Figure 1.1.
Cantor’s Dust fractal object
1.3 GEOMETRIC PROPERTIES OF FRACTALS
Geometric properties of fractals are utilized in a
number of different sciences and numerous models. For
example, in geology fractals can be used for identifying
fractures in rocks, which threaten their structural
integrity (Velde et al., 1990). In economics, fractals are
used to predict complex random fluctuations in the stock
market (Mandelbrot, 1982). In computer sciences, fractals
are used to retrieve patterns in image processing (Liangbin
et al., 2005). In medicine, fractals can predict a
patient’s susceptibility to osteoporosis based on their
bone mineral density structure (Harrar and Hamami, 2007).
In chemistry, they are used to design new materials. The
fractal nature of these materials allows them extraordinary
properties, such as high thermal cooling power (Fleurant et
al., 2009).
It is important to understand there are two different
categories of fractals; theoretical fractals, and real
fractals. Theoretical fractals, such as the Peano curve and
Cantor’s dust mentioned prior, exhibit self similarity and
the dimensions can be mathematically calculated to infinite
(Mandelbrot, 1982). Real fractals, such as objects found in
nature, are not self similar, and do not continue to
7
infinite. To determine the dimensions of real fractals, one
must employ an estimation process such as the box-counting
method (Foroutan-Pour et al., 1999).
1.4 FRACTAL DIMENSIONS
In Euclidian geometry, the point has a dimension of 0.
Line and curves have a dimension of 1. Areas have a
dimension of 2, such as a triangle or circle. Volumes have
a dimension of 4, such as a cylinder or sphere. Fractal
objects also have dimensions (Mandelbrot, 1982).
Fractal dimensions have values which cannot be
expressed by a simple point or line. Objects such as those
found in nature cannot be explained by Euclidian geometry,
but can be expressed using fractals. Barnsley (1993)
affirms this idea by stating,
“Fractal dimensions can be attached to clouds, trees,
coastlines, feathers, networks of neurons in the body,
dust in the air at an instant in time, the clothes you
are wearing, the distribution of frequencies of light
reflected by a flower, the colors emitted by the sun,
and the wrinkled surface of the sea during a storm.”
Fractal dimensions attempt to quantify a subjective feeling
which we have about how densely the fractal object fills
the space in which it lies. They also provide a means for
comparing the complexity of different fractals (Fleurant et
al., 2009).
To demonstrate fractal dimensions, reconsider the
Brittany coastline. If one were to calculate a 1 m length
of a relatively straight line with a 20 cm ruler, the ruler
will be used 5 times, 10 times for a 10 cm ruler, or 20
times with a 5 cm ruler. If one were to measure the same
distance along the coastline, the total length will be
underestimated due to the irregular pattern of the coast.
The smaller the ruler used to measure the coast the more
accurate the estimated length. To evaluate this phenomenon
mathematically, one can declare that the result is more
accurate when using a smaller ruler that fits the curvature
of the line. If one can divide the length of the ruler of
an infinite small size by “n”, one has to use this ruler
“n” times more. This property can define the topological
dimension of the curve (Figure 1.2):
log(n)
Dtopological = m =
Figure 1.2 Equation for the fractal dimension of a line
Replicating this process again using a surface, one can use
a square where the length of the side is L. To measure its
area, one can use a smaller square where the length of one
side is L/2, then one will need 4 squares, 16 squares using
L/4, and so on. If the length of the side of the measuring
square is divided by “n”, the number of such squares used
is multiplied by “n” (Figure 1.3):
log(n2) log (n)
Dtopological = m = m =
Figure 1.3 Equation for the fractal dimension of an area
Similar results can be obtained for volumes and the
topological dimension of a Euclidian geometric object with
a fractal dimension of 3 (Fleurant et al., 2009).
