DISTRIBUTIVE CONTINUOUS FRACTAL ANALYSIS FOR URBAN TRANSPORTATION NETWORK

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1 DISTRIBUTIVE CONTINUOUS FRACTAL ANALYSIS FOR URBAN TRANSPORTATION NETWORK Zhuo SUN PhD Student Graduate School of Environmental Studies Nagoya University Furo-cho Chikusa-ku Nagoya , Japan Fax: Hirokazu KATO Associate Professor Graduate School of Environmental Studies Nagoya University Furo-cho Chikusa-ku Nagoya , Japan Fax: Peng JIA PhD Student Graduate School of Engineering Nagoya University Furo-cho Chikusa-ku Nagoya , Japan Fax: Yoshitsugu HAYASHI Professor and Dean Graduate School of Environmental Studies Nagoya University Furo-cho Chikusa-ku Nagoya , Japan Fax: Abstract: A city has a very complex transportation network. It is very hard for city planners to evaluate it in both micro scale and macro scale and so far most of these works have been done empirically. In this study a new method named Distributive Continuous Fractal Analysis will be introduced to evaluate the road networks. Previous researches treat a city as a whole and did fractal analyses between cities. This study tries to treat a city as distributive continuous space and deploys the fractal analysis on every piece of space. With the power of high end computer and GIS platform this analysis can be done in few hours and shows the results to the planners visually. Comparing with the other subsystems will make the policy maker or planner see the detailed situations of a city more clearly and intuitively and they can easily make some decisions or predictions. Key Words: Fractal Analysis, Road Network, City Planning 1. INTRODUCTION 1.1 Fractals and Fractal Dimension Classical Euclidean geometry describes the visible world with integer dimensions in which an object in the space could be treated as an ideal, regular entity or a combination of some such entities. For example, in CAD platform we can build the whole world by drawing lines, curves, rectangles, boxes, cylinders and other simple geometric forms based on Euclidean geometry. It will be much complicated when we want to represent some natural objects such as mountains, clouds, trees and snowflakes. Those irregular objects could be infinitely magnified and every part of an object shows similarity to the whole one. It is impossible to describe every detail of such objects in CAD. In practice the more polygons are used the more details could be represented. The number of polygons depends on how exact it was expected for viewing and measuring. In 1967, Mandelbrot published a paper How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension in which a paradox that the measured length of a stretch of coastline depends on the scale of measurement has been examined. Empirically the smaller the increment of measurement, the longer the measured length becomes. (Fig.1.) It has been also observed that the measured length L(G) is the function of measured scale G.

2 1 D L( G) = MG (1) In this equation M is a constant and D is defined as the fractal dimension. This relationship has been found not only in the natural phenomena but also manmade infrastructures which have been deeply influenced by the nature. Then the fractals are defined as objects of any kind whose spatial form is nowhere smooth, hence termed "irregular", and whose irregularity repeats itself geometrically across many scales. (Fractal Cities, 1994) The fractal dimension which was deduced from this relationship became to an essential parameter to express how rough an object is or how fast the length increases from one measured scale to another measured scale. The fractal dimension is not an integer other than the dimension in Euclidean geometry. Figure 1 the Smaller the Increment of Measurement, the Longer the Measured Length Becomes (Source: wikipedia.org) 1.2 The Characteristics of Fractals and Cities There are many kinds of fractals and in general a fractal has such characteristics: A fractal is not smooth and shows rugged everywhere, which means the fractal could not be exactly measured in Euclidean geometry. A fractal is self-similar. Parts of a fractal look like the whole, remain the similar form of irregularity from scale to scale. In this way fractals can also be described in terms of a hierarchy of self-similar components. There are three types of self-similarity found in fractals: exact self-similarity, quasi-self-similarity and statistical self-similarity. A fractal poses to be infinitely complex. That means zooming in will bring up more and more details until infinity. Some fractals have been developed through iterations. Figure 2 illustrates a typical fractal developed by Mandelbrot and named as Mandelbrot set. The above characteristics could be found in it clearly. A fractal can be measured by the fractal dimension. This non-integer parameter often used to

