Identifying the correct stochastic model in GNSS time series is essential to study geophysical parameters such as site velocities, and hence enhancing their accuracy. The rate uncertainty is a critical aspect in GNSS time series analysis. The variance component estimation (VCE) methods commonly utilize unconstrained estimation principles. Simulating 1000-time series for 4 different noise combinations with 10 years’ time span, we have investigated the performance of non-negative least squares VCE (NNLS-VCE) method for identifying an appropriate noise model. Our results are provided for both univariate and multivariate analysis. As the noise model's complexity increases, the significance of employing multivariate analysis is prominent in contrast to univariate analysis. After thorough analysis, we have determined that treating the false-positive model as a stochastic model in time series yields significant insights. Specifically, if the accumulative spectral index is lower than the true value, it results in an underestimation of the rate uncertainty. Conversely, if the index is higher than the actual value, it leads to an overestimation. Additionally, we observed that as the noise model complexity increases, the number of false-positive models also increases. However, the implementation of multivariate analysis mitigates this increase, offering a more realistic and reliable approach. In case of four distinct noise models, the detection power percentages of 98.5%, 90.5%, 69.5%, 29.3% of univariate analysis increased to 99.5%, 99.8%, 88.4% and 83.7% for multivariate analysis.

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.