In the moderately simple case of self-similar fractal
objects (meaning they appear the same no matter which
zooming factor is used), resulting in a constant iterative
factor “k”, the fractal dimension is (Figure 1.4):
10
log (n)
D =-————-'
fractal log (k)
Figure 1.4 Equation for the fractal dimension of self-
similar objects
Where:
n = the number of subsets counted during the scaling
process using a factor l/k (self-similarity factor).
k = number of iterations
Cantor’s Dust illustrates how to calculate the fractal
dimension of self-similar fractal objects. Consider a
single line with a length of L. If one were to remove the
middle 1/3rd of that line they would be left with two lines
where L equals 1/3. One can continue to remove the middle
l/3rd of every line formed by the previous division (the
dust presents an infinite number of “lines” with each
iteration). This process can be carried on indefinitely.
Then, using the same reasoningone can calculate the
fractal geometry of Cantor’s Dust (Figure 1.5):
1092
Dfractal = 59—3 = 0.6309
Figure 1.5 Equation for the fractal dimension of Cantor’s
Dust
11
Therefore, one can conclude that the fractal dimension
of this strange curve is not 1 as any of the classic linear
geometrical curves. Cantor’s Dust has a topological
dimension equal to 0 (it’s a broken line), but has a
fractal dimension of greater than 0, which is not an
integer but a real number.
The previous equations (figures; 1.2, 1.3, 1.4, and
1.5) are utilized to calculate the dimension of theoretical
fractals. These equations cannot be used to determine the
dimension of real fractals due to random elements present
in the natural setting (Foroutan-Pour et al., 1999).
Instead one must employ a more appropriate method to
estimate the fractal dimension. One such method that is
commonly utilized is the box-counting method (Foroutan—Pour
et al., 1999).
1.5 INVERSE BOX-COUNTING METHOD: A TOOL FOR REPLICATING
LANDSCAPES
The fractal dimension is not easy to calculate but can
be estimated using several methods. The box-counting method
is one of the simpler and most popular methods to utilize.
The box-counting method was developed by Duchesne et al.
12
(2002) and computed by Durandet (2003) in the Landscape
Department of the National Institute of Horticulture and
Landscape Angers, France, now the Unite de
RecherchePaysage; AgroCampusOuest. The natural object is
covered with a grid of size r and one counts the number of
boxes, N(r) that contain some part of the object. The value
of “r" is progressively reduced and N(r) is similarly re-
measured. As “r” tends to be very small values (0 in a
log (N (r))
1 becomes the fractal
108 (',:)
theoretical way) one finds that
dimension of the object (Fleurant et al., 2009).
The box-counting method is a simple tool to calculate
the complexity of a landscape using the value of its
fractal dimension. The greater an objects fractal dimension
(2 is the maximum value in a plane), the less complex the
arrangement of the planting pattern (in terms of scale,
structure, alignment, etc.)(Fleurant et al., 2009). By
utilizing this method, one is able to control the
randomness of plantings or other landscape features with
certain parameters: the fractal dimension (D), the average
minimum distance between two trees (€Mﬁ) and the average
maximum size of the boxes (6mm).
13
1.6 PLANNING AND DESIGN APPLICATIONS
There is a belief that fractals may have the ability
to re-create complex landscape patterns that are hard to
replicate with Euclidian geometry because the landscape is
full of fractals: rivers, trees, landscape networks in
general (Barnsley, 1993). Fractals are extremely detailed,
complex geometric shapes and a measure of their complexity
is the fractal dimension (Mandelbrot 1982). Accordingly, a
number of professionals have examined fractals in landscape
planning and design including studies by Diaz-Delgado et
al., (2005); DiBari (2007); Griffith et al., (2000); Li
(2000); Milne (1991); Palmer (1988); and Thomas et al.,
(2007). However, the use of fractals seems to be looking
for a more practical application. For example, in
landscapes it has always been relatively simple to describe
an existing pattern, but hard to replicate that pattern.
Presented in this paper is an approach to replicate
landscape patterns and a practical approach towards the use
of fractals.
Application of the inverse box-counting method to a
reclaimed surface mine has the potential to accurately
14
depict the natural vegetation patterns for a Northern
Michigan rock outcrop.