3 examine the degree of ruggedness of a fractal. A coastline could have a fractal dimension between 1 and 2. If it is smooth like a line the fractal dimension will be closed to 1. If it winds everywhere and is complex enough to occupy the plane the fractal dimension will approach to 2. Figure 2 Even 2000 Times Magnification of the Mandelbrot Set Uncovers Fine Detail Resembling the Full Set (Source: wikipedia.org) The characteristics of fractals can also be found in natural objects such as clouds, mountains and trees. Therefore natural objects can be treated as fractals. Fractal analysis can be preformed on natural objects to explore the connections inside nature. A city has been built and planned by man and deeply affected by nature and human s activities. Researches recovered a city that comprises many subsystems, such as pipe network, road network, land use and human settlements, also has those characteristics of fractals. (Figure 3.) Imagine that a city has its artery network and every artery connects degraded road networks and again and again, even in a house contains an invisible footpath network. Statistical self-similarity drives this phenomenon in background, which is the characteristic of nature. Figure 3 Transportation Network in a City Takes on the Characteristics of Fractals

4 1.3 Fractal Analyses of Cities and Transportation Networks A city has very complex subsystems such as population, land use and transportation network. When a policy maker or a city planner wants to conduct some projects or plans on the city, he should consider the harmonies between those subsystems. It will be a hard work to evaluate the situation in both micro scale and macro scale and so far there is no effective method available for such quantitative analysis. Most of the work will be done empirically. Transportation network and other subsystems in a city show the properties of the fractal named irregular, scale invariant and self similar. Therefore fractal analysis will be an effective tool for evaluation and representation. However performing the fractal analysis in a city highly depends on the vast data and powerful computer, as the modern computer developed very fast in the last decade the fractal analysis have been heated up. Cities and transportation networks have attracted many attentions and been examined for their irregular form and organic growth. Some researchers used the fractal property to analyse the city's growth (Batty et al, 1989; Manrubia et al, 1999; Peterson, 1996; Shen, 2002). Others focused on a larger spatial scale and investigate the fractal nature of the patterns of urban systems and human settlements (Appleby, 1996; Longley et al, 1991; Sambrook and Voss, 2001). As to certain subsystems of the city many researches are available, such as land-use patterns (Longley and Batty, 1989), population distributions (Batty and Xie, 1996), or transportation networks (Chen and Luo, 1998; Rodin and Rodina, 2000; Shen, 1997). Fractal analyses of urban transportation networks fall into two groups those aimed at revealing the regularity and selfsimilarity of road systems (Benguiguiand Daoud, 1991; Rodin and Rodina, 2000) and those that go one step further by trying to link the fractal properties of transportation networks with cities' properties and functions (Chen and Luo, 1998; Shen, 1997). The present study follows the logic of the second group and goes beyond describing the fractal property of road systems in human settlements. In contrast to other researches which almost treat the city as a whole or conduct discrete fractal analysis over the city, this study tries to develop a new distributive continuous fractal analysis for evaluating the road network of a city in a spatial standpoint. Chapter two will describe the deduction of the theory and the methodology in practice. A case study will be discussed in chapter three. 2 RESEARCH PROBLEMS AND METHODOLOGY 2.1 The Difficulty of Policymaking and City Planning Complexity of cities makes the policymaking and city planning very difficult. Cities have been built for so many years. Many policymakers and planners conducted many projects there. The residents and nature have also altered the city from time to time. Facing those sophisticated hierarchical sub systems and referring several isolated analyses most decisions will be made empirically. The policymakers and planners need to test the harmonies between those subsystems or predict them. Unfortunately existing methods based on spatial analyses can only provide some results in a certain scale. They cannot evaluate a system from macro to micro as the fractal analysis. 2.2 Fractal Analysis in Continuous Space Previous researches did some analyses between cities or discrete analyses in one city. Different fractal dimensions had been compared with isolated geographic information. These analyses are too obscure for planners. They can reveal the situation and trends of the whole city but sometime planners want to know which part of the city should be change. This study

