Geometry modelling
Most important physical models for determining the geometry of the Earth from geodetic positioning are those of the atmosphere and the oceans. Concerning the vertical component of positioning (heights), the physical ocean surface and the gravity field serve as a reference and have precisely to be modelled.
KANIUTH and HUBER (2004) use atmospheric pressure models to estimate the loading effects causing height variations in Europe and compare them with GPS observa-tions. The tidal effect of ocean loading is studied by ZAHRAN et al. (2006). Corresponding height variations are presented by KANIUTH and VETTER (2005) at coastal sites from GPS measurements.
Problems related to physical models in the definition of vertical reference frames are discussed by HECK (2004).
IHDE and SÁNCHEZ (2005) present an approach to globally unify height systems by uniform physical models of the sea surface and the gravity field of the Earth.
Modelling of the Earth’s orientation
Variations of the Earth’s orientation in space and Earth’s rotation are caused by changes of the angular momentum or torque, respectively. These are generated by mass displacements in the solid Earth, the atmosphere and the hydrosphere including the oceans and continental water storage. They have to be represented by physical models for consideration in geodetic parameter estimation.
The influences of core processes on Earth’s rotation para-meters are studied by GREINER-MAI et al. (2003). The effect of water storage variations on polar motion is presented by FERNANDEZ et al. (2007). STUCK et al. (2005) model the physical mechanism of atmospheric forces in polar motion, and THOMAS et al. (2005) concentrate on the contribution of the oceans. Combined modelling of atmospheric and oceanic effects from coupled physical models is published in a series of papers by SEITZ (2004,2005), SEITZ et al.
(2004), and SEITZ and SCHMIDT (2005). MARCHENKO and SCHWINTZER (2003) combine Earth rotation parameters and the Earth gravity field by a combined parameter estimation.
Gravity field modelling
The physical modelling of the gravity field parameters concentrated in the last years on the use of observation data from the satellite gravity field missions CHAMP and GRACE. A number of papers deals with these issues. A new physical approach of gravity field modelling from these missions using the energy integral from kinematic orbits is presented by GERLACH et al. (2003).
Physical models of the solid Earth include isostatic models presented by KABAN et al. (2004) for the entire lithosphere as well as by WILD and HECK (2005). WZIONTEK (2003)
parameterisesglobal density models. The continental hydro-sphere is probably the most investigated physical aspect in gravity field modelling from space missions. RAMILLIEN
et al. (2004,2005) present an approach for global time variations from GRACE. HARNISCH and HARNISCH (2006) provide the ground truth values from gravimetric data, and NEUMEYER et al. (2006) combine both terrestrial and space observations with hydrology models.
BOSCH (2005) reports about errors in the shipborne marine gravity representation found from GRACE models. The de-aliasing of short-term atmospheric and oceanic gravity variations from GRACE is published by FLECHTNER et al.
(2006).
The atmosphere effects are in principle reduced from the geodetic observations by physical models. An approach for atmoshere pressure reduction from gravimetry is given by NEUMEYER et al. (2004).
The reliability of the regional models of mass variations derived from GRACE data processing is discussed by HORWARTH and DIETRICH (2006), where some errors in the modelling are demonstrated.
Relativity
As far as relativistic aspects in geodetic modelling are concerned work concentrated upon three main topics:
– astronomical reference systems, – dynamical equations of motion and – relativity tests.
MÜLLER et al. (2007b) present a comprehensive overview on this subject.
Astronomical reference systems
Of great importance for high precision geodetic modelling is the introduction of two fundamentally different celestial reference systems: the Barycentric Celestial Reference System (BCRS) with coordinate time TCB and the Geo-centric Celestial Reference System (GCRS) with TCG as coordinate time. SOFFEL et al. (2003) present a detailed discussion of the BCRS and the GCRS. Here, also the relativistic forces acting on a satellite are discussed. Special aspects of local relativistic reference systems are treated in KLIONER (2004). The problem of representation of the cosmic expansion in the BCRS is treated in KLIONER and SOFFEL (2004), SOFFEL and KLIONER (2004a) and in CARRERA and GIULINI (2006). In these papers it was found that the influence of the Hubble expansion of the universe upon physics in the solar system is completely negligible.
