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Chronometric Geodesy: Methods and Applications

  • Pacome DelvaEmail author
  • Heiner Denker
  • Guillaume Lion
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 196)

Abstract

The theory of general relativity was born more than one hundred years ago, and since the beginning has striking prediction success. The gravitational redshift effect discovered by Einstein must be taken into account when comparing the frequencies of distant clocks. However, instead of using our knowledge of the Earth’s gravitational field to predict frequency shifts between distant clocks, one can revert the problem and ask if the measurement of frequency shifts between distant clocks can improve our knowledge of the gravitational field. This is known as chronometric geodesy. Since the beginning of the atomic time era in 1955, the accuracy and stability of atomic clocks were constantly ameliorated, with around one order of magnitude gained every ten years. Now that the atomic clock accuracy reaches the low \(10^{-18}\) in fractional frequency, and can be compared to this level over continental distances with optical fibres, the accuracy of chronometric geodesy reaches the cm level and begins to be competitive with classical geodetic techniques such as geometric levelling and GNSS/geoid levelling. Moreover, the building of global timescales requires now to take into account these effects to the best possible accuracy. In this chapter we explain how atomic clock comparisons and the building of timescales can benefit from the latest developments in physical geodesy for the modelization and realization of the geoid, as well as how classical geodesy could benefit from this new type of observable, which are clock comparisons that are directly linked to gravity potential differences.

Notes

Acknowledgements

The authors would like to thank Jérôme Lodewyck (SYRTE/Paris Observatory) for providing Fig. 1, and Martina Sacher (Bundesamt für Kartographie und Geodäsie, BKG, Leipzig, Germany) for providing information on the EVRF2007 heights and uncertainties, the associated height transformations, and a new UELN adjustment in progress. This research was supported by the European Metrology Research Programme (EMRP) within the Joint Research Project “International Timescales with Optical Clocks” (SIB55 ITOC), as well as the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Centre 1128 “Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q)”, project C04. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. We gratefully acknowledge financial support from Labex FIRST-TF and ERC AdOC (Grant No. 617553).

References

  1. 1.
    R.V. Pound, G.A. Rebka, Resonant absorption of the 14.4-kev \(\gamma \) ray from 0.10-\(\mu \)sec \({\rm fe}^{57}\). Phys. Rev. Lett. 3(12), 554–556 (1959)ADSCrossRefGoogle Scholar
  2. 2.
    R.V. Pound, G.A. Rebka, Gravitational red-shift in nuclear resonance. Phys. Rev. Lett. 3(9), 439–441 (1959)ADSCrossRefGoogle Scholar
  3. 3.
    R.V. Pound, G.A. Rebka, Apparent weight of photons. Phys. Rev. Lett. 4(7), 337–341 (1960)ADSCrossRefGoogle Scholar
  4. 4.
    R.V. Pound, J.L. Snider, Effect of gravity on gamma radiation. Phys. Rev. 140(3B), B788–B803 (1965)ADSCrossRefGoogle Scholar
  5. 5.
    Norman F. Ramsey, History of early atomic clocks. Metrologia 42(3), S1 (2005)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Sigfrido Leschiutta, The definition of the ‘atomic’ second. Metrologia 42(3), S10 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    J. Terrien, News from the international bureau of weights and measures. Metrologia 4(1), 41 (1968)ADSCrossRefGoogle Scholar
  8. 8.
    Leonard S. Cutler, Fifty years of commercial caesium clocks. Metrologia 42(3), S90 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    B. Guinot, E.F. Arias, Atomic time-keeping from 1955 to the present. Metrologia 42(3), S20 (2005)CrossRefGoogle Scholar
  10. 10.
    J.C. Hafele, R.E. Keating, Around-the-world atomic clocks: predicted relativistic time gains. Science 177(4044), 166–168 (1972)ADSCrossRefGoogle Scholar
  11. 11.
    J.C. Hafele, R.E. Keating, Around-the-world atomic clocks: observed relativistic time gains. Science 177, 168–170 (1972)ADSCrossRefGoogle Scholar
  12. 12.
    L. Briatore, S. Leschiutta, Evidence for the Earth gravitational shift by direct atomic-time-scale comparison. Nuovo Cim. B 37(2), 219–231 (1977)ADSCrossRefGoogle Scholar
  13. 13.
    Jean-Marc Lévy-Leblond, One more derivation of the Lorentz transformation. Am. J. Phys. 44(3), 271–277 (1976)ADSCrossRefGoogle Scholar
  14. 14.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973)Google Scholar
  15. 15.
    Jeffrey M. Cohen, Harry E. Moses, New test of the synchronization procedure in noninertial systems. Phys. Rev. Lett. 39(26), 1641–1643 (1977)ADSCrossRefGoogle Scholar
  16. 16.
    J.M. Cohen, H.E. Moses, A. Rosenblum, Clock-transport synchronization in noninertial frames and gravitational fields. Phys. Rev. Lett. 51(17), 1501–1502 (1983)ADSCrossRefGoogle Scholar
  17. 17.
    J.M. Cohen, H.E. Moses, A. Rosenblum, Electromagnetic synchronisation of clocks with finite separation in a rotating system. Class. Quantum Grav. 1(6), L57 (1984)ADSCrossRefGoogle Scholar
  18. 18.
    M.F. Podlaha, Note on the Cohen, Moses and Rosenblum letter about the slow-clock transport synchronization in noninertial reference systems. Lett. Nuovo Cim. 40(7), 223–224 (1984)ADSCrossRefGoogle Scholar
  19. 19.
    N. Ashby, D.W. Allan, Coordinate time on and near the Earth. Phys. Rev. Lett. 53(19), 1858–1858 (1984)ADSCrossRefGoogle Scholar
  20. 20.
    N. Ashby, D.W. Allan, Coordinate time on and near the Earth (erratum). Phys. Rev. Lett. 54(3), 254–254 (1985)ADSCrossRefGoogle Scholar
  21. 21.
    M. Born, Einstein’s Theory of Relativity (Dover Publications, New York, 1962)Google Scholar
  22. 22.
