IERS TECHNICAL NOTE 21. IERS Conventions (1992) Dennis D. McCarthy. July 1996

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1 IERS TEHNIAL NOTE IERS onventions (99) Dennis D. Mcarthy US Naval Observatory July 996 entral Bureau of IERS - Observatoire de Paris 6, avenue de l'observatoire F-75 PARIS - France

2 IERS Technical Notes This series of publications gives technical information related to the IERS activities, e.g. reference frames, excitation of the Earth rotation, computational or analysis aspects, modeis, etc. It also contains the description and results of the analyses performed by the IERS Analysis entres for the Annual Report global analysis. Back issues still available No : No 5: No 6: No 7: No 8: No 9: No : P. harlot (ed.) Earth orientation, reference frames and atmospheric excitation functions submitted for the 99 IERS Annual EeportlSuperseded by T.N. 7]. Boucher, Z. Altamimi and L. Duhem. ITRF9 and its associated velocity field J.O. Dickey and M. Feissel (eds.). Results from the SEARH'9 ampaign. P. harlot (ed.). Earth orientation, reference frames and atmospheric excitation functions submitted for the 99 IERS Annual Report.[Superseded by T.N. 9]. Boucher, Z. Altamimi, L. Duhem. Results and Analysis of the ITRF9 P. harlot (ed.). Earth orientation, reference frames and atmospheric excitation functions submitted for the 99 IERS Annual Report.. Boucher, Z. Altamimi, M. Feissel, P. Sillard. Results and Analysis of the ITRF9 No : D.D. Mcarthy (ed.). IERS onventions (996). Future issues No : No : P. harlot (ed.). Earth orientation and reference frames submitted for the 995 IERS Annual Report.. Boucher, P. Sillard. Evolution of realization of global reference Systems from space techniques and their links with ITRF.

3 T A B L E O F O N T E N T S INTRODUTION Differences Between This Document and IERS Technical Note. ONVENTIONAL ELESTIAL REFERENE SYSTEM Equator Origin of Right Ascension Precision and Accuracy 5 Availability of the Frame 5. ONVENTIONAL DYNAMIAL REFERENE FRAME 8. ONVENTIONAL TERRESTRIAL REFERENE SYSTEM Definition Realization Transformation Parameters of World oordinate Systems and Datums Plate Motion Model. NUMERIAL STANDARDS 8 5. TRANSFORMATION BETWEEN THE ELESTIAL AND TERRESTRIAL SYSTEMS oordinate Transformation Referred to the Equinox The IAU 98 Theory of Nutation The IERS 996 Theory of Precession/Nutation 5 The Multipliers of the Fundamental Arguments of Nutation Theory onversion to Prograde and Retrograde Nutation Amplitudes oordinate Transformation Referred to the Nonrotating Origin Geodesic Nutation 7 6. GEOPOTENTIAL Effect of Solid Earth Tides Pole Tide 6 Treatment of the Permanent Tide 6 Effect of the Ocean Tides 7 7. SITE DISPLAEMENT 5 Local Site Displacement due to Ocean Loading 5 Ocean Loading 5 Effects of the Solid Earth Tides 56 Displacement due to degree tides 6 Displacement due to degree tides 6 ontributions to the transverse 6 Out of phase contributions 6 orrection for frequency 6 Permanent deformation 65 Rotational Deformation Due to Polar Motion 65 Antenna Deformation 67 Atmospheric Loading 67 Postglacial Rebound TIDAL VARIATIONS IN THE EARTH'S ROTATION 7 9. TROPOSPHERI MODEL 78 Optical Techniques 78 Radio Techniques 78

4 . RADIATION PRESSURE REFLETANE MODEL 8 Global Positioning System 8. GENERAL RELATIVISTI MODELS FOR TIME, OORDINATES AND EQUATIONS OF MOTION 8 Equations of Motion for an Artificial Earth Satellite 8 Equations of Motion in the Barycentric Frame 8 Scale Effect and hoice of Time oordinate 8. GENERAL RELATIVISTI MODELS FOR PROPAGATION 87 VLBI Time Delay 87 Gravitational Delay 89 Geometrie Delay 9 Observations lose to the Sun 9 Summary 9 Propagation orrection for Laser Ranging 9 GLOSSARY 95

5 INTRODUTION This document is intended to deiine the Standard reference system to be used by the International Earth Rotation Service (IERS). It is based on the Project MERIT Standards (Melbourne et al, 98) and the IERS Standards (Mcarthy, 989; Mcarthy, 99) with revisions being made to reflect improvements in modeis or constants since the IERS Standards were published. If contributors to IERS do not fully comply with these guidelines, they will carefully identify the exceptions. In these cases, the institution is obliged to provide an assessment of the effects of the departures from the onventions so that its results can be referred to the IERS Reference System. ontributors may use modeis equivalent to those specified herein. Different observing methods have varying sensitivity to the adopted Standards and reference Systems. No attempt has been made in this document to assess the sensitivity of each technique to the adopted reference Systems and Standards. The recommended system of astronomical constants corresponds closely to those of the previous IERS Standards with the exception of the changes listed below. The units of length, mass, and time are in the International System of Units (SI) (Le Systeme International d'unites (SI), 99) as expressed by the meter (m), kilogram (kg) and second (s). The astronomical unit of time is the day containing 86 SI seconds. The Julian entury contains 655 days. Differences Between This Document and IERS Technical Note Most chapters of IERS Technical Note have been revised, and known typographical errors contained in that work have been corrected in this edition. There are some major differences between the current work and the past IERS Standards. The following is a brief list of the major modifications by chapter. HAPTER IERS Dynamical Reference Frame The JPL DE ephemeris (Standish et a/., 995) replaces the DE model of IERS Technical Note. HAPTER IERS Terrestrial Reference System The NUVEL NNR-A Model (DeMets et al, 99) for plate motion has replaced the Nuvel NNR- Model of IERS Technical Note. HAPTER Numerical Standards Numerical values are now given only for the most fundamental constants along with their uncertainties and references. onstants which have been changed include the astronomical unit in seconds and meters, precession, obliquity, equatorial radius, flattening factor and dynamical form factor of the Earth, constant of gravitation, geocentric and heliocentric gravitational constant. HAPTER 5 Transformation Between elestial and Terrestrial Reference Systems An empirical model to be used to predict the difference in the celestial pole coordinates between those published by the IERS and those given by the IAU model is added.

6 HAPTER 6 Geopotential The JGM model replaces the GEM-T. Love Numbers are revised. HAPTER 7 Local Site Displacements The printed table of the components of site displacement due to ocean loading is no longer included. References to machine- readable files are given. Love Numbers are revised. Atmospheric loading and postglacial rebound are included. HAPTER 8 Tidal Variations in UT The subdaily and daily tidal variations in Earth orientation due to the effect of ocean tides has been added. HAPTER General Relativistic Models for Propagation The formulation has been modified to be consistent with IAU/IUGG resolutions. ontributors Z. Altamimi B. Archinal E. F. Arias S. Bettadpur K. Borkowski. Boucher P. Brosche M. Bursa N. apitaine M. S. arter P. ook V. Dehant S. R. Dickman R. Eanes T. M. Eubanks M. Feissel H. Fliegel T. Fukushima J. Gipson R. Gross E. Groten T. A. Herring G. H. Kaplan J. Kovalevsky R. Langley B. Luzum. Ma P. M. Mathews D. N. Matsakis V. Mendes A. E. Niell R. Noomen T. Otsubo S. Pagiatakis E.. Pavlis W. R. Peltier J. R. Ray R. Ray J. Ries M. Rothacher H.-G. Scher neck B. E. Schutz P. Schwintzer P. K. Seidelmann E. M. Standish J. A. Steppe T. vandam J. Verheijen Errata from Technical Note p. 9: GPS Block II dz =.959m vice.9m. p. 8: Table., first line, last value should be -.7 vice.7. p. 8: Hobart Ql EW tangential phase should be 6. vice -6.. p. 89: Malibu M EW tangential phase should be -. vice.. p. 98: Pearblossum M radial phase should be. vice.. p. 7: paragraph, read...of the International Astronomical Union. p. 7: the formula reads ds = c dr = -(l - ^)(dz ) + (l + ~)[{dx*f + (dx ) + (dz ) )]. (the minus sign was omitted in the second line)

7 p., note : the formulae giving TB-TG read TB - TG = c~ [ [ (v e/ + U ext (x e )dt + v e.(x - x e )]. Jt TB - TG = L c x (JP -.5) x 86 + c~ v e.{x - s e ) + P. (the "." in the vector product v e.(x x e ) was omitted) References DeMets,, Gordon, R. G., Argus, D. F., and Stein, S., 99, "Effect of Recent Revisions to the Geomagnetic Reversal Time Scale on Estimates of urrent Plate Motions," Geophys, Res. LeL,, pp Fricke, W., 98, "Determination of the Equinox and Equator of the FK5," Astron. Astrophys., 7, pp. L-6. Le Systeme International d'unites (SI), 99, Bureau International des Poids et Mesures, Sevres, France. Melbourne, W., Anderle, R., Feissel, M., King, R., Mcarthy, D., Smith, D., Tapley, B., Vicente, R., 98, Project MERIT Standards, U.S. Naval Observatory ircular No. 67. Mcarthy, D. D., 989, IERS Standards, IERS Technical Note, Observatoire de Paris, Paris. Mcarthy, D. D., 99, IERS Standards, IERS Technical Note, Observatoire de Paris, Paris. Standish, E. M., Newhall, X X, Williams, J. G., and Folkner, W. F., 995, "JPL Planetary and Lunar Ephemerides, DE /LE," JPL IOM.-7, to be submitted to Astron. Astrophys.

8 HAPTER ONVENTIONAL ELESTIAL REFERENE SYSTEM In 99 the International Astronomical Union (IAU) decided that the IAU celestial reference System would be realized by a celestial reference frame defined by the precise coordinates of extragalactic radio sources. The related IAU recommendations (see Mcarthy, 99) specify that the origin is to be at the barycenter of the solar system and the directions of the axes are to be fixed with respect to the quasars. In compliance with this recommendation, the IERS elestial Reference System (IRS) is realized by the IERS elestial Reference Frame (IRF) defined by the J. equatorial coordinates of extragalactic objects determined from Very Long Baseline Interferometry (VLBI) observations. It is a frame whose directions are consistent with those of the FK5 (Fricke et al., 988). The origin is located at the barycenter of the solar system through appropriate modeling of observations in the framework of General Relativity (see hapters and ). The rotational stability of the frame is based on the assumption that the sources have no proper motions. hecks are performed regularly to ensure the validity of this constraint (Ma and Shaffer, 99; Eubanks et al, 99). The HIPPAROS reference frame is planned to be linked astrometrically to the IRF to unify the radio and optical coordinate Systems at the level of ±'/5 in direction and ±'/5/year in rotation (Lindegren and Kovalevsky, 995). The IRS is recommended for the IAU conventional celestial system by the IAU Working Group on Reference Frames under the name "International elestial Reference System." See Arias et al. (995). Equator The IAU recommendations call for the principal plane of the conventional reference frame to be close to the mean equator at J.. The VLBI observations which are used to establish the extragalactic reference frame also provide the monitoring of the motion of the celestial pole in the sky (precession and nutation). In this way, the VLBI analyses provide corrections to the conventional IAU modeis for precession and nutation (Lieske et al, 977; Seidelmann, 98) and the accurate estimation of the shift of the mean pole at J. relative to its conventional one, to which the pole of the IRS is attached. Based on the VLBI observations available in early 995 and the IERS correction model for precession and nutation (see hapters and 5), one can estimate the shift of the pole at J. relative to the IERS celestial pole to be 7. ±. mas in the direction h and 5. ±. mas in the direction 8 h. On the other hand, the IAU recommendations stipulate that the direction of the new conventional celestial pole be consistent with that of the FK5. The uncertainty in the direction of the FK5 pole can be estimated by considering () that the systematic part is dominated by a correction of O'.'5/cy to the precession constant imbedded in the FK5 System, and () by adopting Fricke's (98) estimation of the accuracy of the FK5 equator (±'/), and Schwan's (988) estimation of the limit of the residual rotation (±'.'7/cy), taking the epochs of observations from Fricke et al. (988). Assuming that the error in the precession rate is absorbed by the proper motions of stars, the uncertainty in the FK5 pole Position relative to the mean pole at J. estimated in this way is ± 5 mas. The IERS celestial pole is therefore consistent with that of the FK5 within the uncertainty of the latter. Origin of Right Ascension The IAU recommends that the origin of right ascensions of the new celestial reference system be close to the dynamical equinox at J.. The x axis of the IERS celestial system was implicitly defined in the initial realization (Arias et al., 988) by adopting the mean right ascension of radio

9 sources in a group of catalogs that were compiled by fixing the right ascension of 7B to the usual (Hazard et al, 97) conventional FK5 value ( h 9 m 6!6997 at J.) (Kaplan et al, 98). The uncertainty in the FK5 origin of right ascensions can be derived from the quadratic sum of the accuracies given by Fricke (98) and Schwan (988), considering a mean epoch of 955 for the proper motions in right ascension. The uncertainty thus obtained is ± 8 mas. Folkner et al. (99), comparing VLBI and LLR Earth orientation and terrestrial frames and the estimated frame tie of planetary ephemerides, show that the mean equinox of J. is shifted from the right ascension origin of the IERS system by 78 ± mas (direct rotation around the polar axis). This shows that the IRS origin of right ascension complies with the IAU recommendations. Precision and Accuracy The estimation of coordinates performed by the entral Bureau of the IERS is based on individual frames contributed by the IERS Analysis enters. Several extragalactic frames are produced each year by independent VLBI groups. Selected realizations are used to form the IRF. The algorithm used for the compilation is designed primarily to maintain the three directions of axes fixed for successive realizations while the precision of coordinates of individual sources is improved. Successive realizations produced up to now have maintained the initial definition of the axes within ±7. The inaccuracy of the conventional IAU 976 Precession and IAU 98 Theory of Nutation would give rise to systematic errors in the source positions and to misorientation of the axes of the frames, both at the level of a few milliarcseconds, in the analysis of VLBI observations. Therefore, the usual practice in catalog work is to estimate additional parameters which describe the motion of the celestial pole relative to its conventional position (see Sovers, 99). Another type of possible systematic error is related to low elevation observations in cases where the observing network has a modest north-south extension. This effect can be modelled as a linear dependence of declination errors on declination which can reach. mas per degree and create a tilt of the frame equator of up to.6 mas (Feissel et al, 995). However, for recent global analyses these systematic differences are at the level of. mas per degree and. mas respectively (IERS, 995). No other type of systematic differences can be found above this level in the present-day VLBI celestial reference frames (Feissel et al., 995, Eubanks et al., 99). After taking into account the small systematic errors mentioned above, the precision of the source coordinates is a white noise function of the number of observations, with a typical value of ±. mas for observations (Arias et al, 995). Avaüability of the Frame The catalog of source coordinates published in the 99 IERS Annual Report (July 995) provides access to the IRS. It includes a total of 68 objects, among which 6 with good observational histories attach the frame to the IRS. In the future, based on new observations and new analyses, the stability of the source coordinates will be monitored, and the appropriate warnings and Updates will appear in IERS publications. The direct access to the quasars is most precise through VLBI observations, a technique which is not widely available to users. Therefore, while VLBI is used for the maintenance of the frame, the tie of the IRF to the major practical reference frames may be obtained through use of the IERS Terrestrial Reference Frame (ITRF, see hapter ), the HIPPAROS Galactic Reference Frame, and the JPL ephemerides of the solar system (see hapter ).

10 The principles on which the ITRF is established and maintained are described in hapter. The IERS Earth orientation parameters provide the permanent tie of the IRF to the ITRF. They describe the orientation of the elestial Ephemeris Pole in the terrestrial system and in the celestial system (polar coordinates x, y; nutation offsets dt/ 7, de) and the orientation of the Earth around this axis (UT-TAI), as a function of time. This tie is available daily with an accuracy of ±.5 mas in the IERS publications. The other ties to major celestial frames are established by differential VLBI observations of solar system probes, galactic stars relative to quasars and other ground- or space-based astrometry projects. The tie of the solar system ephemerides of the Jet Propulsion Laboratory (JPL) is described by Standish et al. (995). Its estimated precision is ± mas, according to Folkner et al. (99). The tie of the galactic frame to IRS is a part of the HIPPAROS project. It is described in some detail by Lindegren and Kovalevsky (995). Its expected accuracy is ±.5 mas at the HIPPAROS mean epoch of Observation (99.5) and ±.5 mas/year for the time evolution. References Arias E. F., harlot P., Feissel M., Lestrade J.-F., 995, "The Extragalactic Reference System of the International Earth Rotation Service, IRS," Astron. Astrophys.,, pp Arias, E. F., Feissel, M., and Lestrade, J.-F., 988, "An Extragalactic elestial Reference Frame onsistent with the BIH Terrestrial System (987)," BIH Annual Rep. for 987, pp. D--D-. Eubanks, T. M., Matsakis, D. N., Josties, F. J.,Archinal, B. A., Kingham, K. A., Martin, J.., Mcarthy, D. D., Klioner, S. A., Herring, T. A., 99, "Secular motions of extragalactic radio sources and the stability of the radio reference frame," Proc. IAU Symposium 66, J. Kovalevsky (ed.). Feissel, M., Ma,, Sovers, O., and Jacobs,, 995, "Sub-milliarcsecond VLBI celestial frames: problem areas," (unpublished contribution). Folkner, W. M., harlot, P., Finger, M. H., Williams, J. G., Sovers,. J., Newhall, X X, and Standish, E. M., 99, "Determination of the extragalactic-planetary frame tie from Joint analysis of radio interferometric and lunar laser ranging measurements," Astron. Astrophys., 87, pp Fricke, W., 98, "Determination of the Equinox and Equator of the FK5," Astron. Astrophys., 7, p. L-6. Fricke, W., Schwan, H., and Lederle, T., 988, Fifth Fundamental atalogue, Part I. Veroff. Astron. Rechen Inst., Heidelberg. Hazard,, Sutton, J., Argue, A. N., Kenworthy,. N., Morrison, L. V., and Murray,. A., 97, "Accurate radio and optical positions of 7B," Nature Phys. Sei.,, p. 89. IERS, 995, 99 International Earth Rotation Service Annual Report, Observatoire de Paris, Paris. Kaplan, G. H., Josties, F. J., Angerhofer, P. E., Johnston, K. J., and Spencer, J. H., 98, "Precise Radio Source Positions from Interferometric Observations," Astron. J., 87, pp

11 Lieske, J. H., Lederle, T., Fricke, W., and Morando, B., 977, "Expression for the Precession Quantities Based upon the IAU (976) System of Astronomical onstants," Astron. Astrophys., 58, pp. -6. Lindegren, L. and Kovalevsky, J., 995, "Linking the Hipparcos catalogue to the extragalactic reference frame," Astron. Astrophys.,, pp Ma,. and Shaffer, D. B., 99, "Stability of the extragalactic reference frame realized by VLBI," in Reference Systems, J. A. Hughes,. A. Smith, and G. H. Kaplan (eds), U. S. Naval Observatory, Washington, pp. 5-. Mcarthy D.D. (ed.), 99, "IERS Standards," IERS Technical Note, Observatoire de Paris, Paris. Schwan, H., 988, "Precession and galactic rotation in the system of FK5," Astron. Astrophys., 98, pp. 6-. Seidelmann, P. K., 98, "98 IAU Nutation: The Final Report of the IAU Working Group on Nutation," elest. Mech., 7, pp Sovers, O. J., 99, "onsistency of Nutation Modeling in Radio Source Position omparisons," in Reference Systems, J. A. Hughes,. A. Smith, and G. H. Kaplan (eds), U. S. Naval Observatory, Washington, pp. -. Standish, E. M., Newhall, X X, Williams, J. G., and Folkner, W. F., 995, "JPL Planetary and Lunar Ephemerides, DE/LE," JPL IOM.-7, to be submitted to Astron. Astrophys.

