Discussion on How to Express a Regional GPS Solution in the ITRF

Transcription

1 162 Dscusson on How to Express a Regonal GPS Soluton n the ITRF Z. ALTAMIMI 1 Abstract The usefulness of the densfcaton of the Internatonal Terrestral Reference Frame (ITRF) s to facltate ts access as a global frame by users nterested on small (regonal or local) networks. Although a regonal GPS soluton of staton postons s usually derved usng IGS products (orbts, clocks,...), whch are nomnally expressed n the ITRF, ts correspondng datum defnton could be far from that of ITRF due to manly the network confguraton. The man queston to be answered here s how to optmally express staton postons of a regonal network n the global frame of ITRF? Ths could be acheved by manly (1) constranng coordnates of a subset of statons to ther ITRF values or (2) algnng the regonal soluton to ITRF usng a transformaton formula. Ths paper wll focus on the second method, based on mnmum constrants approach, yeldng an optmal datum defnton together wth preservng the orgnal characterstc of the regonal soluton. 1. Introducton The ITRF s a result of combnaton of global terrestral reference frames (statons postons and veloctes) provded by 5 space geodesy technques: Very Long Baselne Interferometry (VLBI), Lunar and Satellte Laser Rangng (LLR and SLR), Global Postonng System (GPS) and Doppler Orbtography Rado-postonng Integrated by Satellte (DORIS). From the geodetc pont of vew, densfcaton of the ITRF s meant the expresson of staton postons (and veloctes) of a regonal or local network n the ITRF. The GPS, compared to the other technques, has the advantage of beng the most effcent one for the ITRF densfcaton purpose, gven ts ease use, low cost and the avalablty of the IGS products for all users. A densfcaton part has been acheved recently n the latest ITRF verson, namely the ITRF2000, by ncludng n the global combnaton some regonal GPS solutons LTAMIMI (A et al., 2002a). In terms of Terrestral Reference Frame (TRF) defnton, all the IGS products are expressed n the ITRF: ITRF91 from the begnnng of IGS actvtes untl the end of 1993; ITRF92 durng 1994; ITRF93 durng 1995 untl md-1996; ITRF94 snce md-1996 untl the end of Aprl 1998; ITRF96 startng on March 1, 1998; ITRF97 startng on August 1, 1999 and ITRF2000 snce December 20, Startng wth ITRF96, the expresson of IGS products n the ITRF s ensured by algnng the global IGS TRF combned soluton of staton postons and veloctes to ITRF. Ths algnment s performed usng 14 transformaton parameters between the IGS TRF and the ITRF, estmated over about 50 statons globally dstrbuted (FERLAND et al., 2001). Staton coordnates of a regonal network estmated usng IGS products are theoretcally expressed n the ITRF. Whle ths statement could be vald for the underlyng TRF orentaton (through the orbt fxng), the TRF orgn and scale are generally far from those of the ITRF. Consequently, for varous Earth Scence applcatons, GPS solutons of staton postons of a regonal or local network need to be optmally expressed n the same frame as (and to be consstent wth) the global ITRF. In the followng we dscuss some techncal ssues related to the varous geodetc methods allowng to ntegrate regonal network nto the ITRF. Dsregardng the selected method, and to acheve optmal estmate, t s recommended to ensure the lnk between the ITRF and the regonal solutons through a selecton of ITRF statons of hgh qualty. Among the crtera selecton, t s advsed to select statons havng: an optmal dstrbuton over the regonal network. In case that none of the network statons s already avalable n the ITRF, a certan number of ITRF statons of hgh qualty, surroundng the mpled network, should be ncluded n the GPS processng a long observng hstory (at least 3 years) the ITRF resduals should be less than 5 (eventually 10) mm for postons and 3 mm/y for veloctes for at least 3 dfferent solutons contrbuted to ITRF generaton. The are manly two major methods allowng the expresson of the regonal network soluton n the ITRF: 1. constranng the coordnates of the selected ITRF subset of statons to ITRF values at the central epoch of the mpled observatons used to generate the regonal soluton. The constrants should be easly removable (F m). Ths s for example the current procedure appled by EUREF for ther weekly soluton, where about 12 ITRF statons are constraned to ITRF2000 values. 2. algnng the regonal soluton to the ITRF usng transformaton parameters whch should be estmated usng the selected subset of ITRF statons. In ether case, the followng ponts should be observed: the selected subset of statons should be under constant survellance to detect/dentfy possble dscrepancy between ITRF and the regonal soluton. If sgnfcant dscrepancy occurs (whch s sometmes the case when some staton equpment changes), the correspondng staton should be excluded from the constrant/algnment process. the advantage of method (1) s that the regonal soluton s well expressed n the ITRF frame, whle ts dsadvan- 1 Zuher Altamm, Géographque Natonal (IGN), Ecole Natonale des Scences Géographques, 6 et 8 avenue Blase Pascal, F Marne la Vallée Cedex, France, e-mal altamm@ensg.gn.fr

