Anna Klos

On the Handling of Outliers in the GNSS Time Series by Means of the Noise and Probability Analysis

The data pre-analysis plays a significant role in the noise determination. The most important issue is to find an optimum criterion for outliers removal, since their existence can affect any further analysis. The noises in the GNSS time series are characterized by spectral index and amplitudes that can be determined with a few different methods. In this research, the Maximum Likelihood Estimation (MLE) was used. The noise amplitudes as well as spectral indices were obtained for the topocentric coordinates with daily changes from few selected EPN (EUREF Permanent Network) stations. The data were obtained within the EPN re-processing made by the Military University of Technology Local Analysis Centre (MUT LAC). The outliers were removed from the most noisy 12 EPN stations with the criteria of 3 and 5 times the standard deviations (3σ, 5σ) as well as Median Absolute Deviation (MAD) to investigate how they affect noise parameters. The results show that the removal of outliers is necessary before any further analysis, otherwise one may obtain quite odd and unrealistic values. The probability analysis with skewness and kurtosis was also performed beyond the noise analysis. The values of skewness and kurtosis show that assuming a wrong criterion of outliers removal leads to the wrong results in case of probability distribution. On the basis of the results, we propose to use the MAD method for the outliers removal in the GNSS time series.

Error analysis for European IGS stations

Each of the GPS time series that describes the changes of topocentric components consists of a deterministic and a stochastic part, whose character influences the errors of the deterministic parameters. As to the uncertainties of reliable velocities of permanent satellite station systems, surveys that estimate and take into account any dependencies that may affect subsequent operational efficiency are very important. For this analysis, we used 42 stations from the IGS (International GNSS Service) network from Europe, processed at the Military University of Technology EUREF Permanent Network Local Analysis Centre (MUT LAC). The deterministic part of the GPS time series was removed using the least squares method. The seasonal periods in topocentric components were determined assuming the existence of the residual Chandler oscillation (1.67 cpy), as well as the annual tropical (1 cpy) and draconitic (1.04 cpy) oscillations with their harmonics up to 4th. We assumed the character of the residue as a combination of white and powerlaw noise. The obtained results show, that in the case of the European sub-network of IGS stations we are dealing with the coloured noise between white and flicker noise with the amplitudes between 3 to 6 mm/year-k/4 for horizontal components and between 6 to 15 mm/year-κ/4 for the vertical ones, where κ is a spectral index. Finally, we showed that the amplitudes and spectral indices of noise are reduced after performing a spatio-temporal filtering. All the elicited results referred to the uncertainties of velocities by estimating them before and after filtration and the simulation of their values for different lengths of the time series.

Modelling the Velocity Field in a Regular Grid in the Area of Poland on the Basis of the Velocities of European Permanent Stations

The paper presents the results of testing the various methods of permanent stations’ velocity residua interpolation in a regular grid, which constitutes a continuous model of the velocity field in the territory of Poland. Three packages of software were used in the research from the point of view of interpolation: GMT (The Generic Mapping Tools), Surfer and ArcGIS. The following methods were tested in the softwares: the Nearest Neighbor, Triangulation (TIN), Spline Interpolation, Surface, Inverse Distance to a Power, Minimum Curvature and Kriging. The presented research used the absolute velocities’ values expressed in the ITRF2005 reference frame and the intraplate velocities related to the NUVEL model of over 300 permanent reference stations of the EPN and ASG-EUPOS networks covering the area of Europe. Interpolation for the area of Poland was done using data from the whole area of Europe to make the results at the borders of the interpolation area reliable. As a result of this research, an optimum method of such data interpolation was developed. All the mentioned methods were tested for being local or global, for the possibility to compute errors of the interpolated values, for explicitness and fidelity of the interpolation functions or the smoothing mode. In the authors’ opinion, the best data interpolation method is Kriging with the linear semivariogram model run in the Surfer programme because it allows for the computation of errors in the interpolated values and it is a global method (it distorts the results in the least way). Alternately, it is acceptable to use the Minimum Curvature method. Empirical analysis of the interpolation results obtained by means of the two methods showed that the results are identical. The tests were conducted using the intraplate velocities of the European sites. Statistics in the form of computing the minimum, maximum and mean values of the interpolated North and East components of the velocity residuum were prepared for all the tested methods, and each of the resulting continuous velocity fields was visualized by means of the GMT programme. The interpolated components of the velocities and their residua are presented in the form of tables and bar diagrams.

Irregular variations in GPS time series by probability and noise analysis

Abstract The character of the topocentric components in ETRF2000(R08) from the Polish ASG-EUPOS system was analysed using skewness and kurtosis derived from the data probability density function. The data from 115 permanent GPS stations with a time span of more than 5 years were used. The main goal of this research was to show that any unmodelled systematics can disrupt the results. The obtained median values of skewness and kurtosis clearly indicate the discrepancies between the assumed normality of the GPS time series distribution and the reality, mainly due to stochastic and/or deterministic parts that are still present in the data. The quadratic relationship between skewness and kurtosis was developed with the empirically determined constants. The noise analysis with the Maximum Likelihood Estimation was also performed with the assumptions of white and white plus power-law noise. The estimated spectral indices for power law noises are close to −1 (flicker noise). The uncertainties of the intraplate velocities with white and white plus power law noise assumptions were calculated. It was shown that these uncertainties can be underestimated up to 5 mm/year. Additionally we made 1000 simulation which were aimed at showing how the values of skewness and kurtosis are changed when some mismodelled part of the data remains in the time series. We assumed different values of trend (1 and 6 mm/year), seasonal amplitudes (1 and 6 mm) and offsets (3 and 10 mm) with flicker noise of 1 mm amplitude to see how the metrics we use are biased.

