## 1 Introduction
### Technological Advances in Vertical Urban Growth Monitoring  
The impact of high-rise buildings on urban environments necessitates advanced tools for analyzing VUG that can provide accurate 3D measurements consistently over large areas and time periods. Methodologies to study height data, vertical development, and temporal changes have evolved, but significant challenges remain.  
Photogrammetry and LiDAR: While Light Detection and Ranging (LiDAR) technology has significantly advanced the extraction of building height information and 3D city modeling, outperforming traditional photogrammetric approaches, its reliance on dedicated acquisition campaigns often limits temporal frequency and increases costs [11; 12; 13; 14]. LiDAR's limited availability can lead to gaps in spatial and temporal analysis needed for continuous VUG monitoring. Photogrammetry, whether aerial or satellite-based, faces limitations due to weather/illumination conditions [15].  
Optical Satellite Approaches: Alternatives like the Google Earth Engine and the European Copernicus program offer the potential for nationwide mapping of building heights using the Sentinel-1 Synthetic Aperture Radar (SAR) and Sentinel-2 optical satellite constellations [16; 17]. In cities like Guangzhou, novel methodologies employing shadow analysis from medium-resolution imagery have improved the mapping of both horizontal and vertical urban density [18]. Additionally, semantic segmentation with Landsat data and deep convolutional encoder-decoder networks using aerial imagery for height estimation has emerged, requiring careful handling of issues like lighting variations and spectral ambiguities [19; 20; 21].  
SAR-Based Techniques: SAR data are invaluable owing to their ability to overcome limitations related to weather and daylight and have been used for estimating building heights [22]. SAR-based approaches such as TomoSAR [23; 24] provide three-dimensional insights by utilizing multiple SAR images to identify persistent scatterer candidates (PSC) in urban landscapes, but they are computationally expensive. Interferometric SAR (InSAR) [25] is pivotal in height estimation, but it faces significant challenges, particularly in the phase unwrapping process. In urban environments, the presence of tall buildings, discontinuous surfaces, and layover effects can lead to significant phase gradients and discontinuities, making phase unwrapping a complex task [26].  
Persistent Scatterer Interferometry (PSInSAR) improves upon conventional InSAR by focusing on temporally stable point scatterers (PS), thus mitigating decorrelation issues. While highly effective for monitoring deformation in existing, stable structures, standard PSInSAR assumes scatterer stability throughout the observation period. This makes it un suitable for monitoring VUG in its current form, as it cannot reliably track the emergence or height evolution of new structures.  
To address this limitation, we develop a novel stack-processing PSInSAR approach using dense time series of high-resolution (3 m) COSMO-SkyMed X-band SAR data, specifically designed to quantify the appearance, disappearance, and height changes of structures over extended periods in complex urban environments. This choice capitalizes on SAR's all-weather, day/night capabilities -- crucial for consistent long-term monitoring-- as well as its high spatial resolution for capturing urban details and frequent revisit potential over large areas. While standard PSInSAR assumes stability, this research adapts the technique to monitor dynamic VUG.