Eikonal equation for modelling urban boundary evolution using Huygens principle and machine learning

The study presents the model of dynamic urban boundary evolution using the Eikonal equation arising in the wave Partial Differential Equation (PDE) using Huygens principle. It is shown how the mathematical model may be used in analyzing urban morphology through the formulation of wave equations that can be solved to represent the development of an urban area. The technique uses the urban boundary obtained through satellite data and estimates the parameters used in the ordinary differential equations (ODE) that arise when solving the eikonal PDE using the method of characteristics. The proposed mathematical framework is implemented for a small study area to demonstrate its functionality. The boundary of the study area for three years is extracted using satellite remote sensing data. The initial year boundary is discretized into fifty subdivisions. The Huygens principle is used to determine the parameters - the rate of change in magnitude and direction of urban boundaries at these selected boundary points. Once the parameters are estimated, they are related to various potential urban boundary expansion drivers using neural networks. The calibrated model shows a high degree of accuracy and may be used to further interpolate or extrapolate the boundary data.