The Stochastic Characterization of Noise and Signals in GNSS Coordinate Time Series: Insights from Over 10 Years of Stud

1. The stochastic characterization of noise and signals in GNSS coordinate time series : What do we know after 10+ years of study? Simon Williams (Proudman Oceanographic Laboratory, UK) Matt King (University of Newcastle, UK) John Langbein (United States Geological Survey)

2. 10+ years? • Sufficiently long CGPS time series • Publications : • Zhang et al [1997] • Mao et al [1999] • King et al [1995] • Previously : Short- and Long- term precision estimates

3. Assumptions for (campaign) GPS data [Zhang et al., 1997] • Rate of change of position is linear in time • Position estimates are statistically uncorrelated • Difficult to achieve in practice. GPS antenna always recentered and releveled over the mark for each measurement period • Multipath • Signal scattering • Antenna phase center variations • Geodetic monuments are stable Still applicable for Continuous Measurements!

5. Techniques • Maximum Likelihood estimation • Alan Variance • Power Spectrum • Ad-hoc and heuristic methods • Wavelets • Found temporally correlated noise but NOT random-walk • Flicker noise plus white noise was more appropriate (from MLE study) • Could not rule out RW at the level seen in other geodetic techniques • Hint of some latitude dependence in the noise amplitudes (particularly white noise)

6. Five years later…Five years more data • Used a more general power-law noise model in MLE • Confirmed flicker noise stochastic signal • Latitude dependence evident • Decrease in amplitude with time • Peaks at annual, semi-annual and 13.66 days • Monument stability analysis

7. Monument Stability However : Beavan [2005] Studying concrete pillar geodetic monuments in New Zealand found no discernable difference in noise between them and the deep drilled braced monuments in U.S. Ranking : • Deep Braced • Roof/Chimney • Metal Tripod • Rock Pin • Steel Tower • Concrete Slab • Concrete Pier • Oil Platform Increase in noise level

8. Monument Studies revisited courtesy of John Langbein [2008]

9. The location of the GPS site is probably the biggest predictor of its stability; sites located in the dry deserts will perform best. Sites located near areas of active pumping will perform worst.

10. A Few more years later…… • 800+ sites with 500+ epochs (cf. 268 in last JPL solution) • Langbein [2004] introduced more noise models : FOGM, Band Pass • Development of Fast Error Analysis [Bos et al. 2008] • “Discovery” of anomalous harmonics relating to the GPS year [Ray et al. 2007; Amiri-Simkooei et al 2007]

11. Power-Law vs FOGM • Langbein [2008] found • Half as flicker or random walk noise • Rest a combination of : • Flicker plus random walk • Power-law • FOGM plus random walk • Power-law plus broadband seasonal

12. Effect of GPS year Harmonics

13. Formal Uncertainties and Latitude Dependence • Little latitude dependence in flicker noise amplitudes • Using formal uncertainties generally increases the MLE log-likelihood • 99% vertical • 97% east • 66% north • Formal uncertainties too big to represent white noise component

14. Baseline Studies : Pin1 Pin2

15. 10 Long Running Short Baselines courtesy Matt King • 30s solutions • 7o and 20o elevation cut-off angle solution • Linear trends > 0.25 mm/yr @ 5 baselines

16. Annual > 0.5 mm @ 6 sites Sub daily signals at all sites. Increasing elevation angle decreases magnitude Simulations : Not all annual signal could be accounted for by multipath.

17. Thermal expansion? Linear thermal expansion model only explains large portion of the annual signal at one site METS-METZ Reasonable correlation with temperature at some sites

18. Spectral Indices ranged from -0.7 to -2.0 • Average somewhere between -1 and -2 • Power is up to two orders of magnitude smaller than single site power-spectra • Top baselines : • BOGO-BOGI • HERS-HERT • PIN1-PIN2 • Worst baselines : • JOZE-JOZ2 • ZIMM-ZIMJ • OHI2-OHI3

19. Monument Instability Conclusions • Either we are not seeing RW noise in the series yet or the estimated indices are currently biased low. Power law spectral indices are high for flicker noise but low for random walk • Assuming monument stability is characterized by random walk noise (as seen in other geodetic datasets) then : • The good monument pairs are more stable (deeper bracing) than most two-color EDM • Random walk seen in the EDM data is not due to monument stability but due to another noise source in the two-color technique • The site conditions/geology are better at good monument sites (true for Parkfied, Pearblossom?) • There is a spatial correlation between geodetic monuments over a distance of 50m (10 m short baseline strain data have random walk in the ranges between 0.1 and 0.2 mm/yr1/2)