## 3 Methodology
### PSInSAR Full Stack Processing
#### 3.2.3 Reference Point Selection  
The selection of a stable reference point is crucial in PSInSAR processing, particularly in our multi-stack approach. This selection process is fundamentally guided by the ASDrack, which quantifies the temporal consistency of radar reflectivity. Standard PSInSAR methodologies rely on a global temporal stability approach, expressed mathematically as:  
\[AS\ Index_{global}=1-\frac{\sigma_{amp_{total}}}{\mu_{amp_{total}}} \tag{1}\]  
where \(\sigma_{amp\ total}\) and \(\mu_{amp\ total}\) are computed over all N acquisitions. However, this global approach has significant limitations when applied to dynamic urban environments, particularly in areas experiencing rapid vertical growth. Our methodology addresses these limitations by introducing a stack-wise stability criterion:  
\[AS\ Index_{k}=1-\frac{\sigma_{amp_{k}}}{\mu_{amp_{k}}},\forall k\in[1,K] \tag{2}\]where \(\sigma_{\text{ampk}}\) and \(\mu_{ampk}\) are computed for the k-th sub-stack. This segmented approach enables the detection of PS points with \(AS\)\(Index_{k}\)\(>\) 0.85 in any sub-stack, allowing us to capture transient scatterers that emerge during urban development phases. The reliability of our methodology is maintained through three key innovations:  
Reference Point Consistency: We maintain a carefully selected reference point across all stacks, ensuring the comparability of height measurements between temporal segments. This consistency is crucial for tracking vertical growth over time. Adaptive PS Selection: Rather than applying amplitude stability criteria across the entire time series, we evaluate stability within each stack independently. The complete set of PS is defined through a union operation across all stacks:  
\[PS_{new}=\sum\limits_{k=1}^{K}\{PS|AS\ Index_{k}>0.85\} \tag{3}\]  
Integration Framework: Our post-processing methodology robustly combines results from multiple stacks, creating a coherent timeline of urban vertical growth. While conventional PSInSAR processing typically recommends an AS-Index threshold of 0.75 [42], we have implemented a more stringent threshold of 0.85 for reference point selection. This decision was based on the observation that potential PSCs with AS-Index values below 0.85, while potentially stable within individual stacks, often exhibited appearance/disappearance across the 13 stacks due to inherent amplitude fluctuations (illustrated in Figure 5). Selecting a reference point requires consistent stability throughout the entire observation period to serve as a reliable anchor for relative height estimations across all stacks. The 0.85 threshold provided this necessary margin, ensuring the chosen reference point maintained high stability and consistent presence across the dynamic temporal segments, thus guaranteeing the integrity of the multi-stack comparison. Figure 5 illustrates the amplitude values of each image and AS-Index measurements over time for our chosen reference point at coordinates (30.554607\({}^{\circ}\)N, 114.29445\({}^{\circ}\)E) across thirteen stacks. Each stack represents a series of SAR images processed together, and the color-coded bars indicate the amplitude stability within each stack. The scatter plot shows fluctuations in amplitude and AS-Index values for each image within the stacks. While most stacks maintain AS-Index values above the 0.85 threshold, there are instances of significant amplitude variability, particularly in later stacks. This variability could be  
Figure 5: Amplitude and AS-Index analysis for the reference point. attributed to factors such as atmospheric disturbances, including water vapor, or changes in satellite orbit, which can affect the propagation of radar signals. This approach, being the first attempt to process PSInSAR data in stacks for VUG, introduced unique challenges, particularly in maintaining a consistent reference point due to amplitude discontinuity. The selected reference point generally exhibited a high AS-Index, indicating substantial temporal stability.