Relativistic equations of motion
The problem of relativistic equations of motion of astro-nomical bodies has been persued into two different direc-tions. In a series of papers Xu and collaborators (XU et al., 2003,2005) laid the foundation for a relativistic description of elastic deformable astronomical bodies by means of a displacement field. However, this formalism is extremely complex and the relation with observables, e.g., in the field
of Earth's rotation is unclear. Another approach, specially designed for the problem of Earth's rotation, starts with a rigidly rotating multipole formalism that is described in detail in KLIONER et al. (2003). This formalism forms the basis for the present post-Newtonian approach to improve Newtonian nutation series. The problem of a relativistic description of Earth's rotation is discussed in SOFFEL and KLIONER (2004).
Relativistic tests
Geodetic space techniques such as SLR, LLR or VLBI are able to provide tests of relativity, both for Special Relativity and Einstein's theory of gravity. Such tests concern the Lorentz-invariance, Newton's law of gravity (the 5th force, GC/ G), various forms of the equivalence principle, the determination of post-Newtonian parameters, the geodetic precession and Lense-Thirring effects (frame dragging due to the rotation of the Earth). MÜLLER et al. (2007b) give an overview over such tests. MÜLLER et al. (2006a,2006b) and MÜLLER (2006) discuss the use of LLR data for such tests of relativity.
References
BOSCH W.: Using the EIGEN-GRACE02S gravity field to investi-gate defectiveness of marine gravity data. Springer, IAG Symposia, Vol. 129, 89-94, 2005.
DREWES H.: Integration von Geometrie und Gravimetrie: Das Globale Geodätische Observations-System (GGOS). Wiss.
Arb. Fachr. Verm., Univ. Hannover, Nr. 263, 159-168, 2006a.
DREWES H.: Zum Wandel in der Zielsetzung geodätischer For-schung (The changing objectives in geodetic research).
Zeitschr. für Verm. (131) 292-298, 2006b.
DREWES H.: Science rationale of the Global Geodetic Observing System (GGOS). Springer, IAG Symposia, Vol. 130, 703-710, 2007.
CARRERA M.,GIULINI D.: On Doppler tracking in cosmological Spacetimes, Classical and Quantum Gravity (23) 7483, 2006.
FERNANDEZ L.I.,SCHUH H.,SCHMIDT M.,SEITZ F.: Effects of inter-annual water storage variations on polar motion.
Geophys. J. Int. (169) 12-18, 2007.
FLECHTNER F.,SCHMIDT R.,MEYER U.: De-aliasing of short-term atmospheric and oceanic variations for GRACE. Springer In: J. Flury, R. Rummel, C. Reigber, M. Rothacher, G.
Boedecker, U. Schreiber (Eds.): Observation of the Earth System from Space, 83-97, 2006.
GERLACH CH.,FÖLDVARY L.,SVEHLA D.,GRUBER T.,WERMUTH
M.,SNEEUW N.,FROMMKNECHT B.,OBERNDORFER H., PETERS TH.,ROTHACHER M.,RUMMEL R.,STEIGENBERGER
P.: A CHAMP-only gravity field model from kinematic orbits using the energy integral. Geophys. Res. Lett. (30) No. 20, 7, 5pp, 2003.
GREINER-MAI H.,JOCHMANN H.,BARTHELMES F.,BALLANI L.:
Possible influences of core processes on the Earth's rotation and the gravity field. J. Geodynamics (36) 343-358, 2003.
HARNISCH G.,HARNISCH M.: Hydrological influences in long gravimetric data series. J. Geodynamics (41) 276-287, 2006.
HECK B.: Problems in the definition of vertical reference frames.
Springer, IAG Symposia, Vol. 127, 164-173, 2004.
H. Drewes, M. Soffel: Physical Aspects of Geodetic Modelling, Relativity 155
HORWATH M.,DIETRICH R.: Errors in regional mass variations inferred from GRACE monthly solutions. Geophys. Res.
Lett. (33) L07502, 5pp, doi:10.1029/2005GL025550, 2006.