15
METHODOLOGY
This study examines the application of fractals in the
planting pattern of trees in the Upper Peninsula of
Michigan in Dickinson County. The area selected for the
study, located in Dickinson County (Figure 2.1), was
selected on a rocky and dry xeric northern forest (Figure
2.2), an environment similar to waste rock piles on a
surface mine where the fractal planting plan might be
appropriate (Curtis, 1959). Trees equaling 3 inches dbh
(diameter at breast height) or greater were recorded by a
remote gps (global positioning system) unit.
16
Dickinson
Study site \
F
0 Iron Mountain
Figure 2.1. Location of the study area in Michigan
Figure 2.2. Forest stand at the study site (notice the
rocky terrain and exposed bedrock)
17
To further express the relationship between the study
site and the application site a pre-settlement image of the
study area (figure 2.3) has been created from an
interpretation of the 1816—1856 general land office surveys
by Albert and Comer (2008).
++++++
++++++.
+++++++++
+ + + + + + + + + -
Sugar Maple-Hemlock Forest 1- +
+++
Figure 2.3. Pre-settlement vegetation map (study area is
outlined by the rectangle in the upper right of image)
The map of pre-settlement vegetation suggests that the
study area was originally a Sugar Maple-Hemlock forest.
This forest type was the most predominant upland system in
the Upper Peninsula, and also consisted of large numbers of
White Pine. Soils associated with this cover type can be
18
steep and rocky, including exposures of basalt and granite
bedrock (Albert & Comer, 2008).
Another examination of the study area revealed a more
detailed analysis of the soil conditions. According to the
Soil Survey of Dickinson County, Michigan (United States
Department of Agriculture, Soil Conservation Service) the
study area consists of a Pemene-rock outcrop complex
(Linsemier, 1989). This complex consists of 35—65% Pemene
soil and 15-20% rock outcrop on slopes of 18—35%. Trees to
be planted on this complex include (but are not limited
to); White Pine, White Spruce, and Sugar Maple (Linsemier,
1989).
The location of trees can be placed on a map derived
from remote sensing field survey. This set of points
(location of trees) can be viewed as a complex and fractal
object in nature.
Points were gathered as X, Y data (latitude and
longitude) by a remote global positioning system (gps)
unit. Points were collected on an entire rock outcrop, and
mapped (globally) using ArcGIS software. The map of points
was then projected into UTM's (universal transverse
Mercator) for the application of trial grids to a two-
dimensional surface. The resulting map was then exported to
19
an Autocadd (computer aided drafting) program for creation
of the trial grids.
Selection criteria for the size of the trial grids
were determined by the size of the rock outcrop. The entire
outcrop measured approximately 60 meters by 80 meters. Thus
a trial grid of 50 meters was selected to encompass the
entire site with a series of trials occurring at random
placements within the study site. A total of five different
trials were completed for each species of tree on the rock
outcrop, resulting in 30 (50 by 50 meter) trials (see
Appendix). Each trial was then subject to the box-counting
method.
The box-counting process starts with the pairs of
values r and the number of boxes N(r), the starting value
of r is 50 meters, and the starting value of N(r) is one.
Then r is divided in half and the value of r becomes 25
meters, while N(r) can range from 1 to 4, depending on the
number of boxes which contain trees. The pairs of numbers
for the regression analysis includes the first pair where
at least one box becomes empty, and continues with
successive pairs at smaller sizes until every box contains
either one or no trees (Fleurant et al., 2009). In total
there were five 50 meter by 50 meter boxes for every
20
species of tree recorded in the study area with a count of
greater than one.
21
RESULTS
The tree species tallied on site include; Eastern
Hemlock (Tsuga canadensis carriers), Northern White Cedar
(Thuja occidentalis L.), Red Maple (Acer rubrum L.), Sugar
Maple (Acer saccharum Marshall), White Pine (Pinus strobus
L.), and White Spruce (Picea glauca (MCench) VOss).
