Geographic space is the large-scale and continuous space that encircles the earth and in which human activities occur. The study of geographic space has drawn attention in many different fields and has been applied in a variety of studies, including those on cognition, urban planning and navigation systems. A scaling property indicates that small objects are far more numerous than large ones, i.e., the size of objects is extremely diverse. The concept of scaling resembles a fractal in geometric terms and a power law distribution from the perspective of statistical physics, but it is different from both in terms of application. Combining the concepts of geographic space and scaling, this thesis proposes the concept of the scaling of geographic space, which refers to the phenomenon that small geographic objects or representations are far more numerous than large ones. From the perspectives of statistics and mathematics, the scaling of geographic space can be characterized by the fact that the sizes of geographic objects follow heavy-tailed distributions, i.e., the special non-linear relationships between variables and their probability.

In this thesis, the heavy-tailed distributions refer to the power law, lognormal, exponential, power law with an exponential cutoff and stretched exponential. The first three are the basic distributions, and the last two are their degenerate versions. If the measurements of the geographic objects follow a heavy-tailed distribution, then their mean value can divide them into two groups: large ones (a low percentage) whose values lie above the mean value and small ones (a high percentage) whose values lie below. This regularity is termed as the head/tail division rule. That is, a two-tier hierarchical structure can be obtained naturally. The scaling property of geographic space and the head/tail division rule are verified at city and country levels from the perspectives of axial lines and blocks, respectively.

In the study of geographic space, the most important concept is geographic representation, which represents or partitions a large-scale geographic space into numerous small pieces, e.g., vector and raster data in conventional spatial analysis. In a different context, each geographic representation possesses different geographic implications and a rich partial knowledge of space. The emergence of geographic information science (GIScience) and volunteered geographic information (VGI) greatly enable the generation of new types of geographic representations. In addition to the old axial lines, this thesis generated several types of representations of geographic space: (a) blocks that were decomposed from road segments, each of which forms a minimum cycle such as city and field blocks (b) natural streets that were generated from street center lines using the Gestalt principle of good continuity; (c) new axial lines that were defined as the least number of individual straight line segments mutually intersected along natural streets; (d) the fewest-turn map direction (route) that possesses the hierarchical structure and indicates the scaling of geographic space; (e) spatio-temporal clusters of the stop points in the trajectories of large-scale floating car data.

Based on the generated geographic representations, this thesis further applies the scaling property and the head/tail division rule to these representations for urban studies. First, all of the above geographic representations demonstrate the scaling property, which indicates the scaling of geographic space. Furthermore, the head/tail division rule performs well in obtaining the hierarchical structures of geographic objects. In a sense, the scaling property reveals the hierarchical structures of geographic objects. According to the above analysis and findings, several urban studies are performed as follows: (1) generate new axial lines based on natural streets for a better understanding of urban morphologies; (2) compute the fewest-turn and shortest map direction; (3) identify urban sprawl patches based on the statistics of blocks and natural cities; (4) categorize spatio-temporal clusters of long stop points into hotspots and traffic jams; and (5) perform an across-country comparison of hierarchical spatial structures.

The overall contribution of this thesis is first to propose the principle of scaling of geographic space as well as the head/tail division rule, which provide a new and quantitative perspective to efficiently reduce the high degree of complexity and effectively solve the issues in urban studies. Several successful applications prove that the scaling of geographic space and the head/tail division rule are inspiring and can in fact be applied as a universal law, in particular, to urban studies and other fields. The data sets that were generated via an intensive geo-computation process are as large as hundreds of gigabytes and will be of great value to further data mining studies.