5 aims to use continuous space to represent a city. Fractal analysis is performed continuously over the space and associated with the geographic information. Then policymakers and planners can clearly see the distribution of the fractal dimensions and separate different areas intuitively. Results can also be compared with other subsystems in a city. Collecting all of the data with geographical information the harmonies between subsystems can be evaluated visually. To achieve these goals traditional methods for fractal analysis should be redesigned. 2.3 The Use of Raster Maps and GIS Maps As to the input data before carrying out fractal analysis, there are two major types of data in practice, which are raster maps and GIS maps. When talking to raster maps, it s always saying the digitized satellite maps. Those maps contain everything on the ground but can t be used directly until image processing. Firstly, a monocolor should be used to fill the maps; and then remove useless objects such as trees and cars; thirdly, find edges on all objects; finally, classify those edges in different category. A raster map have a resolution of display, it will be affect the exact of fractal analysis. GIS maps are the ideal input because they consist of resolution-independent vectors. Although GIS maps originate in raster maps, the users do not need to care about the resolution. They offer rich-featured and user-friendly interfaces, which can be easily operated in fractal analysis. 2.4 Create Road Network in GIS In this study a road network firstly had been built in GIS based on a raster map. The most important thing in tracing roads is precision. That means trace step needs to be set as small as possible. The width of each road also needs to be set precisely. In raster maps the width of each road can t be measured precisely so real surveyed data had been used in this study. Figure 4 shows the road network without widths (left) and with widths (right). Figure 4 Use Real Width of Each Road 2.5 Comparison of Different Conventional Analysis Methods Fractal geographical entities in fractal geometry exhibit the following dimensional relationship: L Where L is the length of a geographical entity, S is the area, V is the volume, M is any mass measurement, and D is the fractal geometry or dimension of M. The subsystems in a city shown in GIS are linear features. There are three kinds of fractal analysis methods for linear features: the line-walk method, the length-area relation method, the box-counting method. The line-walk method measures a linear feature by starting from a point and walking along the object with different step length. The length-area method assumes a center in an object and reveals the relationship of its length and area it covers. These two methods are not 1/ / D S 1/ V 1/ M (2)

6 practical in modern cities which contain not continuous, strict hierarchical and exact selfsimilar structure. The complex subsystems in a city show a statistical self-similarity as the characteristic of fractals. In such circumstance the box-counting method had been developed. In this method a mesh firstly is created to cover the study area (Figure 5), and then count the boxes that overlap with objects. Change the mesh cell size and count again. Do this step finite times and plot the number of boxes ( N s) and the mesh cell size (s) in a log-log graph. The relationship between these two variables could be described by following equations: 1 ln N s = A + D ln + E s (3) s ln N = s D lim (4) s 0 ln s Where A is the intercept, Es is the error term and D is the fractal dimension. Although D is deduced in equation 4 theoretically, the size of the mesh cell (s) can t approach to zero in finite steps. Practically D is calculated by regression method by using pairs of N and s. 2.6 The Renewed Box-counting Method The conventional box-counting method calculates the fractal dimension of the whole city while this study renewed the algorithm to calculate the distribution of the fractal dimension in the continuous space of a city. There are three major steps: 1) A mesh is created to cover the part of study area. 2) Do the conventional box-counting method to get the fractal dimension of this area and record it in the center point of this area. 3) Move the mesh a little distance and do the second step until the whole area has been covered. The right part of figure 5 illustrated the process. There are four parameters need to be set before run this algorithm: the mesh size, the initial lattice size, the final lattice size and the move step. They will deeply affect the precision of the results and the calculation time. It s a very intensive calculation. Slight changes will get finer data but increase the calculation time exponentially. It s always a dilemma between the precise results and calculation time. In this study a balance point has been found. Correspondingly they have been set to 1280m, 320m, 10m and 200m. It still took hours in a high end computer for calculating but it s endurable for getting good results. s Figure 5 the Conventional Box-counting method (left) and the Renewed One (right)

7 Why using the proposed method instead of the conventional methods? The conventional methods could not reveal the relationship between the fractal dimension and geography information. The system in nature could not be even on spatial distribution. The conventional methods just calculate the whole fractal dimension. It s hard to judge parts of the system. The proposed method could present fractal dimension in terms of geography information. The proposed method extends the conventional methods to two dimensions and provides much more information of a system. 2.7 Results Processing After performing fractal analysis in the continuous space of a city, an interpolation should be performed between fractal dimensions. Then a colored map can be created in which different colors are according to different fractal dimensions. High fractal dimension means that the transportation network in this area is mature enough while the low fractal dimension means low maturity. Compare the distribution of fractal dimension with other subsystems, such as human settlement. Some more interesting results could appear. With the power of GIS platform all these results associated with the geographic information and could be shown to planner visually. The policymaker or planner can estimate the situation in different area and consider about transferring some resource with neighborhood to lower the high fractal dimension or raise the low fractal dimension for harmony. 3 CASE STUDY 3.1 Profile of the City: Dalian This study tried to apply Distributive Continuous Fractal Analysis on city Dalian. Dalian is the marine gateway of northeast China, North China, East China and the whole world. It is also an important port, and a trade, industry and tourism city. Dalian covers an area of square kilometers, among which 2415 square kilometers of area is the old city. This area abounds with mountains and hills, while plains and lowlands are rarely seen. The terrain, high and broad on the north, low and narrow on the south, tilts to the Yellow Sea on the southeast and the Bohai Sea on the northwest from the center. The city administers 6 districts, 3 county-level cities, and 1 county. Ganjingzi, Zhongshan, Xigang, and Shahekou make up the urban centre. Changhai County is made up entirely of islands east of the peninsula. There are 74 sub-districts and 127 town/townships (11 of which are ethnic). Dalian is the one of the few cities in China where there are not many bicycles and the number of cars on Dalian streets has increased dramatically in recent years. Traffic jams during rush hour are now commonplace. The city has a comprehensive and efficient bus and light rail mass transit system. Table 1 Dalian City Administration Type sub-provincial city City Seat Xigang District Area 13,237 km² (land 12,574) Coastline 1,906 km (excluding islands) Population 6,200,000 (2005) The Number of Roads 223 Total Length of Roads km

