Advertisement

Space Science Reviews

, Volume 155, Issue 1–4, pp 247–291 | Cite as

An Introduction to Data Assimilation and Predictability in Geomagnetism

  • Alexandre FournierEmail author
  • Gauthier Hulot
  • Dominique Jault
  • Weijia Kuang
  • Andrew Tangborn
  • Nicolas Gillet
  • Elisabeth Canet
  • Julien Aubert
  • Florian Lhuillier
Article

Abstract

Data assimilation in geomagnetism designates the set of inverse methods for geomagnetic data analysis which rely on an underlying prognostic numerical model of core dynamics. Within that framework, the time-dependency of the magnetohydrodynamic state of the core need no longer be parameterized: The model trajectory (and the secular variation it generates at the surface of the Earth) is controlled by the initial condition, and possibly some other static control parameters. The primary goal of geomagnetic data assimilation is then to combine in an optimal fashion the information contained in the database of geomagnetic observations and in the dynamical model, by adjusting the model trajectory in order to provide an adequate fit to the data.

The recent developments in that emerging field of research are motivated mostly by the increase in data quality and quantity during the last decade, owing to the ongoing era of magnetic observation of the Earth from space, and by the concurrent progress in the numerical description of core dynamics.

In this article we review briefly the current status of our knowledge of core dynamics, and elaborate on the reasons which motivate geomagnetic data assimilation studies, most notably (a) the prospect to propagate the current quality of data backward in time to construct dynamically consistent historical core field and flow models, (b) the possibility to improve the forecast of the secular variation, and (c) on a more fundamental level, the will to identify unambiguously the physical mechanisms governing the secular variation. We then present the fundamentals of data assimilation (in its sequential and variational forms) and summarize the observations at hand for data assimilation practice. We present next two approaches to geomagnetic data assimilation: The first relies on a three-dimensional model of the geodynamo, and the second on a quasi-geostrophic approximation. We also provide an estimate of the limit of the predictability of the geomagnetic secular variation based upon a suite of three-dimensional dynamo models. We finish by discussing possible directions for future research, in particular the assimilation of laboratory observations of liquid metal analogs of Earth’s core.

Keywords

Geomagnetic secular variation Dynamo: theories and simulations Earth’s core dynamics Inverse theory Data assimilation Satellite magnetics Predictability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.M. Alexandrescu, D. Gibert, J.L. Le MouëL, G. Hulot, G. Saracco, An estimate of average lower mantle conductivity by wavelet analysis of geomagnetic jerks. J. Geophys. Res. 104(B8), 17735–17745 (1999). doi: 10.1029/1999JB900135 ADSCrossRefGoogle Scholar
  2. H. Amit, J. Aubert, G. Hulot, P. Olson, A simple model for mantle-driven flow at the top of Earth’s core. Earth Planets Space 60, 845–854 (2008) ADSGoogle Scholar
  3. J. Aubert, H. Amit, G. Hulot, Detecting thermal boundary control in surface flows from numerical dynamos. Phys. Earth Planet. Inter. 160(2), 143–156 (2007). doi: 10.1016/j.pepi.2006.11.003 ADSCrossRefGoogle Scholar
  4. J. Aubert, H. Amit, G. Hulot, P. Olson, Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity. Nature 454, 758–761 (2008). doi: 10.1038/nature07109 ADSCrossRefGoogle Scholar
  5. G.E. Backus, Kinematics of geomagnetic secular variation in a perfectly conducting core. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 263(1141), 239–266 (1968) ADSCrossRefGoogle Scholar
  6. G.E. Backus, Application of mantle filter theory to the magnetic jerk of 1969. Geophys. J. R. Astron. Soc. 74(3), 713–746 (1983) Google Scholar
  7. G. Backus, R. Parker, C. Constable, Foundations of Geomagnetism (Cambridge University Press, Cambridge, 1996) Google Scholar
  8. C.D. Beggan, K.A. Whaler, Forecasting change of the magnetic field using core surface flows and ensemble Kalman filtering. Geophys. Res. Lett. 36, L18303 (2009). doi: 10.1029/2009GL039927 ADSCrossRefGoogle Scholar
  9. A. Bennett, Inverse Modeling of the Ocean and Atmosphere (Cambridge University Press, Cambridge, 2002) zbMATHCrossRefGoogle Scholar
