GPS Solutions

, Volume 12, Issue 1, pp 55–64 | Cite as

Anomalous harmonics in the spectra of GPS position estimates

  • J. RayEmail author
  • Z. Altamimi
  • X. Collilieux
  • T.  van  Dam
Original Article


Prior studies of the power spectra of GPS position time series have found pervasive seasonal signals against a power-law background of flicker noise plus white noise. Dong et al. (2002) estimated that less than half the observed GPS seasonal power can be explained by redistributions of geophysical fluid mass loads. Much of the residual variation is probably caused by unidentified GPS technique errors and analysis artifacts. Among possible mechanisms, Penna and Stewart (2003) have shown how unmodeled analysis errors at tidal frequencies (near 12- and 24-hour periods) can be aliased to longer periods very efficiently. Signals near fortnightly, semiannual, and annual periods are expected to be most seriously affected. We have examined spectra for the 167 sites of the International GNSS (Global Navigation Satellite Systems) Service (IGS) network having more than 200 weekly measurements during 1996.0–2006.0. The non-linear residuals of the weekly IGS solutions that were included in ITRF2005, the latest version of the International Terrestrial Reference Frame (ITRF), have been used. To improve the detection of common-mode signals, the normalized spectra of all sites have been stacked, then boxcar smoothed for each local north (N), east (E), and height (H) component. The stacked, smoothed spectra are very similar for all three components. Peaks are evident at harmonics of about 1 cycle per year (cpy) up to at least 6 cpy, but the peaks are not all at strictly 1.0 cpy intervals. Based on the 6th harmonic of the N spectrum, which is among the sharpest and largest, and assuming a linear overtone model, then a common fundamental of 1.040 ± 0.008 cpy can explain all peaks well, together with the expected annual and semiannual signals. A flicker noise power-law continuum describes the background spectrum down to periods of a few months, after which the residuals become whiter. Similar sub-seasonal tones are not apparent in the residuals of available satellite laser ranging (SLR) and very long baseline interferometry (VLBI) sites, which are both an order of magnitude less numerous and dominated by white noise. There is weak evidence for a few isolated peaks near 1 cpy harmonics in the spectra of geophysical loadings, but these are much noisier than for GPS positions. Alternative explanations related to the GPS technique are suggested by the close coincidence of the period of the 1.040 cpy frequency, about 351.2 days, to the “GPS year”; i.e., the interval required for the constellation to repeat its inertial orientation with respect to the sun. This could indicate that the harmonics are a type of systematic error related to the satellite orbits. Mechanisms could involve orbit modeling defects or aliasing of site-dependent positioning biases modulated by the varying satellite geometry.


GPS Positions Reference frames VLBI SLR Geophysical surface loads 



Data from the IGS, IVS, and the ILRS have been indispensable. The suggestion of Urs Hugentobler (Technical University of Munich) concerning the GPS constellation repeat period of ∼350 days is greatly appreciated. We also thank Volker Tesmer (Deutsches Geodätisches Forschungsinstitut) and Sarah Böckmann (University of Bonn) for helpful discussions regarding IVS results. The constructive comments of Danan Dong (Jet Propulsion Laboratory), two anonymous reviewers, and Jake Griffiths (National Geodetic Survey) improved our presentation.


