Permanent GNSS stations continuously monitor Earth's crust movements in horizontal and vertical directions. The recorded data include deterministic variations, including linear trends, periodic signals, and offsets, alongside stochastic variations represented by various noise models. Accurately detecting deterministic behaviors depends on a realistic estimation of the observation noise model. A new multivariate algorithm based on Monte Carlo singular spectrum analysis (MCSSA) is developed to analyze the multiple channels of time-series data simultaneously (e.g., different position components or data from multiple stations), considering noise correlations without being limited to a specific noise model. Testing on simulated GNSS data showed that, by increasing the number of channels, the algorithm could accurately identify dominant annual and semiannual components in the presence of colored noise. The results also indicated that unrealistic assumptions about the GNSS position time-series noise model can be misleading in the MCSSA hypothesis testing. Applying the algorithm to real Greenland GNSS data confirmed the significance of annual and semiannual harmonic patterns when white plus flicker noise (FLWN) combinations were considered as the stochastic behavior of data. In the univariate analysis of the vertical position time series contaminated with random walk noise, none of the annual and semiannual signals were interpreted to be significant. At the same time, the proposed multivariate algorithm successfully identified the annual signal but lacked sufficient channels (stations) to confirm the significance of the semiannual signal in the presence of random walk noise. The multivariate analysis has confirmed the significance of semiannual signal across 21 time series contaminated with FLWN, which univariate analysis missed due to a high level of colored noise.

Synthetic aperture radar interferometry (InSAR) measures the projection of three-dimensional (3D) ground displacement in the range direction and in the azimuth direction through image processing. The incapability of InSAR in measuring the 3D displacements restricts its capability for assessing real Earth surface deformation. The near-polar orbiting characteristics of InSAR missions reduce the sensitivity of line-of-sight (LOS) displacements significantly to the north-south components of the real 3D displacement fields and weaken the geometric strength of a given configuration. Applying range measurements from various missions to address 3D displacement leads to an ill-posed inverse problem that needs to be regularized. Moreover, it needs appropriate weighting of the observations to give proper estimates of the parameters. In this study, we propose Tikhonov regularization (TR) and least-squares variance component estimation (LS-VCE) methods for retrieving 3D displacement vectors from range and azimuth displacements. Depending on the functional degree of freedom (DoF) of the inverse problem, the TR and LS-VCE methods are applied in determined and overdetermined equation systems, respectively, to stabilize the ill-conditioned models and estimate the variance components of observations. These methods were evaluated by two synthetic data sets and a real data set from the Sentinel-1 terrain observation by progressive scan (TOPS) and ALOS-2 phased array type L-band synthetic aperture radar (PALSAR-2) missions in 2015 of the MW = 8.3 Illapel earthquake in Chile. Results indicate more than 40% improvement in both the precision and accuracy of retrieving 3D deformation fields when the regularized LS-VCE (RLS-VCE) is adopted instead of the conventional method (CM) that considers primary weighting for observations. Applying the range and azimuth InSAR displacements together with adopting the LS-VCE method reveal a north-south convergent borderline near 31.2° S in the 2015 Illapel earthquake.