1. Wiesław Kosek 1,2, Agnieszka Wnęk 1, Maria Zbylut 1, Waldemar Popiński 3 1) Environmental Engineering and Land Surveying Department, University of Agriculture in Krakow, Poland 2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland 3) Central Statistical Office of Poland, Warsaw, Poland 17th International Symposium on Earth Tides „Understand the Earth” 15-19 April, 2013, Warsaw, Poland Polarization of signals in Earth centre of mass time series observed by satellite techniques

2. The accuracy of the ITRF geocenter has a significant impact on the orbit determination and station coordinates accuracy centre of figure centre of mass ITRF GGOS Requirement (2020): <1 mm TRF accuracy < 0.1 mm/yr TRF stability

3. DATA SLR weekly geocenter data computed by Astronomical Institute of the University of Bern in 1994 – 2013. (Sosnica et al.. 2011) http://www.bernese.unibe.ch/publist/2011/pres/ks_Geod_Woch e.pdf GNSS weekly combined solutions delivered by International GNSS Service (IGS) in years 1994 – 2013. ftp://igs-rf.ign.fr/pub/sum/5-4_igs.sum DORIS geocenter time series available at Crustal Dynamics Data Information System (CDDIS) from 1994.0 to 2013 (Willis et al., 2005) ftp://cddis.gsfc.nasa.gov/pub/doris/products/geoc/

4. DORIS SLR GNSS CoM data Standard deviations [mm] SLRx3.4 y3.3 z6.2 GNSSx3.0 y4.4 z5.7 DORISx5.5 y6.9 z24.8

5. WAVELET SPECTRO-TEMPORAL SEMBLANCE, between and, complex-valued time series is defined for as: where: - spectro-temporal coherence, - spectro-temporal phase synchronization, - the wavelet spectra and the wavelet cross spectrum of time series The spectro-temporal semblance of the order - dilation and translation parameters.

6. The wavelet transform coefficients of complex-valued time series are defined: - Discrete Fourier Transforms of and time series, - Continuous Fourier Transform of the modified Morlet wavelet function given by the time domain formula (Schmitz-Hübsch and Schuh 1999): where

7. The wavelet polarization and the mean wavelet polarization functions of complex- valued time series are defined as: retrograde prograde ellipticcircular the shape of ellipse degenerates to a line - the wavelet spectrum and the mean wavelet spectrum WAVELET POLARIZATION

8. The mean wavelet spectra of Earth centre of mass complex-valued time series computed from SLR and GNSS observations

9. The mean wavelet polarization functions in XY, YZ and ZX planes of complex- valued Earth centre of mass time series determined by satellite techniques XY YZ ZX SLRGNSSDORIS

10. Spectro-temporal wavelet polarization functions in XY plane of Earth centre of mass time series determined by SLR, GNSS and DORIS techniques

11. Spectro-temporal wavelet polarization functions in YZ plane of Earth centre of mass time series determined by SLR, GNSS and DORIS techniques

12. Wavelet polarization functions in ZX plane of Earth centre of mass time series determined by SLR, GNSS and DORIS techniques

13. The mean semblance functions in XY, YZ and ZX planes between Earth centre of mass time series determined by different techniques SLR - GNSS GNSS - DORIS SLR - DORIS

14. Spectro-temporal semblance functions in XY equatorial plane between Earth centre of mass time series determined by different techniques

15. Spectro-temporal semblance in YZ plane between Earth centre of mass time series determined by different techniques

16. Spectro-temporal semblance in ZX plane between Earth centre of mass time series determined by different techniques

17. THE WAVELET BASED SEMBLANCE FILTERING DWT wavelet semblance filtering Thresholding of WT coefficients WT coefficients

18. The common oscillations in Earth centre of mass time series computed by the wavelet semblance filtering assuming threshold equal to 0.9

19. The model center of mass time series computed as the average of GNSS and SLR common oscillations composed of only 6 lower frequency components

20. Conclusions The most energetic oscillation in Earth centre of mass time series determined by SLR, GNSS and DORIS techniques is the annual one with amplitude of the order of few millimeters. The spectro-temporal wavelet semblance with application of the modified Morlet wavelet function enables computation of correlation coefficients between two complex-valued time series as a function of time and frequency. The highest positive semblance values occur in the equatorial xy plane for the retrograde annual oscillation in the GNSS and SLR data. The semblance functions between the GNSS and DORIS as well as the SLR -DORIS geocenter data in the annual frequency band are negative in the equatorial XY plane data. The wavelet based semblance filtering with application of the Shannon wavelet function enables computation of a common signal in GNSS and SLR geocenter time series. This common signal enables determination of the smoothed model geocenter time series as the average of the GNSS and SLR time series reconstructed using lower frequency components.