    C. Møller, Theory of Relativity, 2nd edn. (Oxford University Press, Oxford, 1976)Google Scholar
  23. 23.
    N. Ashby, D.W. Allan, Practical implications of relativity for a global coordinate time scale. Radio Sci. 14(4), 649–669 (1979)ADSCrossRefGoogle Scholar
  24. 24.
    D.W. Allan, N. Ashby, Coordinate Time in the Vicinity of the Earth, vol. 114 (1986), pp. 299–312CrossRefGoogle Scholar
  25. 25.
    A.J. Skalafuris, Current theoretical attempts toward synchronization of a global satellite network. Radio Sci. 20(6), 1529–1536 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    N. Ashby, Relativity in the global positioning system. Living Rev. Relativ. 6, 1 (2003)ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    B. Coll, A Principal Positioning System for the Earth, vol. 14 (2003), pp. 34–38, arXiv:gr-qc/0306043
  28. 28.
    Carlo Rovelli, GPS observables in general relativity. Phys. Rev. D 65(4), 044017 (2002)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Marc Lachièze-Rey, The covariance of GPS coordinates and frames. Class. Quantum Grav. 23(10), 3531 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    B. Coll, J.A. Morales, Symmetric frames on Lorentzian spaces. J. Math. Phys. 32(9), 2450–2455 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Blagojević, J. Garecki, F.W. Hehl, Y.N. Obukhov, Real null coframes in general relativity and GPS type coordinates. Phys. Rev. D 65(4), 044018 (2002)Google Scholar
  32. 32.
    P. Delva, U. Kostić, A. Čadež, Numerical modeling of a Global Navigation Satellite System in a general relativistic framework. Adv. Space Res. 47(2), 370–379 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    D. Bini, A. Geralico, M.L. Ruggiero, A. Tartaglia, Emission versus Fermi coordinates: applications to relativistic positioning systems. Class. Quantum Grav. 25(20), 205011 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    A. Tartaglia, Emission coordinates for the navigation in space. Acta Astronaut. 67(5), 539–545 (2010)ADSCrossRefGoogle Scholar
  35. 35.
    N. Puchades, D. Sáez, Relativistic positioning: four-dimensional numerical approach in Minkowski space-time. Astrophys. Space Sci. 341(2), 631–643 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    D. Bunandar, S.A. Caveny, R.A. Matzner, Measuring emission coordinates in a pulsar-based relativistic positioning system. Phys. Rev. D 84(10), 104005 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    V.A. Brumberg, General discussion, in Relativity in celestial mechanics and astrometry, proceedings of the IAU symposium No.114, ed. by J. Kovalesky and V.A. Brumberg (D. Reidel publishing company, 1986)Google Scholar
  38. 38.
    B. Guinot, Is the International Atomic Time TAI a coordinate time or a proper time? Celest. Mech. 38, 155–161 (1986)ADSCrossRefGoogle Scholar
  39. 39.
    B. Guinot, P.K. Seidelmann, Time scales - their history, definition and interpretation. Astron. Astrophys. 194, 304–308 (1988)ADSGoogle Scholar
  40. 40.
    T.-Y. Huang, B.-X. Xu, J. Zhu, H. Zhang, The concepts of International Atomic Time (TAI) and Terrestrial Dynamic Time (TDT). Astron. Astrophys. 220, 329–334 (1989)ADSGoogle Scholar
  41. 41.
    V.A. Brumberg, S.M. Kopeikin, Relativistic time scales in the solar system. Celest. Mech. Dyn. Astron. 48, 23–44 (1990)ADSzbMATHCrossRefGoogle Scholar
  42. 42.
    S.A. Klioner, The problem of clock synchronization: a relativistic approach. Celest. Mech. Dyn. Astron. 53(1), 81–109 (1992)ADSCrossRefGoogle Scholar
  43. 43.
    N. Ashby, B. Bertotti, Relativistic perturbations of an earth satellite. Phys. Rev. Lett. 52(7), 485–488 (1984)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    T. Fukushima, The Fermi coordinate system in the post-Newtonian framework. Celest. Mech. 44, 61–75 (1988)ADSzbMATHCrossRefGoogle Scholar
  45. 45.
    N. Ashby, B. Bertotti, Relativistic effects in local inertial frames. Phys. Rev. D 34(8), 2246–2259 (1986)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    S.M. Kopejkin, Celestial coordinate reference systems in curved space-time. Celest. Mech. 44, 87–115 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    M.H. Soffel, Relativity in Astrometry, Celestial Mechanics, and Geodesy (Springer, Berlin, 1989)CrossRefGoogle Scholar
  48. 48.
    V.A. Brumberg, S.M. Kopejkin, Relativistic theory of celestial reference frames, in Reference Frames, Astrophysics and Space Science Library, vol. 154, ed. by J. Kovalevsky, I.I. Mueller, B. Kolaczek (Springer, Netherlands, 1989), pp. 115–141CrossRefGoogle Scholar
  49. 49.
    V.A. Brumberg, S.M. Kopejkin, Relativistic reference systems and motion of test bodies in the vicinity of the earth. Nuovo Cim. 103(1), 63–98 (1989)ADSCrossRefGoogle Scholar
  50. 50.
    T. Damour, M. Soffel, C. Xu, General-relativistic celestial mechanics. I. Method and definition of reference systems. Phys. Rev. D 43(10), 3273–3307 (1991)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    S.M. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley, New York, 2011)zbMATHCrossRefGoogle Scholar
  52. 52.
    U. Kostić, M. Horvat, A. Gomboc, Relativistic positioning system in perturbed spacetime. Class. Quantum Grav. 32(21), 215004 (2015)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    D. Bini, B. Mashhoon, Relativistic gravity gradiometry. Phys. Rev. D 94(12), 124009 (2016)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    M. Soffel, S.A. Klioner, G. Petit, P. Wolf, S.M. Kopeikin, P. Bretagnon, V.A. Brumberg, N. Capitaine, T. Damour, T. Fukushima, B. Guinot, T.-Y. Huang, L. Lindegren, C. Ma, K. Nordtvedt, J.C. Ries, P.K. Seidelmann, D. Vokrouhlický, C.M. Will, C. Xu, The IAU 2000 resolutions for astrometry, celestial mechanics, and metrology in the relativistic framework: explanatory supplement. Astron. J. 126(6), 2687 (2003)ADSCrossRefGoogle Scholar
  55. 55.
