For decades, researchers have been interested in identifying mathematical and spatial patterns within and between cities. One strand of these studies has focused on patterns of city sizes within urban systems and on how cities of different sizes function differently. Along these lines, some recent research suggests that larger cities have a predictable and disproportionately higher aggregate output for variables representing (mostly) desirable attributes, like income and innovation, when compared to smaller cities. Put differently, certain indicators of socio-economic well-being tend to scale superlinearly with population. Other important variables scale sublinearly with city size. However, the analysis of these patterns is still in its infancy. More work must be done before findings of urban scaling can be generalized and relied upon to further our understanding of cities and inform urban planning.

This thesis addresses three major questions in an effort to further our understanding of urban scaling patterns. First, what happens when the definition of the "city" is changed to represent larger and smaller geographic areas? Until now, little consideration has been given to the spatial extent at which data may be aggregated for this kind of analysis to be reliable. Second, under which kind of "city systems" do these scaling patterns emerge? Typically in urban scaling research, the city system considered is that of an entire nation. However, it is still not clear whether similar scaling patterns also appear in other, smaller systems of cities within a country. Third, how do selected variables behave under changing geographical extents of both the urban system under consideration, and of the cities within it?

The thesis seeks to answer these three questions for two subnational-scale urban systems: the state of California, and Southern California. Census data are collected for all populated areas within these two systems, using three different kinds of "city" delineations: (a) official administrative designations, (b) census designations of different spatial extents, and (c) systematic modifications of spatial extents based purely on population densities. Finally, for each combination of spatial extents generated, the scaling behavior of seven non-spatial variables is studied. Results partly confirm findings reported in the literature, but also reveal circumstances under which widely discussed urban scaling laws appear to fail.