Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

摘要。最近，已经开发出各种模型，包括分数布朗运动（fBm），以分析大地时间序列的随机属性以及估计的地球物理信号。通常将这些时间序列的噪声频谱建模为混合频谱，包括白噪声和彩色噪声之和。在这里，我们有兴趣在确定性地从观测中减去地球物理信号后对剩余时间序列建模。然后，假定此剩余时间序列是三个随机过程的总和，包括Lévy过程家族。第三随机项的引入对剩余的残余信号和其他相关过程进行建模。通过仿真和实时序列，我们确定了Lévy过程的三类，即高斯，分数阶和稳定。在第一种情况下，残差主要由短存储过程构成。在存在长期相关性和自相似性的情况下，分数Lévy过程可以作为fBm的替代模型。在此，稳定过程受制于无限方差的特殊情况，只有在大地时间序列应用中只有重尾分布的情况下才能满足。因此，该模型暗示了功能模型选择中的潜在焦虑，其中丢失的地球物理信息会生成此类剩余时间序列。仅在大地时间序列应用中出现重尾分布的情况下才能满足。因此，该模型暗示了功能模型选择中的潜在焦虑，其中丢失的地球物理信息会生成此类剩余时间序列。仅在大地时间序列应用中出现重尾分布的情况下才能满足。因此，该模型暗示了功能模型选择中的潜在焦虑，其中丢失的地球物理信息会生成此类剩余时间序列。