    S.A. Klioner, N. Capitaine, W.M. Folkner, B. Guinot, T.-Y. Huang, S.M. Kopeikin, E.V. Pitjeva, P.K. Seidelmann, M.H. Soffel, Units of relativistic time scales and associated quantities, in Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis, Proceedings of the International Astronomical Union, vol. 5 (2009), pp. 79–84CrossRefGoogle Scholar
  56. 56.
    E.F. Arias, G. Panfilo, G. Petit, Timescales at the BIPM. Metrologia 48(4), S145 (2011)ADSCrossRefGoogle Scholar
  57. 57.
    A. Bjerhammar, Discrete approaches to the solution of the boundary value problem in physical geodesy. Bolletino di geodesia e scienze affini 2, 185–241 (1975)Google Scholar
  58. 58.
    M. Vermeer, Chronometric levelling, Technical report, Finnish Geodetic Institute, Helsinki (1983)Google Scholar
  59. 59.
    A. Bjerhammar, On a relativistic geodesy. Bull. Geodesique 59(3), 207–220 (1985)ADSCrossRefGoogle Scholar
  60. 60.
    H.S. Margolis, R.M. Godun, P. Gill, L.A.M. Johnson, S.L. Shemar, P.B. Whibberley, D. Calonico, F. Levi, L. Lorini, M. Pizzocaro, P. Delva, S. Bize, J. Achkar, H. Denker, L. Timmen, C. Voigt, S. Falke, D. Piester, C. Lisdat, U. Sterr, S. Vogt, S. Weyers, J. Gersl, T. Lindvall, M. Merimaa, International timescales with optical clocks (ITOC), in European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC), 2013 Joint (2013), pp. 908–911Google Scholar
  61. 61.
    J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F.N. Baynes, H.S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G.A. Costanzo, C. Clivati, F. Levi, D. Calonico, Geodesy and metrology with a transportable optical. Nat. Phys. 14(5), 437 (2018)Google Scholar
  62. 62.
    V.A. Brumberg, E. Groten, On determination of heights by using terrestrial clocks and GPS signals. J. Geod. 76(1), 49–54 (2002)ADSzbMATHCrossRefGoogle Scholar
  63. 63.
    R. Bondarescu, M. Bondarescu, G. Hetényi, L. Boschi, P. Jetzer, J. Balakrishna, Geophysical applicability of atomic clocks: direct continental geoid mapping. Geophys. J. Int. 191(1), 78–82 (2012)ADSCrossRefGoogle Scholar
  64. 64.
    C.W. Chou, D.B. Hume, T. Rosenband, D.J. Wineland, Optical clocks and relativity. Science 329(5999), 1630–1633 (2010)ADSCrossRefGoogle Scholar
  65. 65.
    G. Lion, I. Panet, P. Wolf, C. Guerlin, S. Bize, P. Delva, Determination of a high spatial resolution geopotential model using atomic clock comparisons. J. Geod. (2017), pp. 1–15Google Scholar
  66. 66.
    A. Bjerhammar, Relativistic geodesy. Technical Report NON118 NGS36, NOAA Technical Report (1986)Google Scholar
  67. 67.
    M. Soffel, H. Herold, H. Ruder, M. Schneider, Relativistic theory of gravimetric measurements and definition of the geoid. Manuscr. Geod. 13, 143–146 (1988)ADSGoogle Scholar
  68. 68.
    B. Hofmann-Wellenhof, H. Moritz, Physical Geodesy (Springer Science & Business Media, 2006)Google Scholar
  69. 69.
    S.M. Kopejkin, Relativistic manifestations of gravitational fields in gravimetry and geodesy. Manuscr. Geod. 16, 301–312 (1991)ADSGoogle Scholar
  70. 70.
    J. Müller, M. Soffel, S.A. Klioner, Geodesy and relativity. J. Geod. 82(3), 133–145 (2007)ADSzbMATHCrossRefGoogle Scholar
  71. 71.
    S.M. Kopeikin, E.M. Mazurova, A.P. Karpik, Towards an exact relativistic theory of Earth’s geoid undulation. Phys. Lett. A 379(26–27), 1555–1562 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    S.M. Kopeikin, W. Han, E. Mazurova, Post-Newtonian reference ellipsoid for relativistic geodesy. Phys. Rev. D 93(4), 044069 (2016)ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    S.M. Kopeikin, Reference ellipsoid and geoid in chronometric geodesy. Front. Astron. Space Sci. 3 (2016)Google Scholar
  74. 74.
    J. Guéna, S. Weyers, M. Abgrall, C. Grebing, V. Gerginov, P. Rosenbusch, S. Bize, B. Lipphardt, H. Denker, N. Quintin, S.M.F. Raupach, D. Nicolodi, F. Stefani, N. Chiodo, S. Koke, A. Kuhl, F. Wiotte, F. Meynadier, E. Camisard, C. Chardonnet, Y. Le Coq, M. Lours, G. Santarelli, A. Amy-Klein, R. Le Targat, O. Lopez, P.E. Pottie, G. Grosche, First international comparison of fountain primary frequency standards via a long distance optical fiber link. Metrologia 54(3), 348 (2017)ADSCrossRefGoogle Scholar
  75. 75.
    C. Lisdat, G. Grosche, N. Quintin, C. Shi, S.M.F. Raupach, C. Grebing, D. Nicolodi, F. Stefani, A. Al-Masoudi, S. Dörscher, S. Häfner, J.-L. Robyr, N. Chiodo, S. Bilicki, E. Bookjans, A. Koczwara, S. Koke, A. Kuhl, F. Wiotte, F. Meynadier, E. Camisard, M. Abgrall, M. Lours, T. Legero, H. Schnatz, U. Sterr, H. Denker, C. Chardonnet, Y. Le Coq, G. Santarelli, A. Amy-Klein, R. Le Targat, J. Lodewyck, O. Lopez, P.-E. Pottie, A clock network for geodesy and fundamental science. Nat. Commun. 7, 12443 (2016)ADSCrossRefGoogle Scholar
  76. 76.