Out of the 30 trials, 113 dependent and independent
variables for the regression analysis were derived (Table
3.1). The regression analysis revealed an adjusted r—square
of 0.444, with a significant p-value of O. The slope of the
line expressed in the regression equation is 0.578. This
suggests that the fractal dimension is between a point and
a line in typology (Figure 3.1).
Ln(N(r)) = 0.578Ln G) + 3.107
Figure 3.1. Fractal dimension equation for all species
Where: N(r): number of boxes with trees
r = length of one side of the box
22
Of the 30 trials, 6 species were identified as having their
own fractal dimension (see Table 3.2 for statistical
information). Each species of tree has the same number of
trials (5) but they have differing pairs of numbers.
23
Table 3.1. Dependent and independent variables for
regression analysis
Species Plot Ln(1/r) Ln(N(r))
Eastern Hemlock 1 -3.219 0.693
-2.526 1.386
‘ -1.833 1.609
-1.139 1.792
Eastern Hemlock 2 -3.219 1.099
-2.526 1.386
-1.833 1.386
-1.139 1.792
Eastern Hemlock 3 -3.219 0.693
-2.526 0.693
-1.833 1.099
Eastern Hemlock 4 -3.219 0.693
-2.526 1.099
-1.833 1.099
-1.139 1.386
—0.447 1.609
Eastern Hemlock 5 -3.219 1.099
Northern White Cedar 1 -3.219 1.099
-2.526 2.079
-1.833 2.639
-1.139 2.833
Northern White Cedar 2 -2.526 2.079
-1.833 2.565
-1.139 2.708
Northern White Cedar 3 -2.526 2.197
-1.833 2.398
-1.139 2.773
Northern White Cedar 4 -3.219 1.099
-2.526 1.792
-1.833 2.303
-1.139 2.565
Northern White Cedar 5 -2.526 1.946
-1.833 2.303
-1.139 2.639
Red Maple 1 -2.526 2.197
-1.833 2.773
-1.139 3.091
—0.447 3.296
24
TABLE 3 . l CONT .
Red Maple 2
Red Maple 3
Red Maple 4
Red Maple 5
Sugar Maple 1
Sugar Maple 2
Sugar Maple 3
Sugar Maple 4
Sugar Maple 5
White Pine 1
White Pine 2
25
-2.526
-1.833
-1.139
-0.447
-2.526
-1.833
-1.139
-2.526
-1.833
-1.139
-0.447
-3.219
-2.526
-1.833
-1.139
-0.447
3.219
-2.526
-1.833
-1.139
-0.447
-3.219
~2.526
-1.833
-1.139
-3.219
-3.219
-2.526
-1.833
-1.139
-3.219
~2.526
-1.833
-1.139
-2.526
-1.833
-1.139
0.447
-2.526
-1.833
-1.139
-0.447
2.485
2.890
3.258
3.367
2.079
2.565
2.944
2.398
2.944
3.219
3.367
1.099
1.792
2.565
2.773
2.944
1.099
1.609
1.792
1.792
1.946
1.099
1.386
1.792
1.946
1.099
1.099
1.386
1.386
1.792
0.693
1.099
1.099
1.609
2.197
2.485
2.565
2.639
2.303
2.708
3.091
3.135
TABLE 3 . 1 CONT .
White Pine 3
White Pine 4
White Pine 5
White Spruce 1
White Spruce 2
White Spruce 3
White Spruce 4
White Spruce 5
26
-2.526
-1.833
-1.139
-0.447
-2.526
-1.833
-1.139
-0.447
-3.219
-2.526
-1.833
-1.139
-3.219
-2.526
-1.833
-1.139
-0.447
-3.219
-2.526
—1.833
-1.139
-3.219
-2.526
-1.833
-1.139
-0.447
-2.526
-1.833
-1.139
-3.219
-2.526
-1.833
-1.139
2.485
3.135
3.135
3.219
2.485
2.833
2.890
2.944
1.099
2.079
2.485
2.565
1.099
1.946
2.303
2.303
2.398
1.099
1.609
1.792
1.946
1.099
1.099
1.099
1.386
1.609
1.946
2.079
2.303
0.693
1.792
1.792
1.946
Species
Mean
Standard
Deviation
E_|0_t§
E__ ch RM s_ v_vE ms
1 6 17 27 7 14 11
2 6 15 29 7 23 7
3 3 16 19 3 25 5
4 5 13 29 6 19 1o
5 3 14 19 5 13 7
4.6 15 24.6 56 18.8 8
1.52 1.53 5.18 167 5.31 2.45
Table 3.2. Mean and standard deviations for each species
trial. EH-Eastern Hemlock, NWC—Northern White Cedar, RM—Red
Maple, SM-Sugar Maple, WP-White Pine, WS-White Spruce
1” Eastern Hemlock consists of 17 pairs of numbers. The
regression analysis revealed an adjusted r-square of
0.580, with a significant p-value of 0, and a
significant t-value of 4.807. The slope of the line
expressed in the regression equation is 0.329,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.2).