8 Total area of Roads 8.95 km² 3.2 Conventional Fractal Analysis for Dalian as a Whole First, the road network of Dalian has been built in GIS platform. Then create a mesh covering all area of Dalian. Box-counting method can be applied in this mesh. The result can be seen in Figure 6. The fractal dimension of the whole transportation network of Dalian (1.497) is gotten from regression method. The parameter R 2 which represent the deviation of the data is quite good (0.99). The whole transportation network in Dalian has the property of fractal. When put this fractal dimension in other researches, it can be found that the transportation network in Dalian has been developed well (fractal dimension of road network is from 1 to in some US cities). But it s not mature enough for severing the population. City planners can follow the fractal dimension to construct more infrastructures in Dalian. The problems are where and how many. Following sections will try to answer these problems. Figure 6 the Fractal Dimension of Dalian as a Whole 3.2 The Distribution of Fractal Dimensions and R 2 Conventional methods as been shown in figure 6 treat the city as a whole. It s useful to compare with other cities by using the single fractal dimension. But the single value can not represent very part of the city. The proposed method then had been used for this sake. As mentioned in 2.6 the renewed method first built an appropriate window at the corner of the whole map of Dalian. Then calculate the fractal dimension of the area which the window covers. Move the window by a small step until the whole map has been covered. Every movement has its fractal dimension that has been saved in the center point of the window. After all of the point being calculated a gradient map could be gotten by using a normal interpolation method. Figure 7 display the result of the distributive continuous fractal analysis. The value range is from 0 to A colored map has been created based on the distribution of the value. It s ease to tell that which area has a mature transportation network and which area needs to be developed much more.

9 Figure 7 the Distribution of Fractal Dimensions Figure 8 shows the distribution of R 2 that in regression method. The periphery and the center of the city have good values. That means those areas have been well affected by human activities and more like fractals or haven t been affected at all. Figure 8 the Distribution of R The Relationship between Distribution of Population Density and Fractal Dimension Figure 9 has been calculated from the survey data. It shows the pattern of human settlement of this city. For displaying the relationship between the distribution of population density and fractal dimension, Figure 7 and Figure 9 had been unified to the range from 0 to 1. In other words, every value in Figure 7 divided by the maximum value in Figure 7 and so did in

10 Figure 9. After subtract unified Figure 9 from unified Figure 7, Figure 10 will be gotten. Unit: people/km 2 Figure 9 the Distribution of Population Density The high values in Figure 10 indicate those areas have been well served by transportation network or over planed. Planners can focus on those low values in Figure 10, especially in the north-east and north-west area. These areas have been planned to be residential locations, but the road network in these places do not appear to be mature enough. To get the balance planners can construct more infrastructures or move the people to the areas which have high values. The easiest way may be using the policy to change the price of the land in the low value areas to guide people s movement. Figure 10 Unify and Subtract Population Density from Fractal Dimension

11 Other information that we can get from figure 10 is the highest value and the lowest value appeared at same time in some places. That means the most matured road network and the densest population could not occupy the same area sometimes. That rooted in the limitation of the land. If the road network in an area is too complex and occupies most of the space, there is no enough room for resident to live. The reverse situation could also happen when the resident occupy most of the land. It s not difficult to tell that if other activities are put into analysis similar consequence will be gotten. It seems there is a resource can t be overused. It s useful for planners to plan land use and predict future trend in an overheated area. 3.3 The Relationship between Distribution of Road Network Density and Fractal Dimension Although the distribution of fractal dimension can present the maturity of the transportation network, another indicator can do it too. Figure 11 shows the distribution of road network density. It s a similar map compared to Figure 7. In this study a same method has been used to do a subtraction between Figure 7 and Figure 11. An Interesting map Figure 12 then appears. % % % % Figure 11 the Distribution of Road Network Density Figure 12 Different Increasing Speeds of Fractal Dimension and Density

