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New Methods for Estimating Fractal Dimensions of Coastlines 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
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