Non-parametric estimation of seasonal variations in GNSS-derived time series

Military University of Technology, Poland (marta.gruszczynska@wat.edu.pl) Seasonal variations in the frame sites can bias the frame realization. I would like to invite you to click on each of the four steps outlined on this slide in order to obtain more information on the particular subject. Figures can be enlarged by clicking on that figure.

The details concerning the processing of the GNSS observations can be found at DATA https://gipsy-oasis.jpl.nasa.gov. DATA We analysed 294 globally distributed IGS stations with the minimum data length of 6.5 years. The longest (DRAO, GRAZ, STJO, YELL) were even 22 years long (1992-2014). We used position obtained by the Jet Propulsion Laboratory (JPL) using the GIPSY- OASIS software in a Precise Point Positioning (PPP) mode.

DATA Daily position changes from PPP (Precise Point Positioning) solution obtained by JPL (Jet Propulsion Laboratory) processed in the GIPSY-OASIS software. DATA Daily position changes from PPP (Precise Point Positioning) solution obtained by JPL (Jet Propulsion Laboratory) processed in the GIPSY-OASIS software.

DATA The seasonal variations in GNSS station s position may arise from geophysical excitations, thermal changes combined with hydrodynamics or various errors which, when superimposed, cause the seasonal oscillations. It are not exactly of real geodynamical origin, but still have to be included in modelling. These variations with different periods included in frequency band from Chandler up to quarter-annual ones will all affect the reliability of permanent station s velocity, which in turn, strictly influences the quality of kinematic reference frames. The annual (dominant) sine curve, has the amplitude and phase that both change in time due to different reasons. In this research we focused on the determination of annual changes in GNSS-derived of North, East and Up components.

DATA The figure shows a Lomb-Scargle periodogram of the GLSV station s.

DATA x( t) x n m 0 vx t i i i x j j x t i 1 j 1 off A sin( t ) O p x ( ) The potential contributors to seasonal variations can be grouped into: 1. gravitation excitation (Sun and Moon) 2. impact of environment: hydrodynamics, atmosphere, antenna thermal changes 3. system errors: satellite orbital models, phase center variation models, local multipath...

DATA The outliers from were removed by means of the median absolute deviation criterion (MAD). The offsets were removed by STARS algoritm (Sequential t-test Analusis of Regime Shifts), which is a combianation of the t-student and the standard deviation test. The linear trend was removed from the using least squares estimation (LSE). Figure. Blue curve represents the without outlieres, offsets and linear trend for the North, East and Up component of BRAZ station.

Non-parametric estimation of DATA The linear trend was removed from the time series using least squares estimation (LSE).

STACKED Each of the topocentric (North, East and Up) was divided into years (from January to December), then the observations gathered in the same days of year were stacked and the weighted medians obtained for all of them such that each of was represented by matrix of size 365xn where n is the data length. Figure. Blue dots represent the stacked data.

STACKED We obtainted the weighted median for each day of year with their WMAD (the weighted median absolute deviation). Figure right. Blue dots represent the stacked data and black curve represents the daily medians. Figure left. Blue curve with red dots represents the computed weighted median for each day of year. Grey lines represent the weighted median absolute deviation.

Non-parametric estimation of

Wavelet decomposition The wavelet transform allows us to decompose the original time series into a number of new, each with a different degree of resolution. We used a symmetric and orthogonal Meyer s wavelet for this task, because it is compact in the frequency domain. Figures. The Meyer wavelet decomposition frequency levels for the North, East and Up component of the BRAZ station. Red curve represents an original signal, green curve represents a wavelet quasi-annual approximation of 7 levels and blue curves represent the detail levels D1 - D7, respectively.

Wavelet decomposition The wavelet transform provides a framework to decompose original into a number of new, each one of them with a different degree of resolutions.

Wavelet decomposition Transformation into different frequency bands was done using wavelet decomposition with Meyer wavelet. We assumed here 7 levels of decomposition, with annual curve as the last approximation of it. The signal approximations made us obtain the seasonal peaks that prevail in North, East and Up data for globally distributed stations. Figure. Blue dots represent the stacked data, black curve represents the daily medians; red curve represents a wavelet quasi-annual approximation.

Non-parametric estimation of Wavelet decomposition

CLUSTERING The vast majority of stations is characterized by amplitudes of 2 to 4 mm. The maximum vertical amplitude was noticed to be at the level of 9 mm with the minimum of it equal to -9 mm, what gives the position change of 18 mm when peak-to-peak changes are considered. The amplitudes and phases of quasi-annual curves in the vertical direction. The length of the arrow means the value of the amplitude, while the azimuth-like angle stays for phase of the quasiannual maximum.

CLUSTERING The analysis of annual curves (approximations), led to divide the stations into clusters for which the similar signal were noticed. The amplitudes and phases of quasi-annual curves in the vertical direction. The length of the arrow means the value of the amplitude, while the azimuth-like angle stays for phase of the quasi-annual maximum.

