Project

A novel approach to model the geodetic time series with Adaptive Wiener Filter and correlations in a stochastic part

Goal: Observations recorded by the Global Positioning System (GPS) permanent stations are currently an essential source of data about large-scale and local phenomena that occur on the Earth’s surface and in the atmosphere as well. During the re-processing of GPS data, we estimate the GPS position time series or the Zenith Total Delay (ZTD) time series. All series have to be modelled with attention paid to trend and seasonal signals with changing/constant amplitude. Stochastic part or residuals are recognized to be close to flicker noise for GPS series or to autoregressive process for ZTD data. Time series may be affected much by employed method. It may totally change a character of residuals from coloured noise into less correlated character of white noise. In this way, the errors of trend and amplitudes of seasonal signals will be underestimated and may lead to wrong interpretations. Most time series contain a seasonal signal in a form of annual and semi-annual period that are routinely modelled by two periodic signals with constant amplitudes using the Weighted Least-Squares (WLS). However, in reality the amplitude of these seasonal signals varies slightly over time and this should be taken into account. Although this variability reach few millimetres at maximum, it may introduce additional temporal correlation and will lead to overestimation of trend. The time-varying signals have been so far modelled with: Kalman Filter (KF), Chebyshev Polynomials (CP), Singular Spectrum Analysis (SSA) or Wavelet Decomposition (WD). Each of above-mentioned methods have been already marginally described for amount of power that is artificially removed from stochastic part (or noise) together with the seasonal signal. The proposed research is aimed at answering the question: how should we model seasonal signals to keep at the same time the stochastic properties of data intact. In this proposal, we will verify the main research hypothesis which reads as follows: “seasonal signals should be modelled as time-varying and removed from geodetic time series with no artificial loss in power”. Moreover, we are taking the working hypothesis that: “a combination of a modified Wiener Filter and a character of coloured noise allows to separate noise from a real geophysical signal that changes over time”. This hypothesis will be verified by 1) comparison of commonly used approaches to model the seasonal part assuming its constancy and variability in time (WLS, KF, SSA & WD) and by 2) employing an innovative approach, which we will introduce within project, based on Adaptive Wiener Filter (AWF). In the following project, we assume that a way which AWF is being constructed must provide the optimum separation between real geophysical signal from observational noise. In this turn, no artificial loss in power and no artificial underestimation of trends and their uncertainties will be assured.

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Project log

Anna Klos
added a research item
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.
Anna Klos
added 3 research items
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.
Anna Klos
added a project goal
Observations recorded by the Global Positioning System (GPS) permanent stations are currently an essential source of data about large-scale and local phenomena that occur on the Earth’s surface and in the atmosphere as well. During the re-processing of GPS data, we estimate the GPS position time series or the Zenith Total Delay (ZTD) time series. All series have to be modelled with attention paid to trend and seasonal signals with changing/constant amplitude. Stochastic part or residuals are recognized to be close to flicker noise for GPS series or to autoregressive process for ZTD data. Time series may be affected much by employed method. It may totally change a character of residuals from coloured noise into less correlated character of white noise. In this way, the errors of trend and amplitudes of seasonal signals will be underestimated and may lead to wrong interpretations. Most time series contain a seasonal signal in a form of annual and semi-annual period that are routinely modelled by two periodic signals with constant amplitudes using the Weighted Least-Squares (WLS). However, in reality the amplitude of these seasonal signals varies slightly over time and this should be taken into account. Although this variability reach few millimetres at maximum, it may introduce additional temporal correlation and will lead to overestimation of trend. The time-varying signals have been so far modelled with: Kalman Filter (KF), Chebyshev Polynomials (CP), Singular Spectrum Analysis (SSA) or Wavelet Decomposition (WD). Each of above-mentioned methods have been already marginally described for amount of power that is artificially removed from stochastic part (or noise) together with the seasonal signal. The proposed research is aimed at answering the question: how should we model seasonal signals to keep at the same time the stochastic properties of data intact. In this proposal, we will verify the main research hypothesis which reads as follows: “seasonal signals should be modelled as time-varying and removed from geodetic time series with no artificial loss in power”. Moreover, we are taking the working hypothesis that: “a combination of a modified Wiener Filter and a character of coloured noise allows to separate noise from a real geophysical signal that changes over time”. This hypothesis will be verified by 1) comparison of commonly used approaches to model the seasonal part assuming its constancy and variability in time (WLS, KF, SSA & WD) and by 2) employing an innovative approach, which we will introduce within project, based on Adaptive Wiener Filter (AWF). In the following project, we assume that a way which AWF is being constructed must provide the optimum separation between real geophysical signal from observational noise. In this turn, no artificial loss in power and no artificial underestimation of trends and their uncertainties will be assured.