Surveys in Geophysics

, Volume 35, Issue 4, pp 959–982 | Cite as

A Wavelet-Based Assessment of Topographic-Isostatic Reductions for GOCE Gravity Gradients

  • Thomas Grombein
  • Xiaoguang Luo
  • Kurt Seitz
  • Bernhard Heck
Article

Abstract

Gravity gradient measurements from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE) contain significant high- and mid-frequency signal components, which are primarily caused by the attraction of the Earth’s topographic and isostatic masses. In order to mitigate the resulting numerical instability of a harmonic downward continuation, the observed gradients can be smoothed with respect to topographic-isostatic effects using a remove–compute–restore technique. For this reason, topographic-isostatic reductions are calculated by forward modeling that employs the advanced Rock–Water–Ice methodology. The basis of this approach is a three-layer decomposition of the topography with variable density values and a modified Airy–Heiskanen isostatic concept incorporating a depth model of the Mohorovičić discontinuity. Moreover, tesseroid bodies are utilized for mass discretization and arranged on an ellipsoidal reference surface. To evaluate the degree of smoothing via topographic-isostatic reduction of GOCE gravity gradients, a wavelet-based assessment is presented in this paper and compared with statistical inferences in the space domain. Using the Morlet wavelet, continuous wavelet transforms are applied to measured GOCE gravity gradients before and after reducing topographic-isostatic signals. By analyzing a representative data set in the Himalayan region, an employment of the reductions leads to significantly smoothed gradients. In addition, smoothing effects that are invisible in the space domain can be detected in wavelet scalograms, making a wavelet-based spectral analysis a powerful tool.

Keywords

GOCE Topographic-isostatic reduction Rock–Water–Ice (RWI) decomposition Forward modeling Tesseroids Continuous Morlet wavelet transform 

