On the maintenance of a proper reference frame for VLBI and GPS global networks

Transcription

1 O he maeace of a proper referece frame for VLB a GPS global ewors Ahaasos Dermas. rouco he use of a approprae erresral referece frame orer o escrbe he poso of pos o he earh, as well as s emporal varao, s a problem of boh heorecal a praccal mporace. hs s he zero orer opmal esg problem he ermology of Grafare 974, eee from he space o he space-me oma. A erresral referece frame ss of a parcular po O, s org a a ra of orhoormal vecors e e e, a s relae o a quas-eral orhoormal frame e e e e wh org a [ e he geoceer C. A po P of he earh has poso vecor OP e + e + e e a erresral cooraes [ ; s eral poso vecor s [ e CP e + e + e a has eral cooraes [. o relae he wo frames, we ee he splaceme vecor CO e e a he compoes Q of he elemes of he erresral ra wh respec o he eral oes,.e. e e Q, or e e Q mar form. he mar Q s a proper orhogoal mar Q Q, Q + as a equece of he orhoormaly a commo oreao rghhae of e a e. he relao bewee he wo frames s epresse by or compoe form e CP CO+ OP + e + e e + e Q, + Q, Q Q. he moo of ay earh po eral space escrbe by he fuco, where eoes me, shoul be eerme by observaos. he correspog moo, wh respec o a erresral frame, epes ao o he more or less arbrary choce of he erresral frame,.e. of he fu Q a. f he earh was rg or applcaos where rgy s a val appromao here are choces of Q a, such ha he erresral cooraes Q are a. For a eformable earh, where eformaos are ow o be small, s reasoable o esablsh a erresral frame a way ha he emporal varaos of Q appear o be "as small as possble",.e., such ha he larges par of he eral moos s absorbe by he roao a poso of he erresral frame. he opmal choce of he erresral frame epes recly o he specfc opmaly crero, whch gves cocree mahemacal meag o he loose epresso "as small as possble". he soluo of he problem requres, apar from he choce of he opmaly crero, he owlege of he moo wh respec o he eral frame, of every po of he earh. Operaoally, hs s possble oly for pos o he surface of he earh, whle he moo of eral pos has o be euce from heorecal argumes. 79

2 he geeral form of a opmaly crero s F F, m, where F s a approprae ow fuco a egrao s carre ou over he earh a he me erval [, F for whch observaoal aa are avalable. A more moes problem s he maeace of a referece frame for a se of scree pos P,,,, whch are he posos of observao saos srbue all over he worl a egage a collecve aalyss of he acqure aa. Formerly, he problem of frame efo was solve a scree way, correspog o scree aa P,, collece repeae campags over shor me ervals, whch coul effcely ere as "saaeous" aa correspog o a sgle epoch. he mos popular approach sars wh a more or less arbrary frame efo a he al epoch a he fs he cooraes of each epoch o hose alreay for he prevous epoch, by applyg he opmaly crero [ [ m. 4 Seg, 5 a roucg he oao,, δ, he opmaly crero δ δ m ca be corporae a learze ajusme moel l Aδ + v, where learzao s base o he appromae values, by roucg a se of er ras δ, where he rows of form a bass for he ull space N A { A } of he mar A. Nowaays, observao saos are egage couous aa colleco, operag as permae saos wh he framewor of eraoal orgazaos, such as he eraoal GPS Servce GS a prove he meas for he esablshme of a offcal eraoal erresral eferece Sysem S wh he care of he eraoal arh oao Servce S. emar: he erm referece frame use here correspos o he erm S use by S, whch preserves he erm eraoal erresral eferece Frame F o he se of saos egage he realzao of he S. Wh he avalably of couously avalable aa he sepwse approach of he pas s o more esrable or praccal from he mplemeao po of vew. sea, we propose o vsualze he aa whch are scree bu wh a very hgh rae of repeo as couous a o see a me-couous soluo o he referece frame choce problem. Such a soluo may be eveually screze or mplemee a scree appromao. Furhermore, he opmaly crero o be rouce for he frame choce he scree po ewor, mus coce wh or a leas aemp o mae he opmaly crero rouce for he whole earh o he bass of heorecal eraos. We wll sgush bewee wo ypes of ewors, whch we call for he sae of coveece VLBa GPS-ype ewors. VLB-ype ewors where observaos are raslao-vara, he poso of he geoceer C cao be eerme a he org of he frame O mus be also rouce. hus we mus eerme boh fu Q a. GPS-ype ewors observaos are le o he geoceer hrough he use of saelle orbs, whch case O C s alreay ow a oly Q shoul be eerme. 8