    M. Schioppo, R.C. Brown, W.F. McGrew, N. Hinkley, R.J. Fasano, K. Beloy, T.H. Yoon, G. Milani, D. Nicolodi, J.A. Sherman, N.B. Phillips, C.W. Oates, A.D. Ludlow, Ultrastable optical clock with two cold-atom ensembles. Nat. Photonics 11(1), 48–52 (2017)ADSCrossRefGoogle Scholar
  77. 77.
    N. Huntemann, C. Sanner, B. Lipphardt, Chr. Tamm, E. Peik, Single-ion atomic clock with \(3\times {}{10}^{-18}\) systematic uncertainty. Phys. Rev. Lett. 116(6), 063001 (2016)Google Scholar
  78. 78.
    H.S. Margolis, P. Gill, Least-squares analysis of clock frequency comparison data to deduce optimized frequency and frequency ratio values. Metrologia 52(5), 628 (2015)ADSCrossRefGoogle Scholar
  79. 79.
    H.S. Margolis, P. Gill, Determination of optimized frequency and frequency ratio values from over-determined sets of clock comparison data. J. Phys.: Conf. Ser. 723(1), 012060 (2016)Google Scholar
  80. 80.
    P. Wolf, G. Petit, Relativistic theory for clock syntonization and the realization of geocentric coordinate times. Astron. Astrophys. 304, 653 (1995)ADSGoogle Scholar
  81. 81.
    H. Denker, L. Timmen, C. Voigt, S. Weyers, E. Peik, H.S. Margolis, P. Delva, P. Wolf, G. Petit, Geodetic methods to determine the relativistic redshift at the level of \(10^{-18}\) in the context of international timescales – a review and practical results. J. Geod. 92(5), 487–516 (2018)Google Scholar
  82. 82.
    W. Torge, Geodesy, 2nd edn. (Berlin; New York, W. de Gruyter, 1991)Google Scholar
  83. 83.
    A. Bauch, Time and frequency comparisons using radiofrequency signals from satellites. Comptes Rendus Phys. 16(5), 471–479 (2015)CrossRefGoogle Scholar
  84. 84.
    E. Samain, Clock comparison based on laser ranging technologies. Int. J. Mod. Phys. D 24(08), 1530021 (2015)ADSCrossRefGoogle Scholar
  85. 85.
    G. Petit, A. Kanj, S. Loyer, J. Delporte, F. Mercier, F. Perosanz, \(1\times 10^{-16}\) frequency transfer by GPS PPP with integer ambiguity resolutio. Metrologia 52(2), 301 (2015)ADSCrossRefGoogle Scholar
  86. 86.
    S. Droste, C. Grebing, J. Leute, S.M.F. Raupach, A. Matveev, T.W. Hänsch, A. Bauch, R. Holzwarth, G. Grosche, Characterization of a 450 km baseline GPS carrier-phase link using an optical fiber link. New J. Phys. 17(8), 083044 (2015)ADSCrossRefGoogle Scholar
  87. 87.
    J. Leute, N. Huntemann, B. Lipphardt, C. Tamm, P.B.R. Nisbet-Jones, S.A. King, R.M. Godun, J.M. Jones, H.S. Margolis, P.B. Whibberley, A. Wallin, M. Merimaa, P. Gill, E. Peik, Frequency comparison of \(^{171}\rm Yb^+\) ion optical clocks at ptb and npl via GPS PPP. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63(7), 981–985 (2016)CrossRefGoogle Scholar
  88. 88.
    P. Dubé, J.E. Bernard, M. Gertsvolf, Absolute frequency measurement of the 88 Sr + clock transition using a GPS link to the SI second. Metrologia 54(3), 290 (2017)ADSCrossRefGoogle Scholar
  89. 89.
    C.F.A. Baynham, R.M. Godun, J.M. Jones, S.A. King, P.B.R. Nisbet-Jones, F. Baynes, A. Rolland, P.E.G. Baird, K. Bongs, P. Gill, H.S. Margolis, Absolute frequency measurement of the optical clock transition in with an uncertainty of using a frequency link to international atomic time. J. Mod. Opt. 65(2), 221–227 (2018)Google Scholar
  90. 90.
    D. Kirchner, Two-way time transfer via communication satellites. Proc. IEEE 79(7), 983–990 (1991)ADSCrossRefGoogle Scholar
  91. 91.
    D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, O. Koudelka, Time transfer with nanosecond accuracy for the realization of International atomic time. Metrologia 45(2), 185 (2008)ADSCrossRefGoogle Scholar
  92. 92.
    M. Fujieda, T. Gotoh, J. Amagai, Advanced two-way satellite frequency transfer by carrier-phase and carrier-frequency measurements. J. Phys.: Conf. Ser. 723(1), 012036 (2016)Google Scholar
  93. 93.
    H. Hachisu, M. Fujieda, S. Nagano, T. Gotoh, A. Nogami, T. Ido, St Falke, N. Huntemann, C. Grebing, B. Lipphardt, Ch. Lisdat, D. Piester, Direct comparison of optical lattice clocks with an intercontinental baseline of 9000 km. Opt. Lett. 39(14), 4072–4075 (2014)ADSCrossRefGoogle Scholar
  94. 94.
    F. Meynadier, P. Delva, C. le Poncin-Lafitte, C. Guerlin, P. Wolf, Atomic clock ensemble in space (ACES) data analysis. Class. Quantum Grav. 35(3), 035018 (2018)ADSCrossRefGoogle Scholar
  95. 95.
    L. Cacciapuoti, Ch. Salomon, Space clocks and fundamental tests: The ACES experiment. Eur. Phys. J. Spec. Top. 172(1), 57–68 (2009)CrossRefGoogle Scholar
  96. 96.
