MFGsoft. Software User Manual

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1 ISSN

2 MFGsof Muli-Fucioal GPS/Galileo Sofwae Sofwae Use Maual Vesio of 004 Guochag Xu GeoFoschugsZeum Posdam Depame : Geodesy ad Remoe Sesig Telegafebeg A7, 4473 Posdam, Gemay Ocobe 004 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

3 MFGsof Muli-Fucioal GPS/Galileo Sofwae Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

4 Sofwae Use Maual Coes. Ioducio. Chaaceisics of MFGsof 3. Ru of MFGsof 4. Ipu Paamee File 4. Defiiios of Ipu Paamee File 4. Sadad Ipu Paamee File 4.3 Sadad Debug Swiches i Ipu Paamee File 5. Muli-Fucioal GPS Daa Pocessig 5. Global Newok Moioig ad Dyamic Obi Deemiaio 5. Global Newok Moioig ad Kiemaic/Dyamic Obi Deemiaio 5.3 Regioal Newok Moioig wih Obi Coecio 5.4 Local Newok Saic ad Kiemaic Posiioig 5.5 Oboad Kiemaic/Dyamic Obi Deemiaio 5.6 Ohe Fucios 6. Sucue ad Diagam of MFGsof 7. Saegies ad Piciples Used 7. Equivale GPS Daa Pocessig Algoihm 7. Diagoalisaio Algoihm 7.3 Opimal Ambiguiy Seach Cieio 7.4 Idepede Paameeisaio Mehod 7.4. Sequeial Daa Dealig ad Geomeic Illusaio 7.4. Coelaio Aalysis i Case of Phase-Code Combiaio 7.5 Diffeeial Soluio of he Vaiace Equaio 7.6 Adjusme Models of he Sola Radiaio ad Amospheic Dag 7.6. Ioducio of he models 7.6. The disubace coodiae sysem ad eo models Numeical simulaio ad models aalysis Summay ad Coclusios 7.7 Iellige Kalma Fileig Techique 7.7. Ioducio of Kalma File 7.7. Kalma Fileig Usig Velociy Ifomaio The Theoeical Poblem vs. Pacical Requieme 3 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

5 7.7.4 Algoihm of Iellige Kalma File 7.8 Idepede Baselie Newok ad Exeded Double Diffeecig Fomig 7.9 Sadad Models ad Algoihms 8. Numeical Examples 8. Daa ad Saio Newok 8. Numeical Examples of Ieal Tes 8.. Soluios i ECEF ad ECSF coodiae Sysems 8.. Equivale Popeies 8..3 Daa Codiios ad Idepede Paameeisaio 8..4 Phase ad Phase-Code Soluios 8..5 Velociy Deemiaio 8..6 Regioal Newok Moioig wih Obi Coecio 8..7 Global Newok Moioig wih Kiemaic/Dyamic Obi Deemiaio 8..8 Global Newok Moioig wih Dyamic Obi Deemiaio 8..9 Time Cosumig 8.3 Exeal Compaisos 9. Summay 0. Ackowledgemes. Refeeces. Appedixes. Appedix : Diagams of he Sofwae. Appedix : Lis of Fucios of he Sofwae 4 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