2 Z. Altamm: Dscusson on How to Express a Regonal GPS Soluton n the ITRF 163 tage s that the selected statons wll have ther coordnates entrely determned by the ITRF selected values. That s why t s recommended to apply removable constrants for possbly later applcatons of unconstraned solutons by some users. gven the nature of a regonal network and ts effect on the estmaton of the transformaton parameters, ts extremely mportant to be very careful when usng method (2). In fact, the most effcent way to use s the transformaton parameter algnment usng mnmum constrants approach as detaled hereafter. 2. Algnng a Regonal Soluton to ITRF Usng Mnmum Constrants Approach In the followng we propose a method allowng to effcently express a regonal soluton of staton postons n the ITRF. Ths method, based on the equatons of mnmum constrants, could of course be appled to any knd of network not only for postons, but also for veloctes, for more detals, see for nstance (ALTAMIMI et al, 2002b). The relaton between a regonal soluton (X R ) and ITRF (X I ), over selected statons, could be wrtten as: X I = X R + A 1 (1) where A and 1 are respectvely the desgn matrx of partal dervatves and the vector of 7 transformaton parameters: xa 0 za ya A = ya za 0 x a za ya xa and 1 = (T x, T y, T z, D, R x, R y, R z ) T Un-weghted least squares adjustment yelds a soluton for 1 as: 647B 48 θ = T 1 T ( AA) A ( XI XR) (2) The approach of mnmum constrants conssts n usng the matrx B = (A T A) -1 A T n such a way that X R wll be expressed n the same frame as the ITRF soluton X I. Therefore to have X R be expressed n the ITRF at a certan E 1 level, a "datum defnton" equaton could be wrtten as: B (X I - X R ) = 0 (E 1 ) (3) where E 1 s the varance matrx at whch equaton (3) s satsfed. E 1 s a dagonal matrx contanng small varances (to be selected at the user level) for each one of the 7 transformaton parameters. It s suggested to use 1 mm for translaton parameters and an equvalent amount for the scale and orentaton parameters. In terms of normal equaton, we then can wrte: B T E 1-1 B(X I - X R ) = 0 (4) Usng IGS products (orbts, clocks, etc.), the ntal normal equaton system of a regonal GPS soluton before addng any knd of constrants could be wrtten as: N unc (DX) = K (5) where DX = X - X apr, wth X beng the unknown vector, X apr s the vector of a pror values, N unc s the unconstraned normal matrx and K s the rghthand sde vector. The normal equaton system (\refeq-n) s nvertble, but the underlyng TRF could be far from that of ITRF,.e. defned at the level of the orbt precson (a few cm). The same normal equaton system could be obtaned also after removng classcal constrants appled to a gven regonal soluton. Selectng a subset of ITRF statons (X I ), the equaton of mnmum constrants (or datum defnton) s: B T E 1-1 B(DX) = B T E 1-1 B (X I - X apr ) (6) Note that the rghthand sde of equaton (6) vanshes f the a pror values are those of ITRF selected soluton. Cumulatng (5) and (6) yelds: (N unc + B T -1 E 1 B) (DX) = K + B T E - 1 B (X I - X apr ) (7) The mnmally constraned soluton, expressed n the ITRF upon the selected statons s then: X = (N unc + B T -1 E 1 B) -1 (K + B T -1 E 1 B (X I - X apr )) + X apr (8) 3. Numercal Applcatons For the purpose of numercal applcatons of the method proposed above, we selected, as an example, the EUREF combned soluton for GPS week In ths soluton, the coordnates of 12 ITRF statons (llustrated n Fgure 1) were constraned to ITRF2000 values. After removng the constrants, 7 transformaton parameters were frst estmated between the unconstraned EUREF soluton and the ITRF2000, upon the subset of 12 statons. The adjusted values of these 7 parameters are lsted n Table 1, dstngushng the weghted and un-weghted estmatons.