    Ph. Laurent, D. Massonnet, L. Cacciapuoti, C. Salomon, The ACES/PHARAO space mission. Comptes rendus de l’Académie des sciences. Physique 16(5), 540–552 (2015)CrossRefGoogle Scholar
  97. 97.
    P. Exertier, E. Samain, C. Courde, M. Aimar, J.M. Torre, G.D. Rovera, M. Abgrall, P. Uhrich, R. Sherwood, G. Herold, U. Schreiber, P. Guillemot, Sub-ns time transfer consistency: a direct comparison between GPS CV and T2L2. Metrologia 53(6), 1395 (2016)ADSCrossRefGoogle Scholar
  98. 98.
    G.D. Rovera, M. Abgrall, C. Courde, P. Exertier, P. Fridelance, Ph. Guillemot, M. Laas-Bourez, N. Martin, E. Samain, R. Sherwood, J.-M. Torre, P. Uhrich, A direct comparison between two independently calibrated time transfer techniques: T2L2 and GPS Common-Views. J. Phys.: Conf. Ser. 723(1), 012037 (2016)Google Scholar
  99. 99.
    E. Samain, P. Vrancken, P. Guillemot, P. Fridelance, P. Exertier, Time transfer by laser link (T2L2): characterization and calibration of the flight instrument. Metrologia 51(5), 503 (2014)ADSCrossRefGoogle Scholar
  100. 100.
    E. Samain, P. Exertier, C. Courde, P. Fridelance, P. Guillemot, M. Laas-Bourez, J.-M. Torre, Time transfer by laser link: a complete analysis of the uncertainty budget. Metrologia 52(2), 423 (2015)ADSCrossRefGoogle Scholar
  101. 101.
    P. Exertier, E. Samain, N. Martin, C. Courde, M. Laas-Bourez, C. Foussard, Ph. Guillemot, Time transfer by laser link: data analysis and validation to the ps level. Adv. Space Res. 54(11), 2371–2385 (2014)ADSCrossRefGoogle Scholar
  102. 102.
    J. Kodet, M. Vacek, P. Fort, I. Prochazka, J. Blazej, Photon Counting Receiver for the Laser Time Transfer, Optical Design, and Construction, vol. 8072 (2011), pp. 80720A (International Society for Optics and Photonics, 2011)Google Scholar
  103. 103.
    I. Prochazka, J. Kodet, J. Blazej, Note: space qualified photon counting detector for laser time transfer with picosecond precision and stability. Rev. Sci. Instrum. 87(5), 056102 (2016)ADSCrossRefGoogle Scholar
  104. 104.
    S.L. Campbell, R.B. Hutson, G.E. Marti, A. Goban, N. Darkwah Oppong, R.L. McNally, L. Sonderhouse, J.M. Robinson, W. Zhang, B.J. Bloom, J. Ye, A fermi-degenerate three-dimensional optical lattice clock. Science 358(6359), 90–94 (2017)ADSCrossRefGoogle Scholar
  105. 105.
    S.B. Koller, J. Grotti, St. Vogt, A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr, Ch. Lisdat, Transportable optical lattice clock with \(7\times 10^{-17}\) uncertainty. Phys. Rev. Lett. 118(7), 073601 (2017)Google Scholar
  106. 106.
    R. Tyumenev, M. Favier, S. Bilicki, E. Bookjans, R. Le Targat, J. Lodewyck, D. Nicolodi, Y. Le Coq, M. Abgrall, J. Guéna, L. De Sarlo, S. Bize, Comparing a mercury optical lattice clock with microwave and optical frequency standards. New J. Phys. 18(11), 113002 (2016)ADSCrossRefGoogle Scholar
  107. 107.
    J. Lodewyck, S. Bilicki, E. Bookjans, J.-L. Robyr, C. Shi, G. Vallet, R. Le Targat, D. Nicolodi, Y. Le Coq, J. Guéna, M. Abgrall, P. Rosenbusch, S. Bize, Optical to microwave clock frequency ratios with a nearly continuous strontium optical lattice clock. Metrologia 53(4), 1123 (2016)ADSCrossRefGoogle Scholar
  108. 108.
    K. Predehl, G. Grosche, S.M.F. Raupach, S. Droste, O. Terra, J. Alnis, Th Legero, T.W. Hänsch, Th Udem, R. Holzwarth, H. Schnatz, A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place. Science 336(6080), 441–444 (2012)ADSCrossRefGoogle Scholar
  109. 109.
    O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, G. Santarelli, Ultra-stable long distance optical frequency distribution using the Internet fiber network. Opt. Express 20(21), 23518–23526 (2012)ADSCrossRefGoogle Scholar
  110. 110.
    N. Chiodo, K. Djerroud, O. Acef, A. Clairon, P. Wolf, Lasers for coherent optical satellite links with large dynamics. Appl. Opt. 52(30), 7342–7351 (2013)ADSCrossRefGoogle Scholar
  111. 111.
    K. Djerroud, O. Acef, A. Clairon, P. Lemonde, C.N. Man, E. Samain, P. Wolf, Coherent optical link through the turbulent atmosphere. Opt. Lett. 35(9), 1479–1481 (2010)ADSCrossRefGoogle Scholar
  112. 112.
    F.R. Giorgetta, W.C. Swann, L.C. Sinclair, E. Baumann, I. Coddington, N.R. Newbury, Optical two-way time and frequency transfer over free space. Nat. Photonics 7(6), 434–438 (2013)ADSCrossRefGoogle Scholar
  113. 113.
    J.-D. Deschênes, L.C. Sinclair, F.R. Giorgetta, W.C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, N.R. Newbury, Synchronization of distant optical clocks at the femtosecond level. Phys. Rev. X 6(2), 021016 (2016)Google Scholar
  114. 114.
    L.C. Sinclair, F.R. Giorgetta, W.C. Swann, E. Baumann, I. Coddington, N.R. Newbury, Optical phase noise from atmospheric fluctuations and its impact on optical time-frequency transfer. Phys. Rev. A 89(2), 023805 (2014)ADSCrossRefGoogle Scholar
  115. 115.
