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 (LOS). However, precise understanding of deformation mechanisms requires three-dimensional (3D) 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 3D inversion. This study proposes a 3D 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta $</tex-math></inline-formula>, the key parameter in the physical constraint. This approach recognizes that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta $</tex-math></inline-formula> varies across locations, contrasting with the conventional assumption of a uniform <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta $</tex-math></inline-formula> for the entire study area. An optimization process with gradient descent is introduced to iteratively refine <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta $</tex-math></inline-formula>, 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 3D inversion and effectively captures the complex dynamics of urban surface deformation.