Case Studies on Fractal and Topological Analyses of Geographic Features Regarding Scale Issues

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.