3 Dermas 995 we rouce a mehoology for he soluo o he space-me aum problem, whch ere also scale rasformaos. Here we wll resrc ourselves o rg rasformaos, sce scale s prove for boh VLB- a GPS-ype ewors, wh he framewor of a orelavsc approach, hrough he assumpo ha mea of he reags of a se of referece clocs oes o accelerae or ecelerae wh respec o Newoa me. Dsace a hus scale s eerg he problem oly mplcly hrough he observao of me ervals. O he oher ha we wll geeralze he approach by loog o alerave opmaly crera a also by roucg "masses" or "weghs" m, for each sao P, whch may reflec eher a measure of he qualy of sao aa, or he egree of parcpao of he sao o he opmaly crero, relao o he par of earh masses closes o he parcular po.. rasformao from a prelmary referece frame o a opmal oe he basc ea of our approach s o mae use of he fac ha he avalable observaos ca very well eerme he shape of he ewor, bu o he aoal formao of s oreao a poso wh respec o a referece frame, whch s coae a se of ewor cooraes. A ay sgle epoch here es a fe umber of cooraes whch gve rse o he same shape for he ewor. f a are wo coorae ses whch boh correspo o he "observe" shape a epoch, here ess a rg rasformao bewee he wo, efe po-wse by b +, 6 where s a proper orhogoal mar. hs meas ha f a prelmary soluo s avalable, we ca swch o a opmal soluo, by applyg a opmzao prcple a eermg he opmal s fu [ a b [ b b b whch rasform o he cooraes sasfyg he opmaly crero. Bu such a soluo s always avalable, because a referece frame mus be rouce for he aalyss of he aa whch lea o he coorae esmaes a every epoch. he oly requreme s ha he fuco s a smooh oe,.e. couous wh couous ervaves up o a cera orer. he referece frame for may be rouce urg he ajusme of he observaos, by a se of mmal ras, whch efe a frame whou ay fluece o he shape of he ewor. he orgal ow roao Q a splaceme Q mus be combe wh he opmal relave roao a relave splaceme b + b, orer o oba he fal oes Q a Q by meas of + b Q + b Q + b Q Q Q & b. 7. Opmal soluos of mmum eergy a mmum legh geoescs A parcular opmaly crero s base o he saaeous relave o he erresral frame ec eergy of he earh v m, where v s he velocy, whch s egrae over a me erval, a he resulg oal eergy s mmze [ F F F v v m m. 8 he scree aalog for a ewor of pos P, a s assge a mass, aes he form m,,, each of whch has opmal cooraes 8

4 M m N, 9 where N M m m v v m v s he ec eergy of he ewor. he mmzao prcple 9 s fac equvale o he mmzao prcple, M δ m, F F F F M M s s m F whch s leag o a soluo whch s a geoesc curve curve of mmum legh s he ewor coorae space X where ay se of ewor cooraes belog, X. Dsace s measure by a eleme of legh s efe by s M m. hs meas ha he "sace" bewee wo ewor coorae ses a s measure by ρ, m m, where s he usual euclea sace bewee he wo posos of po P. he mmzao problem s a saar problem of he calculus of varaos. s soluo, escrbe by meas of he curvlear cooraes s u s b s s epresse as fu of arc legh s, sasfes he uler-lagrage ffereal equaos L s L L L L L,,, b s b s, s b b, s. 4 s he ervao of he eplc form of he uler-lagrage equaos has bee carre ou Dermas 995, for he specal case M, bu hey ca be easly geeralze o he prese case of varyg po masses m. Aoally he geoesc ffereal equaos, correspog o he opmaly crero have bee erve, yelg as epece ecal resuls. he resulg equaos are h C h + C Ω Ω C Ω Ω s h + C Ω [ Ω 5 s 8

5 b s b s s + Ω + C h + Ω Ω Ω h Ω s 6 7 where C s he mome of era mar of he ewor wh respec o he al referece frame C m [ m [ M m, 8 h s s relave agular momeum vecor of he ewor wh respec o he al referece frame h [ 9 m a ΩΩ s a mar efe by [, [ Ω. emar: We mae repeae use of he oao a of he properes Q Q a a a a a [ a a a a a a [ a b[ b a, [ Qa Q[ a Q, [ a [ b ba a b. We have also assume ha referece frame has bee chose a way ha, where are he cooraes of he ceer of mass of he ewor, efe by m m, m. m s always possble o swch from ay referece frame o a "ceere" oe wh, usg. Of he seve equaos 5, 6, 7, he las oe ca be solve for he facor s s s bee legh a os eog ffereao wh respec o me whch replace he frs wo, wll yel a sysem of s o-lear ffereal equaos for he s uow fu [ a b [ b b b. he resulg equaos are very complcae a hey ca be solve oly by umercal mehos. Furhermore ay soluo yels a frame efo where he "curve" s he closes bewee s e pos a F, bu o ecessarly he shores possble. o arrve a a ruly opmal soluo, whch by he way s o uque, we mus selec he opmal amog all al or bouary values whch are ecessary for obag a specfc soluo of he releva ffereal equaos. 8

