Sprawl as Linear Population Density

I've been staring at this graph for days, trying to figure out what it's telling me. The graph is from the paper "Intake fraction of nonreactive vehicle emissions in US urban areas" from Atmospheric Environment. 39 (7), by J D. Marshall, S K. Teoh, and William W. Nazaroff.

What drew me to it is a more recent paper by University of Minnesota civil engineering assistant professor Julian Marshall, that further develops some strange and beautiful mathematical properties of how cities develop. Having such a lovely straight line such as the one on the left is unusual in urban studies, knowing that each city develops in its own unique way. Each point on the graph is a city. The x axis is the population of the city as measured by the US Census. The y axis is something completely new. The paper calls it "linear population density", or the number of people living along an imaginary straight line. It is actually the population of the entire city divided by the square root of the area of the city. Since the area is measured in square metres, its square root is measured in metres, hence the unit of people per metre.

This paper stumbles upon the fact that there is a log-log linear relationship between the population of a town and its "linear population density". With the slope of the graph it means that the linear population density is proportional to p^0.59, where p is the population. Forgive my rusty math, but this tells me that you can solve for the area of the city

p^0.59 = k p a^-0.5
p^1.18 = k² p² a^-1
a = k² p^0.82

That should mean that the area of cities goes up a little bit more slowly than population, that is to say that cities get relatively more compact as they get bigger.

But in a later article, in the September 2007 issue of Urban Studies, Julian Marshall shows that in fact the historical growth path of most cities over the past 50 years is not along that line, but rather as they grow the "linear population density" remains roughly constant. Redoing my calculation above with an exponent of 0 rather than 0.59, that would mean that the area of a city grows with the square of its population. Big difference. Double the population and quadruple its size. But extrapolating a bit, and I don't know whether the model allows this, if a city starts out with a uniform population density, then as it grows it will maintain that uniform density, but if it starts out with a large gradient from high to low density, then as it expands it will maintain that gradient and sprawl more and more.

There is a limit to how the data can be used that way. This analysis is based on US census data that considers a census tract to be urban if it is over a given density threshold. So if a city sprawls quite a lot, the edges will have such low density that they won't even be considered urban at all according to the data, making the city seem smaller, and therefore less dense, than it actually is. But the Brookings Institution study "Who Sprawls Most" corrected for that and came up with similar conclusions: cities with dense cores sprawl most, and cities with more uniform densities sprawl least.

Tags: Housing Sprawl Population Density Urban Planning Smart Growth

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