  1. Agnew DC, Larson KM (2007) Finding the repeat times of the GPS constellation. GPS Solutions 11:71–76CrossRefGoogle Scholar
  2. Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res, in pressGoogle Scholar
  3. Blewitt G, Lavallée D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107(B7):2145.  doi: 10.1029/2001JB000570 CrossRefGoogle Scholar
  4. Blewitt G, Lavallée D, Clarke P, Nurutdinov K (2001) A new global mode of Earth deformation: seasonal cycle detected. Science 294:2342–2345CrossRefGoogle Scholar
  5. Collilieux X, Altamimi Z, Coulot D, Ray J, Sillard P (2007) Spectral and correlation analyses of ITRF2005 VLBI, GPS and SLR height residuals: How well do space geodetic techniques agree? J Geophys Res, submittedGoogle Scholar
  6. Dong D, Fang P, Bock Y, Cheng MK, Miyazaki S (2002) Anatomy of apparent seasonal variations from GPS-derived site position time series. J Geophys Res 107(B4):2075.  doi: 10.1029/2001JB000573 CrossRefGoogle Scholar
  7. Elósegui P, Davis JL, Jaldehag RTK, Johansson JM, Niell AE, Shapiro II (1995) Geodesy using the Global Positioning System: The effects of signal scattering on estimates of site position. J Geophys Res 100(B7):9921–9934CrossRefGoogle Scholar
  8. Farrell WE, (1972) Deformation of the Earth by surface loads. Rev Geophys Space Phys 10:761–797Google Scholar
  9. Ferland R (2004) Reference frame working group technical report. In: Gowey K, Neilan R, Moore A (eds) IGS 2001–2002 technical reports, Jet Propulsion Laboratory Publication 04-017, Pasadena, pp 25–33 (available electronically at Scholar
  10. Hatanaka Y, Sawada M, Horita A, Kusaka M, Johnson J, Rocken C (2001) Calibration of antenna-radome and monument-multipath effect of GEONET—Part 2: Evaluation of the phase map by GEONET data. Earth Planets Space 53:23–30Google Scholar
  11. Hugentobler U (2005) Models in GNSS data analysis. Presentation at “Advances in GPS Data Processing and Modelling for Geodynamics” held at University College London, 9–10 November 2005 (available electronically at Scholar
  12. Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Leetma A, Reynolds R, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Jenne R, Joseph D (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77:437–47CrossRefGoogle Scholar
  13. Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104:2797–2816CrossRefGoogle Scholar
  14. Milly PCD, Shmakin AB (2002) Global modeling of land water and energy balances. Part I: the land dynamics (LaD) model. J Hydrometeorol 3(3):283–299CrossRefGoogle Scholar
  15. Penna NT, Stewart MP (2003) Aliased tidal signatures in continuous GPS height time series. Geophys Res Lett 30(23):2184.  doi: 10.1029/2003GL018828 CrossRefGoogle Scholar
  16. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2001) Numerical recipes in Fortran 77: the art of scientific computing (2nd edn). Vol. 1 of Fortran numerical recipes, Cambridge University Press, Cambridge, pp 569–577Google Scholar
  17. Ray J (2006) Systematic errors in GPS position estimates. Presentation at IGS 2006 workshop, Darmstadt (available electronically at Scholar
  18. Ray J, Gendt G, Ferland R, Altamimi Z (2005) Short-term instabilities in the IGS reference frame. Geophys Res Abstr 7:02864Google Scholar
  19. Ray J, van Dam T, Altamimi Z, Collilieux X (2006) Anomalous harmonics in the spectra of GPS position estimates. Eos Trans AGU 87(52): Fall Meet. Suppl, Abstract G43A-0985Google Scholar
  20. Shmakin AB, Milly PCD, Dunne KA (2002) Global modeling of land water and energy balances. Part III: Interannual variability. J Hydrometeorol 3(3):311–321CrossRefGoogle Scholar
  21. Steblov G, Kogan M (2005) YAKT: Winter position anomaly removed, IGS Station Mail 365. Scholar
  22. van Dam TM, Wahr J (1987) Displacements of the Earth’s surface due to atmospheric loading: Effects on gravity and baseline measurements. J Geophys Res 92:1281–1286CrossRefGoogle Scholar
  23. van Dam TM, Wahr J, Chao Y, Leuliette E (1997) Predictions of crustal deformation and geoid and sea level variability caused by oceanic and atmospheric loading. Geophys J Int 99:507–517CrossRefGoogle Scholar
  24. van Dam T, Wahr J, Milly PCD, Shmakin AB, Blewitt G, Lavallée D, Larson K (2001), Crustal displacements due to continental water loading. Geophys Res Lett 28:651–654CrossRefGoogle Scholar
  25. Vennebusch, M, Böckmann S, Nothnagel A (2007) The contribution of very long baseline interferometry to ITRF2005. J Geodesy  doi: 10.1007/s00190-006-0117-x
  26. Williams SDP, Bock Y, Fang P, Jamason P, Nikolaidis RM, Millar M, Johnson DJ (2004) Error analysis of continuous GPS position time series J Geophys Res 109:B03412.  doi: 10.1029/2003JB002741
  27. Wu X, Heflin MB, Ivins ER, Fukumori I (2006) Seasonal and interannual global surface mass variations from multisatellite geodetic data. J Geophys Res 111:B09401.  doi: 10.1029/2005JB004100
  28. Zhang J, Bock Y, Johnson H, Fang P, Williams S, Genrich J, Wdowinski S, Behr J (1997) Southern California permanent GPS geodetic array: error analysis of daily position estimates and site velocities. J Geophys Res 102(B8):18035–18055CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • J. Ray
    • 1
    Email author
  • Z. Altamimi
    • 2
  • X. Collilieux
    • 2
  • T.  van  Dam
    • 3
  1. 1.National Geodetic Survey, N/NGS6, National Oceanic and Atmospheric AdministrationSilver SpringUSA
  2. 2.Institut Géographique National, Ecole Nationale des Sciences GeographiquesChamps-sur-MarneFrance
  3. 3.Faculté des Sciences, de la Technologie et de la CommunicationUniversity of LuxembourgLuxembourgLuxembourg

Personalised recommendations