12 It s quite difficult to distinguish the distribution of density and the fractal dimension of road network. It seems that they increase and decrease simultaneously. That s true in most circumstances, but they do not change in a same speed. Figure 12 shows an approximate triangle. Consider the extreme situation at the bottom of the triangle; the roads can be treated as theoretical lines without width because almost no area has been calculated. That happens only at the road network just want to be presented by its topological structure. The density of road is almost zero, but the fractal dimension is high. Actually the fractal dimension changes in different rate when the width of road changes according to the Figure 12. Therefore the planner can consider building not wide but high fractal dimension road network to serve people without much money. Another major difference between density and fractal dimension is that density is an aggregated indicator but the fractal dimension isn t. Fractal dimension can be the same in any scale that density can t. In figure 13 the lowest value is 0 which means both the fractal dimension and the network density reach its highest value. These two areas not only collect much more roads than others but also have more complex and mature roads than others. Actually Dalian s two business centers locate the right places highlighted by Figure 13. Even their shapes matched exactly. The areas dyed by light yellow indicate the potential business centers to be come up. It s a new way to find business centers in a city. Figure 13 Unify and Subtract Road Network Density from Fractal Dimension 4 DISCUSSION As the major structure of the urban form, urban transportation network shows the properties of fractals. This study developed a new method that can apply the fractal analysis on the continuous space of a city and get a spatial distribution of fractal dimension to evaluate the transportation network. The Distributive Continuous Fractal Analysis has been applied in Dalian as the case study to explore the urban structure of Dalian. It found that either the whole or the part of the transportation network in a city has the property of self-similar. That could help us better understand the role that the transportation network act in a city. The results of

13 Dalian are very interesting that reveal higher Fractal Dimension relates to the better transportation network and the more developed economy. The distribution of fractal dimension is suitable to be the important indicator for exploring the urban structure, it also become a promising method to identify urban employment center. The combination of GIS and fractal analysis offers many ways to study the function of a subsystem. Planners and policymakers can see more detailed information than conventional analyses and make their decisions more effectively and visually. Further researches can focus on distributive continuous fractal analysis for land use, urban form prediction and transportation network design. In addition, we intend to incorporate GIS technologies into fractal analysis much more to improve the visibility and spatial capability of fractals. REFERENCES Appleby, S. (1996). Multifractal characterization of the distribution pattern of the human population, Geographical Analysis 28, 1996, pp Batty, M., Longley, P. and Fotheringham, A. (1989). Urban growth and form: scaling, fractal geometry,and diffusion-limited aggregation, Environment and Planning A 21, Batty, M., Longley, P. (1985). The fractal simulation of urban structure, Papers in Planning Research 92, Department of Town Planning, University of Wales Institute of Science and Technology, Cardiff 1985 Batty, M., Longley, P. (1994). Fractal Cities (Academic Press, San Diego, CA), 1994 Batty, M., Xie, Y. (1996). Preliminary evidence for a theory of the fractal city, Environment and Planning A 28, Benguigui, L., Daoud, M. (1991). Is the suburban railway system a fractal? Geographical Analysis 23, 1991, pp Chen, Y., Luo, J. (1998). The fractal features of the transport network of Henan Province. Journal of Xinyang Teachers College 11(2), Longley, P., Batty, M. (1989). On the fractal measurement of geographical boundaries, Geographical Analysis 21, Longley, P., Batty, M. and Sheppard, J. (1991). The size, shape and dimension of urban settlements, Transactions of the Institute of British Geographers, New Series 16, Mandelbrot, B. (1967). How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156, 1967, pp Manrubia, S., Zanette, D.H. and Sole, R.V. (1999). Transient dynamics and scaling phenomena in urban growth, Fractals 7, 1-8 Peterson, I. (1996). The shapes of cities: mapping out fractal models of urban growth, Science News 149, 8-9 Rodin, V., Rodina, E. (2000). The fractal dimension of Tokyo's streets, Fractals 8, 2000, pp Sambrook, R., Voss, R. (2001). Fractal analysis of US settlement patterns, Fractals 9, Shen, G.. (1997). A fractal dimension analysis of urban transportation networks, Geographical and Environmental Modelling 1, 1997, pp Shen, G.. (2002). Fractal dimension and fractal growth of urbanized areas, International Journal of Geographical Information Science 16,