12 HAPTER ONVENTIONAL DYNAMIAL REFERENE FRAME The planetary and lunar ephemerides recommended for the IERS Standards are the JPL Development Ephemeris DE and the Lunar Ephemeris LE (Standish et al., 995). The reference frame of these new ephemerides is that of IRF. The ephemerides have been adjusted to all relevant observational data, including recent observations taken with respect to the IERS frame. It is expected that DE/LE will eventually replace DE/LE (Standish, 99) as the basis for the international almanacs. Table. shows the IAU 976 values of the planetary masses and the values used in the creation of both DE/LE and of DE/LE. Also shown in the table are references for the DE set, the current best estimates. Also associated with the ephemerides is the set of astronomical constants used in the ephemeris creation; these are listed in Table.. The constants which are provided directly with the ephemerides should be considered to be an integral part of them; they will sometimes differ from a more Standard set, but the differences are necessary for the optimal fitting of the data. Availability of DE Sections of DE are now available from the anonymous ftp site: "navigator.jpl.nasa.gov" [8.9..8]. For a "navio" version (in-house JPL format), the following "navio" and "nioftp" versions are available, covering 98-: "navigator:/ephem/navio/des" and "navigator:/ephem/navio/des.ftp" For other time-spans, contact F A Mcreary ([email protected]) or E M Standish ([email protected]). An outside user is advised to first get and read the file, "navigator:/ephem/export/read.me"; it should answer most questions about retrieving and using the JPL ephemerides. Table IAU, DE and DE planetary mass values, expressed in reciprocal solar masses. Planet 976 IAU DE DE Reference for DE value Mercury Venus Earth & Moon Mars Jupiter Saturn Uranus Neptune Pluto Anderson et al., 987 Sjögren et al., 99 Williams et al., 995 Null, 969 ampbell and Synott, 985 ampbell and Anderson, 989 Jacobson et al., 99 Jacobson et al., 99 Tholen and Buie, 996

13 Table.. Auxiliary constants from the JPL Planetary and Lunar Ephemerides DE/LE. Scale (km/au) Scale (secs/au) Speed of light (km/sec) Obliquity of the ecliptic 6'l7 Earth-Moon mass ratio GMcere..6 X lo^gmsun GUp allas.5xlo"- GM S un GUvesta.xlO- GM S un density class.8 density c / as55. density ci ass M 5. References Anderson, J. D., olombo, G., Esposito, P. B., Lau, E. L., and Trager, G. B., 987, "The Mass Gravity Field and Ephemeris of Mercury," Icarus, 7, pp ampbell, J. K. and Anderson, J. D., 989, "Gravity Field of the Saturnian System from Pioneer and Voyager Tracking Data," Astron. J., 97, pp ampbell, J. K. and Synott, S. P., 985, "Gravity Field of the Jovian System from Pioneer and Voyager Tracking Data," Astron. J., 9, pp Jacobson, R. A., Riedel, J. E. and Taylor, A. H., 99, "The Orbits of Triton and Nereid from Spacecraft and Earth-based Observations", Astron. Astrophys., 7, pp Jacobson, R. A., ampbell, J. K., Taylor, A. H. and Synnott, S. P., 99, "The Masses of Uranus and its Major Satellites from Voyager Tracking Data and Earth-based Uranian Satellite Data", Astron J., (6), pp Null, G. W., 969, "A Solution for the Mass and Dynamical Oblateness of Mars Using Mariner-IV Doppler Data," Bull. American Astron. Soc,, p. 56. Sjögren, W. L., Trager, G. B., and Roldan G. R., 99, "Venus: A Total Mass Estimate," Res. Let., 7, pp Geophys. Standish, E. M., 99, "The Observational Basis for JPL's DE, the Planetary Ephemerides of the Astronomical Almanac," Astron. Astrophys.,, pp Standish, E. M., Newhall, X X, Williams, J. G. and Folkner, W. F., 995, "JPL Planetary and Lunar Ephemerides, DE/LE," JPL IOM.-7, to be submitted to Astron. Astrophys. Tholen, D. J. and Buie, M. W., 996, "The Orbit of haron. I. New Hubble Space Telescope Observations", submitted to Icarus. Williams, J. G., Newhall, X X, and Standish, E. M., 995, (result from least Squares adjustment of DE/LE).

14 HAPTER ONVENTIONAL TERRESTRIAL REFERENE SYSTEM Definition The Terrestrial Reference System adopted for either the analysis of individual data sets by techniques (VLBI, SLR, LLR, GPS, DORIS, PRARE...) or the combination of individual Solutions into a unified set of data (station coordinates, Earth orientation parameters, etc..) follows these criteria (Boucher, 99): a) It is geocentric, the center of mass being defined for the whole Earth, including oceans and atmosphere. b) Its scale is that of a local Earth frame, in the meaning of a relativistic theory of gravitation. c) Its orientation was initially given by the BIH orientation at 98.. d) Its time evolution in orientation will create no residual global rotation with regards to the crust. Realization A onventional Terrestrial Reference System (TRS) can be realized through a reference frame i.e. a set of coordinates for a network of stations. Such a realization will be specified by cartesian equatorial coordinates A", Y, and Z, by preference. If geographical coordinates are needed, the GRS8 ellipsoid is recommended (a=6787. m, eccentricity =.6698). The TRS which is monitored by IERS is called the International Terrestrial Reference System (ITRS) and is specified by the IUGG Resolution no. adopted at the th IUGG General Assembly of Vienna in 99. Each analysis center should compare its reference frame to a realization of the ITRS. Within IERS, each Terrestrial Reference Frame (TRF) is either directly, or after transformation, expressed as a realization of the ITRS. The position of a point located on the surface of the solid Earth should be expressed by X(t) = X + %(t - t ) + Y, A^(, i where AXi are corrections due to various time changing effects, and XQ and VQ are position and velocity at the epoch o- The corrections to be considered are solid Earth tide displacement (füll correction including permanent effect, see hapter 7), ocean loading, post glacial rebound, and atmospheric loading. Further corrections could be added if they are at mm level or greater, and can be computed by a suitable model. Realizations of the ITRS are produced by IERS under the name International Terrestrial Reference Frames (ITRF), which consist of lists of coordinates (and velocities) for a selection of IERS sites (tracking stations or related ground markers). urrently, ITRF-yy is published annually by the IERS in the Technical Notes (cf. Boucher et al., 996). The numbers (yy) following the designation "ITRF" specify the last year whose data were used in the formation of the frame. Hence ITRF9 designates the frame of coordinates and velocities constructed in 995 using all of the IERS data available through 99. More recently, since 99, other special realizations have been produced, such as Solutions for IGS core stations (ITRF-Py series) (IGS, 995). It is also anticipated that monthly series may be determined.

15 Terrestrial reference frames may be defined within Systems which account for the effect of the permanent tide differently. In the terminology of Ekman (989), Rapp (99) and Poutanen et al. (996) the permanent or "zero-frequency" tide is retained in the "zero-tide" system. In this system the crust corresponds to the realistic time average of the crust which varies because of the action of the luni-solar tides. In a different system referred to as "tide-free," all of the effects of the permanent tide are removed. This is not realistic since the crust in this case cannot be "observed," nor are the Love Numbers required to describe the permanent tide known adequately. No error is associated with either onvention as long as it is clear which system is employed in the analysis of the data. A third "mean-tide" system has been defined in which the crust is equivalent to that of the tide-free system and, in which, the effect of the permanent tide is also removed from the geoid. oordinates of the ITRF are given in a conventional frame where the effects of all tides are removed as recommended in IERS Technical Note (Mcarthy, 99). There, it is recommended that equation 6 on p. 57 be used to account for the permanent tide. Note that this is in contradiction to Resolution 6 of the General Assembly of the IAG in 98. The corresponding equation in this publication is equation 8 of hapter 7. To place the coordinates in a zero-tide system it is necessary to apply equation 8 of IERS Technical Note (Mcarthy, 99) or the corresponding Version, equation 7 of hapter 7 in this publication. In data analysis, Xo and VQ should be considered as solve-for parameters. In particular, if a nonlinear change occurs (earthquake, volcanic event...), a new X should be adopted. When adjusting parameters, particularly velocities, the IERS orientation should be kept at all epochs, ensuring the alignment at a reference epoch and the time evolution through a no net rotation condition. The way followed by various analysis centers depends on their own view of modelling, and on the techniques themselves. For the origin, only data which can be modelled by dynamical techniques (currently SLR, LLR, GPS or DORIS for IERS) can determine the center of mass. The VLBI system can be referred to a geocentric system by adopting for a Station its geocentric position at a reference epoch as provided from external Information. The scale is obtained by appropriate relativistic modelling. Specifically, according to IAU and IUGG resolutions, the scale is consistent with the TG time coordinate for a geocentric local frame. A detailed treatment can be found in hapter. The orientation is defined by adopting IERS (or BIH) Earth orientation parameters at a reference epoch. In the case of dynamical techniques, an additional constraint in longitude is necessary to remove ill-conditioning. The unit of length is the meter (SI). The IERS Reference Pole (IRP) and Reference Meridian (IRM) are consistent with the corresponding directions in the BIH Terrestrial System (BTS) within ±75. The BIH reference pole was adjusted to the onventional International Origin (IO) in 967; it was then kept stable independently until 987. The uncertainty of the tie of the IRP with the IO is ±7. The time evolution of the orientation will be ensured by using a no-net-rotation condition with regards to horizontal tectonic motions over the whole Earth. Transformation Parameters to World oordinate Systems and Various Datums The seven-parameter similarity transformation between any two artesian Systems, e.g., from (x, y, z) to (xs, ys, zs) can be written as D ~ Ä A D ~ y j, () - Rl D

16 where T\, T, T = coordinates of the origin of the frame (xs, ys, zs) in the frame (x, y, z); R\, R, Ä= differential rotations (expressed in radians) respectively, around the axes (xs, ys,zs) to establish parallelism with the (x,y,z) frame; and D = differential scale change. The use of the previous algorithm to transform coordinates from one datum to another must be done with care. At some level of accuracy, each reference system has multiple realizations. In many cases, two such realizations of the same system have nonzero transformation parameters. The usual reason is that such parameters are currently adjusted between two sets of coordinates. In this case, any global systematic error which can be mapped into a scale, shift or rotation will go into these seven parameters. Therefore we give here (Table.) only parameters from ITRF9 to previous ITRF series. These numbers are the responsibility of the IERS. For other transformations, either to global Systems (such as WGS8) or regional (such as NAD8 or ETRS89), or even local Systems, users should contact relevent agencies. For WGS8, the main result is that the most recent realizations of WGS8 based on GPS are consistent with the ITRF realization at the. meter level (Malys and Slater, 99). Otherwise, one can refer to IAG bodies, such as ommission X on Global and Regional Geodetic Networks (Geodesist's Handbook, 996). Table.. Transformation parameters from ITRF9 to past ITRFs. "ppb" refers to parts per billion (IQ 9 ). Rates must be applied for ITRF9. The units for rate are understood to be "per year." oordinate System (datum) Tl (cm) T (cm) T (cm) D (Ppb) Rl (mas) R (mas) R (mas) Epoch ITRF88 ITRF89 ITRF9 ITRF9 ITRF9 ITRF rates Once the artesian coordinates (x,y,z) are known, they can be transformed to "datum" or curvilinear geodetic coordinates (X,(f>,h) referred to an ellipsoid of semi-major axis a and flattening /, using the following code (Borkowski, 989). First compute A = tan"" (^) properly determining the quadrant from x and y ( < A < ir) and r = yjx + y. subroutine GED(r,z,fi,h) c Program to transform artesian to geodetic coordinates c based on the exact Solution (Borkowski,989) c Input : r, z = equatorial [m] and polar [m] components c Output: fi, h = geodetic coord's (latitude [rad], height [m]) implicit real*8(a-h,o-z) c GRS8 ellipsoid: semimajor axis (a) and inverse flattening (fr) data a,fr /6787.d,98.57d/ b «dsign(a - a/fr,z) E = ((z + b)*b/a - a)/r F = ((z - b)*b/a + a)/r

17 Find Solution to: t** + *E*t** *F*t -= P (E*F + I.)*.d/.d Q «(E*E - F*F)*.dO D s P*P*P + Q*Q if(d.ge.odo) then s = dsqrt(d) + Q s * dsign(dexp(dlog(dabs(s))/.d),s) v P/s - s Improve the accuracy of numeric values of v v = -(Q + Q + v*v*v)/(.do*p) eise v =.do*dsqrt(-p)*dcos(dacos(q/p/dsqrt(-p))/.do) endif G =.5dO*(E + dsqrt(e*e + v)) t = dsqrt(g*g + (F - v*g)/(g -»- G - E)) - G fi = datan((l.d - t*t)*a/(.d*b*t)) h (r - a*t)*dcos(fi)» (z - b)*dsin(fi) return end Plate Motion Model One of the factors which affect Earth rotation results is the motion of the tectonic plates which make up the Earth's surface. As the plates move, fixed coordinates for the observing stations become inconsistent with each other. The rates of relative motions for some observing sites are 5 cm per year or larger. The observations of plate motions by modern methods appear to be roughly consistent with the average rates over the last few million years derived from the geological record and other geophysical information. Thus, the NNR-NUVELA model for plate motions given by DeMets et al. (99) is recommended. If a velocity has not yet been determined in the ITRF for a particular Station, the velocity VQ should be expressed as V = V p i ate + V r where V p i a te is the horizontal velocity computed from the NNR-NUVELA model (DeMets et al., 99) and V r a residual velocity. The artesian rotation vector for each of the major plates is given in Table.. A subroutine called ABSMO-NUVEL is also included below. It computes the new site position from the old site position using the recommended plate motion model. Fig.. shows the plates on a map of the world.

18 Figure.. Map of the tectonic plates. Table.. artesian rotation vector for each plate using the NNR-NUVELA kinematic plate model (no net rotation). n x üy n z Plate Name rad/my. rad/my. rad/my. Pacific ocos Nazca aribbean South America Antarctica India Australia Africa Arabia Eurasia North America Juan de Fuca Philippine Rivera Scotia n.nnoin The NNR-NUVELA model should be used as a default, for stations which appear to follow

19 reasonably its values. For some stations, particularly in the vicinity of plate boundaries, users may benefit by estimating velocities or using specific values not derived from NNR-NUVELA. This is also a way to take into account now some non- negligible vertical motions. Published Station coordinates should include the epoch associated with the coordinates. The original subroutine was a coding of the AM- model from J. B. Minster. hanges have been made to represent NNR-NUVELA model (DeMets et a/., 99). SUBROUTINE ABSM_NUVEL(PSIT,TO,XO,YO,ZO,T,X,Y,Z) ABSMO-NUVEL takes a site specified by its initial coordinates X,Y,Z at time TO, and computes its updated positions X,Y,Z at time T, based on the geological "absolute". Original author: J.B. Minster, Science Horizons. DFA: Revised by Don Argus, Northwestern University DFA: uses absolute model NNR-NUVEL Transcribed from USNO ircular 67 "Project Merit Standards" by Tony Mallama with slight modification to the documentation and code. Times are given in years, e.g for Jan i, 988. PSIT is the four character abbreviation for the plate name, if PSIT is not recognized then the new positions are returned as zero. IMPLIIT NONE HARATER* PSIT,PNM(6) REAL*8 MX(6),MY(6),MZ(6) REAL*8 X,Y,Z REAL*8 X,Y,Z,T,T INTEGER* IPSIT.I DFA: NNR-NUVELA DATA (PNM(I), OMX(I), OMY(I), OMZ(I), ft I =,6) & /'PF, -.5,.8, -.997, ft»afr ft 'ANTA' ft 'ARAB' ft 'AUST' ft 'ARB' ft '' ft 'EURA'.89, -.99,.9, -.8, -.7,.76,.6685, -.5,.676,.789,.5,.68, -.78, -.85,.58, -.975, -.65,.95, -.98, -.95,.5, 5

20 & 'INDI',.667,.,.679, & 'NAZ, -.5, ,.969, & 'NOAM',.58, -.599, -.5, & 'SOAM', -.8, -.55, -.87, & 'JUFU\.5,.86, -.58, & 'PHIL',.9, -.76, -.967, & 'RIVR', -.99, -.96, -.5, &, SOT >, -., -.66, -.7/ Initialize things properly IPSIT = - X =.D Y =.D Z = O.ODO Look up the plate in the list. DO I =,6 IF (PSIT.EQ. PNM(I)) IPSIT = I If plate name is not recognized return the new plate position as zero. IF (IPSIT.EQ. -) RETURN ompute the new coordinates X = XO + (RY*Z - RZ*Y) * (T-T)/.OD+6 Y = YO + (RZ*XO - RX*Z) * (T-T)/.OD+6 Z = ZO + (RX*Y - RY*X) * (T-T)/.OD+6 Finish up RETURN END References Borkowski, K. M., 989, "Accurate Algorithms to Transform Geocentric to Geodetic oordinates," Bull. Geod., 6, pp Boucher,., 99, "Definition and Realization of Terrestrial Reference Systems for Monitoring Earth Rotation," Variations in Earth Rotation, D. D. Mcarthy and W. E. arter (eds), pp Boucher,, Altamimi, Z., Feissel, M., and Sillard, P., 996, Results and Analysis of the ITRF9, IERS Technical Note, Observatoire de Paris, Paris. 6

21 DeMets,, Gordon, R. G., Argus, D. F., and Stein, S., 99, "Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions," Geophys. Res. Let.,, pp Ekman, M., 989, "Impacts of Geodetic Phenomena on Systems for Height and Gravity," Bull. 6, pp Geod., GeodesisVs Handhook, 996, International Association of Geodesy, in press. IGS, 995, International GPS Service for Geodynamics 99 Annual Report. IGS entral Bureau, JPL, Pasadena. Malys, S. and Slater, J. A., 99,"Maintenance and enhancement of the World Geodetic System 98," ION GPS 9, Salt Lake ity, Utah. Poutanen, M., Vermeer, M., and Mekinen, J., 996, "The Permanent Tide in GPS Positioning," J. of Geod., 7, pp Rapp, R. H., 989, "The treatment of permanent tidal effects in the analysis of satellite altimeter data for sea surface topography," Man. Geod.,, pp Soler, T. and Hothem, L. D., 989, "Important Parameters Used in Geodetic Transformations," J. Surv. Engrg., 5, pp

22 HAPTER NUMERIAL STANDARDS Table. listing numerical Standards is organized into 5 columns: item, value, uncertainty, reference, comment. All values are given in terms of SI units (Le Systeme International d'unites (SI), 99). The SI second, the basic unit of the TT time scale, is specifically assumed. If the TDB time scale is used, new units of time, ttdb, and length, TDB, are implicitly defined by the expressions (Seidelmann and Fukushima, 99) ttdb = t/(l - ia),and *TDÄ=//(-IB), where t and are the SI units and Lß is a derived constant given in Table.. hapter provides further details on the transformations between time scales. The 976 IAU System of Astronomical onstants (Astronomical Almanac for the Year 98) adopted for all astronomical constants which do not appear in Table.. ls References Astronomical Almanac for the Year 98, U.S. Government Printing Office, Washington, D. [] Bursa, M., 995, Report of the I.A.G. Special ommission S, Fundamental onstants, XXI, I.A.G. General Assembly. [] ohen, E. R. and Taylor, B. N., 97, J. Phys. hem. Ref. Data,, p. 66. [] ohen, E. R. and Taylor, B. N., 986, "The 986 Adjustment of the Fundamental Physical onstants," ODATA Bulletin, Bureau International des Poids et Mesures. [] Fukushima, T., 99, "Time Ephemeris," Astron. Astrophys., 9, pp [5] Herring, T., 996, "KSV- Precession/Nutation," personal communication. Le Systeme International d'unites (SI), 99, Bureau International des Poids et Mesures, Sevres, France. Seidelmann, P. K. and Fukushima, T., 99, "Why New Time Scales?" Astron. Astrophys., 65, pp [6] Standish, E. M., 99, Report of the IAU WGAS Sub-group on Numerical Standards. [7] Standish, E. M., Newhall, X X, Williams, J. G., and Follkner, W. F., 995 "JPL Planetary and Lunar Ephemerides, DE/LE," JPL IOM.-7 to be submitted to Astron. Astrophys. 8

23 Table.. TERS NnTnerical Standards ITEM VALUE c ms x ~ 8 L B Lc LG W r A T A P «O-E // <^ffi G GM e /* u; GM X" X lo" m s" s m 59'.'965/centwy m x " x - n m kg- s x m s" x -trads-.7 x m s- ffe 9.787ms" Äo = GM JW m UNERTAINTY Defining x " 7 x lo" 7 x " 7.m s -.s m '.'/entury '.'5.m.. x " 8.5 x lo-^m ^-^- " [] 8 x 5 m s - [] 5 x - variable x _6 ms -.m REF.. OMMENTS [] [] [] [] [] [7] [7] [5] [7] [] [] [] [7] [] [6] [] [] Speed of light Average value of d(tb)/d(tt)-l Average value of d(tb)/d(tg)-l Average value of d(tg)/d(tt)-l Potential of the geoid Astronomical unit in seconds Astronomical unit in meters General precession in longitude at J. Obliquity of the ecliptic at J. Equatorial radius of the Earth Flattening factor of the Earth Dynamical form-factor of the Earth onstant of gravitation Geocentric gravitational constant Moon-Earth mass ratio Nominal mean angular velocity of the Earth Heliocentric gravitational constant Mean equatorial gravity Geopotential scale factor Some geodetic parameters are affected by tidal variations (see hapter ). The values given in Table. are in the zero-tide system so that they correspond to a realistic time-averaged crust. This is done to be consistent with XVIII IAG General Assembly Resolution 6. 9