6 We wll o pursue hs maer ay furher, bu we wll follow a ffere approach movae by he mehos use he heorecal suy of earh roaos. 4. ssera aes he roao of he earh s govere by he ffereal equao l h, where h v m s he agular momeum, l a m s he acg orque, v are he veloces of earh pos a a he correspog acg acceleraos. he roaoal equao whch he eral frame becomes h smply l, obas a more complcae form whe epresse wh respec o he erresral frame. Dffereao of e e Q, yels e Q Q e eq e[, where e s he saaeous roao vecor of he earh, so ha e e + [ v a smlarly h e + [ h l e l h. he compoes of h e h he erresral sysem become + [ m [ [ m + [ m C h h [ + 4 where C s he era mar a h he relave agular momeum of he earh. eplacg h C+ h he roaoal equaos h + [ h l yels he Louvlle equaos C h C [ C+ h l. 5 he choce of he erresral frame he suy of earh roao s cae by he ee o smplfy he aalycal wor volve solvg he Louvlle equaos. wo choces are uer erao Mu & MacDoal, 96, ch.., p. : he prcpal aes or fgure aes, efe so ha C becomes agoal, a he ssera aes for whch he relave agular momeum vashes, h. he frs choce s more approprae for he heory of roao of a rg earh bu has a serous shorcomg whe a elasc earh moel s use: as a equece of roaoal elasc eformao he hr polar as of fgure ersecg he earh a a po F, uergoes a ural roao arou he correspog poso F of he rg earh moel wh a raus of F F 6 m, whle F uergoes a roao arou he poso O of hr ssera as, wh raus of oly OF m a a Chaler pero of abou 4 ays Morz a Mueller, 987, ch.... For hs reaso he ssera aes are he preferre oes for he escrpo of he roao of he eformable earh. Furhermore he ssera aes have he avaage ha hey mmze he relave o he erresral frame ec eergy of he earh,.e., v m m Morz a Mueller, 987, ch... Boh choces of fgure a ssera aes, cao eerme a splaceme bu oly he roao from a al arbrary referece frame. heory hey are boh ere o be geocerc. he fgure aes are uquely efe for ay boy ha has o as of symmery. hey are herefore well efe for he real earh, bu o for a ellpsoal moel-earh where oly oe reco ha of symmery coces wh oe fgure aes a he poso of he oher wo mus be arbrarly chose. O he corary he ssera aes are o uquely efe. ee f are cooraes wh respec o a se of ssera aes a we er a ew se of aes efe by he rasformao ~ S, where S s a me-epee orhogoal mar he ~ h v S Sv S S Sv S [ ~ ~ m [ m [ m [ v m Sh S 6 84

7 a he ~ aes are also ssera aes. o choose a parcular se of ssera aes we mus f her poso a a al epoch. For a scree ewor of mass pos we may efe a se of "ssera" aes by seg he correspog relave momeum equal o zero h m[ 7 a ry o f he rasformao parameers, b whch cover cooraes a orgally avalable referece frame o "ssera" cooraes + b. Seg [, [ 8 we have + + [ + [ + [ 9 + b + [ + [ + [ [ [ [ + [ [ [ b + b b b [ Ω b + b [ Ω + + herefore, he vashg relave momeum becomes h m[ m b [ + [ b [ Ω

