New Methods for Estimating Fractal Dimensions of Coastlines

by

Jonathan David Klotzbach

A Thesis Submitted to the Faculty of The College of Science in Partial Fulfillment of the Requirements for the Degree of Master of Science

Florida Atlantic University Boca Raton, Florida May 1998 New Methods for Estimating Fractal Dimensions of Coastlines

by

Jonathan David Klotzbach

This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Richard Voss,

Department of Mathematical Sciences, and has been approved by the members of his supervisory committee. It was submitted to the faculty of The College of Science and was accepted in partial fulfillment of the requirements for the degree of Master of Science.

SUPERVISORY COMMITTEE: ~~~y;::ThesisAMo/ ~-=

Mathematical Sciences

Date

11 Acknowledgements

I would like to thank my advisor, Dr. Richard Voss, for all of his help and insight into

the world of fractals and in the world of computer programming. You have given me guidance

and support during this process and without you, this never would have been completed. You have been an inspiration to me. I would also like to thank my committee members, Dr. Heinz-

Otto Peitgen and Dr. Mingzhou Ding, for their help and support. Thanks to Rich Roberts, a graduate student in the Geography Department at Florida Atlantic University, for all of his help converting the data from the NOAA CD into a format which I could use in my analysis. Without his help, I would have been lost.

Thanks to all of the faculty an? staff in the Math Department who made my stay here so enjoyable. I would especially like to thank Dr. Lee Klingler for all of the time and encouragement he has given me over the years. Thanks to Dr. Ron Mullin for his care and commitment to students and his availability to be of help whenever needed.

r would like to thank my parents, David and Barbara Klotzbach, for their advice and encouragement throughout this entire process. I would also like to thank my new parents, Dan and Jean y ankovich, for their continued support. Most of all, I would like to thank my wife

Danielle for the love, encouragement, and patience that she has shown during this time. I love you.

Ill Abstract

Author: Jonathan David Klotzbach

Title: New Methods for Estimating Fractal Dimensions of Coastlines

Institution: Florida Atlantic University

Thesis Advisor: Dr. Richard Voss

Degree: Master of Science

Year: 1998

A coastline is an example of a statistically self-similar fractal. A standard characterization walks

a ruler of fixed size along the coast and estimates fractal dimension from the power-law

relationship between length and ruler size. Multiple intersection can lead to ambiguity in

choosing the next step. The standard method always chooses the fitst intersection along the

curve. Variations Were developed to choose intersections which highlight geographic properties.

The land method measures accessibility to the coast from land at each size while the water

method probes water access. Measurements on sections of the East and West Coasts of the

United States typically showed the land length exceeding water. Jumps in water length as step

size decreased were often caused by narrow rivers or bays which have few corresponding land features. Simple recursive constructions were inadequate to model this asymmetry.

iv Contents

1 Introduction ...... 1

1.1 Overview...... 1

2 Fractal Geometry and Fractal Dimension ...... 4

2.1 What is a fractal?...... 4

2.2 Topological Dimension...... 7

2.3 Hausdorff Dimension...... 8

2.4 Self-Similarity Dimension...... 10

2.5 Divider Dimension...... 12

2.6 Box-Counting Dimension...... 13

3 Fractals Applied to Coastlines ...... 15

3.1 History...... 15

3.2 Mathematics and Coastlines...... 15

3.3 Criticisms...... 17

3.3.1 Last Partial Step Problem...... 18

3.3.2 Starting Point Problem...... 20

3.3.3 Non-linearity in Log-Log Plots...... 20

3.3.4 Limited Range of Measurement...... 21

v 4 A New Approach to Fractal Coastline Characterization...... 22

4.1 Introduction...... 22

4.2 The Four Methods...... 23

4.3 The Data...... 25

4.4 Land vs. Water Results...... 28

5 Modeling Coastlines by Fractal Methods...... 42

6 Future Research...... 48

7 Conclusion...... 50

Bibliography...... 51

VI List of Dlustrations

Figure 2-1 Koch Curve...... 5 Figure 2-2 Modified Koch Curve...... 6 Figure 3-1 San Francisco Bay...... 17 Figure 3-2 Example of Large Partial Step Error...... 19 Figure 3-3 Four methods for the New England Coast...... 21 Figure 4-1 Analysis of the four methods...... 23 Figure 4-2 Land & Water methods on the New England Coast...... 24 Figure 4-3 First & Last methods on the New England Coast...... 25 Figure 4-4 East and West Coast section breakdown...... 26 Figure 4-5 The six different coastlines...... 26 Figure 4-6 Log-log plots for the various coastlines...... 29 Figure 4-7 Land and water methods for Oregon & northern California...... 31 Figure 4-8 Water method in San Francisco Bay (2000 m.)...... 32 Figure 4-9 Water method in San Francisco Bay (700 m.)...... 33 Figure 4-10 Water method in Washington state (walking north to south)...... 34 Figure 4-11 · Water method in Admiralty Inlet, WA...... 34 Figure 4-12 Data from water method in Washington (north to south)...... 35 Figure 4-13 Water method in Washington state (walking north to south)...... 36 Figure 4-14 Data from water method in Washington (south to north) ...... 36 Figure 4-15 Water method in Puget Sound, W A...... 36 Figure 4-16 Data from water method on West Coast...... 37 Figure 4-17 San Francisco Bay...... 39 Figure 4-18 Land vs. Water Difference ...... 41 Figure 5-1 Coastline formed by random Koch generator ...... 43 Figure 5-2 Coastlines using Koch generators of length 0.3 & 0.4 ...... 44 Figure 5-3 Best attempt using Koch generators (overlap at small scales) ...... 45 Figure 5-4 Zig-zag fractal...... 46 Figure 5-5 Best attempt using zig-zag generator (overlap at small scales) ...... 47

vii Chapter 1

Introduction

1.1 Overview

Coastlines, the boundary between land and water, and other natural boundaries have long been a subject of human fascination. For centuries, their irregularity and variation over time have defied attempts at characterization using simple methods based on Euclidean geometry. Many nations throughout history established their borders by using natural features such as mountain ranges and rivers. These boundaries were easily observed, simplifying the determination of a nation's territory. However, attempts at measuring these natural objects often led to frustration because of their irregular shape. Coastlines have also defied such quantification attempts. Early cartographers attempted to capture the shapes and lengths of various coastlines around the world by drawing maps to represent their observations. With the use of satellite imaging and computers, more detailed data has been gathered on coastlines. Some maps are able to locate a position to within a few meters of its true location. Thus, with this abundance of technology and data at our disposal, it would appear to be a simple task to measure the length of a coastline. One of the difficulties in measuring lengths of natural objects is that our intuition based on Euclidean shapes fails when applied to objects like coastlines. Measuring the length of smooth curves is a simple process of successive approximations by line segments. In order to obtain more accurate measurements, smaller segments can be used. As the segment size approaches 0, the sum of the lengths of the segments will approach the length of the curve. However,

1 this naive application of Euclidean geometry to natural curves fails because the limit may not exist. In spite of widespread data on coastline length, apparent contradictions abound. For example, the Encyclopedia Americana states that "Britain has coasts totaling 4650 miles" while Collier's Encyclopedia reports that "The total mileage of the coastline is slightly under 5000 miles." [16] 5000 miles is an unlikely approximation of 4650 miles, which implies that there is an apparent contradiction in the measured length. The border between Spain and Portugal presents another contradiction in measurements. An encyclopedia in Spain lists the border as having length 987 kilometers while a Portuguese encyclopedia states that the border has length 1214 kilometers [16]. There are several possible explanations for the discrepancy. One possibility is that the length measurements were sloppy. Another possibility is that both parties used different data, some of which was erroneous. However, a third possibility is that different measurement techniques were used, leading to differences in measured lengths. There are many techniques for approximating the length of a given coastline. With a digitized map, summing the lengths of the line segments forming the coastline on the map results in one approximation for the length. With a hand-drawn map, a similar approximation can be obtained by covering the curve with line segments smaller than the resolution of the map. However, since a map is a tool for determining the location of an object in relation to other objects, the choice of the map is critical to the result. For ~xample, if a map with an accuracy1 of 1000 meters is used, one answer for the length is obtained. However, if a map with a resolution of 100 meters is used, the coastline will often have a longer length because there are more bays, inlets, and peninsulas which were not distinguishable at the larger scale. The bay may have simply been represented by a straight line across the center of the bay instead of the extra detail showing the curving inwards and then out again. This new bay would cause an increase in the overall length of the coastline which was not visible at the larger resolution. The variation in the lengths of coastlines at different scales intrigued the scientist Lewis Fry Richardson in the 1920s. He examined the coastlines of several countries including Great Britain, South Africa, and Australia as well as the border of Spain and Portugal [22]. He noted

1 By the accuracy or resolution of a map, we mean the maximum error between an object's map position and its actual position.

