Journal of Geodesy

, Volume 86, Issue 7, pp 477–497 | Cite as

Separation of global time-variable gravity signals into maximally independent components

  • E. ForootanEmail author
  • J. Kusche
Original Article


The Gravity Recovery and Climate Experiment (GRACE) products provide valuable information about total water storage variations over the whole globe. Since GRACE detects mass variations integrated over vertical columns, it is desirable to separate its total water storage anomalies into their original sources. Among the statistical approaches, the principal component analysis (PCA) method and its extensions have been frequently proposed to decompose the GRACE products into space and time components. However, these methods only search for decorrelated components that on the one hand are not always interpretable and on the other hand often contain a superposition of independent source signals. In contrast, independent component analysis (ICA) represents a technique that separates components based on assumed statistical independence using higher-order statistical information. If one assumes that independent physical processes generate statistically independent signal components added up in the GRACE observations, separating them by ICA is a reliable strategy to identify these processes. In this paper, the performance of the conventional PCA, its rotated extension and ICA are investigated when applied to the GRACE-derived total water storage variations. These analyses have been tested on both a synthetic example and on the real GRACE level-2 monthly solutions derived from GeoForschungsZentrum Potsdam (GFZ RL04) and Bonn University (ITG2010). Within the synthetic example, we can show how imposing statistical independence in the framework of ICA improves the extraction of the ‘original’ signals from a GRACE-type super-position. We are therefore confident that also for the real case the ICA algorithm, without making prior assumptions about the long-term behaviour or on the frequencies contained in the signal, improves over the performance of PCA and its rotated extension in the separation of periodical and long-term components.


Signal separation PCA ICA GRACE products 


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  1. Aires F, Rossow WB, Chedin A (2002) Rotation of EOFs by the independent component analysis: toward a solution of the mixing problem in the decomposition of geophysical time series. J Atmos Sci 59: 111–123. doi: 10.1175/1520-0469(2002)059<0111:ROEBTI>2.0.CO;2 CrossRefGoogle Scholar
  2. Arendt AA, Luthcke SB, Hock R (2009) Glacier changes in Alaska: can mass-balance models explain GRACE mascon trends?. Ann Glaciol 50: 148–154. doi: 10.3189/172756409787769753 CrossRefGoogle Scholar
  3. Beven K (2001) How far can we go in distributed hydrological modelling?. Hydrol Earth Syst Sci 5: 1–12. doi: 10.5194/hess-5-1-2001 CrossRefGoogle Scholar
  4. Browne MW (2001) An overview of analytic rotation in exploratory factor analysis. Multivar Behav Res 36: 111–150 ISSN:0027-3171CrossRefGoogle Scholar
  5. Cardoso JF (1992) Fourth-order cumulant structure forcing: application to blind array processing, statistical signal and array processing, 1992. In: Conference Proceedings. IEEE Sixth SP Workshop. doi: 10.1109/SSAP.1992.246830
  6. Cardoso JF (1999) High-order contrasts for independent component analysis. Neural Comput 11: 157–192. doi: 10.1162/089976699300016863 CrossRefGoogle Scholar
  7. Cardoso JF, Souloumiac A (1993) Blind beamforming for non-Gaussian signals. In: IEEE proceedings, pp 362–370. doi:
  8. Chambers DP (2006) Observing seasonal steric sea level variations with GRACE and satellite altimetry. J Geophys Res 111: C03010. doi: 10.1029/2005JC002914 CrossRefGoogle Scholar
  9. Chambers DP, Willis JK (2008) Analysis of large-scale ocean bottom pressure variability in the North Pacific. J Geophys Res 113: C11003. doi: 10.1029/2008JC004930 CrossRefGoogle Scholar
  10. Comon P (1994a) Independent component analysis, a new concept?. Signal Process 36: 287–314. doi: 10.1016/0165-1684(94)90029-9 CrossRefGoogle Scholar
  11. Comon P (1994b) Tensor diagonalization, a useful tool in signal processing. In: Proceedings of 10th IFAC symp system identification (IFAC-SYSID), pp 77–82Google Scholar
  12. Cromwell D (2006) Temporal and spatial characteristics of sea surface height variability in the North Atlantic ocean. Ocean Sci 2: 147–159. doi: 10.5194/osd-3-609-2006 CrossRefGoogle Scholar
  13. de Viron O, Panet I, Diament M (2006) Extracting low frequency climate signal from GRACE data. eEarth 1: 9–14CrossRefGoogle Scholar
  14. Dommenget D, Latif M (2002) A cautionary note on the interpretation of EOFs. J Clim 15: 216–225. doi: 10.1175/1520-0442(2002)015<0216:ACNOTI>2.0.CO;2 CrossRefGoogle Scholar
  15. Fenoglio-Marc L (2001) Analysis and representation of regional sea-level variability from altimetry and atmospheric-oceanic data. Geophys J Int 145: 1–18. doi: 10.1046/j.1365-246x.2001.00284.x CrossRefGoogle Scholar
  16. Ferreira P, Magueijo J, Silk J (1997) Cumulants as non-Gaussian qualifiers. Phys Rev D 56: 4592–4603. doi: 10.1103/PhysRevD.56.4592 CrossRefGoogle Scholar
  17. Flechtner F (2007) GFZ Level-2 processing standards document for level-2 product release 0004. GRACE 327-743, Rev 1.0Google Scholar
  18. Frappart F, Ramillien G, Leblanc M, Tweed SO, Bonnet MP, Maisongrande P (2010a) An independent component analysis filtering approach for estimating continental hydrology in the GRACE gravity data. Remote Sens Environ 115(1): 187–204. doi: 10.1016/j.rse.2010.08.017 CrossRefGoogle Scholar
  19. Frappart F, Ramillien G, Maisongrande P, Bonnet MP (2010b) Denoising satellite gravity signals by independent component analysis. Geosci Remote Sens Lett 7: 421–425. doi: 10.1109/LGRS.2009.2037837 CrossRefGoogle Scholar
  20. Golub GH, van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins University Press. ISBN:978-0-470-52833-4Google Scholar
  21. Guan B, Nigam S (2009) Analysis of Atlantic SST variability factoring interbasin links and the secular trend: clarified structure of the Atlantic multidecadal oscillation. J Clim 22: 4228–4240. doi: 10.1175/2009JCLI2921.1 CrossRefGoogle Scholar
  22. Hannachi A, Unkel S, Trendafilov NT, Jolliffe IT (2009) Independent component analysis of climate data: a new look at EOF rotation. J Clim 22: 2797–2812. doi: 10.1175/2008JCLI2571.1 CrossRefGoogle Scholar
  23. Hendricks JR, Leben RR, Born GH, Koblinsky CJ (1996) Empirical orthogonal function analysis of global TOPEX/POSEIDON altimeter data and implications for detection of global sea level rise. J Geophys Res 101: 14131–14145. doi: 10.1029/96JC00922 CrossRefGoogle Scholar
  24. Hyvaerinen A (1999) Survey on independent component analysis. Neural Comput Surv 2: 94–128Google Scholar
  25. Hyvaerinen A, Oja E (2000) Independent component analysis: algorithms and applications. Neural Netw 13: 411–430CrossRefGoogle Scholar
  26. Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. Technical report rep 327. Department of Geodesy and Science and Surveying, Ohio State University, ColumbusGoogle Scholar
