CHAPTER 3 CONVENTIONAL TERRESTRIAL REFERENCE FRAME

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1 HAPTER 3 ONVENTIONAL TERRESTRIAL REFERENE FRAME Definition The Terrestrial Referene System adopted for either the analysis of individual data sets by tehniques (VLBI, SLR, LLR, GPS...) or the ombination of individual Solutions into a unified set of data (Station oordinates, Earth orientation parameters, et..) follows these riteria (Bouher, 1991): a) It is geoentri, the enter of mass being defined for the whole Earth, inluding oeans and atmosphere. b) Its sale is that of a loal Earth frame, in the meaning of a relativisti theory of gravitation. ) Its orientation is given by the BIH orientation at d) Its time evolution in orientation will reate no residual global rotation with regards to the rust. Realization When one wants to realize suh a onventional terrestrial referene system through a referene frame i.e. a network of Station oordinates, it will be speified by artesian equatorial oordinates X, Y, and Z, by preferene. If geographial oordinates are needed, the GRS80 ellipsoid is reommended (see table 3.2). Eah analysis enter ompares its referene frame to a realization of the onventional Terrestrial Referene System (TRS), as desribed above. Within IERS, eah Terrestrial Referene System (TRF) is either diretly, or after transformation, expressed as a realization of the TRS adopted by IERS as its ITRS. The position of a point loated on the surfae of the solid Earth should be expressed by X(t) = X 0 + V Q (t-t 0 ) + AX i (ü), 2 where AXj are orretions to various time hanging effets, and X 0 and V 0 are position and veloity at the epoh t 0. The orretions to be onsidered are solid Earth tide displaement (füll orretion inluding permanent effet, so that the extra orretion whih was originally reommended in order to have zero mean orretion is no longer valid), oean loading, and atmospheri loading. 16

2 Further orretions ould be added if they are at mm level and an be omputed by a suitable model. The veloity V 0 should be expressed as where V put is the horizontal veloity omputed from the NNR-NUVEL-1 model (DeMets, et al., 1990; Argus and Gordon, 1991) and V r a residual veloity. In data analysis, X 0 and V r should be onsidered as solve-for Parameters. In partiular, if a non linear hange ours (earthquake, volani event... ), a new X 0 parameter should be adopted. When adjusting parameters, partiularly veloities, the IERS orientation should be kept at all epohs, ensuring the alignment at a referene epoh and the time evolution through a no net rotation ondition. The way followed by various analysis enters depends on their own view of modelling, and on the tehniques themselves. For the origin, only data whih an be modelled by dynamial tehniques (urrently SLR, LLR or GPS for IERS) an determine the enter of mass. The VLBI system an be referred to a geoentri system by adopting for a Station its geoentri position at a referene epoh as provided from external Information. The sale is obtained by appropriate relativisti modelling. This is partiularly true for VLBI and LLR whih are usually modelled in a baryentri frame. A more detailed treatment an be found in hapter 14. The orientation is defined by adopting IERS (or BIH) Earth orientation parameters at a referene epoh. In the ase of SLR, an additional onstraint in longitude is neessary. The unit of length is the meter (SI). The IERS Referene Pole (IRP) and Referene Meridian (IRM) are onsistent with the orresponding diretions in the BIH Terrestrial System (BTS) within ± The BIH referene pole was adjusted to the onventional International Origin (IO) in 1967; it was then kept stable independently until The unertainty of the tie of the IRP with the IO is ± The time evolution of the orientation will be insured by using a no-net-rotation ondition with regards to horizontal tetoni motions over the whole Earth. Transformation Parameters of World oordinate Systems and Datums The seven-parameter (similarity transformation between any two artesian Systems, e. g., from (u,v,w) to (x,y,z), or in short (u,v,w) -* (x,y,z) an be written (Soler and Hothem, 1989) as 17

