Fast implementation of least squares variance component estimation for diagonal matrices: applications to GNSS time series

The study of long-term GNSS time series provides valuable insights for researchers in the field of earth sciences. Understanding the trends in these time series is particularly important for geodynamic researchers focused on earth crust movements. Functional and stochastic models play a crucial role in estimating trend values within time series data. Various methods are available to estimate variance components in GNSS time series. The least squares variance component estimation (LS-VCE) method stands out as one of the most effective approaches for this purpose. We introduce an innovative method, which streamlines calculations and simplifies equations, and therefore significantly boosting the processing speed for diagonal(ized) cofactor matrices. The method can be applied to the GNSS time series of linear stochastic models consisting of white noise, flicker noise and random walk noise. Moreover, unlike the conventional approaches, our method experiences high computational efficiency even with an increase in the number of colored noise components in time series data. For GNSS time series, this variable transformation has been applied to both univariate and multivariate modes, preserving the optimal properties of LS-VCE. We conducted simulations on daily time series spanning 5, 10, 15, and 20 years, employing two general and fast modes with one and two colored noise components plus white noise. The computation time for estimating variance components was compared between the two modes, revealing a notable decrease in processing time with the fast mode.