24 HAPTER 5 TRANSFORMATION BETWEEN THE ELESTIAL AND TERRES TRIAL SYSTEMS The coordinate transformation to be used from the TRS to the RS at the date t of the Observation can be written as: [RS] = PN(t)R(t)W(t) [TRS], where PN(t), R(t) and W(t) are the transformation matrices arising from the motion of the elestial Ephemeris Pole (EP) in the RS, from the rotation of the Earth around the axis of the EP, and from polar motion respectively. Two equivalent options can be used. These two options have been shown to be consistent within ±.5 milliseconds of are (mas) both theoretically (apitaine, 99) and numerically using existing astrometric data (apitaine and hollet, 99) or simulated data over two centuries (apitaine and Gontier, 99). Option, corresponding to the classical procedure, makes use of the equinox for realizing the intermediate reference frame of date t. It uses apparent Greenwich Sidereal Time in the transformation matrix R(t) and the classical precession and nutation parameters in the transformation matrix PN(t), Option makes use of the onventional Ephemeris Origin (EO) originally referred to as the nonrotating origin (Guinot, 979) to realize the intermediate reference frame of date t: it uses the stellar angle (from the origin in the TRS to the EO in the RS) in the transformation matrix R(t) and the two coordinates of the elestial Ephemeris Pole in the RS (apitaine, 99) in the transformation matrix PN(t). This leads to very simple expressions of the partial derivatives of observables with respect to polar coordinates, UT, and celestial pole offsets. The following sections give the details of these two options as well as the Standard expressions necessary to obtain the numerical values of the relevant parameters at the date of the Observation. Subroutines for options and of the coordinate transformation from the TRS to the RS are available from the entral Bureau on request together with the development of the parameters. The expressions of the precession and nutation quantities have been developed originally as functions of barycentric dynamical time (TDB) defined by IAU recommendations of 976 and 979. In 99 the IAU adopted definitions of other time scales. See hapter for the relationships among these time scales. The parameter t is defined by t = (TT - January ld h TT) in days/655. This definition is consistent with Resolution 7 passed at the 99 Hague General Assembly of the IAU which recommends that J. be defined at the geocenter and at the date. January.5 TT = Julian Date 555. TT. In the following, R\, J? and A denote direct rotations about the axes, and of the coordinate frame. oordinate Transformation Referred to the Equinox Option uses the form of the coordinate transformation [RS] = PN\t)R'(t)W(t) [TRS], ()

25 in which the three fundamental components are (Mueller, 969) W'(t) = R (y p )>R (x p ), x p and y p being the "polar coordinates" of the EP in the TRS; R f (t) = (-GST), GST being Greenwich True Sidereal Time at date t, including both the effect of Earth rotation and the accumulated precession and nutation in right ascension; and PN'(t) = [P][N], with [P] = R (( A ) R(-A) R{ZA) for the transformation matrix corresponding to the precession between the reference epoch and the date t, [N] = Ri(- A ) Ä (A) Ri(e A + Ae) for the transformation matrix corresponding to the nutation at date t. Standard values of the parameters to be used in form () of the transformation are explained below. The Standard polar coordinates to be used for the parameters x p and y p (if not estimated from the observations) are those published by the IERS. Apparent Greenwich Sidereal Time GST at the date t of the Observation, must be derived from the following expressions: (i) the relationship between Greenwich Mean Sidereal Time (GMST) and Universal Time as given by Aoki et al. (98): GMSTOH UTI = $ h m 5f f8866^ + Of97^ - 6? x HT 6 r*, with T' u = d^/655, d f u being the number of days elapsed since January, h UTI, taking on values ±.5, ±.5,..., (ii) the interval of GMST from h UTI to the hour of the Observation in UTI, GMST = GMSToh UTI + r[(utl - UT) + UT], where r is the ratio of universal to sidereal time as given by Aoki et al. (98), r = x HT!^ x l(t 5 r* and the UT-UT value to be used (if not estimated from the observations) is the IERS value. (iii) accumulated precession and nutation in right ascension (Aoki and Kinoshita, 98), GST = GMST + A^cose^ + '/6 sin + 76 sin, where is the mean longitude of the ascending node of the lunar orbit. The last two terms have not been included in the IERS Standards previously. They should not be included in (iii) until January

26 997 when their use will begin. This date is chosen to minimize any discontinuity in UTI. The effect of these terms on the estimation of UTI has been described by apitaine and Gontier (99). The numerical expression for the precession quantities,,, ZA and e A have been given by Lieske et al. (977) as functions of two time parameters t and T (the last parameter representing Julian centuries from J. to an arbitrary epoch). The simplified expressions when the arbitrary epoch is chosen to be J. (i.e. T = ) are d = 678* + 788t t, 9 A = 79t t - 78t, z A = 678t t + 78t, e A = t - 759t + 78t. The nutation quantities At/? and Ac to be used are the nutation angles in longitude and obliquity. For observations requiring values of the nutation angles with an accuracy of ± mas, it is necessary to add (if those quantities are not estimated from the observations) the IERS published values (observed or predicted) for the "celestial pole offsets" (i.e. corrections dpsi and deps). The IAU 98 Theory of Nutation The IAU 98 Theory of Nutation (Seidelmann, 98; Wahr, 98) is based on a modification of a rigid Earth theory published by Kinoshita (977) and on the geophysical model 66A of Gilbert and Dziewonski (975). It therefore includes the effects of a solid inner core and a liquid outer core and a "distribution of elastic parameters inferred from a large set of seismological data." The constants defining the theory are given in Table 5.. VLBI and LLR observations have shown that there are deficiencies in the IAU 976 Precession and in the IAU 98 Theory of Nutation. However, these modeis are kept as part of the IERS Standards and the observed differences (SAip and 6Ae, equivalent to dpsi and deps in the IERS Bulletins) with respect to the conventional celestial pole position defined by the modeis are monitored and reported by the IERS as "celestial pole offsets." Using these offsets the corrected nutation is given by Aip = A^(IAU 98) + 6 Aip, and Ac = Ac(IAU98) + 6Ac. This is practically equivalent to replacing N with the rotation described by Lieske (99), N = ln lau. where 6 Aifr cos e t TZ = «SA^cosc* SAiftsmct SAe

27 e t = e A + Ae, and N\AV represents the IAU 98 Theory of Nutation according to which 6 6 A^ = Y,(Ai + A'i*) sin( ARGUMENT), Ae = ]T( t + B[t) cos( ARGUMENT). i=l t=l Here the N(,(i namely, ARGUMENT = ]T NiF {, = l,--*,5) being integers multiplying the Fundamental Arguments Fi of nutation theory, Fi = / F = /' F = F F = D F5 = Q = Mean Anomaly of the Moon =? t t t - 77t, = Mean Anomaly of the Sun = 57? t - 755t - 76t - 79t, = L - ü = 9? t - 775t - 77t + 77t, = Mean Elongation of the Moon from the Sun = 97? t t t - 769t, = Mean Longitude of the Ascending Node of the Moon = 5? t + 777t + 777t t, where t is measured in Julian enturies of 655 days of 86 seconds of Dynamical Time since J. and where V = 6 = 967 (Simon et al, 99). L is the Mean Longitude of the Moon. The reader is cautioned that there may be an ambiguity in the assignment of the set of five multipliers Nj to any particular term of the nutation series. This ambiguity is illustrated, and a onvention for unique assignment of multipliers for prograde and retrograde circular nutations is presented, in a note at the end of this chapter. Table 5.. IAU 98 Theory of Nutation in longitude Alf) and obliquity Ae, referred to the mean equator and equinox of date, with t measured in Julian centuries from epoch J.. The signs of the fundamental arguments, periods, and coefficients may differ from the original publication. These have been changed to be consistent with other portions of this chapter. 6 6 A^ = ^(Ai + A\t) sin(argument), Ae = ^T(Bi + B[t) cos( ARGUMENT). *=i t=i

28 MULTIPLIERS OF / /' F - - D fi PERIOD (days) LONGITUDE (7) Ai OBLIQUITY (7) Bi Bl

29 e = 6'78 sine = The IERS 996 Theory of Precession/Nutation At the request of the IERS, T. Herring (996) has analyzed the most recent VLBI and LLR data for geophysical parameters required to adjust the rigid Earth theory of nutation to the nonrigid theory. The non-rigid Earth theory coefficients resulting from this procedure are shown in Table 5.. Table 5. contains the planetary nutation terms and the necessary planetary arguments. These coefficients are meant to be used for prediction purposes and by those requiring accurate a priori estimates of nutation. They are not meant to replace the IAU Theory. The IERS will continue to publish observed estimates of the corrections to the IAU 976 Precession and the IAU 98 Theory of Nutation in its publications. The use of the IERS 996 Theory of nutation must also be associated with the use of improved numerical values for the precession rate of the equator in longitude and obliquity: 6tl> A = -.99"/c and 6u A = -."/c 5

30 Table 5.. IERS 996 series for nutation in longitude A^ and obliquity Ae, referred to the mean equator and equinox of date, with t measured in Julian centuries from epoch J.. The signs of the fundamental arguments, periods, and coefficients may differ from the original publication. These have been changed to be consistent with other portions of this chapter. / MULTIPLIERS OF /' F - D A^ = ]T( Ai + A\t) sin( ARGUMENT) + A'/ cos( ARGUMENT), i=i 6 Ae = ^(Bi ft i=l PERIOD + B[t) cos( ARGUMENT) + B'( sin(argument). (days) LONGF rude (. mas) Ai A'i OBLIQU] ITY (O.OOlraas) Bi B\ A'l B>>

31

32

33

34 Table 5.. IERS 996 Series for planetary nutation in longitude Aifi and obliquity Ae, referred to the mean equator and equinox of date, with t measured in Julian centuries from epoch J.. The signs of the fundamental arguments, periods, and coefficients may differ from the original publication. These have been changed to be consistent with other portions of this chapter. 8 8 Aij> = ^(Ai + A, it)sin(argument), Ae = ] ( < + B[t) cos{ ARGUMENT). i=l t=l We ie 'ilfa MULTIPLIERS OF h Isa Pa D F / PERIOD (days) LONGITUDE (O.OOlmas) Ai A\ OBLIQUITY (.raa.s) Bi B\

35 l Ve = 8? ?85678 x t l E =? ?785 x t

36 l Ma = 55?765+ 9?99 x t l Ju =?589+?95676 x t / Sa = 5? ?79 x t p a =?96977 x t +?86 x t The multipliers of the fundamental arguments of nutation theory Each of the lunisolar terms in the nutation series is characterized by a set of five integers Nj which determines the ARGUMENT for the term as a linear combination of the five Fundamental Arguments Fj, namely the Delaunay variables (, ', F,D,tt): ARGUMENT = ]j = JVjFj = N F, where N is the five-vector composed of the values (Ni,, iv 5 ) which characterize the term, and F is the five-vector (Fi,, F$). The Fj are functions of time, and the angular frequency of the nutation described by the term is given by = d( ARGUMENT) dt The frequency thus defined is positive for most terms, and negative for some. Planetary nutation terms differ from the above only in that ARGUMENT = * x TVjFj as noted in Table 5.. Over time scales involved in nutation studies, the frequency u is effectively time-independent, and one may write, for the sth term in the nutation series, ARGUMENT = u s t + a s. Different tables of nutations in longitude and obliquity do not necessarily assign the same set of multipliers Nj to a particular term in the nutation series. ompare, for instance, the.8 day term in Table 5. wherein N = (,,,,) with the same term in Seidelmann (98) where the assignment is N (,,,-,). The differences in the assignments arises from the fact that the replacement Nj Nj accompanied by reversal of the sign of the coefficient of sin(argumentj) in the series for A^> and Ae leaves these series unchanged. This freedom has led to some confusion in the assignment of multipliers of the Delaunay arguments for amplitudes of retrograde and prograde circular nutations. The nutations in longitude and obliquity at a given frequency u> s may be viewed as the superposition of a pair of prograde and retrograde circular nutations whose frequencies have the same absolute value, \u> s \, but with opposite signs. These circular nutation terms may be written as j^pro e -iq s {u) 9 t+a s ) an(j ßret eiq s (w s t+ a s ) where q s is the sign of UJ S : q s u s = K. With the onvention that the exponential factor involving the frequency v is to be written as e~~ %vt, one sees from the above expressions that the frequency of the prograde nutation, u^ro = (q s u 8 ) = \u> s \, is positive, and that of the retrograde nutation is negative (u r s et = ^s )-

37 Now, given the vector N of multipliers of the fundamental arguments for the nutations in longitude and obliquity of frequency u> s, one can choose multipliers N( pr ) and N^ret^ for the corresponding prograde and retrograde nutations in such a manner as to ensure that df -* df pro s Npro. _ = ^ ^ret = Nret. _ = ^ ^ This is accomplished by making the assignments N pro = q s N; N ret = -q s N, a practice that will be adhered to in this work. Note that N^pro^ and N^r t^ are unaffected by whether N or N is used to label the nutations in longitude and obüquity, because on switching from one to the other, the sign of LJ S also is switched simultaneously. onversion to Prograde and Retrograde Nutation Amplitudes With the onvention for the time dependence of the circular nutation amplitudes as stated above, the amplitudes of prograde and retrograde circular nutations are written as A pro = A pro ip pro op ia A ret = A ret ip ret op ia where A pro tp and A pr op are the in-phase and out-of-phase parts of the prograde nutation with angular frequency \u a \ as defined in the foregoing section, and A r s et ip and A r s et op are the in-phase and out-ofphase parts of the retrograde nutation with angular frequency (- u; s ). These amplitudes are related to the coefficients in longitude and obliquity through the following equations (see Defraigne et al, 995): A rt P = _l (A ip_ 9sA^p sin o^ A^ip = ^(Aei P + q s Ai P sme ö ), ^r o p = ~(?sae7 + A P sine ), A r / top = -^(q s Ae o s p -Ar s p sme ). and Ae s are the in-phase (cosine) and out-of-phase (sine) parts of the nutation in obliquity for a nutation of frequency u; s ; A^p and A^p are the in-phase (sine) and out-of-phase (cosine) parts of the nutation in longitude for the same nutation. Ae tp The relations stated above are valid when the onvention regarding the time dependence as stated in the previous section is followed, i.e., when the time dependence of the prograde amplitude is e*""*' ^'* and that of the retrograde amplitude is e*'"*'*. However the opposite onvention is often used in the literature. The amplitudes to be employed in such a case are the complex conjugates of those given above, i.e., (A pro ip + ia pro op ) goes together with c '^*'* for the prograde nutation and (A r 8 et tp + ia r 8 et op ) with e-*l w -l* for the retrograde nutation. Since the real parts are to be eventually taken, switching between mutually complex conjugates versions does not affect the results.

38 oordinate Transformation Referred to the Nonrotating Origin Option uses form () of the coordinate transformation from the TRS to the RS [RS] = PN"(t)R"(t)W"(t)[TRS], () where the three fundamental components of () are given below (apitaine, 99) W"(t) = R (-8')-R (y p )-R (x p ), x p and y p being the "polar coordinates" of the EP in the TRS and s f the accumulated displacement of the terrestrial origin on the true equator due to polar motion. The use of the quantity s f (which is neglected in the classical form ()) provides an exact realization of the "instantaneous prime meridian." R"(t) = R (-e), being the stellar angle at date t due to the Earth's angle of rotation, PN"{t) = Rs(-E) Ä (-d) R (E) R (S), E and d being such that the coordinates of the EP in the RS are A' = sind cos E, Y = sind sin E, Z cosrf and S being the accumulated rotation (between the epoch and the date t) of the celestial EO on the true equator due to the celestial motion of the EP. PN n (t) can be given in an equivalent form involving directly A~ and Y (to which all the observations of a celestial object from the Earth are actually sensitive) as: /-ax -axy X \ PN"(t) = Qi = -axy - ay Y -Ä(«), \ -X -Y l-a(x + Y )J with a = /( + cosrf), which can also be written, with sufficient accuracy as a / + /8(X + Y ). The Standard values of the parameters to be used in the form () of the transformation are detailed below. The Standard pole coordinates to be used for the parameters x p and y p (if not estimated from the observations) are those published by the IERS. The quantity s f (of the order of. mas/cy) is: s' =.5(a /. + a\)t, a c and a a being the average amplitudes (in are seconds) of the handlerian and annual wobbles, respectively in the period considered (apitaine et ai, 986). The stellar angle is obtained by the use of the conventional relationship between the stellar angle 6, the hour angle of the EO and UTI as given by apitaine et ai, (986), (T U ) = TT( T U X 655), where T u = (Julian UTI date )/655, and

39 UTI = UT + (UT-UT), or equivalently (T U ) = TT(UT Julian day number elapsed since T U x 655), the quantity UT-UT to be used (if not estimated from the observations) being the IERS value. The celestial coordinates X and Y of the EP to be used for consistency with the IAU 98 theory of nutation are the Standard values as derived from the series given in Table 5.. These developments of the celestial polar coordinates have been derived (apitaine, 99) from the previous Standard expressions for precession and nutation with a consistency of 5 x ~ 5 see. of are after a entury; such consistency has been numerically checked over two centuries (Gontier, 99). For observations requiring values of the nutation angles with a milliarcsecond accuracy, it is necessary to add (if those quantities are not estimated from the observations) the IERS published values (observed or predicted) for the "celestial pole offsets" (i.e. corrections dx = Aips'meo and dy = Ae). For consistency with the IERS 995 series for nutation as given by Tables 5. and 5., it is necessary to use the following expressions for A' and Y (apitaine, 99): X = sin UJ sin tp, Y = sin eo cosu? + cos eo sin u> cos tp, where eo is the obliquity of the ecliptic at J, u is the inclination of the true equator of date on the fixed ecliptic of epoch and tp is the longitude, on the ecliptic of epoch of the node of the true equator of date on the fixed ecliptic of epoch; these quantities are such that: u = u A + Aei,i' = xp A + Atpi, where tp A and UJ A are the precession quantities in longitude and obliquity (Lieske et al. 977) referred to the ecliptic of epoch and Atp\, Aei are the nutation angles in longitude and obliquity referred to the ecliptic of epoch. A^i, Aei can be obtained with an accuracy better than one micoraresecond after one entury from the nutation angles Atp, Ae in longitude and obliquity referred to the ecliptic of date, following Aoki and Kinoshita (98) by: (Atps'me A cosxa - Acsinx^) Atpi = : -, smu^ Ae! = Atpm\e A sin \ A + Aecosx^, üj A and e^ being the precession quantities in obliquity referred to ecliptic of epoch and ecliptic of date respectively and \ A the precession quantity for planetary precession along the equator (Lieske et al. 977). The Standard value of s compatible with the IAU 98 Theory of Nutation and the Lieske et al. (977) precession can be derived with an accuracy of 5 x ~ 5 " after a entury (apitaine, 99) from the following numerical development and the numerical values of A^ and Y (Table 5.), s = -AT/ + 785t t - '/6 sin ü - 76 sin Ü +77t sin + 76t sin (F - D + ft) 5

40 Table 5. Series for the celestial coordinates A" and Y of the EP referred to the mean equator and equinox of epoch J., with t measured in Julian centuries from epoch J.. The terms following the dotted line are identical in Tables 5. and 5.. The signs of the fundamental arguments, periods, and coefficients may differ from the original publication. These have been changed to be consistent with other portions of this chapter. X= 79* - '.'665< - '.'98656< + 7* +76* cosft + sin {Ei = i,io 6 M«+ A\t)sin(ARGUMENT) + A'ft coa(argument)]} +7* sin Ü + 76* sin (F - D + fi), Y = -'.' - 799* + 786t + 7t + Ei=i,io6K 5 < + ß,'<)cos( ARGUMENT) + B'/t sin( ARGUMENT)] -7* cos ü - 7* cos (F - D + fi) / MULTIPLIERS OF V F D ft PERIOD (days) LONGIT UDE (''.') Ai A[ A'l OBLIQU] TY (.:oooij Bi B\ B?

