Volumetric insights into urban growth analysis: Investigating vertical and horizontal patterns

• Tehran's 1985-2024 growth sprawls outward, surpassing upward expansion. • Early segments (1985) are densely packed, while later ones (2024) spread out with lower density and limited vertical growth. • Spatial analysis reveals disparities in UVG and population distribution near the TMR core. • UVG patterns are shaped by density, elevation, and proximity to transport infrastructure. Urban expansion has traditionally been studied in terms of horizontal sprawl, overlooking the role of vertical densification. This paper adopts a volumetric approach to analyze the spatiotemporal dynamics of urban growth in the Tehran Metropolitan Region (TMR) from 1985 to 2024, addressing two key objectives: (a) to examine border patterns of volumetric growth, demographic shifts, and intra-urban variations, and (b) to identify the underlying drivers of volumetric expansion. Using Landsat imagery, Global Human Settlement Layer (GHSL) data, and population statistics from the Statistical Center of Iran, we apply spatial-statistical methods and the XGBoost-SHAP machine learning framework to assess urban expansion patterns and their driving forces. Our findings reveal a transition from vertical densification to horizontal sprawl, with urban expansion increasingly adopting an outward trajectory. Intra-urban analysis uncovers a diffusion-coalescence pattern, where vertical densification often follows horizontal expansion but can also occur concurrently. While older urban segments continue to densify vertically due to land scarcity, newly developed areas exhibit distinct volumetric characteristics, reflecting a cyclical interplay between centralization and decentralization. The XGBoost-SHAP analysis highlights a shift in driving forces from spatial inertia to population-driven growth and, more recently, to economic and geographic constraints. By integrating remote sensing, spatial analysis, and machine learning, this study provides a systematic and data-driven framework for understanding the spatiotemporal dynamics of urban expansion and its underlying drivers. The findings underscore the importance of integrated urban planning strategies that balance vertical and horizontal growth, optimize land use, and enhance sustainability. Incorporating volumetric analysis into planning frameworks can foster more adaptive, resource-efficient, and resilient urban development.