6 . Ioducio MFGsof, i his fis vesio, is a eal ime muli-fucioal GPS sofwae developed a he GeoFoschugsZeum Posdam fo followig he developme of he EU Galileo sysem ad fo simulaig he measueme of he Galileo sysem. The goal is o develop a sofwae which is able o pocess boh he GPS ad Galileo avigaig ad posiioig daa. The so-called equivale GPS daa pocessig algoihm is used i his sofwae, so ha he udiffeeced ad diffeecig mehods ae uified io oe algoihm ad ca be swiched fom oe o aohe selecively. The sofwae woks i a sequeial way which eables a eal ime epoch-block wise daa pocessig o a oe-sep pos-pocessig. To cove possibly applicaio aeas as moe as possible, he avigaig ad posiioig poblem ca be solved i ECEF coodiae sysem as soo as he obi is cosideed as kow. If obi coecio, dyamic obi deemiaio, o kiemaic/dyamic combied obi deemiaio ae equied, he ECSF coodiae sysem is used. Daa combiaios ae fee selecable. A mehod of idepede paameeisaio of he equivale obsevaio model guaaees a egula omal equaio so ha he poblem ca be solved sably ad he soluio ca be obaied homogeeously. A diagoalisaio algoihm makes i possible o give up he paamees which ae o ay moe acual i case of sequeial daa pocessig. Deails of he fucios ad chaaces of he sofwae ae lised i he ex secio. The documeaio of he heoeical backgoud of he applied algoihms ae published i 003 by Spige Xu 003. Descipios ae also caefully wie i he souce code. A small pa of he souce code ad he feaues of he KSGSof Kiemaic/Saic GPS Sofwae, Xu e al. 998 ae modified ad absobed i his ew developme. The developme of MFGsof was saed i 00, fis yea fo heoeical sudy ad he fo code desig. The sofwae is esed wih may kids of eal daa. Compaisos show good pefomace of he sofwae. Due o he muli-fucioal abiliies of he sofwae, we hope i could be used i a boad aea of applicaios ad sevices fo GPS ad fo simulaio of he Galileo sysem i he fuue, ad a he ed i could be a basis of a excelle muli-fucioal GPS/Galileo sofwae. This maual oulies he chaaceisics ad sucue of he sofwae ad descibes how o use he sofwae. The piciples ad ew feaues ae oulied sysemaically ad efeed paly o exisig efeeces. Numeical examples of muli-fucios ad ieal ess as well as exeal compaisos ae give.. Chaaceisics of MFGsof The mos impoa chaaceisics of MFGsof ae give as follows: MFGsof is developed i C ude Uix opeaig sysem ad ca be diecly implemeed o PCs ude Liux wihou ay chage compue idepede. MFGsof uses he equivale GPS daa pocessig algoihm. The u-diffeeced ad diffeecig mehods ae fee selecable by usig swich. 5 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

7 A idepede paameeisaio algoihm guaaees a egula omal equaio solvabiliy. A diagoalisaio algoihm makes i possible o keep he equaios oly wih acual paamees educe he size of he poblem. The sofwae is able o be used fo a eal ime daa pocessig o pos-pocessig modules. Boad applicaio aeas: ca be used fo daa pocessig of local saic/kiemaic ewok, egioal ewok moioig, global ewok moioig, ad o boad avigaio. Muli-fucios selecable: obi as kow, obi has o be coeced, dyamic obi deemiaio, ad kiemaic/dyamic combied obi deemiaio. I case of obi is kow, soluio i ECEF o ECSF coodiae sysem is selecable. Swiches fo all kids of daa combiaios, all possible opospheic models, IGS o boadcas ephemeides, cycle slips faco, goss eo sadad, debug fucios. Swiches fo clock eo soluio o elimiaio, possible kids of a pioi ifomaio applicaios, fixed ewok, muli-efeece saios, efeece clock. Sigle poi code posiioig ad diffeeial Dopple velociy deemiaio available by usig phase ad/o code measuemes. Abiliies o ead o ceae he ephemeides of he Su, he Moo ad plaes. Equipped wih a exeded sofwae package of adjusme algoihms ad fileig mehods. Opimal ambiguiy seach cieio ad algoihm. Equipped physical models: he Eah/ocea loadig ide displaceme, saellie mass cee coecio, boadcas ioospheic model, elaiviy effecs, sola adiaio, amospheic dag, ec. All possible obi disubace foce models ad obi iegaio ools. The sofwae is use-fiedly desiged fo pacical applicaios ad scieific eseach. The sofwae woks ude bach module ad ca u auomaically. May uiliies pogams. Deailed heoeical descipios ad efeeces. Use maual. 3. Ru of MFGsof To sa he MFGsof, oe jus eeds o ee he followig commad lie: MFGsof <Ipu_paamee_file> whee Ipu_paamee_file is a file ame i which cosiss all ecessay ifomaio ad cool paamees fo uig he sofwae. The Ipu_paamee_file has o be edied befoe uig he pogam. The defiiios ad descipios of he Ipu_paamee_file will be give i he ex secio. The sadad ipu paamee file may be ceaed auomaically. 6 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