References

  1. Abd-Elmotaal H, Seitz K, Abd-Elbaky M, Heck B (2014) Comparison among three harmonic analysis techniques on the sphere and the ellipsoid. J Applied Geod, in print. doi:10.1515/jag-2013-0008
  2. Álvarez O, Gimenez M, Braitenberg C, Folguera A (2012) GOCE satellite derived gravity and gravity gradient corrected for topographic effect in the South Central Andes region. Geophys J Int 190(2):941–959. doi:10.1111/j.1365-246X.2012.05556.x CrossRefGoogle Scholar
  3. Atallah L, Smith PJP, Bates CR (2002) Wavelet analysis of bathymetric sidescan sonar data for the classification of seafloor sediments in Hopvågen Bay - Norway. Mar Geophys Res 23(5–6):431–442. doi:10.1023/B:MARI.0000018239.07561.76 CrossRefGoogle Scholar
  4. Bassin C, Laske G, Masters G (2000) The current limits of resolution for surface wave tomography in North America. EOS, Trans AGU 81, F897Google Scholar
  5. Bouman J, Fiorot S, Fuchs M, Gruber T, Schrama E, Tscherning C, Veicherts M, Visser P (2011) GOCE gravitational gradients along the orbit. J Geod 85(11):791–805. doi:10.1007/s00190-011-0464-0 CrossRefGoogle Scholar
  6. Collin F, Warnant R (1995) Application of the wavelet transform for GPS cycle slip correction and comparison with Kalman filter. Manuscr Geod 20(3):161–172Google Scholar
  7. Debnath L (2001) Wavelet transforms and their applications. Birkhäuser, BostonGoogle Scholar
  8. Fecher T, Pail R, Gruber T (2013) Global gravity field modeling based on GOCE and complementary gravity data. Int J Applied Earth Obs Geoinf, in print. doi:10.1016/j.jag.2013.10.005
  9. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Report 355, 133pp, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, USAGoogle Scholar
  10. Forsberg R, Tscherning C (1997) Topographic effects in gravity field modelling for BVP. In Sansò F, Rummel R (eds), Geodetic boundary value problems in view of the one centimeter geoid, Lecture notes in Earth sciences, IAG Symposia, vol. 65, Springer, 239–272. doi:10.1007/BFb0011707
  11. Gibert D, Holschneider M, Le Mouël JL (1998) Wavelet analysis of the Chandler wobble. J Geophys Res 103(B11):27069–27089. doi:10.1029/98JB02527 CrossRefGoogle Scholar
  12. Goupillaud P, Grossman A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23(1):85–102. doi:10.1016/0016-7142(84)90025-5 CrossRefGoogle Scholar
  13. Grombein T, Seitz K, Heck B (2010) Modelling topographic effects in GOCE gravity gradients. GEOTECHNOLOGIEN Science Report, vol. 17, 84–93. doi:10.2312/GFZ.gt.17.13 Google Scholar
  14. Grombein T, Seitz K, Heck B (2011) Smoothing GOCE gravity gradients by means of topographic-isostatic reductions. In Ouwehand L (ed), Proceedings of the 4th International GOCE User Workshop, ESA Publication SP-696, ESA/ESTECGoogle Scholar
  15. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87(7):645–660. doi:10.1007/s00190-013-0636-1 CrossRefGoogle Scholar
  16. Grombein T, Seitz K, Heck B (2014) Topographic-isostatic reduction of GOCE gravity gradients. In Rizos C, Willis P (eds), Earth on the edge: science for a sustainable planet, Proceedings of the IAG General Assembly, Melbourne, Australia, 2011, IAG Symposia, vol. 139, Springer, 349–356. doi:10.1007/978-3-642-37222-3_46
  17. Gruber T, Rummel R, Abrikosov O, van Hees R (2010) GOCE level 2 product data handbook. GO-MA-HPF-GS-0110, Issue 4.3, GOCE High Level Processing FacilityGoogle Scholar
  18. Hamming RW (1998) Digital filters. Dover Pubn Inc, MineolaGoogle Scholar
  19. Heck B (2003) Rechenverfahren und Auswertemodelle der Landesvermessung, Klassische und moderne Methoden, 3rd edn. Wichmann, HeidelbergGoogle Scholar
  20. Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81(2):121–136. doi:10.1007/s00190-006-0094-0 CrossRefGoogle Scholar
  21. Heisenberg WK (1927) On the perceptual content of quantum theoretical kinematics and mechanics. Zs f Phys 43(2–4):172–198CrossRefGoogle Scholar
  22. Heiskanen WA, Moritz H (1967) Physical geodesy. W. H. Freeman & Co, San FranciscoGoogle Scholar