2 the discrepancies in encyclopedia measurements of length and decided to estimate the length for himself. He chose to measure a coastline length by walking a divider of a specific length E along the coastline to see how many dividers N(E) were needed to cover the entire coastline. He calculated the length of the coastline L(E) by multiplying the number of dividers by the length of each divider. Thus, L(E) = N(E) · E. He explored the relationship between L(E) and E and discovered that, as E decreases, L(E) grows without bound in the form of a power law where L(E) = k · ca for a constant a. For most real coastlines, 0 ::::; a ::::; 0.4. Thus, the differences in the measurements of the coast of Britain and in the Spain-Portugal border may have been a function of which E was used in the estimation of length. Richardson concluded that length is not meaningful when applied to coastlines [22]. The same is also true for many borders between states and countries. Of course, when the border is a straight line like the border of Utah with its surrounding states, then the length of the border measured at any E is the same up to a last partial step. However, when the border is a natural object like a river or a mountain range, border length is not as well-defined. Richardson found in his investigation of the border of Portugal and Spain that the measurements of border length differed by as much as 20% [22] . The same was true in the measurements of the border between the Netherlands and Belgium. All of these inconsistencies could be accounted for by the use of different E's when measuring the length. The observed relationship between L(E) and E and the need for a-clarification of the concept of length inspired the mathematician Benoit Mandelbrot to begin his own study that eventually led to fractal geometry. This is now a rapidly growing subject with applications in mathematics, computer science, graphic design, economics, geography, and physics.

3 Chapter 2

Fractal Geometry and Fractal Dimension

2.1 What is a fractal?

The word fractal was coined by Mandelbrot in 1975. He derived it from the Latin adjective fractus and the corresponding verb frangere which means to break or to create irregular frag­ ments. It has the same root as the word fraction or fractional. Thus, the word fractal refers to irregular or fragmented objects and describes the curves and natural shapes that Mandelbrot was studying, including coastlines and natural borders. He originally defined a fractal as follows: "A fractal1 is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." [16] However, this was not very useful for application to real world shapes since such abstract forms of dimension are not always defined or measurable. Even so, dimension plays an important role in fractal geometry. Many natural objects such as coastlines, clouds, and trees can be viewed as fractals. They are often called natural fractals and have similar properties to some classical mathematical curves over a finite range of spatial scales. The mathematical definition of fractals can be better understood by first examining some examples of fractals and their common properties.

1Mandelbrot later retracted this definition as being too restrictive because it eliminated some interesting cases. He has found it more useful to consider the looser definition of fractals as shapes made of parts similar to the whole in some way [10] [21] .

4 Many classical mathematical curves are fractals. One of the best known is the von Koch curve (Figure 2-1). At stage 0, it is a line segment with length 1. Then, the first stage is constructed by replacing the original segment with a generator. Thus, the new curve has four segments, each of length i· This replacement process is then repeated with each of the four new segments to generate a curve with sixteen segments, each of length i. At stage n, the curve has length (~)n. As n increases without limit, the length diverges.

1/3 1/3

1 1/3 1/3

Stage 0 Stage 1 - Generator 1 segment; length 1 4 segments; length i

Stage 2 Final result 16 segments; length i Dimension=1.26816 ...

Figure 2-1: Koch curve

One striking visual property of the von Koch curve is exact self-similarity. Peitgen defines it this way: "A structure is said to be self-similar if it can be broken down into arbitrarily small pieces, each of which is a small replica of the entire structure." [20] Thus, the von Koch curve can be constructed from smaller copies of itself (e.g. 4 copies scaled by i). Exactly self-similar objects can be assembled from similarity transformed copies of itself. 2 For example, with a point x = (x0 , x 1) in JR , we can define a similarity transformation T : 2 2 JR -+ JR by T(x) = (rxo,rx1) where r is a fixed scaling ratio less than 1 [10]. By applying a series of such transformations to the object and then rotating or shifting the smaller copies, we can construct an exactly self-similar object. As shown above, the Koch curve can be

5 constructed by applying four such transformations. The Koch curve can also be constructed by applying two transformations or by using 4n transformations for any positive integer n. It is possible to construct a variation of the Koch curve with segments of differing lengths corresponding to transformations with differing r. These segments remain exact copies of the original, but some of the copies are smaller than others (Figure 2-2) . This variation of the Koch curve still has exact self-similarity.

1/3 1/3

Stage 1 - Generator Stage 2 Final result 4 segments; length} & ~ 16 segments Dimension = 1.38392 ...

Figure 2-2: Modified Koch curve

Natural fractals do not have exact self-similarity. Even though small parts "look like" the whole, they are not exact reductions. This property is called statistical self-similarity. Statistical self-similarity means that, on average, any one section resembles any other section at different spatial scales. Many fractals are self-affine rather than self-similar. A self-affine fractal set is generated by a series of transformations using different scaling values. We can define an affine transformation 2 2 T: JR -+ JR by T(x) = (r1x0 , r 2x1) where r 1 =F r2 and r 1,r2 < 1 [10]. Objects constructed from a series of affine transformations have exact self-affinity. The scaling factor may be different in each direction, and although pieces of the set look much like the original, directional variations are visually apparent. Some natural fractals have statistical self-affinity rather than exact self-affinity. As in the case of statistical self-similarity, statistical self-affinity means that on average, the object is self-affine. This idea has been useful in the study of Brownian motion [16] [23]. It is also useful in studying surfaces such as that of the earth. The vertical size range is usually much less than the horizontal size range. The relationship of coastlines to self-affine surfaces may

6 play an important role in future research on coastlines. One of the consequences of self-similarity or self-affinity is the presence of a fine structure; detail at arbitrarily small scales. For example, the von Koch curve can be magnified any number of times, and the detail remains the same. Nat ural fractals also have a fine structure, but observation and measurement of natural fractals is usually limited to a finite range of spatial scales. Thus, the study of coastlines is limited by the accuracy of the available maps and data sets that approximate coastlines with a fixed accuracy. Unlike mathematical curves, natural objects such as coastlines are not static in time. There is constant fluctuation of the shoreline due to the action of the water which is influenced by the weather and tides. Thus, although natural objects behave as fractals in many ways, it is impossible to carry the comparison to arbitrarily small sizes. Issues of time and climate create additional challenges. Thus, applying fractal properties to natural objects require extra care.

2.2 Topological Dimension

Dimension is a commonly used term which is often misunderstood. Many variations have been defined, and for some sets, these different dimensions yield different values. One example is fractal dimension. One popular usage [20] is that fractal dimension is a class of different dimensions (such as the box-counting dimension, self-similarity <;_iimension, and the divider dimension) which are all equal with exactly self-similar fractals like the Koch curve. In this paper, unless otherwise noted, fractal dimension will refer to the divider dimension, which will be defined below. Topological dimension is the integer dimension assigned to Euclidean shapes. A point has topological dimension 0, one dimensional objects have length, two dimensional objects have an area, and three dimensional objects have a volume. Thus, lines have topological dimension 1, surfaces have topological dimension 2, and solids have topological dimension 3. However, this definition is not useful for fractal objects. Another definition of topological dimension involves the connectivity of the set. This form of topological dimension is called the large inductive dimension. A simple definition is that an object has topological dimension 1 if it can be separated by removing a countable number of

7 points, has topological dimension 2 if it can be separated by removing a countable number of lines, and has topological dimension 3 if it can be separated by removing a countable number of surfaces and so on [15]. Using this definition of topological dimension, lines, surfaces, and solids have the same dimension as before, but other objects can also have a topological dimension. For example, the Koch curve (see Figure 2-1) has topological dimension 1 because if any point is removed (except the first or last point), then the curve becomes disconnected. Other fractals also have a topological dimension, although this can be difficult to prove. A difficult exercise is proving that the Sierpinski Gasket has topological dimension 1 [8]. A more formal definition involving metric spaces and separability is available [8]. However, all of the definitions of topological dimension are the same when viewed over a separable and compact metric space like IR2 [8]. Coastlines can be viewed as curves2 in a closed and bounded interval over IR2 and hence in a compact, separable metric space.

2.3 Hausdorff Dimension

The Hausdorff (or Hausdorff-Besicovitch) dimension was defined before the term fractal was coined. It was defined by Felix Hausdorff in the early 1900s, and much of the early work on the subject was done by A.N. Besicovitch. It is now considered one of the fractal dimensions because it has been useful in describing the properties of many fl:actal sets. It is based on measures, which are easy to manipulate, but often very difficult to compute. Thus, this is not a practical characterization, but the theory is important for a better understanding of fractals. For a complete treatment of Hausdorff dimension, see [8]. In order to define the Hausdorff dimension, it is necessary to discuss the concepts of mea­ sures and measure spaces. Hausdorff dimension can be defined for any metric space, but the discussion is simplified over IRn. Since coastlines can be viewed as subsets of JR2 , IRn will be used as the basis for defining Hausdorff dimension. If U is a non-empty subset of IRn, then the

2 diameter of U is defined to be lUI= sup{d(x,y): x,y E U} where d(x,y) = JL:7= 0 (xi -yi) is the standard distance metric on IRn. Thus, the diameter is the largest distance _between any two points of the set U. If S is any metric space, then a family U of subsets of S is said to

2We use curve to mean a topologically !-dimensional, connected, non-branching set with no multiple points.

