Journal Article

Spatial scaling of stable night lights

City size distributions, defined on the basis of population, are often described by power laws. Zipf’s Law states that the exponent of the power law for rank-size distributions of cities is near -1. Verification of power law scaling for city size distributions at continental and global scales is complicated by small sample sizes, inappropriate estimation techniques, inconsistent definitions of urban extent and variations in the accuracy and spatial resolution of census administrative units. We attempt to circumvent some of these complications by using a continuous spatial proxy for anthropogenic development and treat it as a spatial complement to population distribution. We quantify the linearity and exponent of the rank-size distribution of spatially contiguous patches of stable night light over a range of brightnesses corresponding to different intensities of development. Temporally stable night lights, as measured by the Defense Meteorological Satellite Program- Operational Line Scanner (DMSP-OLS), provide a unique proxy for anthropogenic development. Brightness and spatial extent of emitted light are correlated to population density (Sutton et al., 2001), built area density (Elvidge et al., 2007c) and economic activity (Doll et al., 2006; Henderson et al., 2009) at global scales and within specific countries. Using a variable brightness threshold to derive spatial extent of developed land area eliminates the complication of administrative definitions of urban extent and makes it possible to test Zipf’s Law in the spatial dimension for a wide range of anthropogenic development. Higher brightness thresholds generally correspond to more intense development while lower thresholds extend the lighted area to include smaller settlements and less intensively developed peri-urban and agricultural areas. Using both Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE) to estimate power law linearity and exponent of the resulting rank-size distributions across a range of upper tail cutoffs, we consistently find statistically significant exponents in the range -0.95 to -1.11 with an abrupt transition to very large, extensively connected, spatial networks of development near the low light detection limit of the sensor. This range of exponents and transition are observed at both continental and global scales. The results suggest that Zipf’s Law also holds for spatial extent of anthropogenic development across a range of intensities at both continental and global scales. The implication is that the dynamics of urban growth and development may be represented as spatial phase transitions when the spatial extent and intensity of development are treated as continuous variables rather than discrete entities.