Various methods have been used to model the time-varying curves within the global positioning system (GPS) position time series. However, very few consider the level of noise a priori before the seasonal curves are estimated. This study is the first to consider the Wiener filter (WF), already used in geodesy to denoise gravity records, to model the seasonal signals in the GPS position time series. To model the time-varying part of the signal, a first-order autoregressive process is employed. The WF is then adapted to the noise level of the data to model only those time variabilities which are significant. Synthetic and real GPS data is used to demonstrate that this variation of theWF leaves the underlying noise properties intact and provides optimal modeling of seasonal signals. This methodology is referred to as the adaptive WF (AWF) and is both easy to implement and fast, due to the use of the fast Fourier transform method.
Most Global Positioning System (GPS) position time series contain annual and semi-annual periods that are routinely modelled as two periodic signals with constant amplitude and phase-lag. However, the amplitudes and phase-lags of seasonal signals vary slightly over time. Also, time series contain specific colored noise. Which methods should we employ to detect time-varying seasonal signal in GPS position time series with different noise levels? Do these methods artificially absorb some part of the power?
Global Positioning System (GPS) position time series contain seasonal signals. Among the others, annual and semi-annual are the most powerful. Widely, these oscillations are modelled as curves with constant amplitudes, using the Weighted Least-Squares (WLS) algorithm. However, in reality, the seasonal signatures vary over time, as their geophysical causes are not constant. Different algorithms have been already used to cover this time-variability, as Wavelet Decomposition (WD), Singular Spectrum Analysis (SSA), Chebyshev Polynomial (CP) or Kalman Filter (KF). In this research, we employed 376 globally distributed GPS stations which time series contributed to the newest International Terrestrial Reference Frame (ITRF2014). We show that for c.a. 20% of stations the amplitudes of seasonal signal varies over time of more than 1.0 mm. Then, we compare the WD, SSA, CP and KF algorithms for a set of synthetic time series to quantify them under different noise conditions. We show that when variations of seasonal signals are ignored, the power-law character is biased towards flicker noise. The most reliable estimates of the variations were found to be given by SSA and KF. These methods also perform the best for other noise levels while WD, and to a lesser extend also CP, have trouble in separating the seasonal signal from the noise which leads to an underestimation in the spectral index of power-law noise of around 0.1. For real ITRF2014 GPS data we discovered, that SSA and KF are capable to model 49-84% and 77-90% of the variance of the true varying seasonal signals, respectively.
The coordinate time series determined with the Global Positioning System (GPS) contain annual and semi-annual periods that are routinely modeled by two periodic signals with constant amplitude and phase-lag. However, the amplitude and phase-lag of the seasonal signals vary slightly over time. Various methods have been proposed to model these variations such as Wavelet Decomposition (WD), writing the amplitude of the seasonal signal as a Chebyshev polynomial that is a function of time (CP), Singular Spectrum Analysis (SSA), and using a Kalman Filter (KF). Using synthetic time series, we investigate the ability of each method to capture the time-varying seasonal signal in time series with different noise levels. We demonstrate that the precision by which the varying seasonal signal can be estimated depends on the ratio of the variations in the seasonal signal to the noise level. For most GPS time series, this ratio is between 0.05 and 0.1. Within this range, the WD and CP have the most trouble in separating the seasonal signal from the noise. The most precise estimates of the variations are given by the SSA and KF methods. For real GPS data, SSA and KF can model 49-84% and 77-90% of the variance of the true varying seasonal signal, respectively.