Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series

摘要

Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods' efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method's efficiency, 360 synthetic daily time series with different lengths N=2048, 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm's outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2-14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods' outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.