Incorporating Physical Constraint in Three-Dimensional Time Series InSAR Inversion for Urban Deformation Monitoring

Spaceborne interferometric synthetic aperture radar (InSAR) has become a key method for remote sensing monitoring of surface deformations, providing millimeter-level accuracy along the satellite’s line-of-sight. However, precise understanding of deformation mechanisms requires three-dimensional (3-D) measurements. Challenges in obtaining long time series and high-density in-situ measurements, along with the limited sensitivity of polar-orbiting SAR satellites to north-south displacements, often lead to suboptimal horizontal accuracy in conventional 3-D inversion. This study proposes a 3-D inversion method for time series InSAR measurements, specifically designed for scenarios with single-track SAR data, common in many areas where free data is limited. The method introduces a physical constraint between the horizontal deformation and the spatial gradients of vertical displacement, concentrated on urban areas where deformations are typically slow, long-term, and small in magnitude. Considering the localized and different drivers of urban deformations, such as diverse settlement patterns from tunneling activities and ocean reclamation, we propose a pixel-by-pixel estimation approach for the proportionality factor $\beta $, the key parameter in the physical constraint. This approach recognizes that $\beta $ varies across locations, contrasting with the conventional assumption of a uniform $\beta $ for the entire study area. An optimization process with gradient descent is introduced to iteratively refine $\beta $, accelerating convergence and achieving a robust solution. Experiments were conducted in representative scenarios, including ocean-reclaimed and metro-tunneling areas, with validation through leveling measurements. The results demonstrate that the proposed method achieves precise 3-D inversion and effectively captures the complex dynamics of urban surface deformation.