Non-parametric Estimation of Seasonal Variations in GPS-Derived Time Series

In this research, we focus on determining the quasi-annual changes in GNSS-derived 3-dimensional time series. We use the daily time series from PPP solution obtained by JPL (Jet Propulsion Laboratory) from more than 300 globally distributed IGS stations. Each of the topocentric time series were stacked into data sets according to year (from January to December) and then decomposed and approximated with a Meyer wavelet. This approach allowed investigating changes of the amplitudes in time. An observed quasi-annual signal for a set of European stations prompted us to divide the stations into different sub-networks called clusters. For Up component seven clusters were established. The signals were then averaged within each cluster and median quasi-annual signal was revealed. The vast majority of the GNSS time series is characterized by vertical changes of 3 mm with their maximum in Summer. The maximum vertical amplitude was at the level of 14 mm with the minimum equal to −13 mm, giving the peak-to-peak position changes up to 27 mm.

Detecting time-varying seasonal signal in GPS position time series with different noise levels

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.

Noise Analysis of Continuous GPS Time Series of Selected EPN Stations to Investigate Variations in Stability of Monument Types

The type of monument that a GPS antenna is placed on plays a significant role in noise estimation for each permanent GPS station. In this research 18 Polish permanent GPS stations that belong to the EPN (EUREF Permanent Network) were analyzed using Maximum Likelihood Estimation (MLE). The antennae of Polish EPN stations are placed on roofs of buildings or on concrete pillars. The analyzed data covers a period of 5 years from 2008 to 2013. The analysis was made on the daily topocentric coordinate changes. Firstly, the existence of the combination of white noise, flicker noise and random-walk on each of the stations was set up before the analysis, secondly – a random-walk plus white noise model was assumed, because monument instability is thought to follow random-walk. The first combination of noises did not yield any conclusions about stability of monuments, probably because of the domination of flicker noise in the time series. The second one, even if not quite correct – noises in GPS time series do not strictly reflect random-walk only-showed that concrete pillars perform better than buildings for GPS antenna locations. Unfortunately, on the basis of this it cannot be clearly stated whether they are better as monuments or not. Moreover, the stacked Power Spectral Densities (PSDs) of topocentric coordinates were obtained with Fast Fourier Transform (FFT) for each monument type. Even though stacked spectra are quite similar and do not really show any differences, PSDs made for certain station are more varied.

Stochastic models in the DORIS position time series: estimates for IDS contribution to ITRF2014

This paper focuses on the investigation of the deterministic and stochastic parts of the Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) weekly time series aligned to the newest release of ITRF2014. A set of 90 stations was divided into three groups depending on when the data were collected at an individual station. To reliably describe the DORIS time series, we employed a mathematical model that included the long-term nonlinear signal, linear trend, seasonal oscillations and a stochastic part, all being estimated with maximum likelihood estimation. We proved that the values of the parameters delivered for DORIS data are strictly correlated with the time span of the observations. The quality of the most recent data has significantly improved. Not only did the seasonal amplitudes decrease over the years, but also, and most importantly, the noise level and its type changed significantly. Among several tested models, the power-law process may be chosen as the preferred one for most of the DORIS data. Moreover, the preferred noise model has changed through the years from an autoregressive process to pure power-law noise with few stations characterised by a positive spectral index. For the latest observations, the medians of the velocity errors were equal to 0.3, 0.3 and 0.4 mm/year, respectively, for the North, East and Up components. In the best cases, a velocity uncertainty of DORIS sites of 0.1 mm/year is achievable when the appropriate coloured noise model is taken into consideration.

Investigation of long-range dependencies in the stochastic part of daily GPS solutions

The long-range dependence (LRD) of the stochastic part of GPS-derived topocentric coordinates change (North, East, Up) results with relatively high autocorrelation values with a focus on self-similarity. One of the reasons for such self-similarity in the GPS time series are noises that are commonly recognised to prevail in the form of the flicker noise model. To prove the self-similarity of the stochastic part of GPS time series we used more than 130 ASG-EUPOS (active geodetic network EUPOS) stations from an area of Poland with a 5-year span of the daily topocentric coordinate changes. The deterministic part of time series was removed by means of the least-squares (LS) method, median absolute deviation (MAD) criterion and the sequential t-test algorithm, respectively. Then the self-similarity of the residue was proved by the results of the Ljung–Box test, whose values close to zero showed the dependence of the stochastic part of the GPS time series. The residue was analysed by means of the rescaled range (R/S) method with the H (Hurst) parameter and the detrended fluctuation analysis (DFA) with the scaling exponent α. Both H and α values ranged within assumed LRD limits of 0.5 and 1. This analysis was followed by noise investigation with a maximum likelihood estimation (MLE). The white plus the power-law noise models were assumed a priori, which gave us a spectral indices κ between − 0.4 and − 1.2 for all of the time series. It proved that fractional white noise outweighs other types of noises in GPS time series. Authors found here, that the LRD methods by omitting the noise amplitude data led to an underestimation of H values and their misinterpretation. The larger the omitted amplitude is, the greater the difference between the noise characters estimated with R/S values in comparison to the reference values of κ are. Some of these differences exceed even the value of 0.6, which may result in the estimation of false noise character in GPS data thus eliciting wrong conclusions.

Noise-Dependent Adaption of the Wiener Filter for the GPS Position Time Series

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 the WF 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.