  10. P. Bergthorsson, B. Döös, Numerical weather map analysis. Tellus 7(3), 329–340 (1955) CrossRefADSGoogle Scholar
  11. J. Bloxham, D. Gubbins, A. Jackson, Geomagnetic secular variation. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci., 415–502 (1989) Google Scholar
  12. J. Bloxham, S. Zatman, M. Dumberry, The origin of geomagnetic jerks. Nature 420(6911), 65–68 (2002). doi: 10.1038/nature01134 ADSCrossRefGoogle Scholar
  13. S.I. Braginsky, Torsional magnetohydrodynamic vibrations of the earth’s core and variations in day length. Geomagnet. Aeron. 10, 1–8 (1970) ADSGoogle Scholar
  14. S.I. Braginsky, Short period geomagnetic variations. Geophys. Astrophys. Fluid Dyn. 30, 1–78 (1984) zbMATHADSCrossRefGoogle Scholar
  15. P. Brasseur, Ocean data assimilation using sequential methods based on the Kalman filter, in Ocean Weather Forecasting: An Integrated View of Oceanography, ed. by E. Chassignet, J. Verron. (Springer, Berlin, 2006), pp. 271–316 CrossRefGoogle Scholar
  16. M. Buehner, Inter-comparison of 4D-Var and EnKF systems for operational deterministic numerical weather prediction, in WWRP/THORPEX Workshop on 4D-VAR and Ensemble Kalman Filter Inter-comparisons, Buenos Aires, Argentina, 2008 Google Scholar
  17. B. Buffett, J. Mound, A. Jackson, Inversion of torsional oscillations for the structure and dynamics of Earth’s core. Geophys. J. Int. 177(3), 878–890 (2009). doi: 10.1111/j.1365-246X.2009.04129.x ADSCrossRefGoogle Scholar
  18. H.P. Bunge, C. Hagelberg, B. Travis, Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophys. J. Int. 152(2), 280–301 (2003). doi: 10.1046/j.1365-246X.2003.01823.x ADSCrossRefGoogle Scholar
  19. E. Canet, A. Fournier, D. Jault, Forward and adjoint quasi-geostrophic models of the geomagnetic secular variation. J. Geophys. Res. 114, B11101 (2009). doi: 10.1029/2008JB006189 ADSCrossRefGoogle Scholar
  20. P. Cardin, P. Olson, Experiments on core dynamics, in Core Dynamics, ed. by P. Olson, G. Schubert. Treatise on Geophysics, vol. 8 (Elsevier, Amsterdam, 2007), pp. 319–343, Chap. 11 CrossRefGoogle Scholar
  21. J.G. Charney, R. Fjortoft, J. Von Neumann, Numerical integration of the barotropic vorticity equation. Tellus 2(4), 237–254 (1950) MathSciNetCrossRefADSGoogle Scholar
  22. E. Chassignet, J. Verron, Ocean Weather Forecasting: An Integrated View of Oceanography (Springer, Berlin, 2006) CrossRefGoogle Scholar
  23. U.R. Christensen, J. Aubert, Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields. Geophys. J. Int. 140, 97–114 (2006). doi: 10.1111/j.1365-246X.2006.03009.x ADSCrossRefGoogle Scholar
  24. U.R. Christensen, A. Tilgner, Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos. Nature 429(6988), 169–171 (2004). doi: 10.1038/nature02508 ADSCrossRefGoogle Scholar
  25. U.R. Christensen, J. Wicht, Numerical dynamo simulations, in Core Dynamics, ed. by P. Olson, G. Schubert. Treatise on Geophysics, vol. 8 (Elsevier, Oxford, 2007), pp. 245–282, Chap. 8 CrossRefGoogle Scholar
  26. U.R. Christensen, V. Holzwarth, A. Reiners, Energy flux determines magnetic field strength of planets and stars. Nature 457(7226), 167–169 (2009). doi: 10.1038/nature07626 ADSCrossRefGoogle Scholar
  27. U.R. Christensen, J. Aubert, G. Hulot, Conditions for Earth-like geodynamo models. Earth Planet. Sci. Lett. (2010). doi: 10.1016/j.epsl.2010.06.009 Google Scholar
  28. A. Chulliat, N. Olsen, Observation of magnetic diffusion in the Earth’s core from Magsat, Oersted and CHAMP data. J. Geophys. Res. 115, B05105 (2010). doi: 10.1029/2009JB006994 CrossRefGoogle Scholar
  29. A. Chulliat, G. Hulot, L.R. Newitt, Magnetic flux expulsion from the core as a possible cause of the unusually large acceleration of the north magnetic pole during the 1990s. J. Geophys. Res. 115, B07101 (2010). doi: 10.1029/2009JB007143 CrossRefGoogle Scholar
  30. S. Cohn, N. Sivakumaran, R. Todling, A fixed-lag Kalman smoother for retrospective data assimilation. Mon. Weather Rev. 122(12), 2838–2867 (1994). doi: 10.1175/1520-0493(1994)122<2838:AFLKSF>2.0.CO;2 ADSCrossRefGoogle Scholar
  31. C. Constable, M. Korte, Is Earth’s magnetic field reversing? Earth Planet. Sci. Lett. 246(1–2), 1–16 (2006). doi: 10.1016/j.epsl.2006.03.038 ADSCrossRefGoogle Scholar