8 4. Ipu Paamee File The esseial wok o u he pogam fo GPS daa pocessig is o wie a ipu paamee file which is defied i a flexible fom. Defiiios of he ipu paamee file, ad sadad ipu paamees as well as sadad defiiios of debug swiches ae give i followig subsecios. 4. Defiiio of Ipu Paamee Files The deailed defiiios of he ipu paamee file fo he GPS daa pocessig ae give below i hee pas: defiiios of ipu paamee file, defiiios of ipu paamees, ad defiiios of debug swiches. Defiiios of he ipu paamee file: Ay lie wih a chaace * sa will be cosideed as a comme lie. Evey o-comme lie should coai he chaace : colo. The ipu coes ae placed befoe he chaace : ad he ideifie of he ipu is placed afe he :. Ipu paamees should be sepaaed fom each ohe by a leas oe blak space. Oly he ideifies defied hee will be acceped. Ay lie wihou kow ideifie will be cosideed as a comme lie. Ay ex afe he ideifie will be cosideed as comme ex. Ay file ame icludig is pah ame is allowed o be up o 80 chaaces. Blak lie will be cosideed as a comme lie. The odes of defiiio lies ae opioal if o specified. Paamees i < > ae defaul o sadad paamees. EOF maks he ed of he ipu paamee file. Defiiios of he ideifies ad hei explaaios: RINEX daa file ame, ipu Riex GPS daa file ame. Coodiaes xyz, sa_po, ipu coodiaes ad popey of he saio a whee above daa ae measued. This lie should be pu afe he elaed lie of Riex daa file ame. Ipu paamee foma: hee floa umbes ad oe iege. Defiiios of saio popeies:, fo efeece saic, saic, 3, 4 fo kiemaic, kiemaic o he ai, 6 fo LEO oboad, 7, 8 fo IGS efeece, IGS. Above Riex daa file ame ad Coodiaes lie could be epeaed uil files ad coodiaes of all saios ae lised. 7 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

9 IGSobi daa file ame, ipu file ame of IGS pecise obi daa. Moe files ae allowed ad ca be give by seveal lies. Boadcas ephemeides file ame, ipu file ame of boadcas obi daa. Moe files ae allowed ad ca be give by seveal lies. Docume file ame fo oupu also used o ecod he used ipu paamee file. IGSsa file ame, ipu file of IGS saio ame lis. IGSsieeceive file ame, ipu file of IGS sie eceive ype lis. Sada file ame, ipu file of GPS saellie paamees. Polada file ame, ipu daa file of pola moio. SelecSa file ame, ipu file of seleced saios fo daa pocessig. Oloadfile ame, ipu file of ocea loadig paamees. GPS_da pah ame, he pah ame of he IGS GPS daa, evey day oe lie. Geopoe file ame, he file ame of geopoeial model. IDaaFile, daa files used,,, 3 fo usig he GPS daa give i his file, SelecSa file, boh. Begiig dae Yea moh day, measueme sa dae, foma: hee ieges. DelaDay, Days of oal daa, foma: oe iege defaul ipu foma. Begiig ime hou miue secod, sa epoch of he daa pocessig, foma: hee ieges. Ed ime hou miue secod, ed epoch of he daa pocessig, foma: hee ieges. DelaT, used daa samplig ae, i secods, foma: oe floa umbe. Mi_da_wi, miimum daa widow legh, i secods, foma: oe floa umbe. DaaIeval, daa ieval ype,,, 3 fo >0sec., sec., 0.sec. SKD, swich of kiemaic/saic/dyamic daa pocessig modules //3. Icoodiaes, <,, 3, 4>, coodiaes used by givig above, i Riex file, by sigle poi posiioig, i saio file. 3 is he sadad opio fo kiemaic daa pocessig, 4 is fo IGS ewok daa pocessig. Dc, daa combiaio ype, <,, 3, 4, 5, 6, 7>, - L, - L, 3 L3, 4 - Lc io-fee, 5 i- io-fee 77%*FiL-FiL*fL/fL, 6 - geeal combiaio: Faco*FiL + Faco*FiL, 7 code. DualF combiaio Faco Faco, foma : wo floa umbes. Cycleslipf, faco fo cycle slips es, foma : oe floa umbe. M_cyces, cycleslip es mehods,, fo usig ioosphee esidual, diff. Dopple fiig. Igs_v, igs daa vesio umbe <0,, >, 0 - old vesio wih x,y,z,xdo,ydo,zdo, - ew vesio wih x,y,z,d_clock, - GFZ ieal vesio wih x,y,z,d_clock,xdo,ydo,zdo. Top_coecio, <0,, > fo <o, yes, as ukow>, oly used i saic case, defaul <>. Toposphee model used, <0,,, 3, 4, 5, 6, 7, 8> fo <o, 8 models>. 8 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

