Case Studies on Fractal and Topological Analyses of Geographic Features Regarding Scale Issues Zheng Ren 2017- 02-25 Degree Project thesis, Master, 15hp Geomatics Degree Project in Geomatics & Land Management Geomatics Supervisor: Prof. Dr. Bin Jiang Examiner: Dr. Julia Åhlén Co-examiner: Mr. Ding Ma 1 Abstract Scale is an essential notion in geography and geographic information science (GIScience). However, the complex concepts of scale and traditional Euclidean geometric thinking have created tremendous confusion and uncertainty. Traditional Euclidean geometry uses absolute size, regular shape and direction to describe our surrounding geographic features. In this context, different measuring scales will affect the results of geospatial analysis. For example, if we want to measure the length of a coastline, its length will be different using different measuring scales. Fractal geometry indicates that most geographic features are not measurable because of their fractal nature. In order to deal with such scale issues, the topological and scaling analyses are introduced. They focus on the relationships between geographic features instead of geometric measurements such as length, area and slope. The scale change will affect the geometric measurements such as length and area but will not affect the topological measurements such as connectivity. This study uses three case studies to demonstrate the scale issues of geographic features though fractal analyses. The first case illustrates that the length of the British coastline is fractal and scale-dependent. The length of the British coastline increases with the decreased measuring scale. The yardstick fractal dimension of the British coastline was also calculated. The second case demonstrates that the areal geographic features such as British island are also scale-dependent in terms of area. The box-counting fractal dimension, as an important parameter in fractal analysis, was also calculated. The third case focuses on the scale effects on elevation and the slope of the terrain surface. The relationship between slope value and resolution in this case is not as simple as in the other two cases. The flat and fluctuated areas generate different results. These three cases all show the fractal nature of the geographic features and indicate the fallacies of scale existing in geography. Accordingly, the fourth case tries to exemplify how topological and scaling analyses can be used to deal with such unsolvable scale issues. The fourth case analyzes the London OpenStreetMap (OSM) streets in a topological approach to reveal the scaling or fractal property of street networks. The fourth case further investigates the ability of the topological metric to predict Twitter user‟s presence. The correlation between number of tweets and connectivity of London named natural streets is relatively high and the coefficient of determination r2 is 0.5083. Regarding scale issues in geography, the specific technology or method to handle the scale issues arising from the fractal essence of the geographic features does not matter. Instead, the mindset of shifting from traditional Euclidean thinking to novel fractal thinking in the field of GIScience is more important. The first three cases revealed the scale issues of geographic features under the Euclidean thinking. The fourth case proved that topological analysis can deal with such scale issues under fractal way of thinking. With development of data acquisition technologies, the data itself becomes more complex than ever before. Fractal thinking effectively describes the characteristics of geographic big data across all scales. It also overcomes the drawbacks of traditional Euclidean thinking and provides deeper insights for GIScience research in the big data era. Keywords: Scale, fractal geometry, big data, conundrum of length, topological analysis. i Acknowledgement Various people and organizations provided valuable help and suggestions when I was writing this thesis. In this part, those who helped me finish this thesis are acknowledged gratefully. First of all, I would like to appreciate all of the professors and lecturers of the Geomatics program in the University of Gävle. Through one year study, I have gained quite useful and practical skills to conduct the research in the field of Geomatics as well as other relating fields. The good combination of lectures and labs enable me to handle the theoretical and practical problems well. And now I become quite acquainted with the GIS software after several lab studies. What‟s more, the teachers also suggest us to pose questions and communicate with each other regularly. These knowledge and experience facilitate me to conduct the complex experiments for my thesis. The idea of this thesis is inspired by the paper of Prof. Bin Jiang and Dr. Anders Brandt. In their paper, they addressed the future works of the study. Accordingly, my thesis is conducted based on the ideas of their remarkable paper. I would like to address my sincere gratitude to the Prof. Bin Jiang and Dr. Anders Brandt. Without their novel ideas, I would not have chance to be inspired and finish this thesis at all. Personally, I would like to appreciate my supervisor, Prof. Bin Jiang. It is him who gave a lot of valuable suggestions and detailed instructions on my thesis, in terms of the structure, the figures and the typesetting etc. Besides, he also advised us to learn some programming skills. I also really appreciate my examiner, Dr. Julia Åhlén, who gave very valuable comments on my thesis. She gave me quite useful suggestion on my thesis structure. Mr. Ding Ma gave me some valuable suggestions about how to make good figures and tackle programming problems. He advised me to post my questions on the Internet forums such as StackExchange and learn from others. And I finally solved all the programming problems I met. I really acknowledge Mr. Ding Ma gratefully for his valuable suggestions and kindly concern. I am also grateful to my classmate, Jeroen Grift, who gave me some advices on my English writing and typesetting of the final thesis. I also learned some practical tips when we did the lab homework on the GIS courses together. The volunteered programmers on the web forum who gave me some possible answers for the programming questions should also be thanked. Discussing with other researchers online is really an efficient way to learn new things. I aware the importance of the cooperation from this thesis work. Finally, the staff in the library and IT support department in the University of Gävle also helped me a lot when I was searching materials and books for the thesis. They are sincerely thanked for their kindly support. ii Table of contents 1. Introduction ................................................................................................................................... 1 1.1 Background ............................................................................................................................... 1 1.2 Study Motivation ...................................................................................................................... 3 1.3 Aims of the study ...................................................................................................................... 3 1.4 Structure of the study ................................................................................................................ 5 2. Basic concepts ................................................................................................................................ 7 2.1 Scaling, head/tail breaks and ht-index ...................................................................................... 7 2.2 Natural cities ........................................................................................................................... 10 2.3 Natural streets and topological representation ........................................................................ 11 2.4 Volunteered geographic information and geospatial big data ................................................ 13 3. Fractal theories ........................................................................................................................... 15 3.1 Three definitions of fractal ..................................................................................................... 15 3.1.1 Strict fractals .................................................................................................................... 15 3.1.2 Statistical fractals ............................................................................................................. 16 3.1.3 New definition of fractals ................................................................................................ 17 3.2 Euclidean and fractal ways of thinking ................................................................................... 19 4. Case studies demonstrating scale issues of geographic features ............................................. 21 4.1 Case I: Scale issues on coast length ........................................................................................ 21 4.1.1 Fractal dimension of the British coastline (yardstick method) ........................................ 22 4.1.2 Fractal dimension of British coastline (box-counting method) ....................................... 24 4.1.3 Results of the case study .................................................................................................. 25 4.2 Case II: Scale issues