  32. E. Cosme, J.M. Brankart, J. Verron, P. Brasseur, M. Krysta, Implementation of a reduced-rank, square-root smoother for high resolution ocean data assimilation. Ocean Model. 33(1–2), 87–100 (2010). doi: 10.1016/j.ocemod.2009.12.004 ADSCrossRefGoogle Scholar
  33. P. Courtier, Variational methods. J. Meteorol. Soc. Jpn. 75(1B), 211–218 (1997) Google Scholar
  34. P. Courtier, O. Talagrand, Variational assimilation of meteorological observations with the adjoint vorticity equation. II: Numerical results. Q. J. R. Meteorol. Soc. 113(478), 1329–1347 (1987). doi: 10.1002/gj.49711347813 ADSCrossRefGoogle Scholar
  35. D. Dee, A. Da Silva, Data assimilation in the presence of forecast bias. Q. J. R. Meteorol. Soc. 124(545), 269–295 (1998) ADSCrossRefGoogle Scholar
  36. F. Donadini, M. Korte, C. Constable, Geomagnetic field for 0–3 ka: 1. New data sets for global modeling. Geochem. Geophys. Geosyst. 10, Q06007 (2009). doi: 10.1029/2008GC002295 CrossRefGoogle Scholar
  37. G.D. Egbert, A.F. Bennett, M.G.G. Foreman, TOPEX/POSEIDON tides estimated using a global inverse model. J. Geophys. Res. 99(C12), 24821–24852 (1994). doi: 10.1029/94JC01894 ADSCrossRefGoogle Scholar
  38. A. Eliassen, Provisional report on calculation of spatial covariance and autocorrelation of the pressure field. Institute of Weather and Climate Research, Academy of Sciences, Oslo, Report 5 (1954) Google Scholar
  39. G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99(C5), 10143–10162 (1994). doi: 10.1029/94JC00572 ADSCrossRefGoogle Scholar
  40. G. Evensen, Data Assimilation: The Ensemble Kalman Filter, 2nd edn. (Springer, Berlin, 2009). doi: 10.1007/978-3-642-03711-5 Google Scholar
  41. C. Eymin, G. Hulot, On core surface flows inferred from satellite magnetic data. Phys. Earth Planet. Inter. 152, 200–220 (2005). doi: 10.1016/j.pepi.2005.06.009 ADSCrossRefGoogle Scholar
  42. A. Fichtner, H.P. Bunge, H. Igel, The adjoint method in seismology I. Theory. Phys. Earth Planet. Inter. 157(1–2), 86–104 (2006). doi: 10.1016/j.pepi.2006.03.016 ADSCrossRefGoogle Scholar
  43. C.C. Finlay, Historical variation of the geomagnetic axial dipole. Phys. Earth Planet. Inter. 170(1–2), 1–14 (2008). doi: 10.1016/j.pepi.2008.06.029 ADSCrossRefGoogle Scholar
  44. C.C. Finlay, A. Jackson, Equatorially dominated magnetic field change at the surface of earth’s core. Science 300(5628), 2084–2086 (2003). doi: 10.1126/science.1083324 ADSCrossRefGoogle Scholar
  45. C.C. Finlay, M. Dumberry, A. Chulliat, A. Pais, Short timescale dynamics: Theory and observations. Space Sci. Rev. (2010, in revision) Google Scholar
  46. A. Fournier, C. Eymin, T. Alboussière, A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system. Nonlinear Process. Geophys. 14, 163–180 (2007) CrossRefGoogle Scholar
  47. E. Friis-Christensen, H. Lühr, G. Hulot, Swarm: A constellation to study the Earth’s magnetic field. Earth Planets Space 58, 351–358 (2006) ADSGoogle Scholar
  48. L.S. Gandin, Objective Analysis of Meteorological Fields (Objektivnyi Analiz Meteorologicheskikh Polei) (Gidrometeor. Izd.i, Leningrad, 1963) (in Russian). English translation by Israel program for scientific translations, Jerusalem, 1965 Google Scholar
  49. G. Gaspari, S.E. Cohn, Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125(554), 723–757 (1999). doi: 10.1002/qj.49712555417 ADSCrossRefGoogle Scholar
  50. A. Genevey, Y. Gallet, C. Constable, M. Korte, G. Hulot, ArcheoInt: An upgraded compilation of geomagnetic field intensity data for the past ten millennia and its application to the recovery of the past dipole moment. Geochem. Geophys. Geosyst. 9(4), Q04038 (2008). doi: 10.1029/2007GC001881 CrossRefGoogle Scholar
  51. A. Genevey, Y. Gallet, J. Rosen, M. Le Goff, Evidence for rapid geomagnetic field intensity variations in Western Europe over the past 800 years from new French archeointensity data. Earth Planet. Sci. Lett., 132–143 (2009). doi: 10.1016/j.epsl.2009.04.024
  52. M. Ghil, P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography. Adv. Geophys. 33, 141–266 (1991) Google Scholar
  53. R. Giering, T. Kaminski, Recipes for adjoint code construction. ACM Trans. Math. Softw. 24(4), 437–474 (1998) zbMATHCrossRefGoogle Scholar
  54. N. Gillet, D. Brito, D. Jault, H.C. Nataf, Experimental and numerical studies of convection in a rapidly rotating spherical shell. J. Fluid Mech. 580, 83–121 (2007). doi: 10.1017/S0022112007005265 zbMATHMathSciNetADSCrossRefGoogle Scholar