41 Geodesic Nutation Fukushima (99) has pointed out that, if extreme precision is required, the effect of geodesic nutation must be taken into account. For Option () this would require a correction in longitude of Atl>g = -75sin/' - 7sin', 7

42 where /' is the mean anomaly of the Sun. For Option () it would require a correction to X of AX g = (-769sin/' - 78sin/')sine. In both cases, the correction would be added to the uncorrected determination of tp or X. References Aoki, S., Guinot, B., Kaplan, G. H., Kinoshita, H., Mcarthy, D. D., and Seidelmann, P. K., 98, "The New Definition of Universal Time," Astron. Astrophys., 5, pp Aoki, S. and Kinoshita, H., 98, "Note on the relation between the equinox and Guinot's non-rotating origin," elest. Mech., 9, pp apitaine, N., 99, "The elestial Pole oordinates," elest. Mech. Dyn. Astr., 8, pp. 7-. apitaine, N., Guinot, B., and Souchay, J., 986, "A Non-rotating Origin on the Instantaneous Equator: Definition, Properties and Use," elest. Mech., 9, pp apitaine, N. and hollet, F., 99, "The use of the nonrotating origin in the computation of apparent places of stars for estimating Earth Rotation Parameters," in Reference Systems, J. A. Hughes,. A. Smith, and G. H. Kaplan (eds), pp. -7. apitaine, N. and Gontier A. -M., 99, "Procedure for VLBI estimates of Earth Rotation Parameter referred to the nonrotating origin," in Reference Systems, J. A. Hughes,. A. Smith, and G. H. Kaplan (eds), pp apitaine, N. and Gontier A. -M., 99, "Accurate procedure for deriving UTI at a submilliarcsecond accuracy from Greenwich Sidereal Time or from stellar angle," Astron. Astrophys., 75, pp Defraigne, P., Dehant, V., and Päquet, P., 995, "Link between the retrograde-prograde nutations and nutations in obliquity and longitude," elest. Mech. Dyn. Astr., 6, pp Fukushima, T., 99, "Geodesic Nutation," Astron. Astrophys.,, pp. L-L. Gilbert, F. and Dziewonski, A. M., 975, "An Application of Normal Mode Theory to the Retrieval Structure Parameters and Source Mechanisms from Seismic Spectra," Phil. Trans. Roy. Soc. Lond., A78, pp Gontier, A. -M., 99, "Tests pour Tapplication de Torigine nontournante en radioastronomie," in Journees 99 Systemes de Reference spatio-temporels, pp Guinot, B., 979, "Basic Problems in the Kinematics of the Rotation of the Earth," in Time and the Earth's Rotation, D. D. Mcarthy and J. D. Pilkington (eds), D. Reidel Publishing ompany, pp Herring, T., 996, Private ommunication. Kinoshita, H., 977, "Theory of the Rotation of the Rigid Earth," elest. Mech., 5, pp

43 Lieske, J. H., Lederle, T., Fricke, W., and Morando, B., 977, "Expression for the Precession Quantities Based upon the IAU (976) System of Astronomical onstants," Astron. Astrophys., 58, pp. -6. Liekse, J., 99, Private ommunication. Mueller, I. L, 969, Spherical and Practical Astronomy o., Inc. as applied to Geodesy, F. Ungar Publishing Seidelmann, P. K., 98, "98 IAU Nutation: The Final Report of the IAU Working Group on Nutation," elest. Mech., 7, pp Simon, J. L., Bretagnon, P., hapront, J., hapront-touze, M., Francou, G., Laskar, J., 99, "Numerical Expressions for Precession Formulae and Mean Elements for the Moon and Planets," Astron. Astrophys., 8, pp Wahr, J. M., 98, "The Forced Nutations of an Elliptical, Rotating, Elastic, and Oceanless Earth," Geophys. J. Roy. Astron. Soc, 6, pp

44 HAPTER 6 GEOPOTENTIAL The recommended geopotential field is the JGM- model (Tapley et ai, 995). The GM and a^ values reported with JGM- (986.5 km /s and m) should be used as scale parameters with the geopotential coefficients. The recommended GM = should be used with the two-body term when working with SI units (986.5 or should be used by those still working with TDT or TDB units, respectively). Although the JGM- model is given with terms through degree and order 7, only terms through degree and order twenty are required for Lageos. Values for the and coefficients are included in the JGM- model. The and S coefficients describe the position of the Earth's figure axis. When averaged over many years, the figure axis should closely coincide with the observed position of the rotation pole averaged over the same time period. Any differences between the mean figure and mean rotation pole averaged would be due to long-period fluid motions in the atmosphere, oceans, or Earth's fluid core (Wahr, 987, 99). At present, there is no independent evidence that such motions are important. The JGM- values for and isi give a mean figure axis that corresponds to the mean pole position recommended in hapter Terrestrial Reference Frame. This choice for and # is realized as follows. First, to use the geopotential coefficients to solve for a satellite orbit, it is necessary to rotate from the Earth-fixed frame, where the coefficients are pertinent, to an inertial frame, where the satellite motion is computed. This transformation between frames should include polar motion. We assume the polar motion parameters used are relative to the IERS Reference Pole. If x and y are the angular displacements of the Terrestrial Reference Frame described in hapter relative to the IERS Reference Pole, then the values = y/x o, = -VSy, where x. X ~ 6 radians (equivalent to.6 arcsec) and y =.5 X ~ 6 radians (equivalent to.9 arcsec) (Nerem et a/., 99) are those used in the geopotential model, so that the mean figure axis coincides with the pole described in hapter. This gives normalized coefficients of i(iers)= -.87 x " 9, 5 i(iers) =.95 x " 9. JGM- is available via ftp at ftp.csr.utexas.edu on the directory pub/grav in file JGM.GEO.Z. It can also be accessed by World Wide Web at by clicking the "library of data files" selection. Effect of Solid Earth Tides The changes induced by the solid Earth tides in the free space potential are most conveniently modeled as variations in the Standard geopotential coefficients nm and S nm (Eanes et ai, 98). The contributions A nm and AS nm from the tides are expressible in terms of the k Love number. The effects of ellipticity and rotation of the Earth on tidal deformations necessitates the use, in general, of three k parameters, k n m and k nm, to characterize the changes produced in the free space potential by

45 tides of spherical harmonic degree and order (nm) (Wahr, 98). Within the diurnal tidal band, for (mn) = (), these parameters have a strong frequency dependence due to the Nearly Diurnal Free Wobble resonance. Anelasticity of the mantle causes knm and /m to acquire small imaginary parts (reflecting a phase lag in the deformational response of the Earth to tidal forces), and also gives rise to a fürt her Variation with frequency which is particularly pronounced within the long period band. Though modeling of anelasticity at the periods relevant to tidal phenomena (8 hours to 8.6 years) is not yet definitive, it is clear that the magnitudes of the contributions from anelasticity cannot be ignored (see below) onsequently the anelastic Earth model is recommended for use in precise data analysis. The degree tides produce time dependent changes in m and Sm, through k m, which can exceed "" 8 in magnitude. They also produce changes exceeding a cutoff of X ~ in \ m and 5 m through fc^m (The direct contributions of the degree tidal potential to these coefficients are negligible.) The only other changes exceeding this cutoff are in s m and Ss m, produced by the degree part of the tide generating potential. The computation of the tidal contributions to the geopotential coefficients is most efficiently done by a two-step procedure. In Step, the (m) part of the tidal potential is evaluated in the time domain for each m using lunar and solar ephemerides, and the corresponding changes A m and A5 m are computed using frequency independent nominal values &m for the respective fe^. The contributions of the degree tides to ^m and S$ m through k rn and also those of the degree tides to im and S^m through k m may be computed by a similar procedure; they are at the level of ". Step corrects for the deviations of the k of several of the constituent tides of the diurnal band from the constant nominal value & i assumed for this band in the first step. Similar corrections need to be applied to a few of the constituents of the other two bands also. With frequency-independent values k nm (Step ), changes induced by the (nm) part of the tide generating potential in the normalized geopotential coefficients having the same (nm) are given in the time domain by j= w (with S n o = ), where k nm = nominal degree Love number for degree n and order m, R e = equatorial radius of the Earth, GM = gravitational parameter for the Earth, GMj = gravitational parameter for the Moon (j = ) and Sun (j = ), rj = distance from geocenter to Moon or Sun,

46 $j = body fixed geocentric latitude of Moon or Sun, Xj = body fixed east longitude (from Greenwich) of Moon or Sun, and P nm by is the normalized associated Legendre function related to the classical (unnormalized) one *nm =: ^* nm* nmi \^) where (n-m)l(n+l)(-6 om ) Nnm = \ ; ; r; (6) y (n + m)\ orrespondingly, the normalized geopotential coefficients ( nm,s nm ) coefficients ( nm,s nm ) by are related to the unnormalized *-nm = nm^-'nm) ^nm = nm*jnm- \&) Equation () yields A nm and AS nm for both n = and n = for all m, apart from the corrections for frequency dependence to be evaluated in Step. (The particular case (nm) = () needs special consideration, however, because it includes a time-independent part which will be discussed below in the section on the permanent tide.) One further computation to be done in Step is that of the changes in the degree coefficients produced by the degree tides. They are given by A m - ias m = ^f- ^ (^) p m(sin *,>- "*', (m =,,), () 5 ]^ GM r i which has the same form as Equation () for n except for the replacement of k m by k\ m. The parameter values for the computations of Step are given in Table 6.. The choice of these nominal values (which are complex for m = and m = in the anelastic case) has been made so as to minimize the number of terms for which corrections will have to be applied in Step. The nominal value for m = has to be chosen real because the contribution to from the imaginary part of /o The frequency dependent values for use in Step are taken from the results of computations by Mathews and Buffett (private communication) using the PREM elastic Earth model with the ocean layer replaced by solid, and for the evaluation of anelasticity effects, the Widmer et al. (99) model of mantle Q. As in Wahr and Bergen (986), a power law was assumed for the frequency dependence of Q with s as the reference period; the value a =.5 was used for the power law index. The anelasticity contribution (out of phase and in phase) to the tidal changes in the geopotential coefficients is at the level of one to two percent in-phase, and half to one percent out-of-phase, i.e., of the order of -.

47 Table 6.. Nominal values of solid Earth tide externa! potential Love numbers. Elastic Earth Anelastic Earth n m nm ^nm "^ ^nm Am nm ^nn The frequency dependence corrections to the A nm and AS nm values obtained from Step are computed in Step as the sum of contributions from a number of tidal constituents belonging to the respective bands. The contribution to A o from the long period tidal constituents of various frequencies / is Re J (A 6k f H f e ie f)= J^ (A H f (6kf cos f - 6k}sin f ), (5a) /(,) /(,) while the contribution to (A i - *A5 i) from the diurnal tidal constituents and to A - ias from the semidiurnals are given by A m - ia5 m = rim J {A m 6kfH f ) e ib <, (m =,), (56) /(,m) where _ A - t =.8 x KT 8 m~\ (5c) ~~ Ä ev /^ A m = ^-L= = (-l) m (.7 x " 8 ) m", (m ± ), (bd) m = -*\% =, (5c) 6kj = difference between k f = k^ at frequency / and the nominal value k m, in the sense kj - k m, 6k^ Hj = real part, and 6k j = imaginary part, of 6kf, = amplitude (m) of the term at frequency / from the harmonic expansion of the tide generating potential, defined according to the onvention of artwright and Tayler (97), and f = n ß = Eti *ißi, or 9, = m( g + TT) - N. F = m( g + w) - * = JV^-, where

48 ß = six-vector of Doodson's fundamental arguments ßi, (r,s,h,p,n f,p s ), n = six-vector of multipliers ni (for the term at frequency /) of the fundamental arguments, F = five-vector of fundamental arguments Fj (the Delaunay variables l,v,f,d,ü) theory, N five-vector of multipliers Ni of the Delaunay variables for the nutation of frequency of nutation f+d g /dt, and g is the Greenwich Mean Sidereal Time expressed in angle units (i.e. A h 5). = 6 ; see hapter (7r in ( g + 7r) is now to be replaced by 8.) For the fundamental arguments (l,l',f,d,q) of nutation theory and the onvention followed here in choosing their multipliers Nj, see hapter 5. For conversion of tidal amplitudes defined according to different onventions to the amplitude Hf corresponding to the artwright-tayler onvention, use Table 6. given at the end of this hapter. The correction due to the Ä'i constituent, for example, is obtained as follows, given that A m = Ai = -.7 x ~ 8, H f =.687, and f = ( g + TT) for this tide. If anelasticity is ignored, (k { l ) ) I< =.577, and the nominal value chosen is.97. Heiice 6k j is = -.9, and A m (6k)fHj reduces to 7. x ~. The corrections to the () coefficients then become (A i) Kl = 7. x " sm(9 g + TT), (AS i)/,'' = 7. x ~ cos{9 g + TT). With anelasticity included, (k )K = i.8, and on choosing the nominal value as (.98 i.) one obtains the corrections to the coefficients by replacing 6kj in the above calculation by ( i.). In general, if 6kj = 6k^ + i6kj, (A m ) K = A m H f (6kfsm f + 6k I fcos f ), (AS m ) Kl = A m H f (ökfcosof-ik'jsmof). Table 6.a lists the results for all tidal terms which contribute ~ or more, after round-off, to the (nm) = () geopotential coefficient. A cutoff at this level is used for the individual terms in order that accuracy at the level of x ~ be not affected by the accumulated contributions from the numerous smaller terms that are disregarded. The imaginary parts of the contributions are below cutoff and are not listed. Results relating to the () and () coefficients are presented in Tables (6.b) and (6.c), respectively. Table 6.a. Amplitudes (A\6kjHf) of the corrections for frequency dependence of k l ', taking the nominal value k \ for the diurnal tides as.97 for the elastic case, and (.98 i.) for the anelastic case. Units: ~. Multipliers of the Doodson arguments identifying the tidal terms are given, as also those of the Delaunay variables.

49 Name deg/hr Qi p\ O x Nrj LKj NO! \i Tl Pl Si Ki V'i i öi Ji SOi OOi Doodson No. 5,65 5,655 7,55 5,55 5,555 5,655 55,55 55,655 55,665 57,55 6,556 6,55 6,555 6,556 65,55 65,555 65,565 65,575 66,55 67,555 7,655 75,55 75,65 8,555 85,555 85,565 r s h p J ff'. Ps //' FD Ü kf Amp. elas k a f nel Amp. anel Table 6.b. orrections for frequency dependence of k of the zonal tides due to anelasticity. Units: ~. The nominal value k o for the zonal tides is taken as.9. The real and imaginary parts 6k f and 6kj of 6kj are listed, along with the corresponding in phase (ip) amplitude (AoHf6kf) and out of phase (op) amplitude ( AoHf6kj) to be used in equation (5a). In the elastic case, k =.955 for all the zonal tides, and no second step corrections are needed. Name Doodson No. deg/hr TS h p N' p s ti'fdsl 6k? Amp. (ip) 8k) Amp. (op) 55,565 55,575 S a 56,55 ^sa 57,555 57,565 58,55 M sm 6,655 65,

50 M m M.j M f M stm M tm Msqm M qm 65,55 65,65 65,655 7,555 75,55 75,555 75,565 75,575 8,655 85,55 85,65 9,555 95, Table 6.c. Amplitudes (A 6kfHf) of the corrections for frequency dependence of k, taking the nominal value fc for the sectorial tides as.98 for the elastic case, and (. i.) for the anelastic case. Units: ~. The corrections are only to the real part, and are the same in both the elastic and the anelastic cases. fame > Doodson No. deg/hr r shpn'p s li'fdsl 6k? Amp. N M 5,655 55, The total Variation in geopotential coefficient o is obtained by adding to the result of Step the sum of the contributions from the tidal constituents listed in Table 6.b computed using equation (5a). The tidal variations in * m and m for the other m are computed similarly, except that equation (5b) is to be used together with Table 6.a for m = and Table 6.c for m =. Solid Earth Pole Tide The pole tide is generated by the centrifugal effect of polar motion, characterized by the potential AV = -(tt R e/)sm9(x p cos\ - y p sm\). (See the section on Deformation due to Polar Motion in hapter 7 for further details). The deformation which constitutes this tide produces a perturbation fc AF in the external potential which is equivalent to changes in the geopotential coefficients and S. Using for fc the elastic Earth value.977 appropriate to the polar tide yields A = -.9 x " 9 (^p), A5 i =.9 x HT 9 (y p ), 6

51 where x p and y p are in seconds of are as defined in hapter 7. For the anelastic Earth, k has real and imaginary parts kf =. and Ar = -.5, leading to A i = -.8 x -%T P +.i/ p ), A5 i =.8 x " 9 (y p -.a: p ). Treatment of the Permanent Tide The degree zonal tide generating potential has a mean (time average) value which is nonzero. This permanent (time independent) potential produces a permanent deformation which is reflected in the static figure of the Earth, and a corresponding time independent contribution to the geopotential which can be considered as part of the adopted value of o, as in the JGM- model. Therefore, for (nm) = (), the zero frequency part should be exeluded from the expression (). Hereafter the symbol A is reserved for the temporally varying part of the tidal contribution to (7 o; the expression () for (ran) = () will be redesignated as. j Its zero frequency part is (A) = A H ko = (.8 x " 8 )(-.6)^- (6) To represent the tide induced changes in the () geopotential, one should then use only the time variable part A = A * - (A ). (7) In evaluating it, the same value should be used for fe o in both A an d (AÖ o). If the elastic Earth value k =.955 is used, (AÖ ) = -.8 x " 9, while with the value k =.9 of the anelastic case, (A ) = -. x ~ 9. The restitution of the indirect effect of the permanent tide is done to be consistent with the XVIII IAG General Assembly Resolution 6; but to obtain the effect of the permanent tide on the geopotential, one can use the same formula as equation (6) using the fluid limit Love number which is k =.9. Effect of the Ocean Tides The dynamical effects of ocean tides are most easily incorporated by periodic variations in the normalized Stokes coefficients. These variations can be written as A nm - ia5 nm = F nm ^(± m T isf nm )e ±ie ', (8) s(n,m) + where = ng Pw I (n + m)\ (l + k' n \ g )j(n-tny.(n+l)(-6 om ) [7 + )' 7

52 g and G are given in hapter, p w = density of seawater = 5 kg m, k' n = load deformation coefficients (k f = -.75, = -.95, k' = -.,fc = -., fc = -.89), f nm^sf nm = ocean tide coefficients (m) for the tide constituent s S = argument of the tide constituent s as defined in the solid tide model (hapter 7). The summation over + and - denotes the respective addition of the retrograde waves using the top sign and the prograde waves using the bottom sign. The f nm and Sf nm are the coefficients of a spherical harmonic decomposition of the ocean tide height for the ocean tide due to the constituent s of the tide generating potential. For each constituent s in the diurnal and semi-diurnal tidal bands, these coefficients were obtained from the SR. ocean tide height model (Eanes et al., 996), which was estimated from the TOPEX/ Poseidon satellite altimeter data. For each constituent s in the long period band, the seif-consistent equilibrium tide model of Ray and artwright (99) was used. The list of constituents for which the coefficients were determined was obtained from the artwright and Tayler (97) expansion of the tide raising potential. These ocean tide height harmonics are related to the Schwiderski onvention (Schwiderski, 98) according to ± -is ± = -i ± e'f^m+xj (q\ ^snm ikj snm *^ snm^ i \* ) where fnm ~ ocean tide amplitude for constituent s using the Schwiderski notation, e tnm = ocean tide phase for constituent s, and Xs is obtained from Table 6., with H s being the artwright and Tayler (97) amplitude at frequency s. Table 6.. Values of \ s for long-period, diurnal and semidiurnal tides. Tidal Band H s > H s < Long Period TT Diurnal f -f Semidiurnal ir For clarity, the terms in equation are repeated in both onventions: or A nm = F nm ]T [(+ nm + ; nm )cos s + (St nm +S; nm )s'm s ] (a) s(n,m) A nm = F nm ] [tnm sin( s + etnm + X-) + 7nm ^(* + e; nm + Xs% (6) s(n,m) 8

53 or ASnm = F nm ^ i( S fnm + S mm )coso, - (+ m - snm )sill9 s ] (c) s{n,m) A5 nm = F nm ^ föm cos(, + c+ nm + x 5 ) - ; nm cos( 5 + cj nm + Xs)]- (lorf) s(n,m) The orbit element perturbations due to ocean tides can be loosely grouped into two classes. The resonant perturbations arise from coefficients for which the order (ra) is equal to the first Doodson's argument multiplier n\ of the tidal constituent s (See Note), and have periodicities from a few days to a few years. The non-resonant perturbations arise when the order ra is not equal to index n\. The most important of these are due to ocean tide coefficients for which ra = ii\ -f- and have periods of about day. ertain selected constituents (e.g. S a and S ) are strongly affected by atmospheric mass distribution (hapman and Lindzen, 97). The resonant harmonics (for ra = ni) for some of these constituents were determined by their combined effects on the orbits of several satellites. These multisatellite values then replaced the corresponding values from the SR. altimetric ocean tide height model. Based on the predictions of the linear perturbation theory outlined in asotto (989), the relevant tidal constituents and spherical harmonics were selected for several geodetic and altimetric satellites. For geodetic satellites, both resonant and non-resonant perturbations were analyzed,whereas for altimetric satellites, only the non-resonant perturbations were analyzed. For the latter, the adjustment of empirical parameters during orbit determination removes the errors in modeling resonant accelerations. The resulting selection of ocean tidal harmonics was then merged into a single recommended ocean tide force model. With this selection the error of Omission on TOPEX is approximately 5 mm along-track, and for Lageos it is rara along-track. The recommended ocean tide harmonic selection is available via anonymous ftp from ftp.csr.utexas.edu. For high altitude geodetic satellites like Lageos, in order to reduce the required omputing time, it is recommended that out of the complete selection, only the constituents whose artwright and Tayler amplitudes H s is greater than.5 rara be used, with their spherical harmonic expansion terminated at maximum degree and order 8. The Omission errors from this reduced selection on Lageos is estimated at approximately cm in the transverse direction for short arcs. NOTE: The Doodson variable multipliers (n) are coded into the argument number (A) after Doodson (9) as: A = m(n + 5)(n + 5).(n + 5)(n 5 + 5)(n 6 + 5). onversion of tidal amplitudes defined according to different onventions The definition used for the amplitudes of tidal terms in the recent high-accuracy tables differ from each other and from artwright and Tayler (97). Hartmann and Wenzel (995) tabulate amplitudes in units of the potential (m s~ ), while the amplitudes of Roosbeek (996), which follow the Doodson (9) onvention, are dimensionless. To convert them to the equivalent tide heights Hf of the artwright-tayler onvention, multiply by the appropriate factors from Table 6.. The following values are used for the constants appearing in the conversion factors: Doodson constant Di = m s" ; g e = g at the equatorial radius = (from GM =.9865 x m s", R e = m). 9