3 164 EUREF Permanent Network - Developments and applcatons Fgure 1. EUREF network underlyng the 12 statons whose coordnates are constraned to ITRF2000 values n the combned soluton for GPS week Table 1. Transformaton parameters from ITRF2000 to the Unconstraned EUREF Soluton for GPS Week T1 T2 T3 D R1 R2 R3 cm cm cm " 0.001" 0.001" Un-weghted L.S. adjustment ±.44 ± 0.83 ±.40 ±.061 ±.244 ±.148 ±.179 Weghted L.S. adjustment ±.66 ±1.08 ±.42 ±.045 ±.306 ±.236 ±.201 As seen n Table 1, the two sets of the 7 parameters are not the same (although they would be equvalent, see below) snce these parameters are correlated due to the network geometry. The unconstraned soluton was then transformed usng the two sets of transformaton parameters. The coordnate dfferences between the two unconstraned \& transformed solutons and the orgnally constraned one are llustrated n Fgure 2. As shown n ths fgure, the vertcal resduals do not have zero mean, reflectng the network effect on the scale factor of ths example of network. The algnment to ITRF2000 was then appled to the EUREF unconstraned soluton usng the mnmum constrants approach dscussed above, upon the 12 selected statons. The coordnates dfferences between the mnmally constraned and the constraned solutons are llustrated n Fgure 3. Comparng resduals of ths fgure wth those of fgure 2, demonstrate that the proposed method of mnmum constrant algnment s more effcent than the classcal one.

4 Z. Altamm: Dscusson on How to Express a Regonal GPS Soluton n the ITRF 165 Un-weghted Transformaton Weghted Transformaton Fgure 2. Coordnate dfferences (mm) between the (unconstraned \& transformed) and the orgnally constraned EUREF soluton. Fgure 3. Coordnate dfferences (mm) between the mnmally constraned and the orgnally constraned EUREF soluton.

5 166 EUREF Permanent Network - Developments and applcatons In order to nvestgate whether t s equvalent to use staton poston values drectly from ITRF2000 or from the IGS realzaton of ITRF2000, fgure 4 plots the 12 coordnate dfferences between respectvely, IGS weekly and cumulatve solutons, IGS weekly and ITRF2000 and between IGS cumulatve and ITRF2000. Ths fgure s a "perfect" llustraton of the network effect on the datum defnton. Whle the IGS weekly and cumulatve solutons are algned to the ITRF2000 over 54 statons globally dstrbuted and havng, by constructon, a zero resdual mean, pckng out 12 statons produces a TRF shft at the regonal level of a few mllmeters: It could be very easly observed from the 3 plots shown n Fgure 4 that the regonal resduals do not have zero mean. In order to assess ths effect on the EUREF regonal soluton, we appled the procedure of mnmum constrants over IGS weekly (week 1149) and cumulatve solutons. Fgure 5 shows the poston dfferences between solutons derved usng IGS weekly (resp. cumulatve) and ITRF2000. The TRF shft due the network effect predcted from Fgure 4 s well transferred to Fgure 5. IGS(Wkly mnus Cum.) IGS-Wkly mnus ITRF00 IGS-Cum. mnus ITRF00 Fgure staton Coordnate dfferences (mm) between IGS weekly and cumulatve, IGS weekly and ITRF2000 and between IGS cumulatve and ITRF2000.

6 Z. Altamm: Dscusson on How to Express a Regonal GPS Soluton n the ITRF 167 IGS-Wkly mnus ITRF2000 IGS-Cum. mnus ITRF2000 Fgure 5. Coordnate dfferences (mm) between the EUREF mnmally constraned solutons usng ITRF2000 and IGS weekly or cumulatve solutons. Concluson Ths paper demonstrates that mnmum constrants approach s an effcent method to optmally express a regonal soluton n a global frame such as the ITRF, mnmzng so the well known network effect. It s therefore suggested to use ths method nstead of the classcal constrants whch hde the orgnal characterstc of the regonal soluton. All staton dscrepances (whch very often occur after some staton equpment changes) between the ITRF and the regonal soluton are then dentfed and, n the same tme, the soluton tself s fully expressed n the global frame. Bblography ALTAMIMI, Z., SILLARD P., BOUCHER C.: ITRF2000: A New Release of the Internatonal Terrestral Reference Frame for Earth Scence Applcatons, jgr, Vol. 107, B10, 2002a. ALTAMIMI, Z., SILLARD P., BOUCHER C.: ITRF2000: From Theory to Implementaton, V Hotne Marus symposum, Matera, Italy, 2002b. FERLAND, R., ALTAMIMI Z., BRUYNINX C., CRAYMER M., HABRICH H., KOUBA J.: Regonal Networks Densfcaton, 2002 IGS Workshop "Towards Real-Tme" Aprl 8-11; Ottawa, Canada.