  55. N. Gillet, A. Pais, D. Jault, Ensemble inversion of time-dependent core flow models. Geochem. Geophys. Geosyst. 10, Q06004 (2009). doi: 10.1029/2008GC002290 CrossRefGoogle Scholar
  56. N. Gillet, D. Jault, E. Canet, A. Fournier, Fast torsional waves and strong magnetic field within the Earth’s core. Nature 465, 74–77 (2010a). doi: 10.1038/nature09010 ADSCrossRefGoogle Scholar
  57. N. Gillet, V. Lesur, N. Olsen, Geomagnetic core field secular variation models. Space Sci. Rev. (2010b, in press). doi: 10.1007/s11214-009-9586-6
  58. G.A. Glatzmaier, Numerical simulations of stellar convective dynamos. I—The model and method. J. Comput. Phys. 55(3), 461–484 (1984). doi: 10.1016/0021-9991(84)90033-0 ADSCrossRefGoogle Scholar
  59. G.A. Glatzmaier, P.H. Roberts, A three-dimensional self-consistent computer simulation of a geomagnetic reversal. Nature 377, 203–209 (1995). doi: 10.1038/377203a0 ADSCrossRefGoogle Scholar
  60. R.S. Gross, I. Fukumori, D. Menemenlis, P. Gegout, Atmospheric and oceanic excitation of length-of-day variations during 1980–2000. J. Geophys. Res. 109, B01406 (2004). doi: 10.1029/2003JB002432 CrossRefGoogle Scholar
  61. D. Gubbins, A formalism for the inversion of geomagnetic data for core motions with diffusion. Phys. Earth Planet. Inter. 98(3), 193–206 (1996). doi: 10.1016/S0031-9201(96)03187-1 ADSCrossRefGoogle Scholar
  62. D. Gubbins, N. Roberts, Use of the frozen flux approximation in the interpretation of archeomagnetic and palaeomagnetic data. Geophys. J. R. Astron. Soc. 73(3), 675–687 (1983). doi: 10.1111/j.1365-246X.1983.tb03339.x Google Scholar
  63. D. Gubbins, P.H. Roberts, Magnetohydrodynamics of the Earth’s core, in Geomagnetism, vol. 2, ed. by J.A. Jacobs (Academic Press, London, 1987) Google Scholar
  64. D. Gubbins, A.L. Jones, C.C. Finlay, Fall in Earth’s magnetic field is erratic. Science 312(5775), 900–902 (2006). doi: 10.1126/science.1124855 ADSCrossRefGoogle Scholar
  65. N. Gustafsson, Discussion on ‘4D-Var or EnKF?’. Tellus 59A(5), 774–777 (2007). doi: 10.1111/j.1600-0870.2007.00262.x ADSGoogle Scholar
  66. J.R. Heirtzler, The future of the South Atlantic anomaly and implications for radiation damage in space. J. Atmos. Sol.-Terr. Phys. 64(16), 1701–1708 (2002). doi: 10.1016/S1364-6826(02)00120-7 CrossRefGoogle Scholar
  67. H. Hersbach, Application of the adjoint of the WAM model to inverse wave modeling. J. Geophys. Res. 103(C 5), 10469–10487 (1998). doi: 10.1029/97JC03554 ADSCrossRefGoogle Scholar
  68. R. Hide, Free hydromagnetic oscillations of the Earth’s core and the theory of the geomagnetic secular variation. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 259, 615–647 (1966) ADSCrossRefGoogle Scholar
  69. R. Holme, Large-scale flow in the core, in Core Dynamics, ed. by P. Olson, G. Schubert. Treatise on Geophysics, vol. 8 (Elsevier, Amsterdam, 2007), pp. 107–130, Chap. 4 CrossRefGoogle Scholar
  70. L. Hongre, G. Hulot, A. Khokhlov, An analysis of the geomagnetic field over the past 2000 years. Phys. Earth Planet. Inter. 106(3), 311–335 (1998). doi: 10.1016/S0031-9201(97)00115-5 ADSCrossRefGoogle Scholar
  71. G. Hulot, J.L. Le Mouël, A statistical approach to the Earth’s main magnetic field. Phys. Earth Planet. Inter. 82(3), 167–183 (1994). doi: 10.1016/0031-9201(94)90070-1 ADSCrossRefGoogle Scholar
  72. G. Hulot, M. Le Huy, J.L. Le Mouël, Secousses (jerks) de la variation séculaire et mouvements dans le noyau terrestre. C. R. Acad. Sci. Sér. 2, Méc. Phys. Chim. Sci. Univers Sci. Terre 317(3), 333–341 (1993) Google Scholar
  73. G. Hulot, A. Khokhlov, J.L. Le Mouël, Uniqueness of mainly dipolar magnetic fields recovered from directional data. Geophys. J. Int. 129(2), 347–354 (1997). doi: 10.1111/j.1365-246X.1997.tb01587.x ADSCrossRefGoogle Scholar
  74. G. Hulot, C. Eymin, B. Langlais, M. Mandea, N. Olsen, Small-scale structure of the geodynamo inferred from Oersted and Magsat satellite data. Nature 416(6881), 620–623 (2002). doi: 10.1038/416620a ADSCrossRefGoogle Scholar
  75. G. Hulot, T. Sabaka, N. Olsen, The present field, in Geomagnetism, ed. by M. Kono, G. Schubert. Treatise on Geophysics, vol. 5 (Elsevier, Amsterdam, 2007), Chap. 2 Google Scholar
  76. G. Hulot, N. Olsen, E. Thebault, K. Hemant, Crustal concealing of small-scale core-field secular variation. Geophys. J. Int. 177(2), 361–366 (2009). doi: 10.1111/j.1365-246X.2009.04119.x ADSCrossRefGoogle Scholar