8 cover a set A if A is contained in the union of all of the sets of U. A family which covers a set is known as a cover of the set. IfF is a subset of IRn, and {Ui} is a collection of sets with diameter at most 6 which cover F, then {Ui} is called a 6-cover of F. Then, let

Hl(F) =in£ { ~ IU;I'' (U;} is a 6- cover ofF} (2.1)

This sum can be either finite or infinite. As 6 decreases, the number of possible covers decreases since covers with diameter bigger than 6 are eliminated from consideration. This means that the inf must either increase or remain constant, but it cannot decrease. It must then approach some limit from below (that limit could be infinite). Then, the s-dimensional Hausdorff measure of F is defined to be (2.2)

With some work, it is possible to show that H 8 (F) is actually a measure on the space IRn [8]. This means that:

2. H 8 (A) ~ H 8 (B) if A~ B with A, B Borel measurable

Now, this measure is (up to some constant) the same as then-dimensional Lebesgue mea­ sure (i.e. n-dimensional volume) [8]. Thus, H 1(F) is the same as the length of the curve, H 2 (F) is the same as the area, and H 3 (F) is the same as the volume (up to some constant). The scaling properties of n-dimensional Hausdorff measure are the same as for n-dimensional volume. When an object is scaled by a factor of 6, then the length is scaled by 8, the area

2 3 is scaled by 8 , and the volume is scaled by 8 . The same holds for n-dimensional Hausdorff measure which scales with a factor of 6n. This is one of the fundamental properties in the theory of fractals. Hausdorff also proved that there exists a number dimH F such that

oo if s < dimH F } H'(F) = { (2.3) 0 if s > dimH F

9 dimH F is called the Hausdorff (or Hausdorff-Besicovitch) dimension of the set F [8]. Another definition is that

dimH F =in£ {s: H 8 (F) = 0} =sup {s: H 8 (F) = oo} (2.4)

Now, if s = dimH F, then H 8 (F) can be 0, infinite, or some real number. This dimension has certain fundamental properties:

1. IfF~ 1Rn, then dimH F ~ n.

2. If F ~ 1Rn and F is open, then dimH F = n (since F contains a ball of positive n-dimensional volume).

3. If E ~ F, then dimH E ~ dimH F (by the measure property that H 8 (E) :::; H 8 (F)).

4. If F is a countable set, then dimH F = 0.

It is desirable for these properties to be satisfied by most definitions of dimension. For example, (1.) means that a real object's Hausdorff dimension cannot exceed the topological dimension of the space in which it is embedded. The Hausdorff dimension is also the most abstract version of the fractal dimensions. It is often difficult to calculate except for some simple fractals. For natural fractals, this dimension is impossible to calculate because there is no data available as 8-+ 0, and other forms of dimension are necessary.

2.4 Self-Similarity Dimension

The self-similarity (or similarity) dimension is a fractal dimension which is defined for exactly self-similar objects. An exactly self-similar object can be broken into a fixed number of arbi­ trarily small pieces, each of which is a scaled down replica of the whole. Thus, at a specifically chosen scaling factor r, there will be N(r) scaled down replicas of the original. For example, in the construction of the Koch curve in Figure 2-1, a line segment of length 1 is replaced by four line segments of length i. Thus, the original Koch curve can be constructed by pasting together four copies of the original scaled down by i. Similarly, at a scale size of ~, the orig-

10 inal curve can be constructed from sixteen of them pasted together. Thus, when r = (k)n, N(r) = 4n. Lines, squares, and cubes are also exactly self-similar, and it is interesting to note that

1 N(r) = (2.5) 0r where D is the standard topological dimension. Thus, when a line is scaled down by ~, two copies are necessary to reconstruct the original. Similarly, it would take four squares scaled down by ~ to obtain the original, and 8 cubes scaled down by ~ to obtain the original cube. It would be advantageous to define a dimension for all exactly self-similar objects that would satisfy equation (2.5). This implies that log N(r) = Dlog(~), leading to the definition of self-similarity dimension as

D = log N(r) (2.6) log(~)

This is sometimes called Ds to avoid confusion with other dimensions. For the Koch curve constructed from 4n pieces with r = (k)n,

log4n log4 Ds = - = - = 1.26186 ... (2.7) 1og 3n 1og 3

Hence, for fractal objects, it is possible that the self-similarity dimension will not be an inte­ ger. For self-similar Euclidean objects, the topological dimension will equal the self-similarity dimension. There is another definition [10] which is more inclusive and defines the self-similarity di­ mension for exactly self-similar objects with different scaling factors. Let n be the number of scaled down pieces in the construction of an exactly self-similar fractal and let r1, ... , r n be the scaling factors (some of them can be equal). Then, the self-similarity dimension Ds is the solution to

(2.8)

11 For an example like the modified Koch curve in Figure 2-2, the self-similarity dimension is the solution to

1)D. (2)D. 1=2· ( 3 +2· 5 (2.9) which is Ds = 1.38392 ... Under this definition, the standard Koch curve retains the same similarity dimension.

2.5 Divider Dimension

The divider (or compass) dimension is another of the fractal dimensions. This dimension is defined for all curves, not just exactly self-similar ones, but if a curve is exactly self-similar, then the divider dimension is the same as the self-similarity dimension. This dimension examines both the relationship between scaling size and the length of the curve and the relationship between scaling size and the number of segments at that size needed to cover the curve. It is measured as follows. A pair of dividers is opened to a step length E and is walked along the curve until the end is reached after N(t:) steps. The length is measured as L(t:) = E · N(t:). When E is large, this method skips many of the irregularities, but as E decreases, the finer features of the curve are also included, and so the overall length increases. Richardson's research on coastlines used the divider method and showed that this behavior is a power law with L(t:) = k · Ea where k, a are constants [22]. Mandelbrot [16]1ater discovered with exactly self-similar fractals that a = 1 - D where D is the similarity dimension. Thus, L(t:) = k · ~: 1 -D. Upon taking the logarithm of both sides, we get log L( ~:) = log k + (1- D) log E. A plot of logE on the x-axis and log L( E) on the y-axis results in a line which has approximate slope 1 - D. Thus, power law behavior results in linearity on a double logarithmic plot (sometimes called a log-log plot). Of course, the data from length measurements of natural fractals is not exactly linear, but the approximation is often good enough to use the least squares method for a close linear fit. Richardson used this technique when he observed the power-law behavior with the coastline of Britain [22]. The double logarithmic plots of coastline length versus the step length are called Richardson plots in honor of him. The resulting value of D is called the divider dimension.

12 It also follows that f.· N(€) = k · f.l-D and so N(€) = k · cD. Then, by taking the logarithm of both sides, we get logN(€) = logk- Dlogf.. When using a double logarithmic plot, the slope of the resulting line will be approximately -D. Thus, there are two ways to calculate the divider dimension, and the values of D are still roughly the same although there can be some deviation because of the different methods used. A major difficulty with the divider dimension is ambiguity in how to cover the curve with segments of a particular scaling factor. This problem is principally due to multiple forward intersections at a particular stepsize. The most common solution is to identify the first time a line segment of the step length f. intersects the curve moving sequentially along the curve. An additional problem is how to compensate for the leftover portion of the curve which has length less than f.. When f. is small, so is the error, but for the larger values of f., this leads to greater error. This problem will be addressed in Section 3.3.1.

2.6 Box-Counting Dimension

Box-counting dimension is the last of the fractal dimensions to be discussed here. This is one of the most commonly used dimensions in the sciences because of its simplicity for machines, and because this dimension can be applied to any object in space. It is a simplification of the Hausdorff dimension, and for many fractals like the Koch curve, the box counting dimension will be equal to the other fractal dimensions. However, for some objects, the box-counting dimension will be different from some of the fractal dimensions. For example, it can be shown that the box-counting dimension of the set of rational numbers is 1 whereas the Hausdorff dimension of the rationals is 0 [9]. The box-counting dimension is defined not just for curves but for objects in other dimensions as well. There are several different versions of the box-counting dimension, and the one to be dis­ cussed here relates to curves. A grid size r is chosen with each box r x r, and the curve is drawn on the grid. Then, the number of boxes N(r) which contain a portion of the curve are counted. N(r) will usually increase as r decreases. For fractals, a plot of N(r) vs. r on a double logarithmic plot is typically linear with slope -Db since the values of N(r) increase with r in the form of a power law. Db is called the box-counting dimension. This definition can

13 be extended to other dimensions by replacing two dimensional boxes by n-dimensional boxes [20]. There are several different methods for calculating dimensions of fractals. All of these are equal and well-defined for many exactly self-similar fractals. However, most natural objects do not have exact self-similarity, and so care must be taken when applying dimension to these objects.

14 Chapter 3

Fractals Applied to Coastlines

3.1 History

Richardson was the first scientist to undertake a detailed study of coastline length measurements and document the apparent contradiction between lengths measured at different stepsizes. The term fractals was not coined for another 50 years. However, he observed many of what later came to be known as fractal properties of a coastline including the fine structure over all scales and the power-law relation between coastline length and stepsize. Benoit Mandelbrot spent a great deal of time studying the coastline of Britain and wrote se:reral papers on the power law behavior that Richardson observed [13] [14]. Coastlines are currently a popular example of fractals that people can understand and observe. Various popular books on the subject have helped to bring this subject to the public attention. However, with the rapid propagation of the subject came confusion over the meaning of terminology and how fractal concepts should be applied to natural objects.