  23. Hirt C, Kuhn M, Featherstone WE, Göttl F (2012) Topographic/isostatic evaluation of new-generation GOCE gravity field models. J Geophys Res 117:B05407. doi:10.1029/2011JB008878 Google Scholar
  24. Hirt C, Claessens S, Fecher T, Kuhn M, Pail R, Rexer M (2013) New ultrahigh-resolution picture of Earth’s gravity field. Geophys Res Lett 40(16):4279–4283. doi:10.1002/grl.50838 CrossRefGoogle Scholar
  25. Holschneider M (1995) Wavelets: an analysis tool. Oxford mathematical monographs. Oxford University Press, New YorkGoogle Scholar
  26. Janák J, Wild-Pfeiffer F (2010) Comparison of various topographic-isostatic effects in terms of smoothing gradiometric observations. In Sansò F, Mertikas SPP (eds), Gravity, geoid and Earth observation, IAG Symposia, vol. 135, Springer, 377–381. doi:10.1007/978-3-642-10634-7_50
  27. Janák J, Wild-Pfeiffer F, Heck B (2012) Smoothing the gradiometric observations using different topographic-isostatic models: A regional case study. In Sneeuw et al. (eds), Proceedings of the VII Hotine-Marussi Symposium, Rome, Italy, 2009, IAG Symposia, vol. 137, Springer, 245–250. doi:10.1007/978-3-642-22078-4_37
  28. Kaban MK, Schwintzer P, Reigber C (2004) A new isostatic model of the lithosphere and gravity field. J Geod 78(6):368–385. doi:10.1007/s00190-004-0401-6 CrossRefGoogle Scholar
  29. Keller W (2004) Wavelets in geodesy and geodynamics. Walter de Gruyter, BerlinCrossRefGoogle Scholar
  30. Kuhn M, Seitz K (2005) Comparison of Newton’s integral in the space and frequency domains. In Sansò F (ed), A window on the future of geodesy, IAG Symposia, vol. 128, Springer, 386–391. doi:10.1007/3-540-27432-4_66
  31. Little SA, Carter PH, Smith DK (1993) Wavelet analysis of a bathymetric profile reveals anomalous crust. Geophys Res Lett 20(18):1915–1918. doi:10.1029/93GL01880 CrossRefGoogle Scholar
  32. Liu L, Hsu H, Grafarend EW (2007) Normal Morlet wavelet transform and its application to the Earth’s polar motion. J Geophys Res 112:B08401. doi:10.1029/2006JB004895 Google Scholar
  33. Luo X (2013) GPS stochastic modelling—signal quality measures and ARMA processes. Springer Theses: Recognizing Outstanding Ph.D. Research, 331pp, SpringerGoogle Scholar
  34. Makhloof AA, Ilk K (2008) Effects of topographic-isostatic masses on gravitational functionals at the Earth’s surface and at airborne and satellite altitudes. J Geod 82(2):93–111. doi:10.1007/s00190-007-0159-8 CrossRefGoogle Scholar
  35. Moritz H (1980) Geodetic reference system 1980. Bull Géod 54(3):395–405. doi:10.1007/BF02521480 CrossRefGoogle Scholar
  36. Morlet J, Arens G, Fourgeau E, Giard D (1982a) Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media. Geophys 47(2):203–221. doi:10.1190/1.1441328 CrossRefGoogle Scholar
  37. Morlet J, Arens G, Fourgeau E, Giard D (1982b) Wave propagation and sampling theory—part II: sampling theory and complex waves. Geophysics 47(2):222–236. doi:10.1190/1.1441329 CrossRefGoogle Scholar
  38. Novák P, Tenzer R (2013) Gravitational gradients at satellite altitudes in global geophysical studies. Surv Geophys 34(5):653–673. doi:10.1007/s10712-013-9243-1 CrossRefGoogle Scholar
  39. Novák P, Kern M, Schwarz KP, Heck B (2003) Evaluation of band-limited topographical effects in airborne gravimetry. J Geod 76(11–12):597–604. doi:10.1007/s00190-002-0282-5 Google Scholar
  40. Pavlis N, Factor J, Holmes S (2007) Terrain-related gravimetric quantities computed for the next EGM. In Kiliçoğlu A, Forsberg R (eds), Proceedings of the 1st International Symposium IGFS: Gravity Field of the Earth, Istanbul, Turkey, 2006, Harita Dergisi, Special Issue 18, 318–323Google Scholar
  41. Rummel R, Colombo OL (1985) Gravity field determination from satellite gradiometry. Bull Géod 59(3):233–246. doi:10.1007/BF02520329 CrossRefGoogle Scholar
  42. Rummel R, Rapp RH, Sünkel H, Tscherning CC (1988) Comparisons of global topographic/isostatic models to the Earth’s observed gravity field. Report 388, 33pp, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, USAGoogle Scholar