    L.C. Sinclair, W.C. Swann, H. Bergeron, E. Baumann, M. Cermak, I. Coddington, J.-D. Deschênes, F.R. Giorgetta, J.C. Juarez, I. Khader, K.G. Petrillo, K.T. Souza, M.L. Dennis, N.R. Newbury, Synchronization of clocks through 12 km of strongly turbulent air over a city. Appl. Phys. Lett. 109(15), 151104 (2016)ADSCrossRefGoogle Scholar
  116. 116.
    C. Robert, J.-M. Conan, P. Wolf, Impact of turbulence on high-precision ground-satellite frequency transfer with two-way coherent optical links. Phys. Rev. A 93(3), 033860 (2016)ADSCrossRefGoogle Scholar
  117. 117.
    L. Blanchet, C. Salomon, P. Teyssandier, P. Wolf, Relativistic theory for time and frequency transfer to order. Astron. Astrophys. 370(1), 10 (2001)CrossRefGoogle Scholar
  118. 118.
    C. Le Poncin-Lafitte, B. Linet, P. Teyssandier, World function and time transfer: general post-Minkowskian expansions. Class. Quantum Grav. 21(18), 4463 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    P. Teyssandier, C. Le Poncin-Lafitte, General post-Minkowskian expansion of time transfer functions. Class. Quantum Grav. 25(14), 145020 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    A. Hees, S. Bertone, C. Le Poncin-Lafitte, Frequency shift up to the 2-PM approximation, in SF2A-2012: Proceedings of the annual meeting of the French Society of Astronomy and Astrophysics (2012), pp. 145–148, arXiv:1210.2577
  121. 121.
    J.L. Synge, Relativity: The General Theory, 1st edn. (North-Holland Publishing Company, Amsterdam, 1960)Google Scholar
  122. 122.
    J. Gers̆l, P. Delva, P. Wolf, Relativistic corrections for time and frequency transfer in optical fibres. Metrologia 52(4), 552 (2015)ADSCrossRefGoogle Scholar
  123. 123.
    A. Rülke, G. Liebsch, M. Sacher, U. Schäfer, U. Schirmer, J. Ihde, Unification of European height system realizations. J. Geod. Sci. 2(4), 343–354 (2013)Google Scholar
  124. 124.
    G. Petit, P. Wolf, P. Delva, Atomic time, clocks, and clock comparisons in relativistic spacetime: a review, in Frontiers in Relativistic Celestial Mechanics - Volume 2: Applications and Experiments, ed. by S.M. Kopeikin. De Gruyter Studies in Mathematical Physics (De Gruyter, 2014), pp. 249–279Google Scholar
  125. 125.
    L. Sánchez, Towards a vertical datum standardisation under the umbrella of Global Geodetic Observing System. J. Geod. Sci. 2(4), 325–342 (2012)Google Scholar
  126. 126.
    M. Burs̆a, S. Kenyon, J. Kouba, Z. S̆íma, V. Vatrt, V. Vtek, M. Vojtís̆ková, The geopotential value \(W_0\) for specifying the relativistic atomic time scale and a global vertical reference system. J. Geod. 81(2), 103–110 (2006)Google Scholar
  127. 127.
    N. Dayoub, S.J. Edwards, P. Moore, The Gauss-Listing geopotential value \(W_0\) and its rate from altimetric mean sea level and GRACE. J. Geod. 86(9), 681–694 (2012)ADSCrossRefGoogle Scholar
  128. 128.
    S. Jevrejeva, J.C. Moore, A. Grinsted, Sea level projections to AD2500 with a new generation of climate change scenarios. Glob. Planet. Chang. 80–81, 14–20 (2012)ADSCrossRefGoogle Scholar
  129. 129.
    C. Voigt, H. Denker, L. Timmen, Time-variable gravity potential components for optical clock comparisons and the definition of international time scales. Metrologia 53(6), 1365 (2016)ADSCrossRefGoogle Scholar
  130. 130.
    T. Takano, M. Takamoto, I. Ushijima, N. Ohmae, T. Akatsuka, A. Yamaguchi, Y. Kuroishi, H. Munekane, B. Miyahara, H. Katori, Real-time geopotentiometry with synchronously linked optical lattice clocks. Nat. Photonics 10(10), 662–666 (2016)Google Scholar
  131. 131.
    P. Delva, J. Lodewyck, S. Bilicki, E. Bookjans, G. Vallet, R. Le Targat, P.-E. Pottie, C. Guerlin, F. Meynadier, C. Le Poncin-Lafitte, others, Test of special relativity using a fiber network of optical clocks. Phys. Rev. Lett. 118(22), 221102 (2017)Google Scholar
  132. 132.
    J. Mäkinen, J. Ihde, The permanent tide in height systems, in Observing our Changing Earth, International Association of Geodesy Symposia (Springer, Berlin, 2009), pp. 81–87Google Scholar
  133. 133.
    J. Ihde, J. Mäkinen, M. Sacher, Conventions for the definition and realization of a European Vertical Reference System (EVRS) – EVRS Conventions 2007. EVRS Conventions V5.1, Bundesamt für Kartographie and Geodäsie, Finnish Geodetic Institute, 2008-12-17. Technical report, 2008Google Scholar
  134. 134.
    H. Denker, Regional gravity field modeling: theory and practical results, in Sciences of Geodesy, vol. II, ed. by G. Xu (Springer, Berlin, 2013), pp. 185–291Google Scholar
  135. 135.
    Bull. Geodesique 58(3), 309–323 (1984)Google Scholar
  136. 136.
    W.A. Heiskanen, H. Moritz, Physical Geodesy (W.H. Freeman and Company, San Francisco, London, 1967)Google Scholar
  137. 137.
    W. Torge, J. Müller, Geodesy, 4th edn. (De Gruyter, Berlin, Boston, 2012)Google Scholar
  138. 138.
    F. Condi, C. Wunsch, Gravity field variability, the geoid, and ocean dynamics, in V Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Symposia, vol. 127 (Springer, Berlin, 2004), pp. 285–292Google Scholar
  139. 139.