  77. G. Hulot, F. Lhuillier, J. Aubert, Earth’s dynamo limit of predictability. Geophys. Res. Lett. 37, L06305 (2010a). doi: 10.1029/2009GL041869 CrossRefGoogle Scholar
  78. G. Hulot, C.C. Finlay, C.G. Constable, N. Olsen, M. Mandea, The magnetic field of planet Earth. Space Sci. Rev. 152(1–4), 159–222 (2010b). doi: 10.1007/s11214-010-9644-0 ADSCrossRefGoogle Scholar
  79. K. Ide, P. Courtier, M. Ghil, A.C. Lorenc, Unified notation for data assimilation: Operational, sequential and variational. J. Meteorol. Soc. Jpn. 75, 181–189 (1997) Google Scholar
  80. A. Jackson, The Earth’s magnetic field at the core-mantle boundary. Ph.D. Thesis, Cambridge (1989) Google Scholar
  81. A. Jackson, Time-dependency of tangentially geostrophic core surface motions. Phys. Earth Planet. Inter. 103, 293–311 (1997). doi: 10.1016/S0031-9201(97)00039-3 ADSCrossRefGoogle Scholar
  82. A. Jackson, C.C. Finlay, Geomagnetic secular variation and its application to the core, in Geomagnetism, ed. by P. Olson, G. Schubert. Treatise on Geophysics, vol. 5 (Elsevier, Amsterdam, 2007), pp. 148–193, Chap. 5 CrossRefGoogle Scholar
  83. A. Jackson, J. Bloxham, D. Gubbins, Time-dependent flow at the core surface and conservation of angular momentum in the coupled core–mantle system, in Dynamics of Earth’s Deep Interior and Earth Rotation, ed. by J.L. Le Mouël, D.E. Smylie, T. Herring. (American Geophysical Union, Washington, 1993), pp. 97–107 Google Scholar
  84. A. Jackson, A. Jonkers, M. Walker, Four centuries of geomagnetic secular variation from historical records. Philos. Trans. R. Soc. Ser. A, Math. Phys. Eng. Sci. 358(1768), 957–990 (2000) ADSCrossRefGoogle Scholar
  85. D. Jault, Axial invariance of rapidly varying diffusionless motions in the Earth’s core interior. Phys. Earth Planet. Inter. 166(1–2), 67–76 (2008). doi: 10.1016/j.pepi.2007.11.001 ADSGoogle Scholar
  86. D. Jault, C. Gire, J.L. Le Mouël, Westward drift, core motions and exchanges of angular momentum between core and mantle. Nature 333(6171), 353–356 (1988). doi: 10.1038/333353a0 ADSCrossRefGoogle Scholar
  87. C. Jones, N. Weiss, F. Cattaneo, Nonlinear dynamos: a complex generalization of the Lorenz equations. Physica D 14, 161–176 (1985). doi: 10.1016/0167-2789(85)90176-9 zbMATHMathSciNetADSGoogle Scholar
  88. A. Kageyama, T. Sato, Generation mechanism of a dipole field by a magnetohydrodynamic dynamo. Phys. Rev. E 55(4), 4617–4626 (1997). doi: 10.1103/PhysRevE.55.4617 MathSciNetADSGoogle Scholar
  89. E. Kalnay, Atmospheric Modeling, Data Assimilation, and Predictability (Cambridge University Press, Cambridge, 2003) Google Scholar
  90. E. Kalnay, M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen et al., The NCEP/NCAR 40-year reanalysis project. Bull. Am. Meteorol. Soc. 77(3), 437–471 (1996). doi: 10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2 ADSCrossRefGoogle Scholar
  91. E. Kalnay, H. Li, T. Miyoshi, S. Yang, J. Ballabrera-Poy, 4-D-Var or ensemble Kalman filter? Tellus 59A(5), 758–773 (2007a). doi: 10.1111/j.1600-0870.2007.00261.x ADSGoogle Scholar
  92. E. Kalnay, H. Li, T. Miyoshi, S. Yang, J. Ballabrera-Poy, Response to the discussion on “4D-Var or EnKF?” by Nils Gustaffson. Tellus 59A(5), 778–780 (2007b). doi: 10.1111/j.1600-0870.2007.00263.x ADSGoogle Scholar
  93. A. Kelbert, G. Egbert, A. Schultz, Non-linear conjugate gradient inversion for global EM induction: resolution studies. Geophys. J. Int. 173(2), 365–381 (2008). doi: 10.1111/j.1365-246X.2008.03717.x ADSCrossRefGoogle Scholar
  94. K. Korhonen, F. Donadini, P. Riisager, L.J. Pesonen, GEOMAGIA50: An archeointensity database with PHP and MySQL. Geochem. Geophys. Geosyst. 9, Q04029 (2008). doi: 10.1029/2007GC001893 CrossRefGoogle Scholar
  95. M. Korte, C.G. Constable, Continuous geomagnetic field models for the past 7 millennia: 2. CALS7K. Geochem. Geophys. Geosyst. 6(2), Q02H16 (2005). doi: 10.1029/2004GC000801 CrossRefGoogle Scholar
  96. M. Korte, F. Donadini, C.G. Constable, Geomagnetic field for 0–3 ka: 2. A new series of time-varying global models. Geochem. Geophys. Geosyst. 10, Q06008 (2009). doi: 10.1029/2008GC002297 CrossRefGoogle Scholar
  97. W. Kuang, J. Bloxham, An Earth-like numerical dynamo model. Nature 389(6649), 371–374 (1997). doi: 10.1038/38712 ADSCrossRefGoogle Scholar
  98. W. Kuang, A. Tangborn, W. Jiang, D. Liu, Z. Sun, J. Bloxham, Z. Wei, MoSST-DAS: the first generation geomagnetic data assimilation framework. Commun. Comput. Phys. 3, 85–108 (2008) zbMATHGoogle Scholar