The analysis of annual curves, by means of non-parametric estimation of amplitudes CLUSTERING and phases, led us to divide the IGS stations into different clusters for which the similar signals were noticed. We define a parameter, = 30 days, which specifies the maximum acceptable phase difference for stations classified within a cluster. The next parameter, Y, is defined as the maximum distance between any two stations within a cluster (e.g. Europe Y=2000 km). We obtained 36 clusters for the Up component. The division of IGS stations into a few different clusters led us to obtain the mean seasonal signal for different regions of the world. The quasi-annual signals were then averaged within each cluster and the average annual signal was revealed.

CLUSTERING

CLUSTERING Arctic All stations were grouped into 9 sets North America Europe Asia of clusters in order to present the results better. This division is related to the location of continents (e.g. Pacific Ocean South America Africa Indian Ocean & Australia Africa, North America), the ocean (Pacific Ocean) and the region Antarctica (Indian Ocean and Australia). The division the IGS stations into different clusters was presented for each of the group. The results can be viewed by clicking on the yellow rectangle with the set of cluster name.

CLUSTERING North America Arctic Europe Asia Pacific Ocean South America Africa Indian Ocean & Australia Antarctica The analysis of annual curves (approximations), led to divide the stations into clusters for which the similar signal were noticed. The results can be viewed by clicking on the yellow rectangle with the set of cluster name.

ARCTIC The stations in the Arctic were divided into two clusters: AR1 and AR2. Both of them have the maximum of annual signal in September. These stations can be characterized by vertical changes of 1-3 mm with their maximum in autumn.

NORTH AMERICA Military University of Technology, Poland The stations (mgruszczynska@wat.edu.pl) in the North America were divided into six clusters: NA1 to NA6. For the majority of the clusters, the annual signal in the vertical coordinates was observed to have its maximum in autumn (NA1, NA2, NA4, NA5). The cluster NA3 has the maximum in summer. The cluster NA6 has the maximum in spring and includes the station situated the closest to the ocean.

NORTH AMERICA

The stations in the South America were divided into two clusters: SA1 and SA2. We can Military University SOUTH of Technology, Poland (mgruszczynska@wat.edu.pl) observe the 6-month twist in phase from South to North in the annual signals. The AMERICA cluster SA1 has the maximum in September and the cluster SA2 has in March. It is worth noting that stations situated in the South America (especially ANTC, BRAZ) have the greatest annual amplitudes when compared with the rest of the stations.

SOUTH AMERICA

The stations in the Antarctica were divided into two clusters: AN1 and AN2. The ANTARCTICA cluster AN1 has the maximum in autumn and the cluster AN2 has the maximum in winter. These stations are characterized by amplitudes of 2-4 mm.

SOUTH The annual curves in vertical direction for different clusters.

The stations in the Europe were divided into eight clusters: E1 to E8. For the majority of the clusters, the annual signal in the vertical coordinates has a minimum in Spring EUROPE and a maximum in Autumn. The clusters differ in terms of their quasi-annual amplitude. The clusters E2 and E7 have a clear maximum in Spring and include the stations situated the closest to the ocean. It is worth noting that stations situated in the eastern Europe (especially clusters E5 and E8) have the greatest annual amplitudes when compared with the rest of the clusters.

EUROPE

PACIFIC OCEAN Military University of Technology, Poland The (mgruszczynska@wat.edu.pl) stations in the Pacific Ocean were divided into two clusters: P1 and P2. Both of them have the maximum of annual signal in April. These stations are characterized by mean annual amplitude of 2.5 mm.

PACIFIC OCEAN

ASIA The stations in the Asia were divided into six clusters: AS1 to AS6. All of the Asian Military University of Technology, clusters Poland have (mgruszczynska@wat.edu.pl) significantly different average mean annual signals in the vertical coordinates, whereby the mainly continental clusters AS1 - AS4 have the maximum in Autumn. For the majority of the clusters in the east Asia, the annual signal has its maximum in Spring (AS5, AS6). These stations are situated the closest to the ocean and have the greatest annual amplitudes when compared with the rest of the stations.

AFRICA The stations in the Africa were divided into two clusters: AF1 and AF2. Both of them have the maximum of annual signal in April. These stations are characterized by vertical amplitudes of 1-4 mm.

AFRICA

The stations in the Indian Ocean and the Australia were divided into six clusters. For the majority of the clusters, the annual signal in the vertical coordinates has its INDIAN OCEAN & AUSTRALIA maximum in Autumn (AU2, AU3, AU4, AU5 and IO1). These stations are characterized by vertical changes of 1-2 mm. The cluster AU1 on the north coast of Australia does not have its maximum annual signal in April or May, as the other stations, only twisted 6-months (in Spring).

Indian Ocean & Australia

CLUSTERING Our results show that the station s location has the impact on annual curve character and can be related to ocean s vicinity or climate changes. Here, the greatest seasonal amplitudes were noticed for Up component what may arise from atmospheric and hydrospheric influences. The reason of wavelet decomposition usage is the fact that the majority of determined seasonal curves is far from being a sine wave. The division of IGS stations into a few different clusters led us to obtain the mean seasonal signal for different regions of the world.

This research was financed by the Faculty of Civil Engineering and Geodesy of the Military University of Technology statutory research funds.