10 ToposHeigh, <,, 3> fo <sadad, s_heigh, ipu paamees>. Lagage polyomial ode, sadad value is <7>, less ha 7 is o allowed. Iv_, <, > fo <Cholesky, Gaussjoda> ivese mehod, if oe failed, aohe will be auomaically used, defaul <>. Clock_coecio, <0,,, 3> fo <o, yes,.>, should be <>. ISMeph,, fo compue, ead Smeph Su Moo plaes obi daa. Tide_coecio, <0, > fo <o, yes> wih he Eah ide coecio, defaul <>. Rela_coecio, <0, > fo <o, yes> wih elaiviy coecio, defaul <>. Ioload_coecio, <0, > fo <o, yes> wih ocea loadig coecio. Cood_ce oupu foms, <,, 3> fo oupu i <Caesia, geodeic ellipsoid, boh> coodiaes. CP_combie, <0,, > fo <code, phase, phase-code>. CP,, fo C, P code used i sigle poi posiioig. WcWp code ad phase weigh facos, foma : wo floa umbes. Weigh sadadizaio will be caied ou i pogam. Ieaho, <0, > fo o, yes of he Eah oaioal coecio. Dopple_sig, <-, > fo Dopple daa fom <Bee_decode, Ladau_decode>. Ioload_sig, <-, >, heoeically, sig should be. Iidesig, <-, >, heoeically, sig should be. Iopsig, <-, >, saic esul shows Iopsig mus be. Iclocksig, <-, >, heoeically, sig should be. Iclocksig, <-, >, heoeically, sig should be. Iclocksig3, <-, >, heoeically, sig should be. Ambiguiy fixig, <0,, > fo <o, yes, equivale> fixig. ReceivC, <0, > fo o, yes wih eceive clock eo coecio, mus be <>. Is_aea_c, <0, > fo <o, yes> coec saellie aea cee offse. Ieo_a Ixyz > eaho used i asmiig ime coecio. Iaso Iluisa, <0, > asmiig ime oaio <o, yes>. Ku, fuhe u fo Ru-4 o o, <0, > fo o, yes. Ambiguiy fixig seach seleco, 0 fo o, else aea faco iege. Ifile, <0,,, 3> fo <sequeial, Kalma, adapive Kalma, iellige Kalma> fileig. Equivale mehod swich, 0 - zeo diffeecig mehod, - sigle diffeecig mehod, - double diffeecig mehod, 3 - iple diffeecig mehod, 4 - use defied diffeecig mehod, 5 equivale mehod. I_C0 I_CS, ode ad gade of he geopoeial model, foma: wo ieges. 9 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