  99. W. Kuang, A. Tangborn, Z. Wei, T. Sabaka, Constraining a numerical geodynamo model with 100-years of geomagnetic observations. Geophys. J. Int. 179(3), 1458–1468 (2009). doi: 10.1111/j.1365-246X.2009.04376.x ADSCrossRefGoogle Scholar
  100. W. Kuang, Z. Wei, R. Holme, A. Tangborn, Prediction of geomagnetic field with data assimilation: a candidate secular variation model for IGRF-11. Earth Planets Space (2010, accepted) Google Scholar
  101. A. Kushinov, J. Velímský, P. Tarits, A. Semenov, O. Pankratov, L. Tøffner-Clausen, Z. Martinec, N. Olsen, T.J. Sabaka, A. Jackson, Level 2 products and performances for mantle studies with Swarm. ESA Technical Report (2010) Google Scholar
  102. F.X. Le Dimet, O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus 38(2), 97–110 (1986) Google Scholar
  103. B. Lehnert, Magnetohydrodynamic waves under the action of the Coriolis force. Astrophys. J. 119, 647–654 (1954). doi: 10.1086/145869 MathSciNetADSCrossRefGoogle Scholar
  104. V. Lesur, I. Wardinski, Comment on“Can core-surface flow models be used to improve the forecast of the Earth’s main magnetic field?” by Stefan Maus, Luis Silva, and Gauthier Hulot. J. Geophys. Res. 114, B04104 (2009). doi: 10.1029/2008JB006188 CrossRefGoogle Scholar
  105. V. Lesur, I. Wardinski, M. Rother, M. Mandea, GRIMM: the GFZ reference internal magnetic model based on vector satellite and observatory data. Geophys. J. Int. 173(2), 382–394 (2008). doi: 10.1111/j.1365-246X.2008.03724.x ADSCrossRefGoogle Scholar
  106. V. Lesur, I. Wardinski, S. Asari, B. Minchev, M. Mandea, Modelling the Earth’s core magnetic field under flow constraints. Earth Planets Space (2010). doi: 10.5047/eps.2010.02.010 Google Scholar
  107. L. Liu, M. Gurnis, Simultaneous inversion of mantle properties and initial conditions using an adjoint of mantle convection. J. Geophys. Res. 113, B8405 (2008). doi: 10.1029/2008JB005594 CrossRefGoogle Scholar
  108. D. Liu, A. Tangborn, W. Kuang, Observing system simulation experiments in geomagnetic data assimilation. J. Geophys. Res. 112, B8 (2007). doi: 10.1029/2006JB004691 Google Scholar
  109. L. Liu, S. Spasojevic, M. Gurnis, Reconstructing Farallon plate subduction beneath North America back to the late cretaceous. Science 322(5903), 934–938 (2008). doi: 10.1126/science.1162921 ADSCrossRefGoogle Scholar
  110. P.W. Livermore, G.R. Ierley, A. Jackson, The construction of exact Taylor states. I: The full sphere. Geophys. J. Int. 179(2), 923–928 (2009). doi: 10.1111/j.1365-246X.2009.04340.x ADSCrossRefGoogle Scholar
  111. P.W. Livermore, G.R. Ierley, A. Jackson, The construction of exact Taylor states. II: The influence of an inner core. Phys. Earth Planet. Inter. 178, 16–26 (2010). doi: 10.1016/j.pepi.2009.07.015 ADSCrossRefGoogle Scholar
  112. A.C. Lorenc, Analysis methods for numerical weather prediction. Q. J. R. Meteorol. Soc. 112(474), 1177–1194 (1986). doi: 10.1002/qj.49711247414 ADSCrossRefGoogle Scholar
  113. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963) MathSciNetADSCrossRefGoogle Scholar
  114. S. Maus, S. Macmillan, T. Chernova, S. Choi, D. Dater, V. Golovkov, V. Lesur, F. Lowes, H. Lühr, W. Mai, S. McLean, N. Olsen, M. Rother, T. Sabaka, A. Thomson, T. Zvereva, The 10th-generation international geomagnetic reference field. Geophys. J. Int. 161, 561–565 (2005). doi: 10.1111/j.1365-246X.2005.02641.x CrossRefGoogle Scholar
  115. S. Maus, L. Silva, G. Hulot, Can core-surface flow models be used to improve the forecast of the Earth’s main magnetic field? J. Geophys. Res. 113, B08102 (2008). doi: 10.1029/2007JB005199 CrossRefGoogle Scholar
  116. S. Maus, L. Silva, G. Hulot, Reply to comment by V. Lesur et al. on “Can core-surface flow models be used to improve the forecast of the Earth’s main magnetic field”. J. Geophys. Res. 114, B04105 (2009). doi: 10.1029/2008JB006242 CrossRefGoogle Scholar
  117. H. Meyers, W.M. Davis, A profile of the geomagnetic model users and abusers. J. Geomagn. Geoelectr. 42(9), 1079–1085 (1990) Google Scholar
  118. R.N. Miller, M. Ghil, F. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems. J. Atmos. Sci. 51(8), 1037–1056 (1994). doi: 10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2 MathSciNetADSCrossRefGoogle Scholar
  119. R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin, P. Odier, J.F. Pinton, R. Volk, S. Fauve, N. Mordant, F. Pétrélis, et al., Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98(4), 044502 (2007). doi: 10.1103/PhysRevLett.98.044502 ADSCrossRefGoogle Scholar
