Global Navigation Satellite System (GNSS) coordinate time series contains obvious seasonal signals, which mainly manifest as a superposition of annual and semi-annual oscillations. Accurate extraction of seasonal signals is of great importance for understanding various geophysical phenomena. In this paper, a Weighted Nuclear Norm Minimization (WNNM) is proposed to extract the seasonal signals from the GNSS coordinate time series. WNNM assigns different weights to different singular values that enable us to estimate an approximate low rank matrix from its noisy matrix. To address this issue, the low rank characteristics of the Hankel matrix induced by GNSS coordinate time series was investigated first, and then the WNNM is applied to extract the seasonal signals in the GNSS coordinate time series. Meanwhile, the residuals have been analyzed, obtaining the estimation of the uncertainty of velocity. To demonstrate the effectiveness of the proposed algorithm, a number of tests have been carried out on both simulated and real GNSS dataset. Experimental results indicate that the proposed scheme offers preferable performances compared with many state-of-the-art methods.

Parameter estimation of chirp signal model with time-varying amplitude and additive noise is a critical problem in statistical signal processing. A statistically efficient algorithm is proposed to estimate the nonlinear parameters of frequency rate (FR) and initial frequency (IF). Initial estimates are first deduced from the ambiguity function (AF) and then refined through a statistics-based iterative procedure that improves their convergence rates from $O_p(N^{-1})$ and $O_p(N^{-2})$ to $O_p(N^{-3/2})$ and $O_p(N^{-5/2}),$ respectively. It is worth noting that the convergence rates of FR and IF double while being controlled by each other at each iteration, causing the convergence rates of FR and IF to jump up and down before the algorithm converges. Four iterative processes are considered involving crossovers between two types of initial estimators and two types of iterative coefficients. The asymptotic distribution of the refined estimators is obtained and demonstrated to be bivariate normal, and the Cramér–Rao lower bound (CRLB) is derived for each. Monte Carlo simulations were performed to verify the effectiveness of the proposed algorithm.

Accurate signal extraction from Global Navigation Satellite System (GNSS) coordinate time series is crucial for geophysical studies. Primary focus of this paper lies in the recovery of trend and seasonal signals from observed GNSS coordinate time series. To achieve this, this paper presents a novel Modified L1-norm Regularization Rank-1 Matrix Completion (ML1MC) method, ML1MC leverages the low-rank property of Hankel matrices and incorporates automatic rank estimation and Block Coordinate Descent (BCD) to effectively handle noise and avoid overfitting, which provides a more comprehensive rank-1 matrix representation than Singular Value Decomposition (SVD) based low-rank matrix recovery methods. Extensive experiments on simulated and real GNSS datasets demonstrate ML1MC's superior performance in comparison to several State-Of-The-Art (SOTA) methods. Specifically, compared to the SOTA methods, ML1MC achieves over 7.8\% misfit reduction and reduces trend uncertainty error by more than 45.9\% under normal noise conditions.