3.2 Mathematics and Coastlines

It is important to begin the discussion of coastlines with definitions of some terms which will be frequently encountered and to observe some standard coastline properties. A coastline is a curve, the !-dimensional boundary between two areas of the plane: land and water. Since all continents are islands, a coastline has no beginning or end point. However, the available data

15 for coastline studies divides continents into several sections which must contain a beginning and end point. Thus, for the purposes of this paper, references to the beginning and end of a coastline will refer to the beginning and end of the sections being discussed. As discussed earlier, the length of a curve is not necessarily easy to compute. For many irregular objects like a coastline, the divider method (Section 2.5) yields a length that depends on the stepsize €. When applied to islands, a starting point is chosen, and the divider is "walked" along the coastline until there are no full steps left without reaching the original starting point. The leftover part is considered a partial step, and that is added to the length to give a total length of the coastline at that stepsize. Any starting point along the island can be chosen, and the same procedure can be followed, but the measured lengths may be different. When dealing with sections of coastlines which are viewed as having a beginning and end, there are several possible methods for evaluating the length at a particular €. The walk can begin at either end point and proceed until there are no full steps left at the opposite end of the curve. The last partial step is added to the length to give the total length at that stepsize. Another approach is to start the walk at some point along the coastline and walk in both directions until there are no full steps left at either end. There will be two partial steps left, one on each end of the curve, and each of those is added to the length to give the total length at that stepsize. As with the case of islands, different starting points can give different lengths even at the same stepsize. Many natural objects including coastlines, cloud formations, and mountain skylines also exhibit statistical self-similarity. For a coastline, each section will not be a scaled down copy of the original, but the sections will share the same distribution of irregularities. For example, the smaller section might have the same distribution of large and small bays as the original. Viewing a coastline with greater resolution will often reveal detail that was not observable at the larger scale. Exactly self-similar models of coastlines usually appear highly artificial. Exact self-similar fractals also fail as models for clouds, mountains, and planetary surfaces. Thus, it is important to add randomness into the process of modeling natural fractals. Consequently, some authors refer to natural fractals as statistical fractals [24]. The asymmetry between land and water is another important consideration in coastline studies. There is often a noticeable difference between the land and water sides of a coastline.

16 Figure 3-1: San Francisco Bay

One such difference is that a typical coastline has noticeably more inlets and bays than it has peninsulas (Figure 3-1). Another feature is the number of rivers which flow from deep within the land into the ocean. These rivers are often narrow at their source and widen as they flow into the ocean. There are no such land features to correspond to those rivers. Thus, even without being told, most observers could tell immediately which side of a map is land and which is water. Another observation about coastlines involves the distributions of changes in the coastline length. Over certain ranges of E, the changes in L(E) are quite large, but over other ranges of

E, there is little or no change [16]. For example, if a coastline is measured with E bigger than the size of the rocks (say 30 centimeters), but smaller than the size of any noticeable bays, then there would be little change in L(E) as E is decreased. However, upon reaching an E where E is smaller than the average rock length, L(E) would begin to change again because of the irregularity of the rocks.

3.3 Criticisms

In recent years, several criticisms have been leveled against using the divider dimension to characterize coastlines. There are four sources of problems. One problem is handling the

17 last partial step. A second problem is that different lengths are obtained when starting from different points along the coastline. Another problem is that log-log plots of statistically self­ similar fractals are often non-linear over certain stepsizes, and so no unique divider dimension is defined. A final problem is that there is a limited range of data. These problems have been highlighted by several papers [1][2][3][11][18] over the past few years and need to be addressed before studying coastline length.

3.3.1 Last Partial Step Problem

The process of finding the divider dimension begins by choosing a starting point and walking a divider of length e along the coastline to determine how many segments N(e) are needed to cover the coastline. The length of the coastline at stepsize e is given by (N(e) ·e)+ P(e), where P(e) is the length of the leftover portion of the curve at stepsize e, known as the last partial step. This last partial step has been the subject of debate over recent years because of the possibility for error which is introduced [1 ][18]. There have been three suggestions to handle the last partial step: ignore it, add a full step if it exceeds ~ and ignore it otherwise, or add the length of the last partial step to the total. Various analyses have been performed on each method, and the error has been approximated [1] [18]. Ignoring the last partial step can lead to an error as large as the size of the step, but the mean error is closer to ~. This method is highly inaccurate, and the other two methods are often better. One study [1] showed how the error from the last two methods can still approach

e in certain cases, even though the mean error is around ~. Most of the problems from the last partial step result when there is an odd configuration of coastline at the end of the walk leading to a small partial step even though there is a significant amount of coastline leftover. In Figure 3-2, point A is the previous intersection, point B is the last point on the curve before adding the last partial step, and point D is the endpoint of this segment of coastline data. The last partial step is close to the stepsize e, but the actual coastline length after B is close to 2e since the distance from B to C and the distance from C to D are each slightly less than e. Thus, in this case, the error is roughly e. One suggestion for handling this problem is to add the length based on the configuration of the last step. This can be difficult to carry out in practice because it requires a machine to deviate from a set algorithm and evaluate the configuration of

18 Figure 3-2: Example of Large Partial Step Error a coastline. However, since the error rarely approaches the full length of the step anyway, this will only be a problem for large stepsizes. Another suggestion for handling the last partial step was Richardson's original idea in his coastline study. Richardson only included stepsizes where there was no partial step (or only a very small percentage of the original) [18] . This approach can limit the number of points obtained. It can also be difficult computationally because many. different stepsizes must be tested before a suitable choice is found. It is possible that over a certain range of stepsizes, there is no stepsize which has a small partial step since a decrease in stepsize doesn't ensure that the partial step will increase or decrease. Thus, it is often infeasible to find stepsizes where there is no partial step. It also seems that the last partial step is only important for the largest stepsizes. When the stepsize is a large percentage of the distance between the endpoints, the length is so dependent on the starting point that a better solution is to ignore the largest steps and focus on the dimension over smaller stepsizes. Thus, although Richardson's approach would increase the accuracy of the results, it is very difficult to implement.

19 3.3.2 Starting Point Problem

There can also be a problem in determining coastline length because different starting points on the curve can lead to different lengths. For example, starting at one point might lead to "stepping over" a certain bay while another starting point might lead to "stepping into" the bay and hence adding a considerable amount of length, especially at the larger stepsizes. This can result in large inaccuracy in the calculation of the divider dimension. Several solutions to this problem have been proposed. One solution is to increase the size of the data set by adding a larger portion of the coastline. However, this is often impractical. Another solution is to start from several points along the coastline and obtain a length for each starting point and then average those results together. In one study of city boundaries, each point along the boundary was used as a starting point, and then the results were averaged together [12]. This method is the most accurate, but it can be impractical because of the computation time if the coastline contains hundreds of thousands of segments as in our study. Studies have been done to determine how many points are needed to reduce the standard deviation between calculations of dimension to less than 10-4. Andrle found that 50 randomly chosen starting points were enough to satisfy those conditions [1].

3.3.3 Non-Linearity In Double Logarithmic Plots

Non-linearity in double logarithmic plots is probably the strongest criticism of using fractals as a model for real coastlines. In the case of an exactly self-similar fractal, the behavior of length vs. stepsize on a double logarithmic plot is linear when the appropriate starting points and scaling factors are chosen. However, with natural fractals, this is not always possible, despite hpw carefully starting points and scaling factors are chosen. Since a coastline is not exactly self-similar, the double logarithmic plots may have jumps and gaps at various stepsizes. Some have proposed what is called a fractal element model for coastlines [19]. A fractal element is a range of over which the fractal dimension is constant, and so the length increases as a power law with respect to stepsize. A fractal element is usually followed by a transition region where the length doesn't obey any power law, and then a new fractal element may begin. Over a fractal element, it is assumed that the curve is statistically self-similar, and that some single process is at work which dominates the formation of the coastline over that range of

20 I I

i "i i v • v •V 0 • •

• land oo v water o first - • lost

Stepsize Log-log plot for eost5

Figure 3-3: Four methods for the New England Coast

€ [19][17][11]. Thus, it can be possible to assign a single dimension to each fractal element of the coastline. This model works for many coastlines, but there are still some exceptions. Our research on finding different ways of estimating fractal dimension and studying some of the differences between land and water was stimulated in part by a desire to understand this non-linearity.

3.3.4 Limited Range of Measurement

With a limited range of measurement, the power-law does not extend to arbitrarily small stepsizes € (Figure 3-3). Although a limited amount of data often leads to power law behavior in the middle range of stepsizes, the largest stepsize is limited by the distance between the endpoints. Hence, there is an upper bound on the original stepsize. Similarly, there is a limit to the smallest € from the resolution of the data. Consequently, as € decreases, the length will approach the length of all of the segments added together, and the graph on the double logarithmic plot will level off with limiting slope approaching 0 as € -+ 0. Thus, there is only a finite range where fitting a line on a double logarithmic plot is useful for dimension; a limited range where the power law behavior can be observed. Hence, limited data can lead to inaccurate results if fractal dimension is applied outside of the range of measurement.