54 Table 6. Factors for conversion to artwright-tayler amplitudes from those defined according to Doodson^s and Hartmann and Wenzel's onventions From Doodson From Harmann & Wenzel f = -^fii = -.65 ft = l& =.6788 / = _ ^ ^ i _ f' l = - ^ = / = ^ ^ = / = ^ =.566 / = - ^ ^ - = /, = h& =.6788 / = ~^ %- = & = -^jä = / = * J j ^ =.6888 fi = ^ =.566 / = "^^^ff- = -.6 & = - ^ = References artwright, D. E. and Tayler, R. J., 97, "New omputations of the Tide-Generating Potential," Geophys. J. Roy. Astron. Soc,, pp asotto, S., 989, "Ocean Tide Models for TOPEX Precise Orbit Determination," Ph.D. Dissertation, The Univ. of Texas at Austin. hapman, S. and Lindzen, R., 97, Atmospheric Tides, D. Reidel, Dordrecht. Doodson, A. T., 9, "The Harmonic Development of the Tide-Generating Potential," Proc. R. Soc. A.,, pp Eanes, R. J., Schutz, B., and Tapley, B., 98, "Earth and Ocean Tide Effects on Lageos and Starlette," in Proceedings of the Ninth International Symposium on Earth Tides, J. T. Kuo (ed), E. Sekweizerbart'sehe Verlagabuchhandlung, Stuttgart. Eanes, R. J. and Bettadpur, S. V., 996, "The SR. global ocean tide model," in preparation. Hartmann, T. and Wenzel, H.G., 995, "The HW95 Tidal Potential atalogue," Geophys. Res. Lett.,, pp Mathews, P. M., Buffett, B. A., and Shapiro,. I. I., 995, "Love numbers for a rotating spheroidal Earth: New definitions and numerical values," Geophys. Res. Lett.,, pp Nerem, R. S., Lerch, F. J., Marshall, J. A., Pavlis, E., Putney, B. H., Tapley, B. D., Eanes, R. J., Ries, J., Schutz, B. E., Shum,. K., Watkins, M. M., Klosko, S. M., han, J., Luthcke, S. B., Patel, G. B., Pavlis, N. K., Williamson, R. G., Rapp, R. H., Biancale, R., Nouel, F., 99, "Gravity Model Development for TOPEX/POSEIDON: Joint Gravity Models and," J. Geophys. Res., 99, pp. -7. Ray, R. D. and artwright, D. E., 99, "Satellite altimeter observations of the Mj and M m ocean tides, with simultaneous orbit corrections," Gravimetry and Space Techniques Applied to Geodynamics and Ocean Dynamics, Geophysical Monograph 8, IUGG Volume 7, pp

55 Roosbeek, F., 996, "RATGP95: An Harmonic Development of the Tide Generating Potential Using an Analytical Method," Geophys. J. Int., in press. Schwiderski, E., 98, "Atlas of Ocean Tidal harts and Maps, Part I: The Semidiurnal Principal Lunar Tide M," Marine Geodesy, 6, pp Tapley, B. D., M. M. Watkins, J.. Ries, G. W. Davis, R. J. Eanes, S. R. Poole, H. J. Rim, B. E. Schutz,. K. Shum, R. S. Nerem, F. J. Lerch, J. A. Marshall, S. M. Klosko, N. K. Pavlis, and R. G. Williamson, 995, "The JGM- Gravity Model," to be submitted to J. Geophys. Res. Wahr, J. M., 98, "The Forced Nutations of an Elliptical, Rotating, Elastic, and Oceanless Earth," Geophys. J. Roy. Astron. Soc, 6, pp Wahr, J., 987, "The Earth's i and S i gravity coefficients and the rotation of the core," J. Roy. Astr. Soc, 88, pp Geophys. Wahr, J., 99, "orrections and Update to The Earth's i and S i gravity coefficients and the rotation of the corey Geophys../. Int.,, pp Wahr, J. and Z. Bergen, 986, "The effects of mantle elasticity on nutations, Earth tides, and tidal variations in the rotation rate" Geophys. J. R. Astr. Soc, Widmer, R., G. Masters, and F. Gilbert, 99, "Spherically Symmetrie attenuation within the Earth from normal mode data", Geophys. J. Int.,, pp

56 HAPTER 7 SITE DISPLAEMENT Local Site Displacement due to Ocean Loading Local site displacement is understood as an effect of (visco-) elastic deformation of the Earth in response to time-varying surface loads. The reference point of zero deformation is the Joint mass center of the solid Earth and the load, while the sites are attached to the solid Earth. This onvention implies that the rigid body translation of the solid Earth that counterbalances the motion of the load's mass center is not contained in the local displacement model. This onvention follows strictly Farrell (97). Ocean Loading Three dimensional site displacements due to ocean tide loading are computed using the following scheme. Let Ac denote a displacement component (radial, west, south) at a particular site and time t. Let W denote the tide generating potential (e.g. Tamura, 987; artwright and Tayler, 97; artwright and Edden, 97), W = g ^ ^ P T i ( cos ^) ^os(ujt + Xj + mjx), () i where only degree two harmonics are retained. The Symbols designate colatitude ip, longitude A, tidal angular velocity Uj, amplitude Kj and the astronomical argument \j at t = h. Spherical harmonic order mj distinguishes the fundamental bands, i. e. long-period (m = ), diurnal (m = ) and semi-diurnal (ra = ). The parameters Kj and Uj are used to obtain the most completely interpolated form with Ac = ] a j cm^u; i i + Xi - <l>cj h () j \A ck cos$ ck A e, M - cos$ c?fc + a cj cos<j> cj = hj\ r. ( - p) + jz p\, L &k Afc + l J. \A ck $m$ ck A c, fc+ isin$ c, fc+ a c j sin (ß cj = hj\ y ( - p) + l! -r P ^/c+l For each site, the amplitudes A ck and phases $ A:, < k <, are taken from Table 7.. For clarity symbols written with bars overhead designate tidal potential quantities associated with the small set of partial tides represented in the table. These are the semi-diurnal waves M,5, A^,Ä'? the diurnal waves K\,\,P\,Q\, and the long-period waves Mj,M m, and S sa. Interpolation is possible only within a fundamental band, i.e. we demand m fc = m j = rä fc+ i. () Then If no ü k or ü k +\ &j U>fc P = -=, Uk < Uj < Lü k +i. <*>*+l - U k can be found meeting (), p is set to zero or one, respectively. 5

57 A shorter form of () is obtained if the summation considers only the tidal species of Table 7. and corrections for the modulating effect of the lunar node. Then, Ac = J fja cj cos(ujjt + Xj + Uj- $ cj ), () j where fj and Uj depend on the longitude of the lunar node according to Table 6 of Doodson (98). The astronomical arguments needed in () can be computed with subroutine ARG below. SUBROUTINE ARG(IYEAR,DAY,ANGLE) OMPUTES THE ANGULAR ARGUMENT WHIH DEPENDS ON TIME FOR TIDAL ARGUMENT ALULATIONS ORDER OF THE ANGULAR quantities IN VETOR ANGLE -M -S -N -K 5-K 6-7-P 8-Q 9-Mf -Mm -Ssa TAKEN FROM 'TABLE ONSTANTS OF MAJOR TIDAL MODES' WHIH DR. SHWIDERSKI SENDS ALONG WITH HIS TAPE OF TIDAL AMPLITUDES AND PHASES INPUT IYEAR - EX. 79 FOR 979 DAY - DAY OF YEAR GREENWIH TIME EXAMPLE.5 FOR FEB NOON.5 FOR JAN 6 AM OUTPUT ANGLE - ANGULAR ARGUMENT FOR SHWIDERSKI OMPUTATION *********************************************************** c c AUTION OEAN LOADING PHASES OMPUTED FROM SHWIDERSKI'S MODELS REFER TO THE PHASE OF THE ASSOIATED SOLID EARTH TIDE GENERATING POTENTIAL AT THE ZERO MERIDIAN AORDING TO 5

58 OLJ)R = OL_AMP X OS (SE_PHASE" - OL-PHASE) WHERE OL = OEAN LOADING TIDE, SE = SOLID EARTH TIDE GENERATING POTENTIAL. IF THE HARMONI TIDE DEVELOPMENT OF ARTWRIGHT, ET AL. ( = TE) (97, 97) IS USED, MAKE SURE THAT SE-PHASE" TAKES INTO AOUNT () THE SIGN OF SE_AMP IN THE TABLES OF ARTWRIGHT ET AL. () THAT TE'S SE.PHASE REFERS TO A SINE RATHER THAN A OSINE FUNTION IF (N+M) = (DEGREE + ORDER) OF THE TIDE SPHERIAL HARMONI IS ODD. I.E. SEJPHASE" = TAU(T) NI + S(T) N + H(T) N + P(T) N + N'(T) N5 + PS(T) N6 + PI IF TE'S AMPLITUDE OEFFIIENT < + PI/ IF (DEGREE + NI) IS ODD WHERE TAU... PS = ASTRONOMIAL ARGUMENTS, NI... N6 - TE'S ARGUMENT NUMBERS. MOST TIDE GENERATING SOFTWARE OMPUTE SE_PHASE" (FOR USE WITH OSINES). THIS SUBROUTINE IS VALID ONLY AFTER 97. ****************************************************************** SUBROUTINE ARG(IYEAR,DAY,ANGLE) IMPLIIT DOUBLE PREISION (A-H,-Z) REAL ANGFA(,) DIMENSION ANGLE(),SPEED() SPEED OF ALL TERMS IN RADIANS PER SE EqUIVALENE (SPEED(l),SIGM),(SPEED(),SIGS),(SPEED().SIGN) EQUIVALENE (SPEED(),SIGK),(SPEED(5).SIGK),(SPEED(6).SIGOl) EQUIVALENE (SPEED(7),SIGP),(SPEED(8),SIGQ),(SPEED(9),SIGMF) EqUIVALENE (SPEED(),SIGMM),(SPEED(ll).SIGSSA) DATA SIGM/.59D-/ DATA SIGS/.5D-/ DATA SIGN/.788D-/ DATA SIGK/.58D-/ DATA SIGK/.79D-/ DATA SIGOl/.67598D-/ DATA SIGP/.75D-/ DATA SIGQ/.6959D-/ DATA SIGMF/.5D-/ 5

59 DATA SIGMM/.69D-/ DATA SIGSSA/.98D-/ DATA ANGFA/.E,-.E,.E,.E,*.E,.E,-.E,.E,.E,.E,*.E, l.eo,*o.eo,.5eo,l.e,-.eo,o.e,-.5eo, -i.e,*.e,-.5e,l.e,-.e,l.e,-.5e,.e,.e,*.e,.e,.e,-.e,.e,.e,*.e/ DATA TWOPI/ DO/ DATA DTR/ D-/ DAY OF YEAR ID=DAY FRATIONAL PART OF DAY IN SEONDS FDAY=(DAY-ID)*86.DO IAPD=ID+65*(IYEAR-75)+((IYEAR-7)/) APT*(79.558D+.5D*IAPD)/655.DO MEAN LONGITUDE OF SUN AT BEGINNING OF DAY HO=( D+( D+.D-*APT)*APT)*DTR MEAN LONGITUDE OF MOON AT BEGINNING OF DAY S=(((.9D-6+APT-.D)*APT D)*APT D)*DTR MEAN LONGITUDE OF LUNAR PERIGEE AT BEGINNING OF DAY P=(((-.D-5*APT-.5D)*APT D)*APT D)*DTR DO 5 K-, ANGLE(K)=SPEED(K)*FDAY+ANGFA(,K)*H+ANGFA(,K)*S. +ANGFA(,K)*PO+ANGFA(,K)*TWOPI ANGLE(K)=DMOD(ANGLE(K).TWOPI) IF(ANGLE(K).LT.O.DO)ANGLE(K)=ANGLE(K)+TWPI 5 ONTINUE RETURN END Table 7. is available electronically by anonymous ftp to maia.usno.navy.mil or ftp://gere.oso.chalmers.se/~pub/hgs/oload/readme. For sites not contained in the list the following is recommended: If the distance to the nearest site contained in the table is less than ten km, its data can be substituted. In other cases, coefficients and/or Software can be requested from hgsfloso.chalmers.se (Hans-Georg Scherneck). The Tamura tide potential is available from the International entre for Earth Tides, Observatoire Royal de Belgique, Bruxelles. 55

60 The coefficients of Table 7. have been computed according to Scherneck (98,99). Tangential displacements are to be taken positive in west and south directions. The ocean tide maps adopted are due to LeProvost et al. (99), Schwiderski (98), Schwiderski and Szeto (98), and Fiather (98). Refined coastlines have been derived from the topographic data sets ETP5 and Terrain Base (Row et ai, 995) of the National Geophysical Data enter, Boulder, O. Ocean tide mass budgets have been constrained using a uniform co-oscillating oceanic layer. Load convolution employed a diskintegrating Green's function method (Farrell, 97; Zschau, 98; Scherneck, 99). An assessment of the accuracy of the loading model is given in Scherneck (99). Adoption of more recent ocean tide results drawing from TOPEX/POSEIDON results is currently under consideration. Table 7. Sample of ocean loading table file. Each site record shows a header with the site name, the DP monument number, geographic coordinates and comments. First three rows of numbers designate amplitudes (meter), radial, west, south, followed by three lines with the corresponding phase values (degrees). olumns designate partial tides M,S,N, A',K\,\,Pi,Qi,Mf,M m, and S s $$ NSALA6O 7 $$ $$ omputed by H.G. Scherneck, Uppsala University, 989 $$ ONSALA 7 lon/lat: Effects of the Solid Earth Tides Site displacements caused by tides of sperical harmonic degree and order (nm) are characterized by the Love number h nm and the Shida number l nm. The effective values of these numbers depend on Station latitude and tidal frequency (Wahr, 98). This dependence is a consequence of the ellipticity and rotation of the Earth, and includes a strong frequency dependence within the diurnal band due to the Nearly Diurnal Free Wobble resonance. A further frequency dependence, which is most pronounced in the long period tidal band, arises from mantle anelasticity which leads to corrections to the elastic Earth Love numbers; these corrections have a small imaginary part and cause the tidal displacements to lag slightly behind the tide generating potential. All these effects need to be taken into account when an accuracy of mm is desired in determining Station positions. In order to account for the latitude dependence of the effective Love and Shida numbers, the representation in terms of multiple h and / parameters employed by Mathews et al. (995) is used. In this representation, parameters h^ and l^ play the roles of /^m and / m, while the latitude dependence is expressed in terms of additional parameters h^),h f and / ( \/ (),/'. These parameters are defined through their contributions to the site displacement as given by equations (5) below. Their numerical values as listed in Mathews et al. (995) have since been revised, and the new values, presented in Table 7. are used here. These values pertain to the elastic Earth and anelasticity modeis referred to in hapter 6. 56

61 The vector displacement due to a tidal term of frequency / is given in terms of the several parameters by the following expressions that result from evaluation of the defining equation (6) of Mathews et al. (995): For a long-period tide of frequency /: Afj *,{[««(!*' - ) +,/»' + cos / ( W < -*&' V 5 cos ff + /() sin <fr cos <ß cos j h sin j e \ (5a) For a diurnal tide of frequency /: Af/ = HJ < /i() sin cos sin(/ + A) r TT + /() cos - / () sin + J^-l' sin(9 f + X)h (56) + /() - J^r-l') sin - / () sin cos cos(fy + A)e For a semidiurnal tide of frequency /: Af ; = J ///{[/?()cos cos(, + A)f - 6sincos[/() + / () ]cos(, + A)n - 6cos[/() + / () sin ] sin(/ - A) e). (5c) In the above expressions, h{>) = h {) + /i () [(/) sin - /], /() = /< > + / () [(/) sin - /], (6) Hj = amplitude (m) of the tidal term of frequency /, = geocentric latitude of Station, A = east longitude of Station, Oj = tide argument for tidal constituent with frequency /, 57

62 e = unit vector in the east direction, h = unit vector at right angles to f in the northward direction. The onvention used in defining the tidal amplitude Hf is as in artwright and Tayler (97). To convert amplitudes defined according to other onventions that have been employed in recent more accurate tables, use the conversion factors given in hapter 6, Table 6.. Equations (5) assume that the Love and Shida number parameters are all real. Generalization to the case of complex parameters (anelastic earth) is done simply by making the following replacements for the combinations Lcos(j + mx) and Lsin(f + mx), wherever they occur in those equations: Lcos(f + mx) -* L R cos(f + raa) - L sin(f + mx), (7a) Lsh\( f + mx) L R sm(f + raa) + L cos( f + ma), (76) where L is a generic symbol for ft(</>),ft',/(< ),/* x), and /\ and L R and L stand for their respective real and imaginary parts. Table 7. lists the values of the Love and Shida number parameters. The Earth model on which they are based is the see PREM, modified by replacement of the ocean layer by solid (Dehant, 987, Wang, 99) and by adjustment of the fluid core ellipticity to make the FN period = sidereal days as inferred from the resonance in nutations. The tidal frequencies shown in the table are in cycles per sidereal day (epsd). Periods, in solar days, of the nutations associated with the diurnal tides are also shown. Table 7.. Displacement Love number parameters for degree tides. Quantities with subscripts elas (anelas) are computed ignoring (including) anelasticity effects. Superscripts R and I identify the real and imaginary parts, respectively. Name Period Frequency h. nm h n * lnm Äl * *. ~ * elas anelas anelas Semidiurnal - epsd Ä< >.6 Diurnal Qi 5,55 Oi NOi Pl 65,55 Ki 65, infinity

63 Long period 55,565 "sa M m M s 75, Name Period Frequency,(),()ß,()/ elas anelas anelas IM t() /' Semidiurnal - epsd Diurnal Qi 5,55 Oi Mi Tl Pl 65,55 Ki 65, infinity Long period 55,565 -'so M m Mj 75, omputation of the variations of Station coordinates due to solid Earth tides, like that of geopotential variations, is done most efficiently by the use of a two-step procedure. The evaluations in the first step use the expression in the time domain for the füll degree tidal potential or for the parts that pertain to particular bands (m =,, or ). Nominal values used for the Love and Shida numbers /*m and.m are common to all the tidal constituents involved in the potential and to all stations. They are chosen with reference to the values in Table 7. so as to minimize the computational effort needed in Step. Along with expressions for the dominant contributions from h^ and / () to the tidal displacements, relatively small contributions from some of the other parameters are included in Step for reasons of computational efficiency. The displacements caused by the degree tides are also computed in the first step, using constant values for h$ and %. orrections to the results of the first step are needed to take account of the frequency dependent deviations of the Love and Shida numbers from their respective nominal values, and also to compute 59

64 the out of phase contributions from the zonal tides. The scheme of computation is outlined in the chart below. ORRETIONS FOR THE STATION TIDAL DISPLAEMENTS Step : orrections to be computed in the time domain in phase for degree and Nominal values. for degree eq (8) h h(>) = fc () + /i () [(sin <t> - l)/] l -+ /(</>) = /( ) + /()[( sin <j> - l)/] elastic Ä< > =.66, h^ = -.6; / () =.8, /< > =. anelastic /i () =.678, h {) = -.6; / () =.87, / () =.. for degree - eq (9) h =.9 and / =.5 out-of-phase for degree only Nominal values. diurnal tides eq () h l = -.5 and l l = -.7. semi-diurnal tides -+ eq () h = -. and l = -.7 contribution from latitude dependence Nominal values. diurnal tides -* eq () /W =.. semi-diurnal tides eq () / () =. Step : orrections to be computed in the frequency domain and to be added to results of Step in phase for degree. diurnal tides > eqs (5) * Sum over all the components of Table 7.a. semi-diurnal tides negligible in phase and out of phase for degree. long-period tides - eqs (6) Sum over all the components of Table 7.b Displacement due to degree tides, with nominal values for h m and l m The first stage of the Step calculations employs real nominal values / and / common to all the degree tides for the Love and Shida numbers. It is found to be computationally most economical to choose these to be the values for the semidiurnal tides (which have very little intraband Variation). For the same reason, the nominal values used when anelasticity is included are different from the elastic case. (Anelasticity contributions are at the one percent level, which is about rara in the radial displacement due to the füll degree tide.) The out of phase contributions due to anelasticity are dealt with separately below. On using the nominal values, the vector displacement of the Station due to the degree tides is given by 6

