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.

了解全球定位系统（GPS）坐标时间序列的噪声含量是现实评估和未知参数不确定性的前提。方差分量估计方法[例如，受限最大似然估计器（REML）]用于评估GPS坐标时间序列的噪声含量。对于大规模数据，即在较大的时空范围内，以前的方法的效率可以大大提高。同时，包括对大型矩阵的反复反演在内的估计方法导致了密集的计算和大量的存储需求。通过利用离散小波变换的去相关特性对REML估计器进行量化，当前的研究提供了FREML（快速REML），用于准确快速地估计噪声含量。\（N = 2048 \），4096和8192个观察时期。该时间序列由线性趋势，周期性信号，偏移，瞬态位移，间隙（最大10％）以及白色，闪烁和随机行走噪声的组合组成。将FREML算法的结果与使用最大似然法量化不确定性的现有软件（例如Hector）进行了比较。结果表明，两种方法在噪声分量，未知参数（速率，偏移和瞬态位移）及其不确定性方面均提供了等效的结果。此外，与Hector软件相比，FREML方法将计算时间减少了2-14倍，具体取决于数据量和缺少的时期。要对该方法进行更多评估，FREML方法应用于36个实时序列，其噪声模型为（i）白色加闪烁噪声，以及（ii）白色，闪烁和随机行走噪声的组合。结果表明，两种方法的结果接近，而FREML方法加快了噪声和未知参数的估计。