21 Chapter 4

A New Approach to Fractal Coastline Characterization

4.1 Introduction

We consider the standard divider dimension technique and three variations. Although these may not provide a more accurate method of assigning a dimension to a coastline, the relative comparisons together give additional information about the irregularity of the coastline and the asymmetry between land and water. All four methods attemp! to cover the coastline with line segments of length f. and study how the number of segments N(E) varies with E. Each method begins by choosing a starting point and rotating a line segment of length f. around that point. The initial orientation of the line segment is the same as the initial segment of the coastline curve. This rotated segment may intersect the coastline in more than one point. One intersection will be chosen, and a line segment constructed from the starting point to this new point. The process is iterated using this new point as the initial point. Any intersections with the coastline before this point will be ignored. The four methods will give the same result if there is only one intersection point at each stage, but there is a difference when there are multiple intersections.

As before, for most choices of E, there will be a partial step leftover at the end of length

P(E), and the length of the curve is L(E) = (E · N(E)) + P(E). As f. decreases, the value of P(E) will usually decrease, and this error will be minimal at small stepsizes. The results are then

22 Figure 4-1: Analysis of the four methods plotted on a double logarithmic plot where E is on the x-axis and L( E) is on the y-axis. If the graph is linear, then it behaves as a power-law and the divider dimension D = 1- slope.

4. 2 The Four Metho_ds

In our research, we examined four different methods for selecting segment positions. The first is the land method, which involves choosing the first point that is encountered upon rotating the segment from its initial position over the land (Figure 4-1, 4-2). Thus, the segment is rotated over the land until it hits the coastline for the first time. For an East Coast section of coastline, if the "walk" is from north to south, then this means choosing the first point reached by a counterclockwise rotation of the starting segment. On the other hand, for the West Coast, if the "walk" is from north to south, then this entails choosing the first point obtained by a clockwise rotation of the starting segment. If the direction is reversed from south to north, then the previous cases are reversed. The water method involves choosing the first point that is encountered upon rotating the segment from its initial position over the water (Figure 4-1, 4-2) . Thus, the segment is rotated over the water until it hits the coastline for the first time. The rotations are just the opposite

23 of the case above. For a north to south walk along the East Coast, the first point reached by a clockwise rotation is chosen, and for the same walk along the West Coast, the first point reached by a counterclockwise rotation is chosen. When direction is reversed from south to north, the previous cases are once again reversed.

Stepsize Stepsize 17,300 m. 50km. 17,300 m. 50 km.

Land Water

Figure 4-2: Land & Water methods on the New England Coast

The third method is the traditional divider dimension method which we call the first method.

This method chooses the first point sequentially along the coastline ·which is a distance E away. This is the first point reached by a person walking along the coastline from the initial point (Figure 4-1, 4-3). The fourth method, called the last method, was an unusual variation. Instead of choosing the first point along the coastline a distance E away, the last point is selected (Figure 4-1, 4-3). The goal was to see what the four methods would reveal about the shape and irregularity of the coastline and to see if there was any relationship between these methods which might allow prediction of one based on the others.

24 Stepsize Stepsize 17,300 m. 50km. 17,300 m. 50 km.

Rrst Last

Figure 4-3: First & Last Methods on the New England Coast

4.3 The Data

The data used for this study was obtained on a CD-ROM from the National Ocean and Atmo­ spheric Administration (NOAA). The data is a Medium Resolution, 1:70,000 Digital Vector Shoreline which was generated by the Data Management and Geographic Information Systems Group of the Strategic Environmental Assessment Division. Their data was collected from 270+ National Ocean Service Navigation Charts. The entire coastlines captured on the CD included the East Coast, West Coast, Gulf Coast, and Great Lakes areas. Each major coastline was subdivided into several sections (see Figure 4-4, 4-5). For example, the East Coast data consisted of 6 sections, although we primarily worked with three of them. East3 included the Chesapeake Bay, East4 ran from Chincoteague Inlet Virginia to Block Island Sound in Rhode Island, and East5 ran from Point Judith in Rhode Island to the St. Croix River in northern Maine. The other three files contained data for southern Virginia, the Carolinas, Georgia, and Florida coasts. The West Coast contained 3 sections. Westl ran from Semiahoo Bay in Washington to Yaquina Head in Oregon, West2 from Alsea Bay in Oregon to Point Sur in California, and West3 from Point Sur to the U.S. / Mexico Border. The Gulf Coast was divided into 6 sections and the Great Lakes into 4. The data was stored in an ARC format. One file contained the coordinates for the ARC, typically anywhere from 10 to 300 segments, and another file contained data on the organization of the arcs. The data included

25 both contiguous pieces of coastline as well as islands.

Figure 4-4: East and West Coast section breakdown

East3 SOkm. East4

100km.

Westt West2 200km. West3 -100km. -

Figure 4-5: The six different coastlines

26 All of the coordinates for the coastline were given as latitude and longitude pairs. Thus, an ARC could consist of several hundred latitude and longitude coordinate pairs. Each pair of consecutive points denoted a line segment approximation to the coastline along the surface of the earth. Since our algorithms were written to use a fiat map projection, part of the project consisted of converting the latitude-longitude data into a mercator projection. The spherical data used the Geodetic Reference System of 1980 as the model of the earth with a semi-major axis dimension (equatorial) of 6,378,137 meters, and a semi-minor axis dimension (polar) of 6,356,752.3 meters. The vertical datum for the coastline (i.e. the height where land and water meet) used the North American Vertical Datum of 1929. The data was transformed from spherical coordinates to rectangular coordinates. The PC ARC/INFO program (Version 3.5) converted the data to a mercator projection. In each case, a longitude of central meridian was chosen which was usually the midpoint between the largest and smallest longitude points in each file. Since most of this research focused on the East and West Coasts of the United States, there was rarely much change in longitude between the endpoints of an individual section of coastline. This minimized the error in the projection which can occur under transformations from spherical coordinates to rectangular. The next step was the handling of the islands. Initial processing concatenated coastline segments in a single file and created other files for individual islands. In some cases, "islands" were naturally part of the coast and needed to be "attached". Cape Cod is one example of a man-made island since the Cape Cod canal is what separates it from the mainland. Thus, in order to examine how scaling affects the coastline length in Massachusetts, Cape Cod must be attached. The method for attaching man-made islands was to "remove" the canal by inserting a small segment (less than 500 meters) on each side of the canal and removing the points along the canal. This process was done in several other places including parts of the Delaware coast which was fragmented by canals. Additional "large" islands close to the shore were also attached. Small islands or those far offshore were skipped. One example is Long Island, which was added to New York by attaching a line segment from the mainland to Long Island at its closest point. We also added Manhattan and Staten Island in New York as well as Long Beach and the Mystic Islands off of the coast of New Jersey. There was some additional land that was also separated by canals

27 from the mainland in New Jersey including the Stony Point, Cohansey River, and Finns Point areas. In Delaware, the Chesapeake and Delaware Canal was removed along with the Lewes and Rehoboth Canal. Bethany Beach in southern Delaware was also added. On the West Coast, Whidbey Island, which contains the large town of Oak Harbor, was attached to Washington state. The rest of the West Coast was already intact. It is interesting to note that the East Coast is much more fragmented than the West Coast, and so there was a great deal more work in obtaining an accurate coastline for the East Coast than there was for the West. In fact, this difficulty precluded detailed analysis of the southern part of the East Coast. It was extremely time consuming to attach all of the fragmented pieces of land in the Carolinas, Georgia, and Florida.

4.4 Land vs. Water Results

Our program implemented the four methods (land, water, first, and last) on the six different coastline segments mentioned above (see Figure 4-6). In our study, many of the coastlines spanned a distance of 100,000 meters or more from endpoint to endpoint. The accuracy of the coastline data was about 100 meters which was about the length of the smallest segments. This typically provided 3 decades of measurement for estimating a dimension, which is more

1 than most studies of real world fractal objects. We started wi_th a largest stepsize E equal to the distance between the endpoints of the coastline and usually decreased by a factor of 0. 78, which is roughly -V2. This factor was chosen because it yielded roughly 10 points per decade. Typically, E was reduced at least 30-35 times which usually left the smallest stepsize somewhere between 30-50 meters. Unless otherwise mentioned, all of the "walks" along the coastline proceeded from the northern end of the coastline to the southern end.