  27. Jolliffe IT (1986) Principal component analysis. Springer, New York. ISBN:9780387954424Google Scholar
  28. Jolliffe IT (1989) Rotation of ill-defined principal components. J R Stat Soc Ser C (Appl Stat) 38:139–147.
  29. Jolliffe IT (2003) A cautionary note on artificial examples of EOFs. J Clim 16: 1084–1086. doi: 10.1175/1520-0442(2003)016<1084:ACNOAE>2.0.CO;2 CrossRefGoogle Scholar
  30. Kaiser H (1958) The VARIMAX criterion for analytic rotation in factor analysis. Psychometrika 23: 187–200. doi: 10.1007/BF02289233 CrossRefGoogle Scholar
  31. Karvanen J, Koivunen V (2002) Blind separation methods based on Pearson system and its extensions. Signal Process 82: 663–673. doi: 10.1016/S0165-1684(01)00213-4 CrossRefGoogle Scholar
  32. Kirimoto T, Amishima T, Okamura A (2011) Separation of mixtures of complex sinusoidal signals with independent component analysis. IEICE Trans Commun 94B:215–221. ISSN:1745-1345. doi: 10.1587/transcom.E94.B.215 Google Scholar
  33. Krishnamurthy V, Goswami BN (2000) Indian Monsoon ENSO relationship on interdecadal timescale. J Clim 13: 579–595. doi: 10.1175/1520-0442(2000)013<0579:IMEROI>2.0.CO;2 CrossRefGoogle Scholar
  34. Koch KR (1988) Parameter estimation and hypothesis testing in linear models. Springer, New York. ISBN:9783540652571Google Scholar
  35. Kusche J (2007) Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models. J Geod 81: 733–749. doi: 10.1007/s00190-007-0143-3 CrossRefGoogle Scholar
  36. Kusche J, Schmidt R, Petrovic S, Rietbroek R (2009) Decorrelated GRACE time-variable gravity solutions by GFZ, and their validation using a hydrological model. J Geod 83: 903–913. doi: 10.1007/s00190-009-0308-3 CrossRefGoogle Scholar
  37. Kusche J, Rietbroek R, Forootan E (2010) Signal separation: the quest for independent mass flux patterns in geodetic observations. AGU Fall Meeting, 17 December 2010, San Francisco, USA.
  38. Latif M, Barnett TP (1994) Causes of decadal climate variability over the North Pacific and North America. Science 266: 634–637. doi: 10.1126/science.266.5185.634 CrossRefGoogle Scholar
  39. Legates DR (1991) The effect of domain shape on principal components analyses. Int J Climatol 11: 135–146. doi: 10.1002/joc.3370110203 CrossRefGoogle Scholar
  40. Lombard A, Cazenave A, Le Traon PY, Ishii M (2005) Contribution of thermal expansion to present-day sea-level change revisited. Glob Planet Change 47: 1–16. doi: 10.1016/j.gloplacha.2004.11.016 CrossRefGoogle Scholar
  41. Lorenz E (1956) Empirical orthogonal function and statistical weather prediction. Technical Report Science Report No 1, Statistical Forecasting Project. MIT, CambridgeGoogle Scholar
  42. Mayer-Guerr T, Eicker A, Kurtenbach E (2010)
  43. Mestas-Nunez AM (2000) Orthogonality properties of rotated empirical modes. Int J Climatol 20: 1509–1516. doi: 10.1002/1097-0088(200010)20:12<1509::AID-JOC553>3.0.CO;2-Q CrossRefGoogle Scholar
  44. Mills GF (1995) Principal component analysis of precipitation and rainfall regionalization in Spain. Theor Appl Climatol 50: 169–183. doi: 10.1007/BF00866115 CrossRefGoogle Scholar
  45. Naik G, Kumar D, Weghorn H (2007) Performance comparison of ICA algorithms for isometric hand gesture identification using surface EGM. In: Intelligent sensors, sensor networks and information, 2007. ISSNIP2007, pp 613–618Google Scholar
  46. North GR, Bell TL, Cahalan RF, Moeng FJ (1982) Sampling errors in the estimation of empirical orthogonal functions. Mon Weather Rev 110: 699–706. doi: 10.1175/1520-0493(1982)110<0699:SEITEO>2.0.CO;2 CrossRefGoogle Scholar
