Bivariate Least-Squares Stacking InSAR for Sparse Data Scenarios

Multitemporal InSAR (MT-InSAR) is pivotal for geohazard monitoring but faces convergence challenges in sparse-data scenarios, such as emergency responses or initial satellite missions, where limited imagery hinders time-series analysis. Concurrently, efficient stacking techniques often struggle with digital elevation model error residuals. While robust stacking mitigates topographic effects via sign-flipping, it is constrained by even-sample requirements and suffers from reduced signal-to-noise ratios under wide-baseline conditions due to inverse-baseline weighting. To address these limitations, we propose a bivariate least-squares (BLS) stacking framework. Grounded in the Gauss-Markov theorem, BLS reformulates sparse-data deformation retrieval as a closed-form least-squares estimation problem. Under the adopted linear model, the use of a bivariate spatiotemporal design matrix enables analytical separation of deformation and topographic residual terms while avoiding the strong short-baseline emphasis inherent in inverse-baseline weighting. Simulations based on real advanced land observing satellite-4 (ALOS-4) geometric parameters show that BLS reduces the RMSE to 3.87 mm/yr, corresponding to a 37% reduction in discrepancy relative to robust stacking in the simulated setting. Additional low-redundancy simulations further support the scenario-specific applicability of BLS: under the tested N = 2 and N = 3 interferometric-pair configurations, BLS remained numerically solvable, whereas the SBAS-like model was frequently rank deficient or ill-conditioned. In the real-data experiments, BLS further reduces the discrepancy relative to the adopted reference solutions by 91.7% in the ALOS-4 wide-baseline case and by 80.6% in the Sentinel-1 case. Benchmarking against the European ground motion service calibrated product yields a correlation of 0.776 and an RMSE of 3.33 mm/yr within a predefined quality-controlled comparison subset, indicating that the proposed method can recover deformation trends broadly consistent with an independent operational reference under sparse-data conditions. Supplementary benchmark/leveling validation and runtime comparison further show that BLS maintains strong agreement with independent ground observations while remaining substantially more efficient than SBAS-like inversion under the tested sparse-data configuration. Overall, BLS provides a practical sparse-data stacking option for deformation monitoring in data-constrained environments.