8 m [ [ Ω + [ + [ b + + b m [ b [ Ω + [ b + [ b + + m m [ [ Ω [ m[ b + + m b [ b [ Ω [ b m m [ b b b C Ω + h + m m b [ [ [ Ω + m[ b + m[ b. Uer he feasble assumpo ha, a he eque, he las equao smplfes o b h C Ω + h + m[ b hese are hree equaos s uows, whch s a uereerme sysem. hs reflecs he fac ha he ssera prcple h ca eerme he oreao bu o he poso of he geocerc ssera aes wh respec o he orgal worg frame. For GPS-ype ewors where he orgal frame s alreay geocerc, we have b, by efo. For VLB-ype ewors we wll se aga b o oba he oreao of a ssera frame parallel o he geocerc ssera frame, wh he same org as he orgal frame. We ee a separae opmzao prcple for he eermao of a opmal org of he ewor, sce he poso of he geoceer remas ueermable from he avalable aa. Wh he choce b, he hree rasformao parameers o "ssera" cooraes shoul be eerme from he soluo of he hree ffereal equaos h C Ω + h, whch uer he aoal assumpo ha Ω ae he form hese ca be egrae o oba a soluo Ω C h. + Ω τ τ C τ h τ τ 4 whch epes o he chose al value whch eermes oe ou of all he possble ssera frames. For eample f s chose, he ssera aes wll coce wh he orgal aes a he al epoch, sce. he eplc form of he ffereal equaos epes o he chose paramerzao of he roao mar erms of hree parameers. 86

9 5. quvalece of ssera aes o a space-me geeralzao of Messl's er ras A ffere ype of soluo ca be base o he eeso of he well ow cocep of er ras, rouce by Messl 965, 969. A ay epoch, whe he ewor has cooraes a orgal referece frame, he se of all cooraes + b, resulg as he parameers a b ae ay possble value, form a mafol,.e. a "curve" subspace M of he ewor coorae space X. fac M s he se of all ewor cooraes whch gve rse o he same ewor shape as he oe efe by. Obvously M for ay parcular epoch. he ea s ow o mpose o he curve o be such ha s velocy s perpecular o he mafol M, or more precsely o he "fla" space, whch s age o he cuve mafol M a he po. Sce he parameers a b comprse a se of curvlear cooraes for M, he age space s he se of all ler combaos of he vecors age o he coorae curves, amely,,,,,. he orhogoaly coos b b b, b ae he form M, M,,,, or compac mar oao b M m, M m b b 5 eplacg Ω [, b, a,, we arrve a from, mplemeg he usual assumpo ha Ω C b h,. 6 he frs oe of 6 s equvale o a herefore he er ra or Messl frame s a ssera frame! he seco of 6 yels b a a proves a soluo o he org eermao for VLB-ype ewors: f we chosse b, hs meas ha, relao o he assumpo, he ewor org shoul rema a he "ceer of mass" of he ewor. Ay soluo of 6 sasfes he geoesc or mmum eergy equaos 5, 6 a 7. hus he ssera or Messl frame soluo s a geoesc a eve more s a geoesc of mmum possble legh amog all geoescs, a propery whch follows from he fac ha M a M F F. orer o see how he prese soluo s relae o Messl's cocep of er ras, we mus use [ Ω [ b Ω, b, a, orer o rewre he orhogoaly coos 5 he form m [,. 7 Assume ha he soluo has bee eerme a some epoch a we wa o eerme he soluo a a slghly laer epoch +,.e. + +, usg as 87

10 a sarg appromae value. f s replace by, s appromae by, a we choose, equaos 7 are covere o he well ow er ras: m [, A llusrave eample A parcular choce of roao parameers s,, 9 yelg [ [ [ 4 [ [ [ 4 [ [ [ We may se [ [ Ω q q q Q, 46 q q 47 s q 48 s s s s s q 49 C z h z Q 5 or seg c c c z z c h C 5 88

11 s s c s c c s c c 54 s s + c 55 verso of he mar Q gves s s a a Q s s 56 s s a he ffereal equaos become or eplcly Q C h Q c 57 z z c 58 c + s a c a c s + c 59 s + c c 6 Acowlegeme hs wor has bee complee urg he auhor's say a he Geoec sue of he Uversy of Sugar, wh he facal suppor of he Aleaer vo Humbol Fouao, whch s graefully acowlege. efereces Dermas, A. 995: he Nolear a Space-me Geoec Daum Problem. Presee a he eraoal Meeg "Mahemasche Mehoe er Geoäse", Mahemasches Forschugssu Oberwolfach, -7 Oc Grafare, Opmzao of Geoec Newors. Bolleo Geoesa e Sceze Aff,, 4, Messl, P. 965: Über e ere Geauge remesoaler Puhaufe. Zeschrf für Vermessugswese, 9, 4, 9-8. Messl, P. 969: Zusammefassug u Ausbau er ere Fehlerheore ees Puhaufes. Deusche Geoäsche Kommsso, ehe A, Nr. 6, 8-. Morz, H. &.. Mueller 987: arh oao. heory a Observao. Ugar, New Yor. Mu, W.H. & G.J.F. MacDoal 96: he oao of he arh. A Geophyscal Dscusso. Cambrge Uversy Press. 89

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