Figure 3.2.
Ln(N(r)) = 0.329Ln G) + 1.936
Fractal dimension equation for Eastern
hemlock
27
:2.N0rthern White Cedar consists of 17 pairs of numbers.
The regression analysis revealed an adjusted r—square
of 0.845, with a significant p-value of 0, and a
significant t—value of 9.382. The slope of the line
expressed in the regression equation is 0.674,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.3).
Ln(N(r)) = 0.674Ln G) + 3.582
Figure 3.3. Fractal dimension equation for Northern
white cedar
3.Red.Maple consists of 20 pairs of numbers. The
regression analysis revealed an adjusted r-square of
0.768, with a significant p-value of 0, and a
significant t-value of 7.994. The slope of the line
expressed in the regression equation is 0.607,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.4).
Ln(N(r)) = 0.607Ln (i) + 3.689
Figure 3.4. Fractal dimension equation for Red maple
28
4“ Sugar Maple consists of 18 pairs of numbers. The
regression analysis revealed an adjusted r-square of
0.681, with a significant p-value of 0, and a
significant t-value of 6.106. The slope of the line
expressed in the regression equation is 0.345,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.5).
Ln(N(r)) = 0.345Ln (é) + 2.168
Figure 3.5. Fractal dimension equation for Sugar maple
£5.White Pine consists of 20 pairs of numbers. The
regression analysis revealed an adjusted r-square of
0.554, with a significant p—value of 0, and a
significant t-value of 4.962. The slope of the line
expressed in the regression equation is 0.442,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.6).
Ln(N(r)) = 0.442Ln G) + 3.342
Figure 3.6. Fractal dimension equation for White pine
29
<5.White Spruce consists of 21 pairs of numbers. The
regression analysis revealed an adjusted r—square of
0.387, with a significant p—value of 0.002, and a
significant t-value of 3.689. The slope of the line
expressed in the regression equation is 0.359,
suggesting that the fractal dimension is between a
point and a line in typology (Figure 3.7).
Ln(N(r)) = 0.359Ln (é) + 2.387
Figure 3.7. Fractal dimension equation for White
spruce
30
APPLICATION AND DISCUSSION
To apply the inverse box-counting method to the
:ceclaimed landscape one would follow these procedures:
1” Divide the landscape to be planted in 50 meter grids.
2L Divide each 50 meter grid into grids with sides equal
to 1.563 meters (the size of the smallest boxes).
23.Use a random number generator to fill the grid with
numbers from 1-1024 for each tree species. Fill any
box which contains a number that is less than or equal
to the mean number of trees (from table 3.2) for each
trial. Next, count each box that contains a tree and
make sure the total falls within one standard
deviation (from table 3.2). The resulting grid
represents that particular species planting plan.
Repeat this process for each species of tree recorded,
and then combine the grids of all species onto one
grid of the same size for the overall planting plan
(Figure 4.1). The number of trees per grid can be
increased proportionally if the mortality rate of the
trees is known.
31
EITInis approach is illustrated with figures 4.1, 4.2, 4.3,
4; -4, 4.5, 4.6, and 4.7. This process generated seven
idiifferent fractal patterns, one for each of the tree
sspecies examined and one for all species combined (figure
49.1).