  43. Rummel R, Yi W, Stummer C (2011) GOCE gravitational gradiometry. J Geod 85(11):777–790. doi:10.1007/s00190-011-0500-0 CrossRefGoogle Scholar
  44. Satirapod C, Rizos C (2005) Multipath mitigation by wavelet analysis for GPS base station applications. Surv Rev 38(295):2–10. doi:10.1179/003962605791521699 CrossRefGoogle Scholar
  45. Satirapod C, Ogaja C, Wang J, Rizos C (2001) An approach to GPS analysis incorporating wavelet decomposition. Artif Satell 36(2):27–35Google Scholar
  46. Schuh WD (2010) Filtering of correlated data—stochastical considerations within GOCE data processing. Lecture material for the GOCE Summer School, 31 May–4 June 2010, Herrsching, GermanyGoogle Scholar
  47. Tenzer R, Novák P (2013) Effect of crustal density structures on GOCE gravity gradient observables. Terr Atmos Ocean Sci 24(5):793–807. doi:10.3319/TAO.2013.05.08.01(T) CrossRefGoogle Scholar
  48. Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79(1):61–78. doi:10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2 CrossRefGoogle Scholar
  49. Trauth MH (2007) MATLAB recipes for Earth sciences, 2nd edn. Springer, BerlinGoogle Scholar
  50. Tsoulis D, Kuhn M (2007) Recent developments in synthetic Earth gravity models in view of the availability of digital terrain and crustal databases of global coverage and increased resolution. In Kiliçoğlu A, Forsberg R (eds), Proceedings of the 1st International Symposium IGFS: Gravity Field of the Earth, Istanbul, Turkey, 2006, Harita Dergisi, Special Issue 18, 354–359Google Scholar
  51. Wang J, Wang J, Roberts C (2009) Reducing GPS carrier phase errors with EMD-wavelet for precise static positioning. Surv Rev 41(312):152–161. doi:10.1179/003962609X390067 CrossRefGoogle Scholar
  52. Wild F, Heck B (2005) A comparison of different isostatic models applied to satellite gravity gradiometry. In Jekeli C (ed), Gravity, geoid and space missions, IAG Symposia, vol. 129, Springer, 230–235. doi:10.1007/3-540-26932-0_40
  53. Wild-Pfeiffer F (2007) Auswirkungen topographisch-isostatischer Massen auf die Satellitengradiometrie. C 604, Deutsche Geodätische Kommission, MünchenGoogle Scholar
  54. Wild-Pfeiffer F (2008) A comparison of different mass elements for use in gravity gradiometry. J Geod 82(10):637–653. doi:10.1007/s00190-008-0219-8 CrossRefGoogle Scholar
  55. Wild-Pfeiffer F, Heck B (2007) Comparison of the modelling of topographic and isostatic masses in the space and the frequency domain for use in satellite gravity gradiometry. In Kiliçoğlu A, Forsberg R (eds), Proceedings of the 1st International Symposium IGFS: Gravity Field of the Earth, Istanbul, Turkey, 2006, Harita Dergisi, Special Issue 18, 312–317Google Scholar
  56. Wittwer T, Klees R, Seitz K, Heck B (2008) Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic. J Geod 82(4–5):223–229. doi:10.1007/s00190-007-0172-y CrossRefGoogle Scholar
  57. Wu J, Gao J, Li M, Wang Y (2009) Wavelet transform for GPS carrier phase multipath mitigation. Proceedings of the 1st International Conference on Information Science and Engineering, Nanjing, China, 2009, 1019–1022. doi:10.1109/ICISE.2009.1344
  58. Yi T, Li H, Wang G (2006) Cycle slip detection and correction of GPS carrier phase based on wavelet transform and neural network. Proceedings of the 6th International Conference on Intelligent Systems Design and Applications, Jinan, China, 2006, 46–50. doi:10.1109/ISDA.2006.129
  59. Zhong P, Ding XL, Zheng DW, Chen W, Huang DF (2008) Adaptive wavelet transform based on cross-validation method and its application to GPS multipath mitigation. GPS Solut 12(2):109–117. doi:10.1007/s10291-007-0071-y CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Thomas Grombein
    • 1
  • Xiaoguang Luo
    • 1
    • 2
  • Kurt Seitz
    • 1
  • Bernhard Heck
    • 1
  1. 1.Geodetic InstituteKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Leica Geosystems AGHeerbruggSwitzerland

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