3 X y z AX I Ay > + (1 + <Ss) AZ 1 5w -50 -?o> e 1 (1) where Ax,Ay,Az = oordinates of the origin of the frame (u,v,w) in the frame (x,y,z); Se, 50, So> = differential rotations (expressed in radians) respetively, around the axes (u,v,w) to establish parallelism with the (x,y,z) frame (Positive rotations are ounterlokwise rotations as viewed looking tov/ard the origin of the right handed oordinate system.); and 5 s = differential sale hange (expressed in ppm X 10" 6 ) (see Table 3.1). Table 3.1. Transformation loal datums. parameters from ITRF90 to major global or oordinate System (datum) Ax Tl (m) Ay T2 (m) Az T3 (m) 6s D (ppm) 5 -Rl (") 60 -R2 ) 6o) -R3 WGS WGS NWL9D BTS BTS BTS BTS ITRFO ITRF ITRF One the artesian oordinates (x,y,z) are known, they an be transformed to "datum 11 or urvilinear geodeti oordinates (A,0,h) referred to an ellipsoid of semi-major axis a and flattening f (see Table 3.2), using the following noniterative method (Bowring, 1985): tan X (2) 18

4 tan <p = z(l - f) + e 2 a sin'u (1 - f) (p - e^a OS J /LI) h = p os 0 + z sin <p - a(l - e 7 sin 2 <p) v ', e 2 = 2f - f 2, p = (x 2 + y 2 )'\ r = (p 2 + z 2 )*, tan u = - P (1 f) + e 2 a (3) (4) (5) (6) (7) (8) The preeding equations an be used in onjuntion with Tables 3.1 and 3.2 superseding the ones previously presented in Soler and Hothera (1988). Table 3.2. Parameters of Some Adopted Referene Ellipsoids. oordinate (datum) system AGD ED-79 GEM-8 GEM-9 (or GEM-10) GEM-10B GEM-Tl GEM-T3 NAD-27 NAD-83 NWL-9D = NSW-9Z2 SA-69 WGS-72 WGS-84 Referene ellipsoid used AN (or SA-69) International GEM-8 GEM-9 or (GEM-10) GEM-10B GEM-Tl GEM-T3 larke 1866 GRS-80 WGS-66 SA-69 (or AN) WGS-72 WGS-84 a (m) 6,378,160 6,378,388 6,378,145 6,378,140 6,378,138 6,378,137 6,378,137 6,378, ,378,137 6,378,145 6,378,160 6,378,135 6,378,137 1/f Note: AGD = Australian geodeti datum; AN = Australian national; ED = European datum; GEM = Goddard Earth model; GRS = geodeti referene system; NAD = North Amerian datum; NSW = Naval surfae warfare enter; NWL = Naval Weapons Laboratory; SA = South Amerian; WGS = World geodeti system. In the ase of IERS, these algorithms are used with Table 3.1, where (x,y,z) are oordinates in the speified oordinate system (datum), and (u,v,w,) are identified with ITRF, or a more speifi realization (ITRF90 for table 3.1). In the urrent pratie of the IERS/B and its publiations, (u,v,w) are denoted (X,Y,Z) and the oordinates in an individual system by (XS, YS, ZS), while the 7 parameters are identified as (T,, T 2, T 3 ) refer to (Ax, Ay, Az), D to 5s and (Ri,R 2,R 3 ) to (-S, -5*, -5 b). 19