32
Eastern Hemlock
Northern White Cedar
Red Maple
SS?
Sugar Maple
White Pine
White Spruce
QQI‘QG
Figure 4.1. Fractal based planting plan for all species
33
EASTERN HEMLOCK
The process generated 6 boxes for planting trees
(figure 4.2). 6 boxes are within one standard deviation
(11.52) of the average of 4.6, so the 6 boxes were deemed
aacceptable.
IJFigure 4.2. Fractal based planting plan for Eastern Hemlock
34
NORTHERN WHITE CEDAR
The process generated 14 boxes for planting trees
(figure 4.3). 14 boxes are within one standard deviation
(11.58) of the average of 15, so the 14 boxes were deemed
aacceptable.
Figure 4.3. Fractal based planting plan for Northern White
Cedar
35
RED MAPLE
The process generated 23 boxes for planting trees
(f igure 4.4) . 23 boxes are within one standard deviation
(1:5 .18) of the average 24.6, so the 23 boxes were deemed
ac: c eptable .
Figure 4.4. Fractal based planting plan for Red Maple
36
SUGAR MAPLE
The process generated 4 boxes for planting trees
(figure 4.5). 4 boxes are within one standard deviation
(:1 .67) of the average 5.6, so the 4 boxes were deemed
ac c eptable .
Figure 4.5. Fractal based planting plan for Sugar Maple
37
WHITE PINE
The process generated 15 boxes for planting trees
(figure 4.6). 15 boxes are within one standard deviation
(15.31) of the average 18.8, so the boxes were deemed
acceptable.
Figure 4.6. Fractal based planting plan for White Pine
38
WHITE SPRUCE
The process generated 6 boxes for planting trees
(figure 4.7). 6 boxes are within one standard deviation
($2.45) of the average 8, so the boxes were deemed
acceptable.
Figure 4.7. Fractal based planting plan for White Spruce
In the Upper Peninsula of Michigan, a typical mine
site contains waste rock, with environmental conditions
similar to xeric forest sites in the region (figure 4.8).
The planting method can be completed with seedlings being
39
planted by hand or machine, as long as the tree is planted
in the correct designated box.
Figure 4.8. Example of a waste rock pile in Michigan’s
Upper Peninsula
The composition of trees in the study are similar to
those specified by Curtis, dominant trees however vary from
typical northern xeric forest. This is not a rare condition
as stated by Curtis (1959)
“Vegetationm is a chaotic mixture of communities, each
composed of a random assortment of species, each
independently adapted to a particular set of external
environmental factors. Rather there is a certain
40
pattern to the vegetation, with more or less similar
groups of species re—occurring from place to place."
This explanation from Curtis can also be attributed to
cover change over time. According to Albert and Comer, the
existing tree species composition is different from the
pre—settlement vegetation according to an interpretation of
the 1816—1856 general land office surveys (2008).
Results of the data collection process reveal a
consistent vegetation type by those described by Curtis
(1959), Linsemier (1989), and Albert & Comer (2008). There
were a number trees not indentified by these sources,
however changes to composition and introduction of new
species by humans can attribute these changes. It is also
important to remember that each area has its own unique set
of environmental conditions which can affect the
composition of vegetation within a given cover type
(Curtis, 1959). One constant that holds true throughout
these investigations is soil conditions, rocky and steep
terrain with exposed bedrock. It can be concluded that most
tree species identified by these investigations will be
appropriate for reforestation of surface mine reclamation
projects within the Upper Peninsula of Michigan.
41
Statistical analysis of the fractal dimensions of each
species, and the combined analysis reveal that all species
have similar patterns. The total fractal dimension of all
species revealed a slope of 0.587. The fractal dimension of
each species ranged from 0.329 to 0.674, revealing that
each species has the same Euclidian dimension of 0, but
their own distinctive fractal dimension. The intercept
value of all species was 3.107. The intercept value of each
species ranged from 1.936 to 3.582, revealing that each
species indeed has their own pattern and the overall
species composition falls within the parameters of these
patterns. The investigation of the fractal dimension of
each species reveals numbers which are similar to that of
Cantor's Dust. This result suggests that these fractal
patterns may be expressed at different scales (100 meter by
100 meter, 1 mile by 1 mile, etc.). Further research is
needed to determine if it is possible to apply these
findings to areas larger than 50 meters by 50 meters.
