Using 30 meter by 30 meter satellite and high altitude photography from 1976 and 1992, the authors study the nature of sprawl in the United States by exactly noting what parts of the US are built up or paved. They note that only 1.9% of the continental US is built up or paved, that even within urban areas there is enormous amounts of open space, and that, by one definition, the percent of built up land around a given house in an urban area has not increased from 1976 to 1992, since the new housing built on the urban fringe is evenly weighed when infill development increases the density of existing houses in already built up areas. The authors regress a number of explanatory variables on their density measure, and find that, among other things, geography (the presence of mountains, the ruggedness of the terrain, and the existence of aquifers suitable for wells) cannot be ignored. There is also a citation-rich summary of the extant models of monocentric and polycentric cities.
(As an aside, some claims in this paper should, as the authors note, be taken with a grain of salt; this gives me a good excuse to link to a 2007 paper of my own on density in the US. The “Sprawl from Space” paper suggests that sprawl has not increased from 1976 to 1992, and second, that counter-intuitively, western cities like SF and Phoenix are not particularly sprawling. Our paper, on the contrary, showed declining population density over each decade, over any measure of a city (municipal, urban area, MSA, etc.) you want, and further pointed out that a quirk in the definition of “metropolitan area” means that MSAs are totally unsuitable for work on density. Some rectification on the population density claim results from the fact that sprawl appears to have been much quicker in the 50s-70s than during the 80s.)