5 Transformations to urrent Datums Table 3.1 gives values reommended to onvert ITRF90 oordinates into other datums. Table 3.2 Lists the required two parameters for several adopted referene ellipsoids defining important geodeti datums. Some ellipsoids were introdued by NASA's GSF to referene their Goddard Earth Models (GEM, series 8,9, and 10). They were primarily used to obtain geoid heights (undulations) and depit global geoid maps. These transformation formulae should be used with are. There are several ways to determine suh parameters, either by diret omparison between two realizations of the datums, or by ombining formula through an intermediate datum. This is well illustrated by WGS84 as mentioned below. The numbers given in Table 3.1 have been determined using the following data: a) The transformation between BTS87 and WGS84 derived from Bouher, Altamimi, and Willis (1988). b) Transformation parameters between the Doppler realized frames (i.e., NWL-9D = NSW-9Z2, WGS-72, WGS-84) have been adopted by the Defense Mapping Ageny. Reall that artesian oordinates, derived from using the Global Positioning System (GPS), are also referred to the WGS-84 oordinate system. Nevertheless, the Doppler WGS-84 (GPS) frames are not neessarily oinident (Malys, 1988). Similarly, although by definition, the NAD-83 and WGS-84 realized artesian frames should oinide, small differenes (<0.5 ppm) in shifts, rotations, and sale between the two frames may be disovered. These differenes merely reflet small, random regional distortions still present in the NAD-83 horizontal datum, whih was primarily established by simultaneously adjusting all arhived lassial-geodeti observations(bossler, 1987; Shwarz, 1989). ) Transformations to old BIH or IERS Systems are derived from IERS TN4, 6, and 9. Transformations to major loal datums an be obtained WGS84. through Plate Motion Model One of the fators whih an affet Earth rotation results is the motion of the tetoni plates whih makes up the Earth's surfae. As the plates move, fixed oordinates for the observing stations will beome inonsistent with eah other. The rates of 20

6 relative motions for some regulär observing sites are believed to be 5 m per year or larger. The observations of plate motions so far by Satellite Laser Ranging and Very Long Baseline Interferometry appear to be roughly onsistent with the average rates over the last few million years derived from the geologial reord and other geophysial Information. Thus, in order to redue inonsistenies in the Station oordinates and to make the results from different tehniques more diretly omparable, a model for plate motions given by DeMets, et al. (1990) is reommended. The artesian rotation vetor for eah of the major plates is given in Table 3.3 A subroutine alled ABSMO_NUVEL, provided by J. B. Minster, is also inluded below. It omputes the new site position at time t from the old site position at time to using the reommended plate motion model. Table 3.3. artesian rotation vetor for eah plate using the NUVEL NNR-1 kinemati plate model (no net rotation) Plate Name fix deq/mv. n Y deq/mv. "z deq/mv. Paifi oos Naza aribbean South Ameria Antartia India Australia Afria Arabia Eurasia North Ameria Juan de Fua Philippine The NUVEL model should be used as a default, for stations whih appear to follow reasonably its values. For some stations, partiularly in the viinity of plate boundaries, users may benefit by estimating veloities or using speifi values not derived from NUVEL. This is also a way to take into aount now some nonnegligible vertial motions. Published Station oordinates should inlude the epoh assoiated with the oordinates. The original subroutine is a oding of the AM0-2 model from J. B. Minster. This was made by modifying the earlier subroutine. The hanges were made by Don Argus and verified by Alie Gripp. 21

7 SUBROUTINE ABSMO_NUVEL(PSIT,TO,XO,YO,ZO,T,X,Y,Z) ABSMO_NUVEL take a site speified by its initial oordinates XO,YO,ZO at time TO, and omputes its updated positions X,Y,Z at time T, based on the geologial "absolute", (no net rotation) plate motion model AM0-2 (Minster and Jordan, 1978). Original author: J.B. Minster, Siene Horizons. DFA: Revised by Don Argus, Northwestern University DFA: uses absolute model NNR-NUVEL1 Transribed from USNO irular 167 "Projet Merit Standards" by Tony Mallama with slight modifiation to the doumentation and ode. Times are given in years, e.g for Jan 1, PSIT is the four harater abbreviation for the plate name, if PSIT is not reognized then the new positions are returned as zero. IMPLIIT NONE HARATER*4 REAL*8 REAL*8 REAL*8 REAL*8 INTEGER*2 DFA: DATA NNR-NUVEL1 (PNM(I) 1 = 1 /'PF 'AFR 'ANTA' 'ARAB' 'AUST' 'ARB' 'OO' 'EURA' 'INDI' 'NAZ 'NOAM' 'SOAM' 'JUFU' 'PHIL' Initialize things 14) PSIT,PNM(14) OMX(14),OMY(14),OMZ(14) X0,Y0,Z0 X,Y,Z,T,T0 ORX,ORY,ORZ IPSIT,I OMX(I), , , , , , , , , , , , , , , properly OMY(I), , , , , , 2027, 2944, 1434, 0026, 5138, 2155, 0906, 4805, 4412, OMZ(I), , , , , , 0945, 6544, 1887, 4066, , , , , / 22