  120. H.C. Nataf, T. Alboussière, D. Brito, P. Cardin, N. Gagnière, D. Jault, J.P. Masson, D. Schmitt, Experimental study of super-rotation in a magnetostrophic spherical Couette flow. Geophys. Astrophys. Fluid Dyn. 100, 281–298 (2006). doi: 10.1080/03091920600718426 ADSCrossRefGoogle Scholar
  121. N. Olsen, M. Mandea, Rapidly changing flows in the earth’s core. Nat. Geosci. 1, 390–394 (2008). doi: 10.1038/ngeo203 ADSCrossRefGoogle Scholar
  122. N. Olsen, R. Holme, G. Hulot, T. Sabaka, T. Neubert, L. Tøffner-Clausen, F. Primdahl, J. Jørgensen, J. Léger, D. Barraclough, J. Bloxham, J. Cain, C. Constable, V. Golovkov, A. Jackson, P. Kotze, B. Langlais, S. Macmillan, M. Mandea, J. Merayo, L. Newitt, M. Purucker, T. Risbo, M. Stampe, A. Thomson, C. Voorhies, Ørsted initial field model. Geophys. Res. Lett. 27(22), 3607–3610 (2000). doi: 10.1029/2000GL011930 ADSCrossRefGoogle Scholar
  123. N. Olsen, H. Lühr, T.J. Sabaka, M. Mandea, M. Rother, L. Tøffner-Clausen, S. Choi, CHAOS-a model of the Earth’s magnetic field derived from CHAMP, Ørsted, and SAC-C magnetic satellite data. Geophys. J. Int. 166(1), 67–75 (2006). doi: 10.1111/j.1365-246X.2006.02959.x ADSCrossRefGoogle Scholar
  124. N. Olsen, M. Mandea, T. Sabaka, L. Tøffner-Clausen, CHAOS-2—a geomagnetic field model derived from one decade of continuous satellite data. Geophys. J. Int. 179(3), 1477–1487 (2009). doi: 10.1111/j.1365-246X.2009.04386.x ADSCrossRefGoogle Scholar
  125. A. Pais, G. Hulot, Length of day decade variations, torsional oscillations and inner core superrotation: evidence from recovered core surface zonal flows. Phys. Earth Planet. Inter. 118(3–4), 291–316 (2000). doi: 10.1016/S0031-9201(99)00161-2 ADSCrossRefGoogle Scholar
  126. T. Penduff, P. Brasseur, C. Testut, B. Barnier, J. Verron, A four-year eddy-permitting assimilation of sea-surface temperature and altimetric data in the South Atlantic Ocean. J. Mar. Res. 60(6), 805–833 (2002). doi: 10.1357/002224002321505147 CrossRefGoogle Scholar
  127. K. Pinheiro, A. Jackson, Can a 1-D mantle electrical conductivity model generate magnetic jerk differential time delays? Geophys. J. Int. 173(3), 781–792 (2008). doi: 10.1111/j.1365-246X.2008.03762.x ADSCrossRefGoogle Scholar
  128. L.F. Richardson, Weather Prediction by Numerical Process (Cambridge University Press, Cambridge, 1922) zbMATHGoogle Scholar
  129. P.H. Roberts, S. Scott, On analysis of the secular variation. J. Geomagn. Geoelectr. 17(2), 137–151 (1965) Google Scholar
  130. T. Sabaka, N. Olsen, M. Purucker, Extending comprehensive models of the Earth’s magnetic field with Ørsted and CHAMP data. Geophys. J. Int. 159(2), 521–547 (2004). doi: 10.1111/j.1365-246X.2004.02421.x ADSCrossRefGoogle Scholar
  131. A. Sakuraba, P.H. Roberts, Generation of a strong magnetic field using uniform heat flux at the surface of the core. Nat. Geosci. 2, 802–805 (2009). doi: 10.1038/ngeo643 ADSCrossRefGoogle Scholar
  132. M. Sambridge, P. Rickwood, N. Rawlinson, S. Sommacal, Automatic differentiation in geophysical inverse problems. Geophys. J. Int. 170(1), 1–8 (2007). doi: 10.1111/j.1365-246X.2007.03400.x ADSCrossRefGoogle Scholar
  133. Y. Sasaki, Some basic formalisms in numerical variational analysis. Mon. Weather Rev. 98(12), 875–883 (1970). doi: 10.1175/1520-0493(1970)098<0875:SBFINV>2.3.CO;2 ADSCrossRefGoogle Scholar
  134. L. Scherliess, R.W. Schunk, J.J. Sojka, D.C. Thompson, L. Zhu, Utah State University Global Assimilation of Ionospheric Measurements Gauss-Markov Kalman filter model of the ionosphere: Model description and validation. J. Geophys. Res. 111, A11315 (2006). doi: 10.1029/2006JA011712 ADSCrossRefGoogle Scholar
  135. D. Schmitt, T. Alboussière, D. Brito, P. Cardin, N. Gagnière, D. Jault, H.C. Nataf, Rotating spherical Couette flow in a dipolar magnetic field: experimental study of magneto-inertial waves. J. Fluid Mech. 604, 175–197 (2008). doi: 10.1017/S0022112008001298 zbMATHADSCrossRefGoogle Scholar
  136. Z. Sun, A. Tangborn, W. Kuang, Data assimilation in a sparsely observed one-dimensional modeled MHD system. Nonlinear Process. Geophys. 14(2), 181–192 (2007) ADSCrossRefGoogle Scholar
  137. F. Takahashi, M. Matsushima, Y. Honkura, Scale variability in convection-driven mhd dynamos at low Ekman number. Phys. Earth Planet. Inter. 167, 168–178 (2008). doi: 10.1016/j.pepi.2008.03.005 ADSCrossRefGoogle Scholar