1 Avnir studied 86 fractals papers published between 1990 and 1996 in Phyiscal Review A - E and Phyiscal Review Letters and found that, with few exceptions, the data spanned less than 2 decades with most covering 1.5 decades. [7]

28 - :Jn..t;::::::::····· ..• ••¥f'fvv··r:vv ••• •• :•• •••••• ••

··::..•:vv ···.·· •••. .. ••. ··vv ···•· • - ;.;.;;.;;_.~ 0 •• • land "•'•· ... • water v water • first • first • last • lost

10·~------~------~------L------~~----_J 10·~------~------~------~------~----_J 101 102 103 104 10' 108 101 102 103 104 10' 10° Stepsize Stepsize LO

• land v water v water • first . first • lost . lost

10~oL7,------1~o~2------1~o~,~-----1oL,~-----1~o7•----~1o• 10. 101 102 103 104 105 10" Stepsize Stepsize Log-log plot for West2 Log-log plot for EosH

- -

1 1 ·. ....J 10 ,~-.·•N!!i!;:::······· -: .. - : .. "''"...;;...... _ .. : •land ...... • land .. • water v water .. .. • first • first • lost • lost

10°~------~----~~-----L~----~~----~ 1o•L------~------L ______L ______-J ______-J 101 1o' 103 104 10° 10° 101 102 103 104 105 108 Stepsize Stepsize LO

Figure 4-6: Log-log plots for the various coastlines

In most cases, the length estimate using the land method was less than the length estimate using the first method. The difference between them (typically between 15 and 30 percent of the first length) decreased along with the stepsize. Both were also similar in shape on the

29 double logarithmic plot. On the other hand, the difference in the length estimate from the land method and the water method was often quite large (ranging from 30 to 60 percent of the land length depending on which coast is being studied) with the water method usually less than the land method. The difference in the length estimate from the water method and the last method was usually quite small (usually between 10 and 20 percent of the water length) with the water method slightly larger than the last method. The percentage difference in all cases typically decreased as the stepsize decreased with the exception of the four or five largest stepsizes. The percentage of difference between land and first and between water and last was smaller for the West Coast than it was for the East Coast. Some of the coastline asymmetry is highlighted by the difference in accessibility from land and water. The land method is a covering that must arrive at the coast via land whereas the water method is a covering which must arrive at the coast via water. On a typical coastline, there are few narrqw land masses such as peninsulas protruding out into the water whereas there are many narrow bodies of water (rivers, bays, etc.) protruding into the land. Thus, the land estimate of length adds in the lengths of most rivers and bays, increasing the length of the coastline considerably. However, the water estimate of length will be smaller because there are few features which correspond to the rivers and bays other than an occasional peninsula. The water estimate of length doesn't include the lengths of those rivers and bays until a much smaller stepsize than the land method because of the narrow entranees. However, the estimates of both methods will eventually approach the same value at sufficiently small stepsizes because of the limited resolution of the data. These methods are similar to the method of determining the areas of rough surfaces by molecule adsorption. In a series of studies by Avnir, Pfeifer and Farin [4][5][6], the surface area of rough surfaces was determined by the number of molecules of various sizes adsorbed by the surface. It was observed that the surface area increased with the use of smaller molecules because as molecule size decreases, they can enter the smaller crevices that the larger molecules couldn't enter. Thus, many rough surfaces were determined to have a surface fractal dimension between 2 and 3. With molecules, it is impossible, however, to approach the surface from both sides. Another common feature on the double logarithmic plots was the existence of jumps in

30 the data, particularly from the water side (see Figure 4-7). Often, as the stepsize decreased, relatively smooth linear behavior would be punctuated by large jumps. This is consistent with the fractal element model proposed by some geography researchers who focused on the standard divider dimension technique. Their jumps weren't as well-defined as those obtained from the water method, and they also had larger transition regions.

.... ··. I • ~~~pe -0.2490 ·~·'· '~'

~e -0.1960

~~· ...__slope -0.0765 slope -O.OJ 19 1 ._ slope -0.2162 ~ _ 10 ' I I I I• • • • • • io~,-----,o~2-----,~o3~---,~o.~---,~o.~--~,o' Stepsize Stepsize Land method for Wesl2 Water method for West2

Figure 4-7: Land and water methods for Oregon & northern California

Even so, the slope before and after the transition zones were often nearly identical. This

may give an indication of the relative irregularity of the coastline and at over which range of € these changes occur. In a coastline where jumps were present, they often occurred more than once. Many of the coastline sections had two or three jumps before the coastline length leveled off. These jumps were more evident in the northern California coastline than in any other section of coastline. Clear understanding of the water method can be gained by a study of the northern California

coastline and the San Francisco Bay area. For decreasing € down to about 2000 meters, the length of the coastline remained relatively stable because the water covering never entered San Francisco Bay with its narrow mouth near the Golden Gate Bridge (Figure 4-8). However, around 2000 meters, the segments are small enough to enter the bay. Since the bay encompassed a large area, the length of the coastline increased substantially upon entering the bay (about 250,000 meters which is 20% of the length at that point). This created the first jump in the double logarithmic plot (Figure 4-7). However, this was not the only jump. San Francisco

31 Bay also has "sub-bays", and one of those, Suisun Bay near the Carquinez Bridge, has a very narrow mouth as well (Figure 4-9). Thus, for several more decreasing stepsizes, the length of the coastline barely increases because the segments are not small enough to enter any more sub-bays. However, with a stepsize of around 700-800 meters, the covering now enters Suisun Bay, and there is a large increase in length once again (about 300,000 meters which is about 20% again), corresponding to another jump on the double logarithmic plot. These jumps correspond to the narrow bays and the maximum stepsize needed to finally enter those bays from the water. Even though many of these bays are quite wide, a narrow mouth will cause the length to increase in jumps as the stepsize decreases rather than in the more continuous manner seen in some exactly self-similar fractals.

Stepsize Stepsize 2280 m. 1780m.

30km. 30km.

Figure 4-8: Water Method in San Francisco Bay (2000 m.)

32 Stepslze Stepsize 850m. 650m.

30km. 30km.

Figure 4-9: Water Method in San Francisco Bay (700 m.)

Another example of the same scenario occurs in Washington state. This walk along the coastline began at the northern endpoint and proceeded south. With this approach, the coastline length is increasing with decreasing stepsizes at a gradual rate until a stepsize of about 6183 meters. Then, there is a large jump in the length (about 300,000 meters which is 35% of the length at that point) at this stepsize because the segments are small enough to enter the Puget Sound via Admiralty Inlet, and much of the la!ld inside the sound is added (see Figure 4-10). The length does not always increase as stepsize is decreased. Here, as the stepsize is decreased from 6183 meters to 6153 meters, the length decreases (see Figure 4-11, 4-12). At 6153 meters, the covering from our starting point hits a point at the entrance to the bay, and so the next step actually "steps over" the bay and doesn't enter it. Thus, there is a substantial decrease in length at this point, but for future decreasing stepsizes, we enter the bay, and the length approaches the same value it had at a stepsize of 6183 meters. Thus, the precise position of the jump point is partially determined by the starting point of the walk along the coast.

33 Stepsize Stepsize 6200 m. 6180 m.

West1 West1 100 km. 100 km.

Figure 4-10: Water method in Washington state (walking north to south)

stepsize stepsize 6183 m. 6153 m.

-Skm. -Skm.

Figure 4-11: Water method in Admiralty Inlet, WA

34 l:l I I I x105 .. z.··· 11 1- ... . ····.. ···- -

z:; c;, c: 10 r- - ~

91- - ··--·- - ...... B I I I 5900 6000 6100 6200 630( Stepsize Water method for West2 (walking north to south)

Figure 4-12: Data from water method in Washington (north to south)

A different result is found when starting with the southern endpoint and proceeding north. The change in direction means that land is now on the right and water is on the left. However, it is the change in starting point which impacts the change in the jump for the water method. First of all, the length of the coastline from the water method is longer at most of the stepsizes, and there is significant change in the length of the coastline between 5500 meters and 7500 •. meters with the greatest changes occurring between 5600 and 5800 meters (see Figure 4-13, 4-14). The jumps are less pronounced from this side, but there is a clear transition range in the data at the same approximate stepsize as on the other side. With either starting point, the next significant increase in length occurs when the stepsize decreases to roughly 1375 meters. The length increases about 200,000 meters (about 12% of the length) when the covering enters the Henderson Bay area near Tacoma through the Narrows by the Tacoma Narrows Bridge (see Figure 4-15). Also, when the stepsize decreases to about 600 meters, the coastline increases in length by about 300,000 meters (roughly 12% of the length) because the segments can move beyond a narrow bend in the Columbia River and now can travel much further upstream.

35 Stepsize Stepsize 5870m. 5720 m.

West1 West1 100 km. 100 km.

Figure 4-13: Water method in Washington state (walking south to north)

1J I I I I x105 . --·. 12 r-- •• - Ia ... .

11 r-- - :5 . "'c . ~ 10 r-- ...... - • .. ~· 9f- ...... ••...! ·""· ...... I I I I B 5000 5500 6000 6500 7000 750( Stepsize Water method for West2 {walking south to north)

Figure 4-14: Data from water method in Washington (south to north)

36 stepsize stepsize 1500 m. 1300 m.

.- 30 km. 30 km.

Figure 4-15: Water method in Puget Sound, WA

A similar situation was also found in southern California near San Diego. The dimension for southern California is roughly 1.04 for stepsizes larger than 900 meters (see Figure 4-16). However, there is a substantial increase in the length of the coastline at a stepsize between 600- 900 meters. Here, the length increases by 70,000 meters, which is about 10% of the coastline length, when the covering enters the San Diego Bay area. As the stepsize decreases, the length levels off briefly before there is another jump with an approximate stepsize of 250 meters when the covering enters Mission Bay, a smaller bay just a little north of San Diego Bay.

. ·· ... ·. ······ ...... ··· . ·. ... ·... ·· ·.~ slope -0.0319 slope -0 . 103~ 0 •••• ... ~ ...