IHDE J.,SANCHEZ L.: A unified global height reference system as a basis for IGGOS. J. Geodynamics (40) 400-413, 2005.
KABAN M.K.,SCHWINTZER P.,REIGBER C.: A new isostatic model of the lithosphere and gravity field. J. Geodesy (78) 368-385, 2004.
KANIUTH K.,HUBER S.: Modelling vertical site displacements due to atmospheric pressure loading with the Bernese software -A demonstation using EUREF data. Mitteilungen des BKG (33) 89-95, 2004.
KANIUTH K.,VETTER S.: Vertical velocities of European coastal sites derived from continuous GPS observations. GPS Solutions (9) 32-40, DOI 10.1007/s10291-004-0124-4, 2005.
KANZOW T.,FLECHTNER F.,CHAVE A.,SCHMIDT R.,SCHWINTZER
P.,SEND U.: Seasonal variation of ocean bottom pressure derived from Gravity recovery and Climate Experiment (GRACE): Local validation and global patterns. J. Geophys.
Res. (110) C09001, 1-14, 2005.
KLIONER S.: Physically adequate reference system of a test observer and relativistic description of the GAIA attitude, Physical Review DS 69, 124001, 2004.
KLIONER S.,SOFFEL M.,XU C.,WU X.: Earth's rotation in the framework of general relativity: rigid multipole moments.
In: Proceedings of Les Journees 2001, Brussels, 2003.
KLIONER S.,SOFFEL M.: Refining the relativistic model for GAIA:
cosmological effects in the BCRS. In: Proceedings of the Symposium 'The Three-Dimensional Universe with GAIA', 4-7 October 2004, Observatoire de Paris-Meudon, France (ESA SP-576), 305-308, 2004.
MARCHENKO A.N.,SCHWINTZER P.: Estimation of the Earth's tensor of inertia from recent global gravity field solutions.
J. Geodesy (76) 495-509, 2003.
MÜLLER J.,WILLIAMS J.,TURYSHEV S.,SHELUS P.: Potential Capabilities of Lunar LAser Ranging and Relativity. In:
Dynamic Planet. P.Tregoning, C.Rizos (eds.), IAG Sympo-sium Vol. 130, 903-909, Springer 2006a
MÜLLER J.,J.WILLIAMS,S.TURYSHEV: Lunar Laser Ranging Contributions to Relativity and Geodesy. Proceedings of the Conference on Lasers, Clocks and Drag-free, ZARM, Bremen, 30.5.-1.6.2005, eds. H.Dittus, C.Lämmerzahl, S.Turyshev, p. 357-372, Springer 2006b
MÜLLER J.: Lunar Laser Ranging: A Space Geodetic Technique to test relativity. Proceedings of the 11th Marcel Grossmann Meeting, Berlin, 24-28 July 2006, H. Kleinert, R.Ruffini, R.Jantzen (eds.), World Scientific, 2007a, in press.
MÜLLER J.,SOFFEL M.,KLIONER S.: Geodesy and Relativity.
Journal of Geodesy, 2007b, in press
NEUMEYER J.,HAAGEDOORN J.,LEITLOFF J.,SCHMIDT T.: Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements. J. Geodynamics (38) 437-450, 2004.
NEUMEYER J., BARTHELMES F.,DIERKS O.,FLECHTNER F., HARNISCH M.,HARNISCH G,,HINDERER J.,IMANISHI Y., KRONER C.,MEURERS B., PETROVIC S., REIGBER CH., SCHMIDT R,,SCHWINTZER P.,SUN H.-P.,VIRTANEN H.:
Combination of temporal gravity variations resulting from superconducting gravimeter (SG) recordings, GRACE satellite observations and global hydrology models. J.
Geodesy (79) 573-585, 2006.
RAMILLIEN G.,CAZENAVE A.,BRUNAU O.: Global time variations of hydrological signals from GRACE satellite gravimetry.
Geophys. J. Int. (158) 813-826, 2004.
RAMILLIEN G., CAZENAVE A., REIGBER C.,SCHMIDT R., SCHWINTZER P.: Recovery of global time-variations of surface water mass by GRACE geoid inversion. Springer, IAG Symposia, Vol. 129, 310-315, 2005.
