City distributions across elevation: A maximum-entropy approach

Abstract

Most of the world’s cities are concentrated near sea level, far more than low-lying land availability alone would predict. This study quantifies that relationship by combining the elevations of 47,916 individual cities worldwide, each represented by a single point location, with matched national hypsography and modelling elevation intensity within a maximum-entropy framework. Across six candidate intensity models, the data strongly favour concave-cost specifications over the linear-cost exponential. The best AIC fit is a piecewise exponential with a steep low-elevation regime and a shallow higher-elevation regime, while the preferred smooth mechanistic model is a power-law decay, implying a concave effective cost function C(h) ∝ ln(1 + h/h 0 ). Here h denotes elevation above sea level (in metres) and h 0 is a fitted scale parameter that sets where the curvature of the cost function is concentrated. Zipfian rank-frequency scaling (the inverse rank–size power law familiar from city-size distributions) is rejected for city elevations, and the main inference is robust to data perturbations and country stratification. Transport evidence is retained as physical context rather than the central empirical test: scale economies and maritime access provide a plausible mechanism for the steep near-coastal penalty and flatter higher-elevation gradient, but do not identify the statistical pattern.