Impacts of Period Offset and Period Variation on modeling seasonal signals in GNSS coordinate time series from both theoretical and practical perspectives

Global Navigation Satellite System (GNSS) coordinate time series exhibit significant periodic characteristics. The existing studies mainly focus on obtaining accurate amplitude and phase of periodic signals, while the period lengths are commonly set as constant empirical values. Nevertheless, due to geophysical signals and technical-related systematic errors, the actual period length may have Period Errors (PE), including Period Offset (PO) or Period Variation (PV). Using empirical period length would lead to incorrect periodic modeling and overestimation of velocity uncertainty. This paper establishes a rigorous theoretical harmonic model considering the impact of PE, then uses the Nonlinear Least Squares (NLS) and Unscented Kalman Filter (UKF) to estimate periodic signals in GNSS coordinate time series. Our simulation results show that the NLS model considering PO (NLS_Period) significantly outperforms the traditional harmonic model estimated by Weighted Least Squares (WLS) in modeling period signals. Especially for long time series and large amplitudes, NLS_Period reduces the period amplitude fitting error by up to 8 times compared to the traditional harmonic model when the annual amplitude is 8.2 mm over a 28‑year span. When the period length varies, the UKF model, UKF_Amp_Period, considering PV, PO, and Period Amplitude Variations (PAV) together achieve the best fitting results, with accuracies of 61%, 40.6%, and 40.2% higher than WLS, Singular Spectrum Analysis (SSA), and the model considering only PAV (UKF_Amp), respectively. When applied to modeling periodic signals in real GNSS coordinate time series, the UKF_Amp_Period method improves fitting accuracy by about 9–10% compared to WLS, while the spectral index decreases by 5.6–8.4%. Our work confirms the importance of accurate period length in the harmonic modeling of GNSS coordinate time series. Considering the actual period length of signals not only achieves more accurate periodic modeling results but could also potentially explain the period signals from the perspective of period length variations.