  47. Ogawa R (2010) Transient, seasonal and inter-annual gravity changes from GRACE data: geophysical modelings. PhD Dissertation, Hokkaido University, SapporoGoogle Scholar
  48. Preisendorfer R (1988) Principal component analysis in meteorology and oceanography. Elsevier, Amsterdam. ISBN:0444430148Google Scholar
  49. Preisendorfer R, Barnett TP (1977) Significance tests for EOF. In: 5th Conference on probability and statistics in atmospheric sciences, pp 169–172Google Scholar
  50. Rangelova E, Sideris MG (2008) Contributions of terrestrial and GRACE data to the study of the secular geoid changes in North America. J Geodyn 46: 131–143. doi: 10.1016/j.jog.2008.03.006 CrossRefGoogle Scholar
  51. Rangelova E, van der Wal W, Sideris MG, Wu P (2009) Spatiotemporal analysis of the GRACE-derived mass variations in North America by means of multi-channel singular spectrum analysis. Int Assoc Geod Symposia 2010 135(7): 539–546. doi: 10.1007/978-3-642-10634-7_72 Google Scholar
  52. Richman M (1986) Rotation of principal components. Int J Climatol 6: 293–335. doi: 10.1002/joc.3370060305 CrossRefGoogle Scholar
  53. Rieser D, Kuhn M, Pail R, Anjasmara I, Awange J (2010) Relation between GRACE-derived surface mass variations and precipitation over Australia. Austral J Earth Sci 57: 887–900. doi: 10.1080/08120099.2010.51264 CrossRefGoogle Scholar
  54. Rietbroek R, Brunnabend SE, Dahle C, Kusche J, Flechtner F, Schrter J, Timmermann R (2009) Changes in total ocean mass derived from GRACE, GPS, and ocean modeling with weekly resolution. J Geophys Res 114: C11004. doi: 10.1029/2009JC005449 CrossRefGoogle Scholar
  55. Rietbroek R, Brunnabend SE, Kusche J, Schroter J (2011) Resolving sea level contributions by identifying fingerprints in time-variable gravity and altimetry. In press, , doi: 10.1016/j.jog.2011.06.007
  56. Rodell M, Houser PR, Jambor U, Gottschalck J, Mitchell K, Meng CJ, Arsenault K, Cosgrove B, Radakovich J, Bosilovich M, Entin JK, Walker JP., Lohmann D, Toll D (2004) The global land data assimilation system. Bull Am Meteorol Soc 85: 381– 394CrossRefGoogle Scholar
  57. Schaefli B, Harman CJ, Sivapalan M, Schymanski SJ (2011) Hydrologic predictions in a changing environment: behavioral modeling. J Hydrol Earth Syst Sci 15(2): 635–646. doi: 10.5194/hess-15-635-2011 CrossRefGoogle Scholar
  58. Schmeer M, Bosch W, Schmidt M (2008) Separation of oceanic and hydrological mass variations by simulated gravity observations. DFGI report 83. Accessed 31 March 2011
  59. Schmidt R, Flechtner F, Meyer U, Neumayer KH, Dahle C, Koenig R, Kusche J (2008) Hydrological signals observed by the GRACE satellites. Surv Geophys 29: 319–334. doi: 10.1007/s10712-008-9033-3 CrossRefGoogle Scholar
  60. Schrama EJO, Wouters B, Lavalle DA (2007) Signal and noise in gravity recovery and climate experiment (GRACE) observed surface mass variations. J Geophys Res 112: B08407. doi: 10.1029/2006JB004882 CrossRefGoogle Scholar
  61. Swenson S, Wahr J (2006) Post-processing removal of correlated errors in GRACE data. Geophys Res Lett 33. doi: 10.1029/2005GL025285
  62. Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins MM (2004a) GRACE measurements of mass variability in the Earth system. Science 305: 503–505. doi: 10.1126/science.1099192 CrossRefGoogle Scholar
  63. Tapley BD, Bettadpur S, Watkins M, Reigber C (2004b) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31: L09607. doi: 10.1029/2004GL019920 CrossRefGoogle Scholar