65 A f = E ^ { h ^ ( h Ä r f ) - l ) + h(r J -mj-(k-r)f}^ (8) where h and l of the semidiurnal tides are chosen as the nominal values h<i and /. These values depend on whether anelasticity is included in the computation or not, but no imaginary parts are included at this stage. In equation (8), GMj = gravitational parameter for the Moon (j = ) or the Sun (j = ), GM = gravitational parameter for the Earth, Rj, Rj = unit vector from the geocenter to Moon or Sun and the magnitude of that vector, R e = Earth's equatorial radius, f,r= unit vector from the geocenter to the Station and the magnitude of that vector, /i = nominal degree Love number, - = nominal degree Shida number. Note that the part proportional to /i gives the radial (not vertical) component of the tide-induced Station displacement, and the terms in I represent the vector displacement transverse to the radial direction (and not in the horizontal plane). The computation just described may be generalized to include the latitude dependence arising through h^ by simply adding /^) [(/)sin <j) - (/)] to the constant nominal value given above, with h {) = -.6. The addition of a similar term (with / () =.) to the nominal value of l takes care of the corresponding contribution to the transverse displacement. The resulting incremental displacements are small, not exceeding. rara radially and. rara in the transverse direction. Displacements due to degree tides The Love numbers of the degree tides may be taken as real and constant in computations to the degree of accuracy aimed at here. The vector displacement due to these tides is then given by ^ - t ^ M i ' * J - ' i, - l < * > - ' ) ) + ' > ( T ( * ' - ", - i ) ' * ' - ( * ' - ",9) Only the Moon's contribution (j = ) need be computed, the term due to the Sun being quite ignorable. The transverse part of the displacement (9) does not exceed. rara, but the radial displacement can reach.7 rara. 6

66 ontributions to the transverse displacement due to the l^ term The imaginary part of l^ (anelastic case) is completely ignorable, and so is the difference between the real part and the elastic Earth value, as well as the intra-band frequency dependence; and l^ is effectively zero in the zonal band. In the expressions given below, and elsewhere in this hapter, $j body fixed geocentric latitude of Moon or Sun, and Xj body fixed east longitude (from Greenwich) of Moon or Sun. The following formulae may be employed when the use of artesian coordinates Xj, Yj, Zj of the body relative to the terrestrial reference frame is preferred: P (sin$ j ) = -lqz -iä ), (a) 9V.7. QV.7. Pl (sin $,-) cos A, = ±^, P\ (sin *,) sin A,- = -j^-, (6) P (sin# j )cosa j = ~^(X] - Yj), P (sin$ j )sina j = -^XjYj. (c) ontribution from the diurnal band (with l^ =.): bt= -/ () sinv - ^-^P (sin$ j )[sin^cos(a- Xj) h - cos<risin(a - Xj)e\. () Pl GM R ontribution from the semidiurnal band (with Z* * =.): M- R* bt = --/ () sin^cos^v J J e P (sin^)[cos(a-a J )n-hsin</>sin(a- Aj)e]. () f^ ^M^Rj The contributions of the Z ( * term to the transverse displacements caused by the diurnal and semidiurnal tides could be up to.8 mm and. rara respectively. Out of phase contributions from the imaginary parts of h ^ and l m In the following, h and l stand for the imaginary parts of h ^ and l ^, which do not exist in the elastic case. ontributions ir to radial and bt to transverse displacements from diurnal tides (with h = -.5, l l = -.7): 6

67 GMjRj * r = ~i fef GM & sin^sin<^sin (A - Xj), (a) i= 6t = --l ^ ' D sin$? [cos<ftsin(a- Aj)n + sincos(a - Xj)e\. (6) ~^ rmeit^ ontributions from semidiurnal tides (with ft 7 = -., l = -.7): o M R* br = --h V ^cos $? cos < sin(a- A? ), (a) j = w J o M - R* 6t = ~/ 7^ J * cos $ i [sin</>sin(a - A 7 )n - cos</>cos(a - A^e]. f^gm^r) (6) The out of phase contributions from the zonal tides has no closed expression in the time domain. omputations of Step are to take account of the intraband Variation of /r J and l m. Variations of the imaginary parts (anelastic case) are ignorable except as stated below. For the zonal tides, however, the contributions from the imaginary part have to be computed in Step. A FORTRAN program for omputing the various corrections is available at ftpserver.oma.be, subdirectory /pub/astro/dehant/iers (anonymous-ftp system). orrection for frequency dependence of the Love and Shida numbers (a) ontributions from the diurnal band orrections to the radial and transverse Station displacements br and <üf due to a diurnal tidal term of frequency / are obtainable from equation (5b): where br = br f sm<t>sm(f + A), (5a) bt = bt f [sin (ßcos( f + X)e + cos>sm(f + X)h], (56) 6Rf = -l\h 6 h f H f and «r / = - v ^ w ^ ' (5c) and 6hj = difference of h^ at frequency / from the nominal value of /i, 6 j = difference of Z< ) at frequency / from the nominal value of /. 6

68 Values of ARj and AT/ listed in Tables 7.a and 7.b are for the constituents that must be taken into account to ensure an accuracy of rara. AR e j l and ATf are for the elastic case, and AR a j nel and ATj nel are for use when anelasticity effects are included. It should be noted that different nominal values are used for the two cases in order to minimize the number of terms for which corrections are needed. () f orrections to the out-of-phase parts, arising from Variation of the imaginary parts of h l and l^i /(), are very small. The only one with an amplitude exceeding the cutoff of.5 rara used for the table is.6 rara in the vertical component due to the Ki tide. Its contribution to the displacement is.6sin(j)co$( g + TT + A) mm. Table 7.a. orrections due to frequency Variation of Love and Shida numbers for diurnal tides. Units: rara. All terms with radial correction >.5 rara are shown. Nominal values are h =.66 and / =.8 for the elastic case, and h R =.678 and l R =.87 for the real parts in the anelastic case. Frequencies shown are in degrees per hour. Name Frequency Doodson T s h P N'p s l l' F D ü AR) < \Tf AR a f nel ATp el Qi Oj NOi fi Pi Ki ,655 5,55 5,555 55,655 6,556 6,555 65,55 65,555 65,565 66,55 67, (b) ontributions from the long period band orrections 6r and 6t due to a zonal tidal term of frequency / include both in phase (ip) and out of phase (op) parts. ^From equations (5a) and (7) one finds that 6r= (^mi >-^) (6R { f ip) cos f + 6R { f op) sm f ), (6a) and where St = sin (bt)-<»p) lv ' cos {op) S + t>t) V} sin Bj) n, (66) *R{ip), 5 ShjHj and SR(op) f = -\]-^ShJHf, (6c) ÖT(ip) f = lf^öl " Hf and 6T{op) f = " I ^ F " / * ' - (6d) 6

69 Table 7.b. orrections due to frequency Variation of Love and Shida numbers for zonal tides. Units: rara. All terms with radial correction >.5 rara are shown. Nominal values are h =.66 and / =.8 for the elastic case, and h R =.678 and l R =.87 for the real parts in the anelastic case. For each frequency, the in phase amplitudes ARj and ATj are shown on the first line, and the out of phase amplitudes ARj and ATj on the second line. Frequencies shown are in degrees per hour. Name Frequency Doodson r s h p N' p s t V F D Ü AR) las ATf as AR a f nel ATf nel. 55, S sa.8 57, M m.58 65, M f.98 75, , Permanent deformation The zonal part of the degree potential contains a time-independent constituent of amplitude (-.6) ra. A permanent deformation due to this constituent forms a part of the site displacement computed from Equation (8) using the nominal values /i =.66 and I =.8 when ignoring anelasticity. The permanent part of radial displacement thus computed is ' (.66)(-.6)(-sin < - - J = -.96( - sin - - ] meters, (7a) 7r \ / V Ä and the transverse component, which is in the northward direction, is (O.O8)(-O.6O)cossin= -.7sin meters. (76) This permanent deformation must be removed from the displacement vector computed by the procedure described above in order to obtain the temporally varying part of the tide-induced site displacement. If the nominal values for the anelastic case (/* =.678, =.87) were used in the first step computation, one should have.6 and.5 respectively in the expressions for the radial and northward components (instead of.96 and.7). If the /^-latitude dependence of the Love numbers were accounted for in the first step, i.e. if the change h - h(<f>) = /i () + /i () [(sin <f> - l)/] and l -+ l(>) = / () + / () [(sin - l)/] were made, the values will change accordingly. The restitution of the indirect effect of the permanent tide is done to be consistent with the XVIII IAG General Assembly Resolution 6; but to get the permanent tide at the Station, one should 65

70 use the same formula (equations (7a) and (7b)) replacing the Love numbers by the fluid limit Love numbers which are / = and h = + k with k =.9. Rotational Deformation Due to Polar Motion The Variation of Station coordinates caused by the pole tide is recommended to be taken into account. Let us choose x, y and z as a terrestrial system of reference. The z axis is oriented along the Earth's mean rotation axis, the x axis is in the direction of the adopted origin of longitude and the y axis is orthogonal to the x and z axes and in the plane of the 9 E meridian. The centrifugal potential caused by the Earth's rotation is V= l -[r*\ä\ -(r-ü)% (8) where ft = ü(m\x + ra y + ( + m$)z). SI is the mean angular velocity of rotation of the Earth, ra; are small dimensionless parameters, rai, ra describing polar motion and m$ describing Variation in the rotation rate, r is the radial distance to the Station. Neglecting the variations in m^ which induce displacements that are below the rara level, the m.\ and?n terms give a first order perturbation in the potential V (Wahr, 985) ft r AV(r,, X) = sin^(rai cos A + ra sin A). (9) The radial displacement S r and the horizontal displacements S# and S\ (positive upwards, south and east respectively in a horizon system at the Station) due to AV are obtained using the formulation of tidal Love numbers (Munk and MacDonald, 96): AV S r = /i, 9 S e = -d 9 AV, () 9 S x = ^-^-d x AV. g sin# In general, these computed displacements have a non-zero average over any given time span because m\ and ra, used to find AV, have a non-zero average. onsequently, the use of these results will lead to a change in the estimated mean Station coordinates. When mean coordinates produced by different users are compared at the centimeter level, it is important to ensure that this effect has been handled consistently. It is recommended that m\ and irt used in Equation 9 be replaced by parameters defined to be zero for the Terrestrial Reference Frame discussed in hapter. Thus, define x p = ni\ - räi, () V P = -(m -^), 66

71 where rä\ and rn^ are the values of m\ and ra for the Terrestrial Reference Frame. Then, using Love number values appropriate to the pole tide (h =.67, / =.86) and r = a = 6.78 x 6 m, one finds for x p and y p in seconds of are. S r = sin (x v cos A y p sin A) mm, Sß = 9 cos (x p cos X y p sin A) mm, () S\ = 9cos(x v sin A + y p cos A) mm. Taking into account that x p and y p vary, at most,.8 arcsec, the maximum radial displacement is approximately 5 rara, and the maximum horizontal displacement is about 7 mm. If X, Y, and Z are artesian coordinates of a Station in a right-handed equatorial coordinate system, we have the displacements of coordinates [dx,dy,dz) T = R T [S e,sx,s r ] T, () where cos cos A cos sin A sin R = sin A cos A sin cos A sin sin A cos The deformation caused by the pole tide also leads to time dependent perturbations in the i and 5 i geopotential coefficients (see hapter 6). Antenna Deformation hanges in antenna height and axis offset due to temperature changes can be modeled simply. Let V be the antenna height and A the axis offset. changes due to temperature, T, are then given by dv = kv(t-to), and da = k'a(t -T ö ), where k and k are estimated constants and To is the reference temperature. Typically k and k' are of the order of ~ 6. Atmospheric Loading Temporal variations in the geographic distribution of atmospheric mass load the Earth and deform its surface. Displacement variations are dominated by effects of synoptic pressure Systems; length scales of - km and periods of two weeks. Pressure loading effects are larger at high latitudes due to the larger storms found there. Effects are smaller at low latitude sites (5S to 5N) and at sites within km of the sea or ocean. Theoretical studies by Rabbel and Zschau (985), Rabbel and Schuh (986), vandam and Wahr (987), and Manabe et al. (99) demonstrate that vertical displacements of up to 5 rara are possible with horizontal effects of one-third this amount. All pressure loading analyses make the assumption that the response of the ocean to changes in air pressure is inverse barometric. It is likely that the ocean responds to pressure as an inverted 67

72 barometer at periods of a few days to a few years. (See helton and Enfield (986) or Ponte et al. (99) for a summary of observational evidence for the inverted barometer response.) A local inverted barometer response is probably appropriate for periods as short as or days. On the other hand, the decidedly nonequilibrium diurnal ocean tides imply that the global response is certainly not an inverted barometer at periods close to a day. There are many methods for omputing atmospheric loading corrections. In contrast to ocean tidal effects, analysis of the Situation in the atmospheric case does not benefit from the presence of a well-understood periodic driving force. Otherwise, estimation of atmospheric loading via Green's function techniques is analogous to methods used to calculate ocean loading effects. Rabbel and Schuh (986) recommend a simplified form of the dependence of the vertical crustal displacement on pressure distribution. It involves only the instantaneous pressure at the site in question, and an average pressure over a circular region with a km radius surrounding the site. The expression for the vertical displacement (rara) is Ar= -.5p-.55p (5) where p is the local pressure anomaly with respect to the Standard pressure of. kpa (equivalent to mbar), and p the pressure anomaly within the km circular region mentioned above. Both quantities are in " kpa (equivalent to mbar). Note that the reference point for this displacement is the site location at Standard pressure. Equation 5 permits one to estimate the seasonal displacement due to the large-scale atmospheric loading with an error less than ± rara (Rabbel and Schuh, 986). An additional mechanism for characterizing p may be applied. pressure distribution surrounding a site is described by p(x,y) = A + A x x + A y + A x + A xy + A 5 y, The two-dimensional surface where x and y are the local East and North distances of the point in question from the VLBI site. The pressure anomaly p may be evaluated by the simple Integration giving where R = (x + y ). J J c dx dy p(x,y) I Je dx d V p= A + (A + A b )R /, It remains the task of the data analyst to perform a quadratic fit to the available weather data to determine the coefficients A-5. van Dam and Wahr (987) computed the displacements due to atmospheric loading by performing a convolution sum between barometric pressure data and the mass loading Green's Function. They found that the corrections based on Equation 5 are inadequate for stations close to the coast. For these coastal stations, Equation 5 can be improved by extending the regression equation. A few investigators (Manabe et ai, 99; MacMillan and Gipson, 99; vandam and Herring, 99; vandam et ai, 99) have attempted to calculate site dependent pressure responses by regressing local pressure changes with theoretical, VLBI or GPS height variations. For example vandam and Herring (99) find significant site displacements in some VLBI stations which can be modeled by a simple linear regression. Table 7. shows the relationship between variations in local atmospheric pressure and theoretical radial surface displacements for these sites as found in their analysis. This rate is available by anonymous ftp to gracie.grdl.noaa.gov. 68

73 Table 7.. Regression oefficients Station Slope (mm/mbar) GIL -.5 ±. ONSA -. ±. WETT -.6 ±. WEST -. ±. HAT -. ±. NRAO -. ±. KASH -. ±. MOJA -. ±. GRAS -.55 ±. RIH -.5 ±. KAUA -.6 ±.5 Postglacial Rebound The current state-of-the-art model of the global process of postglacial rebound is that described in Peltier (99). The model consists of a history of the variations in ice thickness from the time of the last glacial maximum (LGM) until the present coupled with a radial profile of viscosity in the planetary mantle. The ice model, called IE-G, is a significant improvement over the previous model of Tushingham and Peltier (99) called IE-G in that it has been constrained to fit the detailed history of relative sea level rise at Barbados that is derivative of dated coral sequences that extend from the LGM to the present. The planet's response to the model deglaciation event is computed by first solving an integral equation to determine the site dependent Variation of ocean bathymetry since the LGM in order to ensure that the ocean surface remains a gravitational equipotential. Time dependent, three dimensional displacement fields are then determined by spectral convolution of the visco-elastic impulse response Green function for the planet with the complete history of surface mass loading (with its distinct ice and ocean components). A succinct review of the complete theory for the three dimensional displacement calculation and a recent application to the computation of VLBI baseline variations will be found in Peltier (995). A file containing a listing of the radial and horizontal displacements for a number of sites is available by anonymous ftp. ftp to maia.usno.navy.mil. hange directories to Standards (cd Standards). The file is called pgr.model. Any comments or corrections to this file should be directed to Prof. Richard Peltier ([email protected]). References artwright, D. E. and Tayler, R. J., 97, "New omputations of the Tide-Generating Potential," Geophys. J. Roy. Astron. Soc,, pp artwright, D. E. and Edden, A., 97, "orrected Tables of Tidal Harmonics," Geophys. Astron. Soc.,, pp J. Roy. helton, D. B. and Enfield, D. B., 986, "Ocean Signals in Tide Gauge Records," J. Geophys., 9, pp

74 Dehant, V., 987, "Tidal parameters for an inelastic Earth," Phys. Earth Plan. Int., 9, pp Doodson, A. T., 98, "The Analysis of Tidal Observations," Phil Trans. Roy. Soc. Lond., 7, pp. Farrell, W. E., 97, "Deformation of the Earth by Surface Loads," Rev. Geophys. Space Phys.,, pp Fiather, R. A., 98, Proc. Norwegian oastal urrent Symp. Geilo, 98, Saetre and Mork (eds), pp Le Provost,, Genco, M. L., Lyard, F., Incent, P., and anceil, P., 99, "Spectroscopy of the world ocean tides from a finite element hydrological model," J. Geophys. Res., 99, pp MacMillan, D.S. and J.M. Gipson, 99, "Atmospheric Pressure Loading Parameters from Very Long Baseline Interferometry Observations," J. Geophys. Res., 99, pp Manabe, ST., T. Sato, S. Sakai, and K. Yokoyama, 99, "Atmospheric Loading Effects on VLBI Observations," in Proceedings of the AGU hapman onference on Geodetic VLBI: Monitoring Global hange, NOAA Tech. Rep. NOS 7 NGS 9, pp. -. Mathews, P. M., Buffett, B. A., and Shapiro, I. L, 995, "Love numbers for a rotating spheroidal Earth: New definitions and numerical values, Geophys. Res. Lett.,, pp Munk, W. H. and MacDonald, G. J. F., 96, The Rotation New York, pp. -5. of the Earth, ambridge Univ. Press, Peltier, W. R, 99, "Ice Age Paleotopography," Science, 65, pp Peltier, W.R., 995, "VLBI baseline variations from the IE-G model of postglacial rebound," Geophys. Res. Lett,, pp Ponte, R. M., Salstein, D. A., and Rosen, R. D., 99, "Sea Level Response to Pressure Forcing in a Barotropic Numerical Model," J. Phys. Oceanogr.,, pp Rabbel, W. and Schuh, H., 986, "The Influence of Atmospheric Loading on VLBI Experiments," J. Geophys., 59, pp Rabbel, W. and Zschau, J., 985, "Statis Deformations and Gravity hanges at the Earth's Surface due to Atmospheric Loading, " J. Geophys., 56, pp Row, L. W., Hastings, D. A., and Dunbar, P. K., 995, "TerrainBase Worldwide Digital Terrain Data," NOAA, National Geophysical Data enter, Boulder O. Scherneck, H. G., 98, rustal Loading Affecting VLBI Sites, University of Uppsala, Institute of Geophysics, Dept. of Geodesy, Report No., Uppsala, Sweden. 7

75 Scherneck, H. G., 99, "Loading Green's functions for a continental shield with a Q-structure for the mantle and density constraints from the geoid," Bull. d'inform. Maries Terr., 8, pp Scherneck, H. G., 99, "A Parameterized Solid Earth Tide Model and Ocean Tide Loading Effects for Global Geodetic Baseline Measurements," Geophys. J. Int., 6, pp Scherneck, H. G., 99, "Ocean Tide Loading: Propagation of Errors from the Ocean Tide into Loading oefficients," Man. Geod., 8, pp Schwiderski, E. W., 98, "Atlas of Ocean Tidal harts and Maps, Part I: The Semidiurnal Principal Lunar Tide M," Marine Geodesy, 6, pp Schwiderski, E. W. and Szeto, L. T., 98, NSW-TR 8-5, Naval Surface Weapons enter, Dahlgren Va., 9 pp. Tamura, Y., 987, "A harmonic development of the tide-generating potential," Bull. dtnform. Terr., 99, pp Marees Tushingham, A. M. and Peltier, W. R., 99, "Ice-G: A New Global Model of Late Pleistocene Deglaciation Based lipon Geophysical Predictions of Post-Glacial Relative Sea Level hange," J. Geophys. Res., 96, pp Tushingham, A. M. and Peltier, W. R., 99, "Validation of the IE-G Model of Wurm-Wisconsin Deglaciation Using a Global Data Base of Relative Sea Level Histories,". Geophys. Res., 97, pp vandam, T. M. and Wahr, J. M., 987, "Displacements of the Earth's Surface due to Atmospheric Loading: Effects on Gravity and Baseline Measurements," J. Geophys. Res., 9, pp vandam, T. M. and Herring, T. A., 99, "Detection of Atmospheric Pressure Loading Using Very Long Baseline Interferometry Measurements," J. Geophys. Res., 99, pp vandam, T. M., Blewitt, G., and Heflin, M. B., 99, "Atmospheric Pressure Loading Effects on Global Positioning System oordinate Determinations," J. Geophys. Res., 99, pp.,97-,95. Verheijen and Schrama, 995, personal communication. Wahr, J. M., 98, "The Forced Nutations of an Elliptical, Rotating, Elastic, and Oceanless Earth," Geophys. J. Roy. Astron. Soc, 6, pp Wahr, J. M., 985, "Deformation Induced by Polar Motion," J. Geophys. Res., 9, pp Wang, R., 99, "Effect of rotation and ellipticity on earth tides," Geophys. J. Int., 7, pp Zschau, J., 98, "Rheology of the Earth's mantle at tidal and handler Wobble periods," Proc. Ninth Int Symp. Earth Tides, New York, 98, J. T. Kuo (ed), Schweizerbart'sehe Verlagabuchhandlung, Stuttgart, pp