A limitation of this study is the scale of application
as specified above. This investigation used 50 by 50 meter
square grids, most reclamation projects are larger than
this. To be able to apply these findings at a larger scale
is an area of further investigation one may choose to
42
explore. The box—counting method expresses this scale based
limitation in the upper and lower limits of the regression
line. As the regression line continues past the boundaries
of the box-counting method, the line is skewed. The upper
limit of the regression line flattens, while the lower
limit of the regression line is abnormally steep. To scale
the results of this study without the proper mathematical
function would yield an unreliable result. Another
limitation of the box—counting method also relates to
scale, specifically the maximum size of the grid. The
fractal dimension estimate is highly correlated to the size
of the largest box (Kenkel & Walker, 1996). Site
limitations which caused this experiment to utilize 50
meter grids may have ultimately affected the fractal
dimension estimated. The estimated fractal dimension of all
species was relatively low when compared to the previous
investigations of Fleurant et al. (2009) and does not meet
the standards for a set of unaligned points in a two
dimensional plane. According to Kenkel & Walker (1996) the
fractal dimension of a two dimensional point pattern should
beZlSZ.
Another limitation of this study is the site of
application. This study focused solely on the vegetation of
43
a Northern Michigan rock outcrop. To apply these findings
anywhere but a Northern Michigan surface mine, one would
have to conduct their own survey of a natural area they
wished to replicate. This investigation determined the
fractal pattern of trees, while this is not a limitation,
further research is needed to determine if this process can
be used for other landscape features such as; topography,
or water networks.
In conclusion, it is determined that the box-counting
method can be used to estimate the fractal dimension of an
individual species of tree within a vegetation stand. The
inverse box-counting method can then be applied to re-
create the fractal patterns found in the landscape.
Successfully replicating natural tree stands is important
to many disciplines outside of mine reclamation. This
method can be applied for many projects including;
reforestation after forest fire (or other natural
disaster), restoration, or any project which attempts to
blend in with the surrounding vegetative community. Why
then choose reclamation to utilize this method? Returning
to a quote from Berger (2008), “reclamation can act as a
laboratory for experimentation.”
44
APPENDIX
Images of each trial of existing tree species
Figure A.1. Eastern Hemlock trial 1
Figure A.2. Eastern Hemlock trial 2
45
Figure A.3. Eastern Hemlock trial 3
Figure A.4. Eastern Hemlock trial 4
Figure A.5. Eastern Hemlock trial 5
46
Figure A.6. Northern White Cedar trial 1
Figure A.7. Northern White Cedar trial 2
Figure A.8. Northern White Cedar trial 3
47
Figure A.9. Northern White Cedar trial 4
Figure A.10. Northern White Cedar trial 5
Figure A.11. Red Maple trial 1
48
Figure A.12. Red Maple trial 2
Figure A.13. Red Maple trial 3
Figure A.14. Red Maple trial 4
49
Figure A.15. Red Maple trial 5
Figure A.16. Sugar Maple trial 1
Figure A.17. Sugar Maple trial 2
50
Figure A.18. Sugar Maple trial 3
Figure A.19. Sugar Maple trial 4
Figure A.20. Sugar Maple trial 5
51
Figure A.21. White Pine trial 1
Figure A.22. White Pine trial 2
Figure A.23. White Pine trial 3
52
Figure A.24. White Pine trial 4
Figure A.25. White Pine trial 5
Figure A.26. White Spruce trial 1
53
Figure A.27. White Spruce trial 2
Figure A.28. White Spruce trial 3
Figure A.29. White Spruce trial 4
54
Figure A.30. White Spruce trial 5
55
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58
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