Eikonal Equation Modeling for 3D Urban Morphology

This study presents a theory of the evolution of three-dimensional (3D) urban morphology based on the Eikonal equation, which arises from the wave Partial Differential Equation (PDE) through Huygens principle. Existing models of 3D urban growth either describe the growth of cities without predicting it or simulate it with grid-based transition rules calibrated for each city; in contrast, the proposed model is, to our knowledge, the first continuous PDE model of 3D urban growth derived from a physical principle, in which horizontal expansion, vertical densification and urban planning are parts of one equation. As an extension of our two-dimensional boundary model, the built volume of a city is treated as a region in space whose bounding surface propagates outwards and upwards with spatially varying, direction dependent speeds. We develop the 3D Eikonal PDE, its characteristic equations for ray tracing, the optimal growth (Fermat) principle and the Hamilton–Jacobi model, and, since cities expand horizontally and grow vertically at markedly different rates, we introduce an anisotropic Eikonal equation with separate horizontal and vertical speeds, in which the Huygens wavelets are ellipsoids rather than spheres. We show that the 3D model reduces exactly to a Hamilton–Jacobi equation for the built-volume height field, which separates growth into vertical densification in place and lateral motion of the edges of the built area, and whose ground slice recovers the two-dimensional model. Urban planning enters the PDE through the speed field rather than as an external constraint: zoning and protected land act as obstacles, infrastructure and land release act as incentives, and density and height regulations act as a capacity factor, for which a comparison principle guarantees that the model city never exceeds the planning envelope. The height field formulation further transforms the inverse problem into a direct regression, so that the speed fields can be identified from multi-temporal 3D data without numerical integration of the PDE. Applied to Las Vegas, Dubai and Shenzhen with the Global Human Settlement Layer and, for Shenzhen, annual building heights, the 27-parameter model reduces the 15-year prediction error relative to persistence and a linear trend in 9 of 12 combinations of dataset and period, by 2–23%; under spatially blocked cross-validation the improvement holds in all held-out areas for Las Vegas and for the building height data, but not on average for the other two datasets. The framework demonstrates that horizontally expanding and intensifying cities are governed by the same model, the morphological character of each city being captured by the ratio of its horizontal to vertical propagation speeds, which a spatial bootstrap identifies with a relative standard deviation of about 10%.