76 HAPTER 8 TIDAL VARIATIONS IN THE EARTH'S ROTATION Periodic variations in UTI due to tidal deformation of the polar moment of inertia were derived by Yoder et al. (98) including the tidal deformation of the Earth with a decoupled core. This model uses effective Love numbers that differ from the bulk value of. because of the oceans and the fluid core giving rise to different theoretical values of the ratio k/ for the fortnightly and monthly terms. However, Yoder et al. recommend the value of.9 for k/ for both cases. Oceanic tides also cause variations in UTI represented by modeis given by Brosche et al. (99, 989), Dickman (99, 99, 99, 989), Gross (99), Herring and Dong (99), and Ray et al. (99). The contribution of the oceanic tides is split into a part which is in phase with the solid Earth tides and an out-of-phase part. Table 8. provides corrections for the tidal variations in UTI UT with periods between five and 5 days. This is identical to tables in IERS Technical Notes and which defined UTR-UT, A - AR, and u; - u;r. Table 8. continues to define UTR-UT, A - AR, and u - u>r. UTl-UTlR=^A;sin& t=i A-AR= ^A-cos& i=l u; - u;r = V" A'l cos & 5 aij = integer multipliers of the <%j (l, V, F, D or fi) for the i ih tide given in the first five columns of Table 8.. Ai, A\, A" are given in columns 7-9 respectively in Table 8.. Table 8. is identical to Table 8. in IERS Technical Note which defined UTS-UT, A - AS, and u;-u;s except that the table below corrects one misprint in the out-of-phase term for the 8.6-year tide. Table 8. provides corrections for the tidal variations with periods from five days to 8.6 years and incorporates the most significant effects of oceanic tides using the model of Dickman (99). 6 UTI - UTS = ]T Bi sin & + { cos & t=i 6 A - AS = ]T B\ cosfc + \ sinfc «=i 6 W - ^S = Ys B 'i S & + * SiU & i=l t are defined the same way as for UTR above except the a^ are given in the first five columns of Table 8.. B {, d, B\, \, B'/, and f are given in columns 7- respectively in Table 8.. 7

77 Table 8. provides corrections for daily and sub-daily tidal variations in the Earth's rotation from Ray (995). Values corrected using Table 8. are referred to as UTD, A - AD, and u - u>d. Those corrected using Table 8. and 8. will then be called UTI DR, A - ADR, and u> - udr, and those corected using Table 8. and 8. will be called UTDS, A - ADS, and u - u;ds. UTI - UTD = ]T Di sin & + Ei cos & i=i 8 A - AD = ]TZ^cos& + i=l 8 E[sink U ~ U;D = S " S & + E " Sln & «= 6 & = i j i i +^ jj = integer multipliers of the 7j (/, /', F, D, ü, or ö) for the i ih 8.. (j>i = the phase given in column 8 of Table 8.. tide given in columns -7 of Table Table 8. provides corrections for daily and subdaily tidal variations in polar motion from Ray (995). Values corrected by using this table are referred to as xd and yd and are obtained by 8 xd - x = _\ Fi sin & + Gi cos &, i=i 8 yd - y = Y^Hi sin & + Ki cos, i=i where Fi, G\, Hi, and h'i are given in the table and 6 i=i c,j = integer multipliers of the 7, (/, /', F, D, Ü, or ) for the i th 8.. <f>i = the phase given in column 8 of Table 8.. tide given in columns -7 of Table 7

78 Table 8.. Zonal Tide Terms With Periods Up to 5 Days. UTR, AR, and ur represent the regularized forms of UTI, the duration of the day A, and the angular velocity of the Earth, u>. The units are IQ" s for UT, " 5 s for A, and ~ rad/s for u. / ARGUMENT* /' F - D ft PERIOD Days UT-UTR oefficient of Sin (Argument) A-AR w ur oefficient of os (Argument)

79 Table 8.. Zonal Tide Terms. UTS, AS, and usl represent the regularized forms of UTI, the duration of the day A, and the angular velocity of the Earth, u. The units are ~ s for UT, ~ 5 s for A, and ~ rad/s for UJ. / ARGUMENT '* /' F - D Ü PERIOD Days UT--UTS Sin os A- -AS oefficient of os Sin u> U>S os Sin O. 75

80 Table 8.. Diurnal and subdiurnal zonal tide terms. The units are s for UT, 5 s for A, and "~ rad/s for u;. Tide Qi Oj Pl K! N M s K / - - /' F D Ü phase Period (deg.) (hours) UT- UTD Sin os A- AD oefficients of os Sin (J - os u;d Sin = 8 + 6?996(MJD-55.5) Earth's rotation angle See hapter 5 for definitions of /, /', F, D, ü. 76

81 Table 8.. Diurnal and subdiurnal tide terms for polar motion. The units are msec. of are for Ax and Ay. Tide Qi Oi Pi Ki N M s K / - - /' F D Ü phase Period (deg.) (hours) Sin A X Ay oefficients of os Sin os References Brosche, P., Seiler, U., Sündermann, J., and Wünsch, J., 989, "Periodic hanges in Earth's Rotation due to Oceanic Tides," Astron. Astrophys.,, pp. 8-. Brosche, P., Wünsch, J., ampbell, J., and Schuh, H., 99, "Ocean Tide Effects in Universal Time detected by VLBI," Astron. Astrophys., 5, pp Dickman, S. R., 989, "A omplete Spherical Harmonic Approach to Luni-Solar Tides," Geophys. 99, pp J., Dickman, S. R., 99, "Experiments in Tidal Mass onservation" (research note), Geophys. J.,, pp Dickman, S. R., 99, "Ocean Tides for Satellite Geodesy," Mar. Geod.,, pp Dickman, S. R., 99, "Dynamic ocean-tide effects on Earth's rotation," Geophys J. Int.,, pp. Gross, R. S., 99, "The Effect of Ocean Tides on the Earth's Rotation as Predicted by the Results of an Ocean Tide Model," Geophys. Res. Let,, pp Herring, T. A. and Dong, D., 99, "Measurement of Diurnal and Semidiurnal Rotational Variations and Tidal Parameters of Earth," J. Geophys. Res., 99, pp Ray, R. D., Steinberg, D. J., hao, B. F., and artwright, D. E., 99, "Diurnal and Semidiurnal Variations in the Earth's Rotation Rate Induced by Oceanic Tides," Science, 6, pp Ray, R. D., 995, Personal ommunication. Yoder,. F., Williams, J. G., Parke, M. E., 98, "Tidal Variations of Earth Rotation," J. Res., 86, pp Geophys. 77

82 HAPTER 9 TROPOSPHERI MODEL Optical Techniques The formulation of Marini and Murray (97) is commonly used in laser ranging. The formula has been tested by comparison with ray-tracing radiosonde profiles. The correction to a one-way ränge is where A + B A R - J M - f(<j>,h) Sjn + B H A + B ) ' Mri J Sini^ -f sin _ _. A =.57P +.e, () m K) B = (.8 x - 8 )P r A' + (.7 x " 8 )^-. Q *., () lo (o - /A ) A = cos< -.T +.Q5P o, () where AÄ = ränge correction (meters), E = true elevation of satellite, Po = atmospheric pressure at the laser site (in "" kpa, equivalent to millibars), To = atmospheric temperature at the laser site (degrees Kelvin), eo = water vapor pressure at the laser site (~ x kpa, equivalent to millibars), f(x) = laser frequency parameter ( A = wavelength in micrometers), and f((j>,h) = laser site function. Additional definitions of these parameters are available. The water vapor pressure, eo, can be calculated from a relative humidity measurement, Rh(%) by The laser frequency parameter, f(x), Rh 7.5(T -7.5) e = - x 6. x 7 +^o- 75 >. is ^xv ^^r^ -6.8 f(x) = jf- + y. f(x) = for a ruby laser, [i.e. /(.69) = ], while /(Ä G ) =.579 and f(x m ) and infrared YAG lasers. = for green The laser site function is f(>,h) = -.6cos^~.if, where <f> is the latitude and H is the geodetic height (km). Radio Techniques The differences between mathematical tropospheric modeis are often less than the errors which would be introduced by the character and distribution of the wet component and by the departures of 78

83 the refractivity from azimuthal symmetry. For this reason it is customary in the analysis of geodetic data to estimate the zenith atomspheric delay and to model only the mapping function, which is the ratio of delay at a given elevation angle to the zenith delay. The mapping function may be for the hydrostatic, wet, or total troposphere delay. Accordingly, the IERS conventional model applies primarily to the mapping functions. For the most accurate a priori hydrostatic delay, desirable when the accuracy of the estimate of the zenith wet delay is important, the formula of Saastamoinen (97) as given by Davis et al. 985) should be used. omparisons of many mapping functions with the ray tracing of a global distribution of radiosonde data have been made by Janes et al. (99) and by Mendes and Langley (99). For observations below elevation, which may be included in geodetic programs in order to increase the precision of the vertical component of site position, the mapping functions of Lanyi (98), Ifadis (986), Herring (99, designated MTT) and Niell (996, designated NMF) are the most accurate. Only the last three were developed for observations below an elevation of 6, with MTT and NMF being valid to and Ifadis to. Each of these mapping functions consists of a component for the water vapor and a component for either the total atmosphere (Lanyi) or the hydrostatic contribution to the total delay (Ifadis, MTT, and NMF). In all cases the wet mapping should be used as the function partial derivative for estimating the residual atmosphere zenith delay. The parameters of the atmosphere that are readily accessible at the time of the Observation are the surface temperature, pressure, and relative humidity. The mapping functions of Lanyi, Ifadis, and Herring were developed to make use of this information. Lanyi additionally allows for parameterization in terms of the height of a surface isothermal layer, the lapse rate from the top of this layer to the tropopause, and the height of the tropopause. Including the surface meteorology without these data results in larger discrepancies from radiosonde data than the Ifadis and Herring modeis. The hydrostatic mapping function of Niell differs from the other three by being independent of surface meteorology. It relies instead on the greater contribution by the conditions in the atmosphere above approximately km, which are strongly seasonal dependent. The RMS Variation is comparable to those using Ifadis and MTT, and all three are less than that from the Lanyi model when only surface data are available. Thus NMF offers comparable precision and accuracy to Ifadis and MTT, when they are provided with accurate surface meteorology data, but with no dependence on externa! measurements. Thus, if information is available on the vertical temperature distribution in the atmosphere, Lanyi is preferred. Otherwise one of the other three mapping functions should be used. References Davis, J. L., Herring, T. A., Shapiro, I. I., Rogers, A. E. E., and Elgered, G., 985, "Geodesy by Radio Interferometry: Effects of Atmospheric Modelling Errors on Estimates of Baseline Length," Radio Science,, No. 6, pp Herring, T. A., 99, "Modeling Atmospheric Delays in the Analysis of Space Geodetic Data," Proeeedings of Refraction of Transatmospheric Signals in Geodesy, Netherlands Geodetic ommission Series, 6, The Hague, Netherlands, pp

84 Ifadis, I. I., 986, "The Atmospheric Delay of Radio Waves: Modeling the Elevation Dependance on a Global Scale," Technical Report No. 8L, halmers U. of Technology, Göteburg, Sweden. Janes, H. W., Langley, R. B., and Newby, S. P., 99, "Analytical Tropospheric Delay Prediction Models: omparison with Ray- tracing and Implications for GPS Relative Positioning," Bull. Geod, 65, pp 5-6. Lanyi, G., 98, "Tropospheric Delay Affecting Radio Interferometry," TDA Progress Report, pp. 5-59; see also Observation Model and Parameter Partiais for the JPL VLBI Parameter Estimation Software 'MASTERFIT-987, 987, JPL Publication 8-9, Rev.. Marini, J. W. and Murray,. W., 97, "orrection of Laser Range Tracking Data for Atmospheric Refraction at Elevations Above Degrees," NASA GSF X Mendes, V. B. and Langley, R. B., 99, "A omprehensive Analysis of Mapping Functions Used in Modeling Tropospheric Propagation Delay in Space Geodetic Data," International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigations, Banff anada. Niell, A. E., 996, "Global Mapping Functions for the Atmospheric Delay of Radio Wavelengths," J. Geophys. Res.,, pp Saastamoinen, J., 97, "Atmospheric orrection for the Troposphere and Stratosphere in Radio Ranging of Satellites," Geophysical Monograph 5, Henriksen (ed), pp

85 HAPTER RADIATION PRESSURE REFLETANE MODEL For a near-earth satellite the solar radiation pressure acceleration, is given by: r = K a R m R where K =.56 x ~ 6 newtons/m (67 watts/m ), A = astronomical unit in meters, R = heliocentric radius vector to the satellite, a cross-sectional area (m ) of the satellite perpendicular to R, m = satellite mass, R = reflectivity coefficient, usually an adjusted parameter. The radiation pressure due to backscatter from the Earth is ignored. The model for the Earth's and Moon's shadows should include the umbra and the penumbra (Haley, 97). Earth Radius Moon Radius Solar Radius 6 km 78 km 696 km Global Positioning System For GPS satellites, the solar radiation pressure modeis T (for Block I and IIA) and T (for Block II) of Fliegel et al. (99), (Gallini and Fliegel, 995) are recommended. These modeis include thermal reradiation. A preliminary model for the forthcoming Block HR satellites, which may begin to replace Block II/IIA as early as January 997, is given here as T. Those modeis provide the A~ and Z components of the total nominal solar pressure force as functions of the angle B between the Sun and the +Z axis of the satellite. For Block I, II, and IIA, < B < 8 degrees, but Block HR, < B < 6 degrees (no "noon turn"). The model formulae for T are (angles in radians): The model formulae for T are: X = -.55 sin B +.8 sin(p +.9) -.6 cos(p +.8) +.8, Z = -.5 cos B +. sin(jb -.) -. sin P. X = sin B +.6 sin P +. sin 5P -.7 sin B, The model formulae for T (Fliegel, 995) are: Z = -8. cos B. X = -.sinp-.sinp +.sin5p, 8

86 In both cases the units are ~ 5 N. References Z = -. cos B +. cosp +.cos5p. Fliegel, H. F., Gallini, T. E., and Swift, E., 99, "Global Positioning System Radiation Force Models for Geodetic Applications," J. Geophys. Res., 97, No. Bl, pp Fliegel, H. F. and Gallini, T. E., 996, "Solar Force Modelling of Block HR GPS Satellites," J. of Space Graft and Rockets, in press. Gallini, T. E. and Fliegel, H. F., 995, The Generalized Solar Force Model, Aerospace Report No. TOR-95(57)-, The Aerospace orporation, El Segundo, A. Haley, D., 97, Solar Radiation Pressure alculations in the Geodyn Program, EG&G Report 8-7, Prepared for NASA Goddard Space Flight enter. 8

87 HAPTER GENERAL RELATIVISTI MODELS FOR TIME, OORDINATES AND EQUATIONS OF MOTION The relativistic treatment of the near-earth satellite orbit determination problem includes corrections to the equations of motion, the time transformations, and the measurement model. The two coordinate Systems generally used when including relativity in near-earth orbit determination Solutions are the solar system barycentric frame of reference and the geocentric or Earth-centered frame of reference. Ashby and Bertotti (986) constructed a locally inertial E-frame in the neighborhood of the gravitating Earth and demonstrated that the gravitational effects of the Sun, Moon, and other planets are basically reduced to their tidal forces, with very small relativistic corrections. Thus the main relativistic effects on a near-earth satellite are those described by the Schwarzschild field of the Earth itself. This result makes the geocentric frame more suitable for describing the motion of a near-earth satellite (Ries et ai, 988). The time coordinate in the inertial E-frame is Terrestrial Time (designated TT) (Guinot, 99) which can be considered to be equivalent to the previously defined Terrestrial Dynamical Time (TDT). This time coordinate (TT) is realized in practice by International Atomic Time (TAI), whose rate is defined by the atomic second in the International System of Units (SI). Terrestrial Time adopted by the International Astronomical Union in 99 differs from Geocentric oordinate Time (TG) by a scaling factor: TG - TT = L G x (MJD -.) x 86 seconds, where MJD refers to the modified Julian date. Figure. shows graphically the relationships between the time scales. See IERS Technical Note, pages 7- for copies of the. IAU Recommendations relating to these time scales. Equations of Motion for an Artificial Earth Satellite The correction to the acceleration of an artificial Earth satellite Aa is A «= -^{[(/* + 7)^-7t> a ]r + ( + 7)(r *?)<?}, () where c = speed of light, /j,7 = PPN parameters equal to in General Relativity, f,v,a= geocentric satellite position, velocity, and acceleration, respectively, G'M^ = geocentric constant of gravitation. The effects of Lense-Thirring precession (frame-dragging), geodesic (de Sitter) precession, and the relativistic effects of the Earth's oblateness have been neglected. Equations of Motion in the Barycentric Frame The n-body equations of motion for the solar system frame of reference (the isotropic Parameterized Post-Newtonian system with Barycentric oordinate Time (TB) as the time coordinate) are required to describe the dynamics of the solar system and artificial probes moving about the solar 8

88 system (for example, see Moyer, 97). These are the equations applied to the Moon's motion for Lunar Laser Ranging (Newhall et ai, 987). In addition, relativistic corrections to the laser ränge measurement, the data timing, and the Station coordinates are required (see hapter ). Scale Effect and hoice of Time oordinate A previous IAU definition of the time coordinate in the barycentric frame required that only periodic differences exist between Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT) (Kaplan, 98). As a consequence, the spatial coordinates in the barycentric frame had to be rescaled to keep the speed of light unchanged between the barycentric and the geocentric frames (Misner, 98; Hellings, 986). Thus, when barycentric (or TDB) units of length were compared to geocentric (or TDT) units of length, a scale difference, L, appeared. This is no longer required with the use of the TG time scale. The difference between TB and TDB is given in seconds by TB - TDB = L B x (MJD -.) x 86. The difference between Barycentric oordinate Time (TB) and Geocentric oordinate Time (TG) involves a four-dimensional transformation, TB - TG = c~ U [^ + U ext (x e )]dt + v e (x - x e )\, where x e and v e denote the barycentric position and velocity of the Earth's center of mass and ifis the barycentric position of the observer. U ex t is the Newtonian potential of all of the solar system bodies apart from the Earth evaluated at the geocenter. to is chosen to be consistent with 977 January, h m s TAI and t is TB. An approximation is given in seconds by (TB - TG) = L c x (MJD -.) x 86 + c~ v e (x - x e ) + P. with MJD measured in TAI. Table. lists the values of the rates Lß, Lc, and LQ. Periodic differences denoted by P have a maximum amplitude of around.6 ms. These can be evaluated by the "FBL" model of Fairhead, Bretagnon and Lestrade (995). A comparison with the numerical time ephemeris TE5 (Fukushima, 995) revealed that this series provides the smallest deviation from TE5 for the years after removing a linear trend. Users may expect the FBL model to provide the periodic part in TB-TT within a few ns for a few decades around the present time. This is sufficient for all precision measurement including timing observations of millisecond pulsars. Software to implement the FBL model is available by anonymous ftp to maia.usno.navy.mil and is located in the directory /conventions/chapterll. The files of interest are called fbl.f and fbl.results. This model is based on earlier works (Fairhead et ai, 988; Fairhead and Bretagnon, 99). 8

89 TDT Terrestrial Dynamical Time TT Terrestrial Time TDT = TT - TAI + Ü8 TG Geocentric oordinate Time TG - TT = L G X AT -dimensional space-time transformation Linear transformation L c x AT TDB TB Barycentric Dynamical Time Barycentric oordinate Time TB = TDB +L B x AT AT = (date in days January, h )TAIx 86 see Fig. Relations between time scales. References Ashby, N. and Bertotti, B., 986, "Relativistic Effects in Local Inertial Frames," Phys. Rev. D, (8), p. 6. Fairhead, L. and Bretagnon, P., 99, "An Analytic Formula for the Time Transformation TB-TT," Astron. Astrophys., 9, pp. -7. Fairhead, L., Bretagnon, P., and Lestrade, J.-F., 988, "The Time Transformation TDB-TDT: An Analytical Formula and Related Problem of onvention," in The Earth's Rotation and Reference Frames for Geodesy and Geophysics, A. K. Babcock and G. A. Wilkins (eds.), Kluwer Academic Publishers, Dordrecht, pp Fairhead, L., Bretagnon, P., and Lestrade, J.-F., 995, Private ommunication. Fukushima, T., 995, "Time Ephemeris," Astron. Astrophys., 9, pp

