Janusz Bogusz

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.

GPS-derived height changes in diurnal and sub-diurnal timescales

This paper describes the research concerning precise short-time GPS solutions conducted in the Centre of Applied Geomatics, Military University of Technology, Warsaw, Poland. The data from ASG-EUPOS (Polish Active Geodetic Network) was processed using Bernese 5.0 software and EPN (EUREF Permanent Network) standards and models. In this study, the adapted 3-hour observation window is shifted every hour for obtaining hourly geocentric coordinates in ITRF2005 reference frame. The adjusted network consisted of over 130 stations from Poland and the neighbouring countries, the period covered observations collected from June 2008 through June 2010. These two years of observations allowed for examining short-period oscillations which were found to be closely related to the tidal (dynamic) frequencies. The analysis of the residua from the IERS2003 tidal model was performed using the least squares method with the Eterna software. It confirmed the existence of significant (several millimetres) oscillations in the frequencies corresponding to S1, K1 and K2. The paper describes the idea of data processing and analysis, presents the results of vertical (the Up component) oscillations in main tidal frequency bands, and also includes discussion of possible explanation of the existence of short-period oscillations in the GPS precise solutions and the possibility of propagation of short-period oscillations into long-period spurious changes in the daily (standard) GPS solutions.

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.

Investigation on Tidal Components in GPS Coordinates

Investigation on Tidal Components in GPS Coordinates This paper presents analyses on the GPS coordinates from sub-diurnal solutions of EPN data provided by Warsaw Military University of Technology. The aim of this research is to investigate the way the tidal models used in Bernese software (solid Earth and ocean tides as well) fit to the individual conditions of EPN stations. The 1-hour solution technique of GPS data processing was utilized to obtain coordinates of above 70 EPN stations. Additionally several Polish permanent sites with clearly seen oscillations were examined. This processing technique allowed us to recognize diurnal and sub-diurnal residual oscillations which could be next utilized for validation of the tidal models.

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.

Unmodelled Effects in the Horizontal Velocity Field Determination: ASG-EUPOS Case Study

Unmodelled Effects in the Horizontal Velocity Field Determination: ASG-EUPOS Case Study Recent space and satellite technologies offer great opportunities to determine crustal movements in the single, global reference frame. The densification of the global network through local area networks leads to increase the resolution of modelled deformations both horizontal and vertical. However, the credibility of the obtained velocity field is limited by several factors associated with unmodelled (or mismodelled) effects at the stage of GNSS data adjustment. Some of them are periodic (the influence of local atmospheric or hydrological impacts), some temporary (natural or anthropogenic seismicity) or related to local influences (snow load or effects associated with the freezing of the ground). This paper presents the usefulness of ASG-EUPOS time series for determination of the regional velocity field. The system has been operating since mid-2008, so the velocities obtained through the processing of 3-year time series are supposed to be reliable. The paper also presents comparison of the velocity determinations to the geological NNR-NUVEL-1A and geodetic APKIM2005 models.

SHORT-PERIOD INFORMATION IN GPS TIME SERIES

Short-Period Information in GPS Time Series This paper presents results of the Polish Active Geodetic Network (ASG-EUPOS) processing. The analyses on the GPS coordinates from sub-diurnal solutions of ASG-EUPOS and EPN data provided by Warsaw Military University of Technology were performed. The aim of this research is to find out how the tidal models used in Bernese software (solid Earth and ocean tides as well) fit to the individual conditions of GPS stations. The 1-hour solution technique of GPS data processing was utilized to obtain coordinates of above 130 Polish and foreign stations. This processing technique allowed us to recognize residual diurnal and sub-diurnal oscillations which could be next utilized for validation of the tidal models used in GPS software.

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.