  138. O. Talagrand, The use of adjoint equations in numerical modelling of the atmospheric circulation, in Automatic Differentiation of Algorithms: Theory, Implementation, and Application, ed. by A. Griewank, G.G. Corliss. (Society for Industrial and Applied Mathematics, Philadelphia, 1991), pp. 169–180 Google Scholar
  139. O. Talagrand, Assimilation of observations, an introduction. J. Meteorol. Soc. Jpn. 75(1B), 191–209 (1997) MathSciNetGoogle Scholar
  140. O. Talagrand, A posteriori validation of assimilation algorithms, in Data Assimilation for the Earth System, ed. by R. Swinbank, V. Shutyaev, W. Lahoz. (Kluwer Academic, Dordrecht, 2003), pp. 85–95 Google Scholar
  141. O. Talagrand, P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Q. J. R. Meteorol. Soc. 113(478), 1311–1328 (1987). doi: 10.1002/gj.49711347812 ADSCrossRefGoogle Scholar
  142. A. Tarantola, Inversion of seismic reflection data in the acoustic approximation. Geophysics 49(8), 1259–1266 (1984). doi: 10.1190/1.1441754 ADSCrossRefGoogle Scholar
  143. A. Tarantola, Theoretical background for the inversion of seismic waveforms including elasticity and attenuation. Pure Appl. Geophys. 128(1), 365–399 (1988). doi: 10.1007/BF01772605 ADSCrossRefGoogle Scholar
  144. J.B. Taylor, The magneto-hydrodynamics of a rotating fluid and the earth’s dynamo problem. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 274(1357), 274–283 (1963) zbMATHADSCrossRefGoogle Scholar
  145. E. Thébault, A. Chulliat, S. Maus, G. Hulot, B. Langlais, A. Chambodut, M. Menvielle, IGRF candidate models at times of rapid changes in core field acceleration. Earth Planets Space (2010). doi: 10.5047/eps.2010.05.004 Google Scholar
  146. Y. Trémolet, Accounting for an imperfect model in 4D-Var. Q. J. R. Meteorol. Soc. 132(621), 2483–2504 (2006). doi: 10.1256/qj.05.224 ADSCrossRefGoogle Scholar
  147. J. Tromp, C. Tape, Q. Liu, Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys. J. Int. 160(1), 195–216 (2005). doi: 10.1111/j.1365-246X.2004.02453.x ADSCrossRefGoogle Scholar
  148. J. Tromp, D. Komatitsch, Q. Liu, Spectral-element and adjoint methods in seismology. Commun. Comput. Phys. 3, 1–32 (2008) zbMATHGoogle Scholar
  149. N.A. Tsyganenko, M.I. Sitnov, Magnetospheric configurations from a high-resolution data-based magnetic field model. J. Geophys. Res. 112, A06225 (2007). doi: 10.1029/2007JA012260 CrossRefGoogle Scholar
  150. F. Uboldi, M. Kamachi, Time-space weak-constraint data assimilation for nonlinear models. Tellus A 52(4), 412–421 (2000). doi: 10.1034/j.1600-0870.2000.00878.x ADSCrossRefGoogle Scholar
  151. R. Waddington, D. Gubbins, N. Barber, Geomagnetic field analysis-V. Determining steady core-surface flows directly from geomagnetic observations. Geophys. J. Int. 122(1), 326–350 (1995). doi: 10.1111/j.1365-246X.1995.tb03556.x ADSCrossRefGoogle Scholar
  152. P. Wessel, W.H.F. Smith, Free software helps map and display data. Trans. Am. Geophys. Union 72, 441–445 (1991). doi: 10.1029/90EO00319 ADSCrossRefGoogle Scholar
  153. K. Whaler, D. Gubbins, Spherical harmonic analysis of the geomagnetic field: an example of a linear inverse problem. Geophys. J. R. Astron. Soc. 65(3), 645–693 (1981). doi: 10.1111/j.1365-246X.1981.tb04877.x zbMATHADSGoogle Scholar
  154. C. Wunsch, Discrete Inverse and State Estimation Problems (Cambridge University Press, Cambridge, 2006) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Alexandre Fournier
    • 1
    Email author
  • Gauthier Hulot
    • 1
  • Dominique Jault
    • 2
  • Weijia Kuang
    • 3
  • Andrew Tangborn
    • 4
  • Nicolas Gillet
    • 2
  • Elisabeth Canet
    • 2
    • 5
  • Julien Aubert
    • 6
  • Florian Lhuillier
    • 7
  1. 1.GéomagnétismeInstitut de Physique du Globe de Paris, Université Paris Diderot, CNRSParis cedex 5France
  2. 2.Laboratoire de Géophysique Interne et TectonophysiqueCNRS, Université Joseph-FourierGrenobleFrance
  3. 3.Planetary Geodynamics LaboratoryGoddard Space Flight CenterGreenbeltUSA
  4. 4.Joint Center for Earth Systems TechnologyUniversity of Maryland Baltimore CountyBaltimoreUSA
  5. 5.Institut für GeophysikETH ZürichZürichSwitzerland
  6. 6.Dynamique des Fluides GéologiquesInstitut de Physique du Globe de ParisParisFrance
  7. 7.Géomagnétisme & Dynamique des Fluides GéologiquesInstitut de Physique du Globe de ParisParisFrance

Personalised recommendations