90 Guinot, B., 99, "Report of the Sub-group on Time," in Reference Systems, J. A. Hughes,. A. Smith, and G. H. Kaplan (eds), U. S. Naval Observatory, Washington, D., pp. -6. Hellings, R. W., 986, "Relativistic Effects in Astronomical Timing Measurement," Astron. J., 9 (), pp Erratum, ibid., p. 6. Kaplan, G. H., 98, The IAU Resolutions on Astronomical onstants, Time Scale and the Fundamental Reference Frame, U. S. Naval Observatory ircular No. 6. Misner,. W., 98, Scale Factors for Relativistic Ephemeris oordinates, NASA ontract NAS5-5885, Report, EG&G, Washington Analytical Services enter, Inc. Moyer, T. D., 97, Mathematical Formulation of the Double-precision Orbit Determination Program, JPL Technical Report -57. Newhall, X X, Williams, J. G., and Dickey, J.., 987, "Relativity Modeling in Lunar Laser Ranging Data Analysis," in Proceedings of the International Association of Geodesy (IAG) Symposia, Vancouver, pp Ries, J., Huang,, and Watkins, M. M., 988, "Effect of General Relativity on a Near-Earth Satellite in the Geocentric and Barycentric Reference Frames," Phys. Rev. Lei., 6, pp

91 HAPTER GENERAL RELATIVISTI MODELS FOR PROPAGATION VLBI Time Delay There have been many papers dealing with relativistic effects which must be accounted for in VLBI processing; see (Robertson, 975), (Finkelstein et al., 98), (Hellings, 986), (Pavlov, 985), (annon et al, 986), (Soffel et ai, 986), (Zeller et ai, 986), (Sovers and Fanselow, 987), (Zhu and Groten, 988), (Shahid-Saless et ai, 99), (Soffel et ai, 99). As pointed out by Boucher (986), the relativistic correction modeis proposed in various articles are not quite compatible. To resolve differences between the procedures and to arrive at a Standard model, a Workshop was held at the U. S. Naval Observatory on October 99. The proceedings of this Workshop have been published (Eubanks, 99) and the model given here is based on the consensus model resulting from that Workshop. Much of this chapter dealing with VLBI time delay is taken directly from that work and the reader is urged to consult that publication for further details. One change from that model has been made in order to adopt the IAU/IUGG onventions for the scale of the terrestrial reference system, in accord with 99 resolutions (see Appendix in Mcarthy, 99). Geodetic lengths should be expressed in "SI units," i.e., be consistent with the second as realized by a clock running at the TAI rate at sea level. The only change needed to the 99 formulation to satisfy the IAU/IUGG Resolutions is to include the Earth's potential, (/$, in the total potential U in the delay equation (Equation 9). The only observable effect of this change will be an increase of the VLBI terrestrial scale by parts in 9. As pointed out by Eubanks, the use of clocks running at the geoid and delays calculated "at the geocenter" ignoring the scale change induced by the Earth's gravitational potential means that terrestrial distances calculated from the consensus model will not be the same as those calculated using the expressions in this chapter which are equivalent to using meter sticks on the surface of the Earth. The accuracy limit chosen for the consensus VLBI relativistic delay model is ~ seconds (one picosecond) of differential VLBI delay for baselines less than two Earth radii in length. In the model all terms of order ~ seconds or larger were included to ensure that the final result was accurate at the picosecond level. Source coordinates derived from the consensus model will be solar system barycentric and should have no apparent motions due to solar system relativistic effects at the picosecond level. The consensus model was derived from a combination of five different relativistic modeis for the geodetic delay. These are the Masterfit/Modest model, due to Fanselow and Thomas (see Treuhaft and Thomas, in (Eubanks, 99), and (Sovers and Fanselow, 987)), the I. I. Shapiro model (see Ryan, in (Eubanks, 99)), the Hellings-Shahid-Saless model (Shahid-Saless et al., 99) and in (Eubanks, 99), the Soffel, Muller, Wu and Xu model (Soffel et al., 99) and in (Eubanks, 99), and the Zhu-Groten model (Zhu and Groten, 988) and in (Eubanks, 99). Baseline results are expressed in "local" or "SF coordinates appropriate for clocks running at the TAI rate on the surface of the Earth, to be consistent with the more general IERS onventions on the terrestrial reference system scale. This means that the gravitational potential of the Earth is now included in U; the scale effects of the geocentric Station velocities can still be ignored as they will at most cause scale changes of order part in (see Zhu and Groten, Soffel et ai, and Fukushima, all in Eubanks (99) and Shahid-Saless et al. (99) for further details on the implications of these choices). As the time argument is now based on TAI, which is a quasi-local time on the geoid, distance estimates from these onventions will now be consistent in principle with "physical" distances. 87

92 The model is designed for use in the reduction of VLBI observations of extra-galactic objects acquired from the surface of the Earth. The delay error caused by ignoring the annual parallax is > psec for objects closer than several hundred thousand light years, which includes all of the Milky Way galaxy. The model is not intended for use with observations of sources in the solar system, nor is it intended for use with observations made from space-based VLBI, from either low or high Earth orbit, or from the surface of the Moon (although it would be suitable with obvious changes for observations made entirely from the Moon). It is assumed that the inertial reference frame is defined kinematically and that very distant objects, showing no apparent motion, are used to estimate precession and the nutation series. This frame is not truly inertial in a dynamical sense, as included in the precession constant and nutation series are the effects of the geodesic precession (~ 9 milli are seconds / year). Soffel et al. (in Eubanks (99)) and Shahid-Saless et al. (99) give details of a dynamically inertial VLBI delay equation. At the picosecond level, there is no practical difference for VLBI geodesy and astrometry except for the adjustment in the precession constant. Although the delay to be calculated is the time of arrival at Station minus the time of arrival at Station, it is the time of arrival at Station that serves as the time reference for the measurement. Unless explicitly stated otherwise, all vector and scalar quantities are assumed to be calculated at t\, the time of arrival at Station including the effects of the troposphere. The notation follows that of Hellings (986) and Hellings and Shahid-Saless in Eubanks (99) as closely as possible. It is assumed that the Standard IAU modeis for precession, nutation, Earth rotation and polar motion have been followed and that all geocentric vector quantities have thus been rotated into a nearly non-rotating celestial frame. The errors in the Standard IAU modeis are negligible for the purposes of the relativistic transformations. The notation itself is given in Table.. The consensus model separates the total delay into a classical delay and a general relativistic delay, which are then modified by relativistic transformations between geocentric and solar system barycentric frames. Table.. Notation used in the model ti the time of arrival of a radiointerferometric signal at the i th VLBI reeeiver in terrestrial time (TAI) Ti the time of arrival of a radiointerferometric signal at the i th VLBI reeeiver in barycentric time (TB or TDB) t 9t the "geometric" time of arrival of a radiointerferometric signal at the i th VLBI reeeiver including the gravitational "bending" delay and the change in the geometric delay caused by the existence of the atmospheric propagation delay but neglecting the atmospheric propagation delay itself t Vi the "vaeuum" time of arrival of a radiointerferometric signal at the i th VLBI reeeiver including the gravitational delay but neglecting the atmospheric propagation delay and the change in the geometric delay caused by the existence of the atmospheric propagation delay tij the approximation to the time that the ray path to Station i passed dosest to gravitating body J btatm t the atmospheric propagation delay for the i th reeeiver = ti t 9i the differential gravitational time delay, commonly known as the gravitational "bending delay" Xi(ti) the geocentric radius vector of the i th reeeiver at the geocentric time ti Atg rav b #(^) x\(h) and is thus the geocentric baseline vector at the time of arrival t\ 6 the a priori geocentric baseline vector at the time of arrival ti 6b 6(ti)-6 (ti) 88

93 Wi the geocentric velocity of the i th reeeiver K the unit vector from the barycenter to the source in the absence of gravitational or aberrational bending ki the unit vector from the i th Station to the source after aberration Xi the barycentric radius vector of the i th reeeiver X($ the barycentric radius vector of the geocenter Xj the barycentric radius vector of the J th gravitating body Rij the vector from the J th gravitating body to the i th reeeiver R&J the vector from the J th gravitating body to the geocenter the vector from the Sun to the geocenter Rq )e Nij the unit vector from the J ih gravitating body to the i th reeeiver VQ the barycentric velocity of the geocenter U the gravitational potential at the geocenter plus the terrestrial potential at the surface of the Earth. At the picosecond level, only the solar and terrestrial potentials need be included in U so that U = GM / Ä e Jc + GM /a#c Mi the mass of the i th gravitating body MQ the mass of the Earth 7 a PPN Parameter = in general relativity c the speed of light in meters / second G the Gravitational onstant in Newtons meters kilograms" a$) the equatorial radius of the Earth Vector magnitudes are expressed by the absolute value sign [.T = (.r )J. Vectors and scalars expressed in geocentric coordinates are denoted by lower case (e.g. x and t), while quantities in barycentric coordinates are in upper case (e.g. X and T). MKS units are used throughout. For quantities such as V, Wi, and U it is assumed that a table (or numerical formula) is available as a function of TAI and that they are evaluated at the atomic time of reeeption at Station, t\, unless explicitly stated otherwise. A lower case subscript (e.g. Xi) denotes a particular VLBI reeeiver, while an upper case subscript (e.g. xj) denotes a particular gravitating body. Gravitational Delay The general relativistic delay, At grav, is given for the J th gravitating body by A v..,. ( i + 7 ) g % I M +!! ' in At the picosecond level it is possible to simplify the delay due to the Earth, At grav(b becomes, which A, M, vgm e fi + K -x x Mgrav* = ( + l) ^ In , () \X\ + A -x The Sun, the Earth and Jupiter must be included, as well as the other planets in the solar system along with the Earth's Moon, for which the maximum delay change is several picoseconds. The major satellites of Jupiter, Saturn and Neptune should also be included if the ray path passes close to them. This is very unlikely in normal geodetic observing but may oeeur during planetary oecultations. 89

94 The effect on the bending delay of the motion of the gravitating body during the time of propagation along the ray path is small for the Sun but can be several hundred picoseconds for Jupiter (see Sovers and Fanselow (987) page 9). Since this simple correction, suggested by Sovers and Fanselow (987) and Hellings (986) among others, is sufficient at the picosecond level, it was adapted for the consensus model. It is also necessary to account for the motion of Station during the propagation time between Station and Station. In this model Rij, the vector from the J th gravitating body to the i th reeeiver, is iterated once, giving t u = min[*i,<! - Ä' (Xj(h) - Xi(ti))], () so that Äi J (*i) = A- (* )-X J (ti J ), () and R j = X (h)~^(k -bo)- Xj(t Xj ). (5) c Only this one iteration is needed to obtain picosecond level accuracy for solar system objects. If more accuracy is required, it is probably better to use the rigorous approach of Shahid-Saless et al. (99). X\(t\) is not tabulated, but can be inferred from X (ti) using Xi(t ) = X (t l ) + x i (t l ), (6) which is of sufficient accuracy for use in equations,, and 5, when substituted into equation but not for use in omputing the geometric delay. The total gravitational delay is the sum over all gravitating bodies including the Earth, L^tgrav /. ^tgravj V / J Geometrie Delay In the barycentric frame the vaeuum delay equation is, to a sufficient level of approximation: T _ Tl = AK. (X (T ) - X X (T X )) + At grav. (8) This equation is converted into a geocentric delay equation using known quantities by performing the relativistic transformations relating the barycentric vectors Xi to the corresponding geocentric vectors ä?,*, thus onverting equation 8 into an equation in terms of if t. The related transformation between barycentric and geocentric time can be used to derive another equation relating T T\ and t h, and these two equations can then be solved for the geocentric delay in terms of the geocentric baseline vector 6. The papers by Soffel et al. in Eubanks (99), Hellings and Shahid-Saless in Eubanks (99), Zhu and Groten (988) and Shahid-Saless et al. (99) give details of the derivation of the vaeuum delay equation. To conserve accuracy and simplify the equations the delay was expressed as much as is possible in terms of a rational polynomial. In the rational polynomial form the total geocentric vaeuum delay is given by,, _ M grav - &&[! - ( + 7)t/ - % - %fr] ~ %^d + K V 9 /c) 9

95 Given this expression for the vaeuum delay, the total delay is found to be K ($ w\) t- - h = t V - t Vl + (t at m - Otatrm ) + 6t atmi. () For convenience the total delay can be divided into separate geometric and propagation delays. The geometric delay is given by K - (Ü) W\) tg lg = ty I^j "T ^^atltll f \ ) and the total delay can be found at some later time by adding the propagation delay: h h = tg t 9l + (6t atm 6t atmi ). () The tropospheric propagation delay in equations and need not be from the same model. The estimate in equation should be as accurate as possible, while the 6t at m model in equation need only be accurate to about an air mass (~ nanoseconds), If equation is used instead, the model should be as accurate as is possible. If the difference, 6b, between the a priori baseline vector &o used in equation 9 and the true baseline vector is less than roughly three meters, then it suffices to add (Ä' 6b)je to f - t\. If this is not the case, however, the delay must be modified by adding to the total time delay / t\ from equation or. Observations lose to the Sun For observations made very close to the Sun, higher order relativistic time delay effects become increasingly important. The largest correction is due to the change in delay caused by the bending of the ray path by the gravitating body described in Richter and Matzner (98) and Hellings (986). The change to At grav is _(l + 7) G' M t? 9raV '~ S {\ä\ h +R h.k)*' which should be added to the At grav in equation. Summarv c b.(n lt +Ii) Assuming that time t\ is the Atomic (TAI) time of reeeption of the VLBI signal at reeeiver, the following steps are recommended to correct the VLBI time delay for relativistic effects.. Use equation 6 to estimate the barycentric Station vector for reeeiver.. Use equations,, and 5 to estimate the vectors from the Sun, the Moon, and each planet except the Earth to reeeiver. 9 (H)

96 . Use equation to estimate the differential gravitational delay for each of those bodies.. Use equation to find the differential gravitational delay due to the Earth. 5. Sum to find the total differential gravitational delay. 6. Add At grav to the rest of the a priori vaeuum delay from equation alculate the aberrated source vector for use in the calculation of the tropospheric propagation delay: j i = K + ^ ± ^. - K k c ' ( ^ c + i8i \ (5) 8. Add the geometric part of the tropospheric propagation delay to the vaeuum delay, equation. 9. The total delay can be found by adding the best estimate of the tropospheric propagation delay h - <i = t 9 - t 9l + [6t atm (h - -± ^,k ) - 6t atmi (k )}. (6). If necessary, apply equation to correct for "post-model" changes in the baseline by adding equation to the total time delay from equation step 9. Propagation orrection for Laser Ranging The space-time curvature near a massive body requires a correction to the Euclidean computation of ränge, p. This correction in seconds, At, is given by (Holdridge, 967) A ( = < i ± ^, ( l ± Ä ± ), (7) 6 \R\ + Ä - PJ where c = speed of light, 7 = PPN parameter equal to in General Relativity, R\ = distance from the body's center to the beginning of the light path, Ä = distance from the body's center to the end of the light path, GM = gravitational parameter of the deflecting body. For near-earth satellites, working in the geocentric frame of reference, the only body to be considered is the Earth (Ries et al., 989). For lunar laser ranging, which is formulated in the solar system barycentric reference frame, the Sun and the Earth must be considered (Newhall et al., 987). In the computation of the instantaneous space-fixed positions of a Station and a lunar reflector in the analysis of LLR data, the body-centered coordinates of the two sites are affected by a scale reduetion and a Lorentz contraction effect (Martin et al., 985). The scale effect is about 5 cm in the height of a tracking Station, while the maximum value of the Lorentz effect is about cm. 9

97 The equation for the transformation of f, the geocentric position vector of a Station expressed in the geocentric frame, is * = < ' - H 6 - y a 8 ) where ffj = Station position expressed in the barycentric frame, $ = gravitational potential at the geocenter (excluding the Earth's mass), V = barycentric velocity of the Earth, A similar equation applies to the selenocentric reflector coordinates; the maximum value of the Lorentz effect is about cm (Newhall et al., 987). References Boucher,, 986, "Relativistic effects in Geodynamics" in Relativity in elestial Mechanics and Astrometry, J. Kovalevsky and V. A. Brumberg (eds), pp. -5. annon, W. H., Lisewski, D., Finkelstein, A. M., Kareinovich, and V. Ya, 986, "Relativistic Effects in Earth Based and osmic Long Baseline Interferometry," ibid., pp Eubanks, T. M., ed, 99, Proceedings of the U. S. Naval Observatory Workshop on Relativistic Models for Use in Space Geodesy, U. S. Naval Observatory, Washington, D.. Finkelstein, A. M., Kreinovitch, V. J., and Pandey, S. N., 98, "Relativistic Reductions for Radiointerferometric Observables," Astrophys. Space Sei., 9, pp. -7. Hellings, R. W., 986, "Relativistic effects in Astronomical Timing Measurements," Astron. J., 9, pp Erratum, ibid., p. 6. Holdridge, D. B., 967, An Alternate Expression for Light Time Using General Relativity, JPL Space Program Summary 7-8, III, pp. -. Martin,. F., Torrence, M. H., and Misner,. W., 985, "Relativistic Effects on an Earth Orbiting Satellite in the Barycentric oordinate System," J. Geophys. Res., 9, p. 9. Mcarthy, D. D., 99, IERS Standards, IERS Technical Note, Observatoire de Paris, Paris. Newhall, X X, Williams, J. G., and Dickey, J.., 987, "Relativity Modelling in Lunar Laser Ranging Data Analysis," in Proceedings of the International Association of Geodesy (IAG) Symposia, Vancouver, pp Pavlov, B. N., 985, "On the Relativistic Theory of Astrometric Observations. III. Radio Interferometry of Remote Sources," Soviet Astronomy, 9, pp Richter, G. W. and Matzner, R. A., 98, "Second-order ontributions to Relativistic Time Delay in the Parameterized Post-Newtonian Formalism," Phys. Rev. D, 8, pp

98 Ries, J., Huang,, and Watkins, M. M., 988, "The Effect of General Relativity on Near-Earth Satellites in the Solar System Barycentric and Geocentric Reference Frames," Phys. Rev. Let., 6, pp Robertson, D. S., 975, "Geodetic and Astrometric Measurements with Very-Long-Baseline Interferometry," Ph. D. Thesis, Massachusetts Institute of Technology. Shahid-Saless, B., Hellings, R. W., and Ashby, N., 99, "A Picosecond Accuracy Relativistic VLBI Model via Fermi Normal oordinates," Geophys. Res. Let., 8, pp. 9-. Soffel, M., Ruder H., Schneider, M., ampbell, J., and Schuh, H., 986, "Relativistic Effects in Geodetic VLBI Measurements," in Relativity in elestial Mechanics and Astrometry, J. Kovalevsky and V. A. Brumberg (eds), pp Soffel M. H., Muller, J., Wu, X., and Xu,, 99, "onsistent Relativistic VLBI Theory with Picosecond Accuracy," Astron. J.,, pp. 6-. Sovers,. J. and Fanselow, J. L., 987, Observation Model and Parameter Partiais for the JPL VLBI Parameter Estimation Software 'Masterfit'-987, JPL Pub. 8-89, Rev.. Zeller, G., Soffel, M., Ruder, H., and Schneider, M., 986, Astr. Geod. Arbeiten, Heft 8, Bay. Akad. d. Wiss., pp Zhu, S. Y. and Groten, E., 988, "Relativistic Effects in the VLBI Time Delay Measurement,"Man. Geod.,, pp

99 GLOSSARY AAM BIH BIPM EP ERGA IR IO ODE SR DORIS DUT EMWF EMR EOP ESO GFZ GMST GPS IAG IAU IERS IRF IGS ITRF IRP IR.M JPL LLR M NEOS NOAA NRan SLR SI SIO TAI TT UKMO USNO UT UTXMO VLBI Atmospheric Angular Momentum Bureau International de l'heure Bureau International des Poids et Mesures elestial Ephemeris Pole entre d'etudes et de Recherches Geodynamiques et Astronomiques International Radio onsultative ommittee onventional International Origin enter for Orbit Determination in Europe enter for Space Research, University of Texas Doppler Orbit determination and Radiopositioning Integrate on Satellite Delft University of Technology European entre for Medium-range Weather Forecasting See NRan Earth Orientation Parameters European Space Operations enter Geo Forschungszentrum Greenwich Mean Sidereal Time Global Positioning System International Association of Geodesy International Astronomical Union International Earth Rotation Service IERS elestial Reference Frame International GPS Service for geodynamics IERS Terrestrial Reference Frame IERS Reference Pole IERS Reference Meridian Jet Propulsion Laboratory Lunar Laser Ranging Russian Mission ontrol National Earth Orientation Service National Oceanic and Atmospheric Administration Natural Resources anada, formerly EMR Satellite Laser Ranging Systeme International Scripps Institution of Oceanography Temps Atomique International Terrestrial Time U.K. Meteorological Office United States Naval Observatory oordinated Universal Time Dept. of Astronomy. The University of Texas at Austin. Very Long Baseline Interferometry 95

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101 Imprime en France par INSTAPRINT S.A. --, levee de la Loire -LA RIHE - B.P TOURS edex Tel Depot legal * trimestre 996