    H. Drewes, F. Kuglitsch, J. Adám, S. Rózsa, The Geodesist’s handbook 2016. J. Geod. 90(10), 907–1205 (2016)ADSCrossRefGoogle Scholar
  140. 140.
    J. Ihde, R. Barzaghi, U. Marti, L. Sànchez, M. Sideris, H. Drewes, C. Foerste, T. Gruber, G. Liebsch, G. Pail, Report of the ad-hoc group on an international height reference system (IHRS), in Reports 2011-2015, Number 39 in IAG Travaux (2015), pp. 549–557Google Scholar
  141. 141.
    M. Burša, J. Kouba, M. Kumar, A. Müller, K. Raděj, S.A. True, V. Vatrt, M. Vojtíšková, Geoidal geopotential and world height system. Stud. Geophys. Geod. 43(4), 327–337 (1999)CrossRefGoogle Scholar
  142. 142.
    L. Sánchez, R. C̆underlk, N. Dayoub, K. Mikula, Z. Minarechová, Z. S̆íma, V. Vatrt, M. Vojtís̆ková, A conventional value for the geoid reference potential \(W_0\). J. Geod. 90(9), 815–835 (2016)Google Scholar
  143. 143.
    M.S. Molodenskii, V.F. Eremeev, M.I. Yurkina, Methods for Study of the External Gravitational Field and Figure of the Earth (Israel Program for Scientific Translations, Jerusalem, 1962)Google Scholar
  144. 144.
    N.K. Pavlis, S.A. Holmes, S.C. Kenyon, J.K. Factor, The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res. 117(B4), B04406 (2012)ADSCrossRefGoogle Scholar
  145. 145.
    T. Mayer-Gürr, N. Zehentner, B. Klinger, A. Kvas, ITSG-Grace2014: a new GRACE gravity field release computed in Graz, 2014. GRACE Science Team Meeting (GSTM), Potsdam, 29 Sept.-01 Oct. 2014Google Scholar
  146. 146.
    T. Mayer-Gürr, G. Team, The combined satellite gravity field model GOCO05s, 2015. EGU General Assembly 2015, Vienna (2015)Google Scholar
  147. 147.
    J.M. Brockmann, N. Zehentner, E. Höck, R. Pail, I. Loth, T. Mayer-Gürr, W.-D. Schuh, EGM\(\_\)TIM\(\_\)RL05: an independent geoid with centimeter accuracy purely based on the GOCE mission. Geophys. Res. Lett. 41(22), 8089–8099 (2014)ADSCrossRefGoogle Scholar
  148. 148.
    D.E. Smith, R. Kolenkiewicz, P.J. Dunn, M.H. Torrence, Earth scale below a part per billion from Satellite Laser Ranging, in Geodesy Beyond 2000, International Association of Geodesy Symposia, vol. 121 (Springer, Berlin, 2000), pp. 3–12CrossRefGoogle Scholar
  149. 149.
    J.C. Ries, The scale of the terrestrial reference frame from VLBI and SLR, 2014. IERS Unified Analysis Workshop, Pasadena, CA, 27–28 June 2014Google Scholar
  150. 150.
    R. Forsberg, Modelling the fine-structure of the geoid: methods, data requirements and some results. Surv. Geophys. 14(4–5), 403–418 (1993)ADSCrossRefGoogle Scholar
  151. 151.
    C. Jekeli, O. Error, Data requirements, and the fractal dimension of the geoid, in VII Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Symposia, vol. 137 (Springer, Berlin, 2012), pp. 181–187Google Scholar
  152. 152.
    R. Rummel, P. Teunissen, Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. Geodesique 62(4), 477–498 (1988)ADSCrossRefGoogle Scholar
  153. 153.
    B. Heck, R. Rummel, Strategies for solving the vertical datum problem using terrestrial and satellite geodetic data, in Sea Surface Topography and the Geoid, International Association of Geodesy Symposia, vol. 104 (Springer, New York, 1990), pp. 116–128CrossRefGoogle Scholar
  154. 154.
    J. Kelsey, D.A. Gray, Geodetic aspects concerning possible subsidence in southeastern England. Phil. Trans. R. Soc. Lond. A 272(1221), 141–149 (1972)ADSCrossRefGoogle Scholar
  155. 155.
    P. Rebischung, H. Duquenne, F. Duquenne, The new French zero-order levelling network – first global results and possible consequences for UELN, 2008, EUREF Symposium, Brussels, June 18-21, 2008 (2008)Google Scholar
  156. 156.
    M. Véronneau, R. Duvai, J. Huang, A gravimetric geoid model as a vertical datum in Canada. Geomatica 60, 165–172 (2006)Google Scholar
  157. 157.
    D. Smith, M. Véronneau, D.R. Roman, J. Huang, Y. Wang, M. Sideris, Towards the unification of the vertical datums over the North American continent, 2010. IAG Comm. 1 Symposium 2010, Reference Frames for Applications in Geosciences (REFAG2010), Marne-La-Vallée, France, 4–8 Oct. 2010 (2010)Google Scholar
  158. 158.
    D.A. Smith, S.A. Holmes, X. Li, S. Guillaume, Y.M. Wang, B. Bürki, D.R. Roman, T.M. Damiani, Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid slope validation survey of 2011. J. Geod. 87(10–12), 885–907 (2013)ADSCrossRefGoogle Scholar
  159. 159.
    Z. Altamimi, X. Collilieux, L. Métivier, ITRF2008: an improved solution of the international terrestrial reference frame. J. Geod. 85(8), 457–473 (2011)ADSCrossRefGoogle Scholar
  160. 160.
    Z. Altamimi, P. Rebischung, L. Métivier, X. Collilieux, ITRF2014: a new release of the International Terrestrial Reference Frame modeling nonlinear station motions. J. Geophys. Res. Solid Earth 121, 6109–6131 (2016)ADSCrossRefGoogle Scholar
  161. 161.
