## 2 Methods
### Global Stitching and Error Adjustment  
To achieve the seamless integration of block-based solution results, we propose a global stitching framework based on overlapping region constraints and graph theory-connected component analysis. The core process includes an initial adjustment estimation, connected component identification, and least squares adjustment. First, the validity criterion for the overlapping region between adjacent sub-blocks \((k,l)\) is defined: if the number of common PS points \(N_{\text{overlap}}\) exceeds a preset threshold \(\tau_{\text{overlap}}\), the overlap is considered valid; otherwise, the number of common points is set to 0 to exclude low-confidence connections. For valid overlapping regions, outliers in the deformation rate differences and elevation residual differences are removed using the three-sigma criterion, and the initial adjustment quantities are calculated based on the remaining valid common points' deformation rate difference \(\Delta v_{kl}^{(m)}\) and elevation residual difference \(\Delta e_{kl}^{(m)}\)[35]:  
\[\Delta v_{kl}=\frac{1}{N_{valid}}\sum_{m\in V}\Delta v_{kl}^{(m)},\Delta e_{kl} =\frac{1}{N_{valid}}\sum_{m\in V}\Delta e_{kl}^{(m)} \tag{6}\]  
where \(V\) is the set of valid common points; \(N_{valid}\) is the number of such points; and \(\Delta v_{kl}^{(m)}=v_{k}^{(m_{1})}-v_{l}^{(m_{2})}\) and \(\Delta e_{kl}^{(m)}=e_{k}^{(m_{1})}-e_{l}^{(m_{2})}\) represent the rate difference and residual difference for common points \((m_{1},m_{2})\), respectively. Next, the connected components between sub-blocks are calculated [20], i.e., the set of sub-blocks is mapped to a graph \(G\), and a depth-first search (DFS) is performed to identify all the connected components \(\mathcal{C}\):  
\[G=(V,E) \tag{7}\]  
\[\mathcal{C}=\{\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{\mathcal{M}}\} \tag{8}\]  
where the node set \(V=\{1,\ 2,\ \ldots,\ K\}\) represents sub-block indices, with \(K\) being the total number of sub-blocks; the edge set \(E=\{(p,q)\}\) represents adjacent sub-block pairs \(p\) and \(q\) having a valid overlapping relationship. Each connected component \(\mathcal{C}_{m}=\{k_{1},k_{2},\ldots,k_{K_{m}}\}\) represents a set of sub-blocks that can be directly or indirectly connected via an overlap, with \(K_{m}\) being the number of sub-blocks in this connected component. To suppress the error propagation, the sub-block with the most connecting edges in each connected component \(\mathcal{C}_{m}\) is selected as the reference sub-block \(k_{\text{base}}\) (its adjustment quantity is set to \(X_{k_{\text{base}}}=0\)), and the coefficient matrix \(B\) and observation vector \(L\) are constructed. The matrix elements are defined as  
\[B_{(\mathbf{a},\mathbf{p})}=\left\{ \begin{array}{ll}1&\text{if edge $a$ corresponds to sub $-block pair $(k,l)$ and $p=k$,}\\ -1&\text{if edge $a$ corresponds to sub $-block pair $(k,l)$ and $p=l$,}\\ 0&\text{otherwise}\end{array} \right. \tag{9}\]  
The weight matrix \(P\) can be set as a diagonal matrix with diagonal elements representing the ratio of the number of valid common points for each sub-block pair, thus enhancing the weight of highly redundant connections. The adjustment quantities \(X=[X_{1},X_{2},\ldots,X_{K_{m}}]^{\top}\) are then solved using a least squares adjustment:  
\[X=\left(B^{\top}PB\right)^{-1}\!\!B^{\top}PL \tag{10}\]  
where \(B\in\mathbb{R}^{L_{m}\times K_{m}}\), \(P\in\mathbb{R}^{L_{m}\times L_{m}}\), \(L_{m}\) is the number of valid edges in connected component \(\mathcal{C}_{m}\), and \(K_{m}\) is the number of nodes (sub-blocks). \(L\in\mathbb{R}^{L_{m}}\) represents the initial deformation rate adjustment \(\Delta v_{kl}\) or elevation residual adjustment \(\Delta e_{kl}\) for each edge \(a\) corresponding to the sub-block pair \((k,l)\). Finally, the adjusted quantities \(X_{k}\) are added to the original solution results of each sub-block, generating a globally consistent deformation rate field and elevation residual field within each connected component.