Stepsize Stepsize Stepsize Woter method for West 1 Woter method for West2 Woter method for West3

Figure 4-16: Data from water method on West Coast

One of the reasons that jumps seem so prevalent along the West Coast is that in general, the

37 West Coast has dimensions over large stepsizes ranging between 1.03 and 1.10 (Figure 4-16). Thus, when substantial increases in length occur, it is very clear on the double logarithmic plot. However, the East Coast is more irregular with dimensions ranging from 1.25 in New England to 1.55 in the Chesapeake Bay, and so there are few stepsizes where a small change in stepsize leads to a large increase in length. Thus, each decrease in stepsize leads to a larger increase in length and so big increases are not as noticeable. Hence, this phenomenon of jumps seems to occur with coastlines that are rather smooth with a few narrow bays or rivers. Since coastline data has a limited resolution, one expected trend is that the coastline length will reach a well-defined limit as the stepsize E decreases below the resolution of the coastline. Thus, it was expected that, with all of the methods, the length would gradually level out on the double logarithmic plot as E decreased below 100 meters, which is the resolution of our data. The results, however, were quite surprising. When using the first method on the coastline, the length starts leveling out as the stepsize decreases below 1000 meters, and by the time the stepsize is 100 meters, it is almost completely fiat on the double logarithmic plot (see Figure 4-6). With the land method, the leveling out as stepsize decreases is similar, although the value where the leveling out begins is usually a bit less than 1000 meters. The final leveling out is usually at a stepsize of less than 100 meters. The apparent cause is that there are only a few peninsulas jutting out into the water, and even fewer of them are narrow enough to increase the land length at a stepsize less than 1000 meters. This is wky the land and first methods seem to level out at a stepsize a bit larger than the resolution of the map. The water method was quite different. The first signs of leveling off as stepsize decreased didn't usually appear until a stepsize of about 100 meters. In some cases, the length leveled off very quickly as stepsize decreased after that, but in one case (the file East4 which runs from Rhode Island to Delaware), the leveling off wasn't seen even as the stepsize decreased to 10 meters (which was as small as could reasonably be carried out with this amount of data). The cause of this is the Hudson River and the Delaware River, which are long and become narrower further up. Thus, as stepsize decreases, the covering travels further up the river and the length increases accordingly. Hence, with the water method, the stepsize where the length levels off as stepsize decreases is important because it is useful in determining whether the coastline has many long and narrowing rivers. The length from the last method usually levels off as stepsize

38 decreases at the same stepsize as the water method. One use of the land method is in providing a good approximation to the real coastline with much less data. For example, Figures 3-1 of San Francisco Bay is actually generated from the land covering at a stepsize of 250 meters. Although this isn't that different from the resolution of 100 meters, the difference in the number of points is quite significant. The original San Francisco Bay file contained about 41 ,000 points whereas this approximation only has 7,300 points, which is 5.6 times less than the original. This is a good compression ratio, and it is difficult to notice the difference with this view (see Figure 4-17). The first method is another possible choice since it will often be a better approximation. However, the problem with the first method is that there are more points than the land method at a given stepsize, and using a larger stepsize often isn't good enough because some small features don't show up. For example, the first method with a stepsize of 250 meters has 8200 points.

Land approximation (250 m.) Original Coastline Data

Figure 4-17: San Francisco Bay

The difference in the length of the coastline according to the land and water methods became a useful characterization for coastlines (see Figure 4-18). The difference in land and water lengths was plotted against the stepsize Eon a semi-log plot (where the log of E is plotted on the x-axis versus the difference in the length on they-axis). Several characteristics became

39 apparent. First of all, since the land method usually gave a longer measurement than water, the difference was almost always positive since we calculated the difference between land length and water length. The land-water difference is near 0 for large f. because the coastline has a finite distance from starting point to end point. That difference increases as f. decreases, reaching a maximum when the stepsize is about 500 meters. Then, as f. decreases further, the difference usually decreases rapidly although it rarely reached 0. This is because the water length was often still growing when we stopped the estimates with a smallest stepsize of roughly

20 meters. The accuracy of the data is only 100 meters, and so the results at smaller f. are unreliable. These results on the differences in the land and water methods have helped to highlight some of the asymmetries which are found in real coastlines. By studying the land and water results, we can gain useful information about the irregularity of a coastline as well as the presence of narrow bays and rivers on a relatively flat coastline. Useful information about the resolution of our data can be obtained based on which stepsizes result in the leveling off of land and water lengths. The stepsize where the length levels off can also help us understand how narrow many of the rivers on the map are. We hope these results can be useful in determining properties of a coastline and giving us information about modeling them.

40 £() b I I I I I I I I x10S x106 15 r- . - - ...... 0 10 I- . - 0 "c: ...... c: ~ .. ~ - " .. .. 51- ... - .. .. ~ rs .. ,• ······· ...... 0 01-- ·······- ···················

-5 I I I I -2 I I I I 1 4 5 1 1 10 102 10J 10 10 10 10 1o2 10J 104 105 101 Stepsize Stepsize Semi-log plot for west1 Semi-log plot for eost5

£() I I I b I I x10S .... x106 15 r- •' .. - .. 4 - . .. .. 0 10 r- . - 0 "c: .. c: .· .····. ~ . ~ . .. . - ~ ... ~ .. . . 0 51- ...... - 5 ...... 0 ··········· . ... - or- . ·····~

-5 I I I -2 I I I I 101 1o2 1oJ 104 105 10' 101 tal 10J 104 105 101 Stepsize Step size Semi-log plot for west2 Semi-log plot for eost4

~u b I I I 4 x10 x106 15 r- - ...... 4 .. •' .. . . 0 10 . . - "0 . "c: . . c: .. . ~ ~ .. ... ~ ~ . 0 5 - 0 ...... 0 0 ~· ..... ············-

-5 I I I I - 2 I I I I 101 1o2 10J 104 105 101 101 1o2 10J 104 105 10' Stepsize Stepsize Semi-log plot for west3 Semi-log plot for east3

Figure 4-18: Land vs. Water Difference

41 Chapter 5

Modeling Coastlines by Fractal Methods

One of the offshoots of this research was an attempt to construct fractal coastlines closely resembling real coastlines based on traditional constructions of fractals like the Koch curve. The goal was to construct a "fractal coastline" with many of the same properties that a real coastline exhibits, especially the observed land and water asymmetry. The use of a Koch-like construction makes computation of similarity dimension possible. A Koch generator is a four segment generator with the general layout of the Koch curve ge!lerator, but the lengths of the segments may differ from l (see Figure 2-1) . In some cases, all of the segments had equal lengths, but some of the generators also had different lengths for different segments. The initial attempt chose randomly between the standard Koch generator and its negative (a generator with four equal segments of length l but oriented down instead of up). The dimension of this curve is the same as the standard Koch curve (D = 1.26186 ... ), which is about the dimension of some parts of the East Coast. With the application of two generators, overlap became a problem because occasionally two generators went inside the curve at the same time. This overwriting made it difficult to use in our real coastline model because we desired a continuous boundary between land and water. Overlap could be interpreted as a lake or an island, but since each overlap would have to be handled separately, it would be difficult to prepare this coastline for our purposes. It was also clear that even if the enclosed pieces of

42 land and bodies of water were treated as discussed, the coastline still didn't closely resemble a real coastline. For example, there were no clear cases of deep rivers running into the land, but instead, the coastline still has most of the symmetry between land and water. Thus, this initial attempt didn't result in an accurate coastline representation.

50% 1/3

1/3 50%

Dimension=1.26186 ...

Figure 5-1: Coastline formed by random Koch- generator

The next attempt was to vary the original Koch generator and use a generator with a length other than i but continue to apply it along with its negative randomly. Several were tried including generators with length 0.3, 0.35, 0.4, 0.45, and 0.48 (see Figure 5-2). For generators with length less than i, the problem of overlap no longer occurred. However, the coastline looked too flat and didn't have enough irregularities to be considered a good model for a real coastline except maybe the southern coast of California. The other generators led to more overlap than the case with length i. We concluded that simple random Koch-like constructions with different r does not model coastline asymmetry.

43 50% 2/5 50% 3/10 - 1\'---

2/5 50% 3/10 50% v Dimension=1.15143 ... Dimension=l.51294 ...

Figure 5-2: Coastlines using Koch generators of length 0.3 & 0.4

Our next attempt combined several different Koch generators in pairs (if we used the positive version, then we also used the negative version). Several combinations of four generators, each applied with equal probability, were tried. We also tested unequal probabilities. Some of the generators had all segments of equal length, and others had differ~t segment length generators. By using unequal segments with varying probabilities, it is no longer clear how to determine the similarity dimension. Thus, estimates of dimension for these curves had to be obtained from the divider technique.

The goal was to use some generators of length longer than ~ (which would make them steep), and some generators with length less than ~ (which would make them flat). The hope was that the combination of a steep generator (imitating the deep rivers), and a flat generator (which would imitate the small amounts of land jutting into the water) would lead to a good model since the model explicitly included asymmetry. Unfortunately, once again, the big problem was overlap. If the generators were too steep, then they overlapped, even if there were a lot of flat generators. Thus, the problem was that steep generators led to overlap, and flat generators led to coastlines which didn't look real.

