Geodetic time series are usually modeled with a deterministic approach that includes trend, annual, and semiannual periodic components having constant amplitude and phase-lag. Although simple, this approach neglects the time-variability or stochasticity of trend and seasonal components, and can potentially lead to inadequate interpretations, such as an overestimation of global navigation satellite system (GNSS) station velocity uncertainties, up to masking important geophysical phenomena. In this contribution, we generalize previous methods for determining trends and seasonal components and address the challenge of their time-variability by proposing a novel linear additive model, according to which (i) the trend is allowed to evolve over time, (ii) the seasonality is represented by a fractional sinusoidal waveform process (fSWp), accounting for possible non-stationary cyclical long-memory, and (iii) an additional serially correlated noise captures the short term variability. The model has a state space representation, opening the way for the evaluation of the likelihood and signal extraction with the support of the Kalman filter (KF) and the associated smoothing algorithm. Suitable enhancements of the basic methodology enable handling data gaps, outliers, and offsets. We demonstrate the advantage of our method with respect to the benchmark deterministic approach using both observed and simulated time series and provide a fair comparison with the Hector software. To that end, various geodetic time series are considered which illustrate the ability to capture the time-varying stochastic seasonal signals with the fSWp.

大地测量时间序列通常采用确定性方法进行建模，其中包括具有恒定幅度和相位滞后的趋势、年度和半年周期分量。虽然简单，但这种方法忽略了趋势和季节性成分的时间变化或随机性，并可能导致解释不充分，例如高估全球导航卫星系统 (GNSS) 站速度的不确定性，甚至掩盖重要的地球物理现象。在这篇文章中，我们概括了以前用于确定趋势和季节性成分的方法，并通过提出一种新颖的线性相加模型来解决其时间变化的挑战，根据该模型，（i）允许趋势随时间演变，（ii）季节性由分数正弦波形过程 (fSWp) 表示，考虑了可能的非平稳循环长记忆，以及 (iii) 额外的串行相关噪声捕获短期变化。该模型具有状态空间表示，为在卡尔曼滤波器 (KF) 和相关平滑算法的支持下评估似然性和信号提取开辟了道路。基本方法的适当增强能够处理数据间隙、异常值和偏移。我们使用观测和模拟时间序列证明了我们的方法相对于基准确定性方法的优势，并与 Hector 软件进行了公平的比较。为此，考虑了各种大地测量时间序列，这些时间序列说明了使用 fSWp 捕获时变随机季节性信号的能力。