Removing Population Size, World Cities Leave on Land a Footprint of Wealth

Abstract: Urban expansion builds on precious arable land, affecting ecosystems and health, despite worldwide measures to alleviate its impacts, in small and large cities. Nonetheless, the link between urban extent and population size is still unclear. Here we uncover a scaling law governing built-up land in 1800+ global urban areas. We observe that world cities have homothetic (or isometric) radial land use profiles, and that built-up footprint is proportional to total population. This spatial scaling law is important for the understanding and the definition of urban areas, to help build a science of cities and make them more sustainable. It implies that small and large cities have similar internal structures, and that built-up area per capita is constant across city sizes. The remaining variations are very strongly correlated to wealth measured by gross domestic product per capita, suggesting a pick between economic development and sustainabiliy.

• The paper establishes that built-up land follows a universal exponential radial profile, with characteristic distance scaling as √N and built-up area scaling linearly with population.
• Using 100-meter satellite data and population figures for more than 1,800 cities, the study estimates a central built-up share of about 35% and a characteristic density scale near 29,000 inhabitants per square kilometer.
• The paper introduces the Urban Built Land Index, which measures residual land consumption and correlates strongly with national GDP per capita (R = 0.952 across 24 countries), highlighting a sustainability trade-off between wealth and land use.

This paper by Vignolles and Lemoy analyzes radial profiles of built-up land across more than 1,800 of the world's largest urban areas (each above 300,000 inhabitants, together exceeding 2.5 billion people) and establishes a simple scaling law for urban land use. Combining the GHSL Built-up Surface dataset at 100 m resolution with United Nations World Urbanization Prospects population figures, the authors show that once city size is removed, world cities are nearly identical in their internal land-use structure — and that the residual variation is overwhelmingly explained by national wealth.

Data and method

The study uses the GHS-BUILT-S R2022A raster (built-up share per 100 m cell), masked by the ESRI–Garmin World Water Bodies layer so that built-up shares are computed over emerged land only. City centers are located manually via city halls or central landmarks using the Global Urban Centres database, and concentric rings of 200 m width are extracted within a maximal disc whose radius scales as $r_{\max} = D/k_N$ with $k_N = \sqrt{N_{\text{Tokyo}}/N}$ and $D = 150$ km for Tokyo. This population-based buffer definition sidesteps the lack of a uniform global definition of cities, a known confounder in comparative urban studies.

Two complementary methods remove the size effect. First, rescaling distances by $\sqrt{N}$ produces a data collapse of all profiles; a signal-to-noise analysis over candidate exponents confirms that $\alpha = 1/2$ is optimal. Second, each profile is fitted with an exponential $s_N(r) = a_N \exp(-r/l_N)$, yielding a median $R^2$ of about 0.9 across all cities.

A universal exponential profile and linear area–population scaling

The rescaled profiles collapse onto a common exponential curve that is not scale-free. The central built-up share $a_N$ is roughly constant at approximately 35%, and the characteristic distance follows

$$l_N \sim l_1 \sqrt{N}, \qquad l_1 \simeq 6 \text{ m}.$$

Fitted exponents across model variants (one- vs. two-parameter fits, water kept or removed, polycentric areas excluded) range from 0.480 to 0.531, tightly bracketing 0.5. Since $S \sim l_N^2$, built-up area is proportional to population: the exponent of 1 settles a long-standing debate in which literature estimates ranged from 2/3 to 1 (2608.18792). The implication is that built-up land per capita is constant across city sizes, so soil-sealing mitigation is needed equally in small and large cities.

The authors distill this into a universal formula $k_N = \sqrt{N_{\text{Tokyo}}/N}$0 with $k_N = \sqrt{N_{\text{Tokyo}}/N}$1 and $k_N = \sqrt{N_{\text{Tokyo}}/N}$2 m, and propose $k_N = \sqrt{N_{\text{Tokyo}}/N}$3 inhabitants/km² as a characteristic "net" density scale. Waterfront cities and polycentric regions introduce only small deviations (a few percentage points), though cities with more water tend to have slightly more built-up land in their peripheries, and polycentric areas show larger characteristic distances on average.

Wealth as the residual driver: the UBLI

The size-independent Urban Built Land Index, $k_N = \sqrt{N_{\text{Tokyo}}/N}$4, captures residual per-capita land consumption (mean ≈ 21 m², median ≈ 12 m²). Its global map closely mirrors GDP per capita. For the countries with the most cities, the correlation between median national UBLI and GDP pc reaches R = 0.952 ($k_N = \sqrt{N_{\text{Tokyo}}/N}$5) over 24 countries, rising to R = 0.984 with 10 countries and R = 0.99987 with 3 countries. Household size correlates negatively ($k_N = \sqrt{N_{\text{Tokyo}}/N}$6), consistent with wealthier, smaller households consuming more built space per capita.

The authors interpret the wealth–land link as plausibly bidirectional: wealth enables larger dwellings, while constructing and conditioning them generates economic activity. They draw a pointed policy conclusion — that GDP growth as an objective directly conflicts with land sustainability, since richer societies seal proportionally more land per capita, particularly in peripheral rings where most urban land lies. This echoes findings on carbon footprints but is notable for the extreme strength of the empirical association.

Limitations and open questions

Several caveats bear on these results. The scaling law's own $k_N = \sqrt{N_{\text{Tokyo}}/N}$7 is moderate (roughly 0.4–0.5 depending on specification), so substantial inter-city variance remains beyond the mean trend. The GDP correlation is computed at national level using median UBLI over cities, and its strength increases as fewer countries are included — the authors themselves note that adding noisier countries weakens it, so the near-unity correlations rest on small samples. Wealth is acknowledged to be multi-faceted (household size, car ownership), and the causal direction of the UBLI–GDP relationship is not identified. The analysis covers only cities above 300,000 inhabitants; whether smaller settlements follow the same law is untested. Finally, the authors suggest similar results should hold for artificial or impervious land, but reliable global datasets for those quantities do not yet exist.

Conclusion

The paper delivers a robust stylized fact for urban science: world cities are homothetic in built-up land use, with exponential radial profiles whose characteristic length scales as $k_N = \sqrt{N_{\text{Tokyo}}/N}$8, implying linear area–population scaling and constant per-capita land consumption. Residual variation is dominated by wealth, quantified through a new dimensioned index (UBLI) correlated with GDP per capita at R ≈ 0.95. The result reframes urban sustainability debates: sprawl control is a universal need across city sizes, but intensifies with economic development.