  64. Tapley BD, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02: an improved Earth gravity field model from GRACE. J Geod 79: 467–478. doi: 10.1007/s00190-005-0480-z CrossRefGoogle Scholar
  65. Thode HC (2002) Testing for normality. Marcel Dekker. ISBN: 9780824796136Google Scholar
  66. Timmen L, Gitlein O, Mueller J, Denker H, Maekinen J, Bilker M, Pettersen B, Lysaker D, Omang O, Svendsen J, Wilmes H, Falk R, Reinhold A, Hoppe W, Scherneck HG, Engen B, Harsson B, Engfeldt A, Lilje M, Strykowski G, Forsberg R (2006) Observing Fennoscandian gravity change by absolute gravimetry. Int Assoc Geod Symposia 131: 193–199. doi: 10.1007/978-3-540-38596-7_23 CrossRefGoogle Scholar
  67. van der Wal W, Wu P, Sideris MG, Shum CK (2008) Use of GRACE determined secular gravity rates for glacial isostatic adjustment studies in North-America. J Geodyn 46: 144–154. doi: 10.1016/j.jog.2008.03.007 CrossRefGoogle Scholar
  68. van den Broeke M, Bamber J, Ettema J, Rignot E, Schrama E, van de Berg WJ, van Meijgaard E, Velicogna I, Wouters B (2009) Partitioning recent Greenland mass loss. Science 326: 984–986. doi: 10.1126/science.1178176 CrossRefGoogle Scholar
  69. Velicogna I (2009) Increasing rates of ice mass loss from the Greenland and Antarctic ice sheets revealed by GRACE. Geophys Res Lett 36: L19503. doi: 10.1029/2009GL040222 CrossRefGoogle Scholar
  70. Velicogna I, Wahr J (2005) Greenland mass balance from GRACE. Geophys Res Lett 32: L18505. doi: 10.1029/2005GL023955 CrossRefGoogle Scholar
  71. Voeroesmarty CJ, Green P, Salisbury J, Lammers RB (2000) Global water resources: vulnerability from climate change and population growth. Science 289: 284–288. doi: 10.1126/science.289.5477.284 CrossRefGoogle Scholar
  72. von Storch H, Zwiers F (1999) Statistical analysis in climate research. Cambridge University Press. ISBN:9780521012300Google Scholar
  73. Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103: 30205–30229. doi: 10.1029/98JB02844 CrossRefGoogle Scholar
  74. Waymire E, Gupta VK (1981) The mathematical structure of rainfall representations 1. A review of the stochastic rainfall models. Water Resour Res 17: 1261–1272. doi: 10.1029/WR017i005p01261 CrossRefGoogle Scholar
  75. Werth S, Guentner A, Schmidt R, Kusche J (2009) Evaluation of GRACE filter tools from a hydrological perspective. Geophys J Int 179: 1499–1515. doi: 10.1111/j.1365-246X.2009.04355.x CrossRefGoogle Scholar
  76. Westra S, Brown C, Lall U, Sharma A (2007) Modeling multivariable hydrological series: principal component analysis or independent component analysis?. Water Resour Res 43: W06429. doi: 10.1029/2006WR005617 CrossRefGoogle Scholar
  77. Wouters B, Schrama EJO (2007) Improved accuracy of GRACE gravity solutions through empirical orthogonal function filtering of spherical harmonics. Geophys Res Lett 34: L23711. doi: 10.1029/2007GL032098 CrossRefGoogle Scholar
  78. Xu J, von Storch H (1990) Predicting the state of the southern oscillation using principal oscillation pattern analysis. J Clim 3: 1316–1329. doi: 10.1175/1520-0442(1990)003<1316:PTSOTS>2.0.CO;2 CrossRefGoogle Scholar
  79. Zavala-Fernaendez H, Sander T, Burghoff M, Orglmeister R, Trahms L (2006) Comparison of IAC algorithms for the isolation of biological artifacts in magnetoencephalography. Independent component analysis and blind signal separation. Lecture notes in computer science, vol 3889, pp 511–518. doi: 10.1007/11679363_64

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute of Geodesy and GeoinformationBonn UniversityBonnGermany

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