8 IPSIT = -1 X = 0.0D0 Y = O.ODO Z = O.ODO Look up the plate in the list. DO 20 I = 1,14 20 IF (PSIT.EQ. PNM(I)) IPSIT = I If plate name is not reognized return the new plate position as zero. IF (IPSIT.EQ. -1) RETURN onvert from degree/my to radians/yr. ORX = OMX(IPSIT) * D-08 ORY = OMY(IPSIT) * D-08 ORZ = OMZ(IPSIT) * D-08 ompute the new oordinates X = X0 + (ORY*Z0 - ORZ*Y0) * (T-TO) Y = Y0 + (ORZ*X0 - ORX*Z0) * (T-TO) Z = ZO + (ORX*Y0 - ORY*X0) * (T-TO) Finish up RETURN END Referenes Argus, D. F., and Gordon, R. G., 1991, fl No-Net-Rotation Model of urrent Plate Veloities Inorporating Plate Motion Model NUVEL-1," Geophys. Res. Let., 18, pp Bossler, J. D., 1987, "Geodesy solves 900,000 equations simultaneously," Eos, Trans. AGU, 68(23), p Bouher,, Altamimi, Z., and Willis, P., 1988, "Relation between BTS87, WGS84 and GPS ativities," Annual Report for 1987, Bureau International de l'heure, Paris, Frane, pp. D D Bouher,., 1990, "Definition and Realization of Terrestrial Referene Systems for Monitoring Earth Rotation," Variations in Earth Rotation, D. D. Marthy and W. E. arter (eds), pp

9 Bouher,., and Altamimi, Z., 1991, ITRF 89 and other realizations of the IERS Terrestrial Referene System for 1989, IERS Tehnial Note 6, Observatoire de Paris, Paris. Bowring, B. R., 1985, "The auray of geodeti latitude and height equations," Survey Review, 28, pp DeMets,., Gordon, R. G., Argus, D. F., and Stein, S., 1990, "urrent Plate Motions," Geophys. J. Int., 101, pp Department of Defense World Geodeti System 1984 Its definition and relationships with loal geodeti Systems," 1987, DMA Tehnial Report , Defense Mapping Ageny, Washington, D.. Kaula W. M., 1975, "Absolute Plate Motions by Boundary Veloity Minimizations," J. Geophys. Res., 80, pp Malys, S., 1988, "Similarity transformation between NAVSAT and GPS referene frames," AGU hapman onf. on GPS measurements for Geodynamis, Amerian Geophysial Union, Ft. Lauderdale, Fla., Sept. Minster, J. B., and Jordan, T. H., 1978, "Present-Day Plate Motions," J. Geophys. Res., 83, pp Soler, T. and Hothem, L. D., 1988, "oordinate Systems used in geodesy: basi definitions and onepts," J. Surv. Engrg., ASE, 114, pp Soler, T. and Hothem, L. D., 1989, "Important Parameters Used in Geodeti Transformations," J. Surv. Engrg., 115, pp Tushingham, A. M. and Peltier, W. R., 1991, "Ie-3G: A New Global Model of Late Pleistoene Deglaiation Based Upon Geophysial Preditions of Post-Glaial Relative Sea Level hange," J. Geophys. Res., 96, pp

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