SEITZ F.: Atmosphärische und ozeanische Einflüsse auf die Rotation der Erde. Dt. Geod. Komm., München, Reihe C, Nr. 578, 2004.
SEITZ F.: Atmospheric and oceanic influences on polar motion -numerical results from two independent model combina-tions. Artificial Satellites (40) 199-215, 2005a.
SEITZ F.: Zur Anregung der Chandler-Schwingung. Zeitschr. für Verm. (130) 166-173, 2005b.
SEITZ F.,STUCK J.,THOMAS M.: Consistent atmospheric and oceanic excitation of the Earth's free polar motion. Geo-phys. J. Int. (157) 25-35, 2004.
SEITZ F.,SCHMIDT M.: Atmospheric and oceanic contributions to Chandler wobble excitation determined by wavelet filtering. J. Geophys. Res. (110) B11406, 11pp, 2005.
SOFFEL M., KLIONER S.,PETIT G.,WOLF P.,KOPEIKIN S., BRETAGNON P.,BRUMBERG V.,CAPITAINE N.,DAMOUR T., FUKUSHIMA T.,GUINOT B.,HUANG T.,LINDEGREN L.,MA
C.,NORDTVEDT K.,RIES J.,SEIDELMANN P.,VOKROUHLICKY
D.,WILL C.,XU C.: The IAU resolutions for astrometry, celestial mechanics and metrology in the relativistic frame-work: explanatory supplement. Astronomical Journal 126(6), 2687-2706, 2003.
SOFFEL M.,KLIONER S.: The BCRS and the large scale structure of the universe. In: Proceedings of Les Journees 2003. A.
Finkelstein and N. Capitaine (eds.), Paris Obervatory, Paris, 297-301, 2004.
SOFFEL M.,KLIONER S.: Relativity in the problems of Earth rotation and astronomical reference systems: status and prospects. In: Proceedings of Les Journees 2004, N.
Capitaine (ed.), Paris Observatory, Paris, 191-195, 2004.
STUCK J.,SEITZ F.,THOMAS M.: Atmospheric forcing mechanisms of polar motion. Cahiers du Centre Europeen de Geodyn.
et de Seismol. (24) 127-133, 2005.
THOMAS M.,DOBSLAW H.,STUCK J., SEITZ F.: The ocean's contribution to polar motion excitation - as many solutions as numerical models? Cahiers du Centre Europeen de Geodyn. et de Seismol. (24) 143-148, 2005.
WILD F.,HECK B.: A comparison of different isostatic models applied to satellite gravity gradiometry. Springer, IAG Symposia, Vol. 129, 230-235, 2005.
WZIONTEK H.: Zur Parametrisierung radialsymmetrischer Dichtemodelle für die Erde. Dt. Geod. Komm., München, Reihe C, Nr. 587, 2005.
XU C.,WU X.,SOFFEL M.,KLIONER S.: Relativistic theory of elastic deformable astronomical bodies: Perturbation equations in rotating spherical coordinates and junction conditions, Physical Review D 68, 064009, 2003.
XU C.,WU X.,SOFFEL M.: General-Relativistic perturbation equations for the dynamics of elastic deformable astro-nomical bodies expanded in terms of generalized spherical harmonics. Physical Review D 71, 024030, 2005.
ZAHRAN K.H.,JENTZSCH G.,SEEBER G.: Accuracy assessment of ocean tide loading computations for precise geodetic observations. J. Geodynamics (42) 159-174, 2006.
1 Wolfgang Keller: Geodetic Institute, Universität Stuttgart, Geschwister-Scholl-Str. 24/D, D - 70174 Stuttgart, Germany, Tel. +49 - 711 - 6858 3459, Fax +49 - 711 - 6858 3285, e-mail wolfgang.keller@gis.uni-stuttgart.de
2 Willi Freeden: AG Geomathematik, Fachbereich Mathematik, Universität Kaiserslautern, Kurt-Schumacher-Str.26, D - 67653 Kaiserslautern, Germany, Tel. +49 - 631 - 205-2852 / -3867, Fax +49 - 631 - 205-4736, e-mail freeden@mathematik.uni-kl.de