Cucci Davide A, Voirol Lionel, Kermarrec Gaël, Montillet Jean-Philippe, Guerrier Stéphane
Geneva School of Economics and Management, University of Geneva, Geneva, Switzerland.
Institute for Meteorology and Climatology, Leibniz University Hannover, Hannover, Germany.
J Geod. 2023;97(2):14. doi: 10.1007/s00190-023-01702-8. Epub 2023 Feb 6.
The global navigation satellite system (GNSS) daily position time series are often described as the sum of stochastic processes and geophysical signals which allow to study global and local geodynamical effects such as plate tectonics, earthquakes, or ground water variations. In this work, we propose to extend the Generalized Method of Wavelet Moments (GMWM) to estimate the parameters of linear models with correlated residuals. This statistical inferential framework is applied to GNSS daily position time-series data to jointly estimate functional (geophysical) as well as stochastic noise models. Our method is called GMWMX, with X standing for eXogenous variables: it is semi-parametric, computationally efficient and scalable. Unlike standard methods such as the widely used maximum likelihood estimator (MLE), our methodology offers statistical guarantees, such as consistency and asymptotic normality, without relying on strong parametric assumptions. At the Gaussian model, our results (theoretical and obtained in simulations) show that the estimated parameters are similar to the ones obtained with the MLE. The computational performances of our approach have important practical implications. Indeed, the estimation of the parameters of large networks of thousands of GNSS stations (some of them being recorded over several decades) quickly becomes computationally prohibitive. Compared to standard likelihood-based methods, the GMWMX has a considerably reduced algorithmic complexity of order for a time series of length . Thus, the GMWMX appears to provide a reduction in processing time of a factor of 10-1000 compared to likelihood-based methods depending on the considered stochastic model, the length of the time series and the amount of missing data. As a consequence, the proposed method allows the estimation of large-scale problems within minutes on a standard computer. We validate the performances of our method via Monte Carlo simulations by generating GNSS daily position time series with missing observations and we consider composite stochastic noise models including processes presenting long-range dependence such as power law or Matérn processes. The advantages of our method are also illustrated using real time series from GNSS stations located in the Eastern part of the USA.
全球导航卫星系统（GNSS）的每日位置时间序列通常被描述为随机过程和地球物理信号的总和，这有助于研究全球和局部地球动力学效应，如板块构造、地震或地下水变化。在这项工作中，我们建议扩展小波矩广义方法（GMWM）来估计具有相关残差的线性模型的参数。这个统计推断框架应用于GNSS每日位置时间序列数据，以联合估计函数（地球物理）以及随机噪声模型。我们的方法称为GMWMX，其中X代表外生变量：它是半参数的，计算效率高且可扩展。与广泛使用的最大似然估计器（MLE）等标准方法不同，我们的方法提供了统计保证，如一致性和渐近正态性，而无需依赖强参数假设。在高斯模型下，我们的结果（理论结果和模拟结果）表明，估计参数与MLE得到的参数相似。我们方法的计算性能具有重要的实际意义。实际上，估计由数千个GNSS站组成的大型网络的参数（其中一些记录了几十年）很快就会在计算上变得令人望而却步。与基于标准似然的方法相比，对于长度为 的时间序列，GMWMX的算法复杂度显著降低。因此，根据所考虑的随机模型、时间序列长度和缺失数据量，GMWMX与基于似然的方法相比，处理时间减少了10 - 1000倍。因此，所提出的方法允许在标准计算机上在几分钟内估计大规模问题。我们通过蒙特卡罗模拟验证了我们方法的性能，生成了带有缺失观测值的GNSS每日位置时间序列，并考虑了复合随机噪声模型，包括呈现长程相关性的过程，如幂律或马特恩过程。我们还使用位于美国东部的GNSS站的实时序列说明了我们方法的优点。