44 This led us to address specifically the problem of overlaps. The problem of overlap resulted from two of the same generator being applied successively. Thus, when the segments meet in a peak, there was potential for a problem. An attempted solution was to ensure that generators were carefully chosen around peaks. The initial attempt was to use just two generators and to assign one as the land generator and one as the water generator (see Figure 5-3). The land generator extended into the land and was generally steeper to model the deep rivers. The water generator extended into the water and was generally fairly flat. An algorithm was implemented which prevented two successive land generators from being applied at a water peak and to prevent two successive water generators from being applied at a land peak. The intended goal was to prevent overlap and see some of the land and water asymmetry. The asymmetry was definitely evident, but upon closer inspection, there was still overlap present when the land generator was too steep. The only way to eliminate the problem was to make the land generator so flat that the coastline didn't look real.

50% 2/5 A

3/10 50%

Figure 5-3: Best attempt using Koch generators (overlap at small scales)

45 Beginning a line segment of length 1, the next attempt was to use the zig-zag generator (see Figure 5-4) along with a Koch generator. At first, this approach appeared to be promising because we were able to get some coastlines that looked very much like a real coastline with land-water asymmetry and bays going into the land without the corresponding land jutting out into the water (see Figure 5-5). This is because the Koch curve variation was somewhat steep and always jutted into the land rather than the water. However, upon closer inspection, even these curves overlapped, and so this approach didn't work. The reason for overlap is that at each stage, the zig-zag generator replaces a rotated segment by the full generator which rotates some more. Eventually, after a certain number of steps, the generator will rotate back over itself. Even though the choice of zig-zag generator versus Koch generator was random, replacing a sl!gment on the Koch curve by this generator will rotate the segments around faster, and they will overlap sooner. Thus, it usually took 8 stages before the curve overwrites itself. Thus, this combination of generators didn't satisfy our goals because of the problem of overlaps and the lack of a proven method to remove them.

1

Stage 0 Stage 1 - Generator 1 segment; length 1 3 segments; length ~-

Final Stage Dimension=1.19898 ...

Figure 5-4: Zig-zag fractal

46 50%

50% 2/5

2/5 2/5

Figure 5-5: Best attempt using zig-zag generator (overlap at small scales)

It appears that the biggest problem with generating realistic coastlines is that in order to get narrow rivers from a Koch-like construction, the chances for overlap are too great. No algorithm to fix this overlap problem has been suggested yet. It appears that modeling a coastline by this ordered processes either results in a curve with too much land and water symmetry, or it overlaps itself.

47 Chapter 6

Future Research

These new methods of land and water measurements of coastline length should be a useful tool for future exploration of real coastline properties. Some of the remaining unanswered questions should be of interest to geographers and geologists as well as mathematicians. The first question is what geological processes might cause the large differences in land and water measurements. There could be different processes working over different scales which cause the jumps over certain stepsizes like in the water method in northern California (see Figure 4-7) . Another question is the cause of the narrow river mouths on much of the West Coast in areas like San Francisco, Seattle, and San Diego, and correspondingly, why the northern parts of the East Coast have wider mouth bays such as the Chesapeake Bay, Delaware Bay, and Massachusetts Bay. These new methods may help researchers to be able to relate information about coastline formation based on the data analysis obtained from these new methods. Another question which still needs to be answered is how much do these methods tell us about a coastline. These methods reveal the presence of narrow bays and rivers on relatively flat coastlines, and as stepsize decreases, the leveling out stepsize can give information about the width of the narrowest rivers. However, it is still unclear whether there is more to be learned. It would be interesting to run this same analysis on other coastlines around the world and also to examine areas like the Gulf Coast or the southern part of the East Coast of the United States. An area like Louisiana could be especially interesting because of the Mississippi River delta and all of the bayous and bays. The land method might also display some of the same properties as the water method because of the small land masses jutting out into the

48 water. Another question is how to define similarity dimension for curves like those generated in our coastline simulation study. Some of the curves had two generators which were applied with varying probabilities, but each segment could have a different length. The similarity dimension is not clearly defined under the standard definition, and defining similarity dimension for all curves constructed by combinations of generators could be an interesting project. A final question is the simulating of land-water asymmetry of a real coastline without having overlaps. Our Koch-like constructions failed to produce obvious asymmetries, and other variations were too complicated to implement. An open question is whether there might be another curve like construction which could give reasonable results, or whether it is necessary to try a different approach. In a self-affine surface approach [23], a surface height h( x, y) is defined for every point in the x,y plane and the coastline is viewed as the intersection of the surface with a plane parallel to the x,y plane. Then, z = H, and if h( x, y) > H, then the point is on the land, and if h(x, y) < H, then the point is in the water. The coastline is the boundary between the two where h(x, y) =H. This approach automatically includes lakes and islands which were a difficulty using our method. It is also easy to change the value of H, which allows a change in the water height. This approach has been tried before by using a rough Poisson-Brown stochastic model for choosing the points [14], and this might be the best way of modeling a coastline. However, this doesn't necessarily address the problem of land-water asymmetry which has been difficult to model using our techniques. More work is needed in this area.

49 Chapter 7

Conclusion

Although a coastline is not an exactly self-similar fractal, it still shares many properties with these curves. The new land and water methods for estimating coastline length are useful in determining some of the properties of coastlines that were not observed with the standard divider dimension technique. As with previous fractal methods, they reveal information about the irregularity of the coastline as well as new information about the land and water asymmetry. The presence of jumps may be the result of several processes at work in the formation of the coastline and usually indicates the presence of narrow bays and rivers along with a fairly flat coastline. More research on these new methods could lead to valuable measuring tools for obtaining information about real world coastlines.

50 Bibliography

[1] R. Andrle, Estimating fractal dimension with the divider method in geomorphology. Ge­ omorphology, 5 : 131 - 141, 1992.

[2] R. Andrle, The West Coast of Britain: Statistical self-similarity vs. characteristic scales in the landscape. Earth Surface Processes and Landforms, 21 : 955 - 962, 1996.

[3] R. Andrle, Complexity and scale in geomorphology: Statistical self-similarity vs. charac­ teristic scales. Mathematical Geology, 28 (3): 275 - 293, 1996.

[4] D. Avnir, P. Pfeifer, Fractal dimension in chemistry. An intensive characteristic of surface irregularity. Nouv. J. Chim., 7 : 71 - 72, 1983.

[5] D. Avnir, D. Farin, P. Pfeifer, Chemistry in noninteger dimensions between two and three. Journal of Chemical Physics, 79 : 3566- 3571, 1983.

[6] D. Avnir, D. Farin, P. Pfeifer, Molecular fractal surfaces. Nature, 308 : 261 - 263, 1984.

[7] D. Avnir, 0. Biham, D. Lidar, 0. Malcai, Scaling range and cutoffs in empirical fractals. Physical Review E, 56 (3) : 2817 - 2828, 1997.

[8] G.A. Edgar, Measure, Topology, and Fractal Geometry, Springer- Verlag, New York, 1990.

[9] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990.

[10] J. Feder, Fractals, Plenum Press, New York, 1988.

[11] N.S. Lam, D. Quattrochi, On the issues of scale, resolution, and fractal analysis in mapping sciences, Professional Geographer, 44 (1) : 88- 98 , 1992.

51 [12] P.A. Longley, M. Batty, Measuring and simulating the structure and form of cartographic lines. Papers in Planning Research, 102.

[13] B.M. Mandelbrot, How long is the coast of Britain? Statistical self- similarity and fractional dimension. Science, 156 (May) : 636- 638, 1967.

[14] B.M. Mandelbrot, Stochastic models for the earth's relief, the shape and the fractal di­ mension of coastlines, and the number- area rule for islands. Proceedings of the National Academy of Sciences, 72 (10) : 3825 - 3828, 1975.

[15] B.M. Mandelbrot, Fractals: Form, Chance, and Dimension, W.H. Freeman, San Francisco, 1977.

[16] B.M. Mandelbrot, The Fractal Geometry Of Nature, W.H. Freeman, San Francisco, 1982.

[17] D.M. Mark, P.B. Aronson, Scale-dependent fractal dimensions of topographic surfaces: an empirical investigation, with applications in geomorphology and computer mapping. Mathematical Geology, 16 (7) : 671 - 683, 1984.

[18] R.T. Mills, Resolving the problem of the last partial step in estimating fractal dimension with the divider method, preprint, 1997.

[19] J.D. Orford, W.B. Whalley, The use of fractal dimension to quantify the morphology of irregular-shaped particles. Sedimentology, 30 : 655 - 668, 1983.

[20] H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer - Verlag, New York, 1992.

[21] H.O. Peitgen, P.H. Richter, The Beauty of Fractals, Springer-Verlag, Berlin, 1986.

[22] L.F. Richardson, The problem of contiguity. General Systems Yearbook, 6 : 139 - 187, 1961.

[23] R.F. Voss, Fractals in nature: From characterization to simulation, in: The Science of Fractal Images, H.O. Peitgen, D. Saupe (eds.), Springer-Verlag, New York, 1988.

[24] T. Xu, I. Moore, J. Gallant, Fractals, fractal dimensions, and landscapes- a review, Geo­ morphology, 8 : 245 - 262, 1993.

52