    M. Seitz, D. Angermann, H. Drewes, Accuracy assessment of the ITRS 2008 realization of DGFI: DTRF2008, in Reference Frames for Applications in Geosciences, International Association of Geodesy Symposia, vol. 138 (Springer, Berlin, 2013), pp. 87–93Google Scholar
  162. 162.
    H. Margolis, Timekeepers of the future (2014), https://www.nature.com/articles/nphys2834
  163. 163.
    F. Riehle, Towards a redefinition of the second based on optical atomic clocks. Comptes Rendus Phys. 16(5), 506–515 (2015)CrossRefGoogle Scholar
  164. 164.
    P. Gill, Is the time right for a redefinition of the second by optical atomic clocks? J. Phys.: Conf. Ser. 723(1), 012053 (2016)Google Scholar
  165. 165.
    F. Riehle, Optical clock networks. Nat. Photonics 11(1), 25–31 (2017)ADSCrossRefGoogle Scholar
  166. 166.
    S. Falke, N. Lemke, C. Grebing, B. Lipphardt, S. Weyers, G. Vladislav, N. Huntemann, C. Hagemann, A. Al-Masoudi, S. Häfner, S. Vogt, S. Uwe, C. Lisdat, A strontium lattice clock with \(3\times 10^{-17}\) inaccuracy and its frequency. New J. Phys. 16(7), 073023 (2014)ADSCrossRefGoogle Scholar
  167. 167.
    H. Denker, A new European Gravimetric (Quasi)Geoid EGG2015, 2015. XXVI General Assembly of the International Union of Geodesy and Geophysics (IUGG), Earth and Environmental Sciences for Future Generations, Prague, Czech Republic, 22 June–02 July 2015 (Poster)Google Scholar
  168. 168.
    M. Sacher, J. Ihde, G. Liebsch, J. Mäkinen, EVRF2007 as realization of the European vertical reference system, 2008, EUREF Symposium, Brussels, Belgium, 18–21 June 2008 (2008)Google Scholar
  169. 169.
    D.E. Cartwright, J. Crease, A comparison of the geodetic reference levels of England and France by means of the sea surface. Proc. R. Soc. Lond. A 273(1355), 558–580 (1963)ADSCrossRefGoogle Scholar
  170. 170.
    M. Greaves, R. Hipkin, C. Calvert, C. Fane, P. Rebischung, F. Duquenne, A. Harmel, A. Coulomb, H. Duquenne, Connection of British and French levelling networks – Applications to UELN, 2007, EUREF Symposium, London, June 6–9, 2007 (2007)Google Scholar
  171. 171.
    A. Kenyeres, M. Sacher, J. Ihde, H. Denker, U. Marti, EUVN densification action – final report. Technical report (2010), https://evrs.bkg.bund.de/SharedDocs/Downloads/EVRS/EN/Publications/EUVN-DA_FinalReport.pdf?__blob=publicationFile&v=1
  172. 172.
    H. Moritz, Geodetic reference system 1980. J. Geod. 74(1), 128–133 (2000)ADSCrossRefGoogle Scholar
  173. 173.
    H. Moritz, Advanced Physical Geodesy (Wichmann, Karlsruhe, 1980)Google Scholar
  174. 174.
    H. Denker, Hochauflösende regionale Schwerefeldbestimmung mit gravimetrischen und topographischen Daten. PhD thesis, Wiss. Arb. d. Fachr. Verm.wesen d. Univ. Hannover, Nr. 156, Hannover, 1988Google Scholar
  175. 175.
    H. Denker, Evaluation and improvement of the EGG97 quasigeoid model for Europe by GPS and leveling data, in Second Continental Workshop on the Geoid in Europe, Proceed., Report of Finnish Geodetic Institute, Masala, vol. 98:4, ed. by M. Vermeer, J. Ádám (1998), pp. 53–61Google Scholar
  176. 176.
    C. Förste, S.L. Bruinsma, O. Abrikosov, J.-M. Lemoine, J.C. Marty, F. Flechtner, G. Balmino, F. Barthelmes, R. Biancale, EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse (2014)Google Scholar
  177. 177.
    D.A. Smith, There is no such thing as “The” EGM96 geoid: subtle points on the use of a global geopotential model. Technical report, IGeS Bulletin No. 8, International Geoid Service, Milan, Italy, 1998Google Scholar
  178. 178.
    S.J. Claessens, C. Hirt, Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid. J. Geophys. Res. Solid Earth 118(11), 5991–6002 (2013)ADSCrossRefGoogle Scholar
  179. 179.
    R. Forsberg, A new covariance model for inertial gravimetry and gradiometry. J. Geophys. Res. 92(B2), 1305–1310 (1987)ADSMathSciNetCrossRefGoogle Scholar
  180. 180.
    R. Forsberg. An overview manual for the GRAVSOFT – Geodetic Gravity Field Modelling Programs. Technical report, DNSC – Danios National Space Center, 2003Google Scholar
  181. 181.
    D. Coulot, P. Rebischung, A. Pollet, L. Grondin, G. Collot, Global optimization of GNSS station reference networks. GPS Solut. 19(4), 569–577 (2015)CrossRefGoogle Scholar
  182. 182.
    J. Cao, P. Zhang, J. Shang, K. Cui, J. Yuan, S. Chao, S. Wang, H. Shu, X. Huang, A compact, transportable single-ion optical clock with \(7.8\times 10^{17}\) systematic uncertainty. Appl. Phys. B 123(4), 112 (2017)Google Scholar
  183. 183.
    M. Yasuda, T. Tanabe, T. Kobayashi, D. Akamatsu, T. Sato, A. Hatakeyama, Laser-controlled cold ytterbium atom source for transportable optical clocks. J. Phys. Soc. Jpn. 86(12), 125001 (2017)ADSCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.SYRTE Observatoire de Paris, Université PSL, CNRS, Sorbonne UniversitéParisFrance
  2. 2.Institut für ErdmessungLeibniz Universität Hannover (LUH)HannoverGermany
  3. 3.LASTIG LAREG IGNENSG, Univ Paris Diderot, Sorbonne Paris CitéParisFrance

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