11 Iobi, ipu paamee fo obi daa selecio., 3 fo IGS i ECEF, boadcas obi i ECEF, wo ieges. Isequeial, 0,, 3 fo epoch-block soluio, sequeial soluio, accumulaed oal soluio. I_sf_ef, fo fomig equaio i ECSF o ECEF coodiae sysems. I_ocood, fo fixed ewok o fee ewok o coodiaes solved, coodiaes solved. Ipospocessig,, 3, 4, 5 see mfgsof.c, pos-pocessig swich. Idck_obi 4, 5, 6, 7 obi kow, deemie,, coecio, obi deemiaio swich. Iapioi,, 3,,, 3, 4, 5, 6, 7, a pioi swich,, fo used fo saio cood., 6 obi paamees, 3 fo boh, > fo IGS mehod: jiapioi-0 fo a*+0.**j. Idaacodiio 0, fo o, yes o use he daa codiios. Iiiial_updae 0, fo o, yes o updae he iiial values of he ukows. Defiiios of he debug swiches: Debug O_ipu <0,, > : <ead daa, epi daa, check> Debug O poe <0, > : <o, yes, check> Debug O oload <0, > : <o, yes check> Debug O_oload_ide <0, > : <o, yes, check> Debug O_igsda <0,,, 3> : <all, o, yes, check> i obi.c defied swich Debug O_lagage <0, else> : <all, o> Debug O_lagagef <0, else> : <all, o> Debug O_igs <0, > : <all, o> used i _igs_d, _igs_h Debug O_boadcas <0, > : <all, o, check> i boad.c Debug O_bobc <0, else> : <all, o> Debug O_eph_check <0, else> : <all, o> Debug O_boadcas_obi <0, else> : <all, o> Debug O eph <0, else> : <all, o> Debug O_ioo <0,, > : <o, yes, check> pi ioo_esidual Debug O_iexd <0,, else> : <all, pa, o> Debug O_pepo <0, else> : <all, o> Debug O_mai <0,, > : <o, yes, check> gpsda, sigl Debug O_ime0 ime swich IO_ome0 ou, else else Debug O_iexh <0, else> : <all, o> swich i iex.c Debug O_efsa <0,,, 3, 4, 5> : <all,..., o> i obi.c 0 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

12 Debug O_geob <0,, > : <all, o, check> Debug O_geda <0,, else> : <all, pa, o> Debug O_gedao <0, else> : <all, o> Debug O_oda_c <0, else> : <all, o> Debug O_sdda <0, else> : <all, o> Debug O_cycle <0,, else> : <all, pa, o> oupu Debug O_gpsda <-, 0,,, 3, 4, 5, 6> : <all, o-, s-, d-, da, cyc, sigl> Debug O_co <0, else> : <all, o> Debug O_o <0, else> : <all, o> Debug O_o <0, else> : <all, o> Debug O_ela <0, else> : <all, o> Debug O_Roa <0, else> : <all, o> Debug O_SL_eph <0,,, 3, 4> : <all,..., o> Debug O_ide <0,,, 3, 4> : <allide_day, alliepo_oa,...o> Debug O_gcsls <0,,, 3> : <all, pa, mii, o> i ad_coe.c Debug O_gcsls <0,,, 3> : <all, pa, mii, o> Debug O_cls <0,,, 3> : <all, pa, mii, o> Debug O_ls <0,,, 3> : <all, pa, mii, o> Debug O_ls <0, > : <all, o> Debug O_chol <0,,, 3, 4, 5> : <all, pa, mii,..., o> Debug O_gaussj <0,,, 3, 4, 5> : <all, pa, mii, o> Debug O_eq <0,,, 3, 4, 5, 6, 7> : i obs. equaio Debug O_om <0,,, 3> : <all, pa, mii, o> i omal equaio Debug O_model <0,, > : <all, pa, o> i model fucio Debug O level <0,,, 3, 4, 5, 6, 7> : <all, pa, mii,..., o> i mai.c Debug O_ime ime swich ed i mai.c Debug O_Sop ime swich sop i mai.c Debug O_om_cp <0,,, 3, 4, 5, 6> : <all, pa,..., o> i om_dd_cp.c Debug O_kalma <0,, > : <all, pa, o> i ykalma.c Debug O_siglep <0,,, 3, 4, 5, 6> : <all, pa,..., o> Debug O_velociy <0,,, 3, 4, 5, 6> : <all, pa,..., o> Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

13 4. Sadad Ipu Paamee file Followig is a example of a sadad ipu paamee file fo a IGS global ewok ecoic moioig wih GPS saellie obi deemiaio. Explaaios will be oulied afe his ipu paamee file. The Ipu_Paamee_File of all ohe umeical ess give i his maual ca be obaied hough mio modificaio fom his sadad ipu paamee file. * o be defied ipu daa file ames ad coodiaes */home/sf/xu/ew/popo/ch_.xxx : RINEX daa file ame * : Coodiaes xyz, sa_po */home/sf/xu/ew/popo/sylrh.0ox : RINEX daa file ame * : Coodiaes xyz, sa_po */home/sf/xu/ew/popo/sylmh.0ox : RINEX daa file ame * : Coodiaes xyz, sa_po */home/sf/xu/ew/popo/sylrh.0ox : RINEX daa file ame * : Coodiaes xyz, sa_po */home/sf/xu/ew/popo/sylmh.0ox : RINEX daa file ame * : Coodiaes xyz, sa_po * sa_po : / saic/_ef, 3/4 kiemaic/_o_ai, 6 LEO, 7/8 IGS ef /IGS */home/sf/xu/commo_daa/igs64.sp3 : IGSobi daa file ame /home/sf/xu/commo_daa/igs643.sp3 : IGSobi daa file ame */home/sf/xu/commo_daa/igs644.sp3 : IGSobi daa file ame */home/sf/xu/d0/ifag00.0 : Boadcas ephemeides file ame */home/sf/xu/d0/bdc00.0 : Boadcas ephemeides file ame /home/sf/xu/d/bdc0.0 : Boadcas ephemeides file ame /home/sf/xu/d/ifag0.0 : Boadcas ephemeides file ame */home/sf/xu/d/ifag0.0 : Boadcas ephemeides file ame */home/sf/xu/d/bdc0.0 : Boadcas ephemeides file ame zou. : Docume file ame fo oupu /home/sf/xu/commo_daa/sako_gps_snx_f_0 : IGSsa file ame /home/sf/xu/commo_daa/igs.sx_acual : IGSsieeceive file ame /home/sf/xu/commo_daa/gpssada_eu : Sada file ame /home/sf/xu/commo_daa/pii.snx_b.0.00_999 : Polada file ame /home/sf/xu/commo_daa/seleced_saio : SelecSa file ame /home/sf/xu/commo_daa/oload.gps : Oloadfile ame */home/sf/xu/d0/ : GPS_da pah ame /home/sf/xu/d/ : GPS_da pah ame */home/sf/xu/d/ : GPS_da pah ame Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

14 /home/sf/xu/commo_daa/potfilet_pgm055.cio : Geopoe file ame * o be defied paamees : IDaaFile, <,,3> daafiles used <above, file, boh> 00 5 : begiig dae Yea moh day 00 5 : DelaDay oal days, i : Begiig ime hou miue sec : Ed ime hou miue sec i : DelaT used samplig ae : Mi_da_wi miimum daa legh i secods double : DaaIeval,,3 fo >0sec, sec, 0.sec : SKD, Saic/Kiemaic/Dyamic 0// / 4 : Icoodiaes, coodiaes used 3 *,,3,4 - use xyz <give above, i Riex files, sigle poi, i saio file> * 3 fo kiemaic by sigel poi posiioig * 4 fo IGS ewok daa pocessig 4 <4>: Dc, daa combiaio ype, <,,3,4,5,6,7>, fo L, L, L3, Lc io-fee, * i-io-fee, geeal combiaio: Faco*FiL + Faco*FiL, code.. -. : DualF combiaio Faco Faco if above dc6 Widelae 4. : Cycleslipf, faco fo cycle slip es, defaul.0 <>: M_cyces, cycleslip es mehods * - ioosphee esiduals * - diffeeial Dopple fiig <>: Igs_v igs daa vesio, <0,,> fo 3 vesios <>: Top_coecio, <0,,> fo <o, yes, as ukow> 7 <>: Toposphee model used, <0,,,3,4,5,6,7,8> fo <o, 8 models> <>: ToposHeigh, <,,3> fo <sadad, s_heigh, ipu_paamees> 7 <7>: Lagage polyom ode used, defaul 7 <>: Iv_, <,> fo <Choleski,Gaussjoda> ivese mehod <>: Clock_coecio, <0,,,3> fo <o,yes,...,...> <>: ISMeph, <,> fo <compue,ead> Su Moo plaes obis <>: Tide_coecio, <0,> fo <o, yes> <>: Rela_coecio, <0,> fo <o, yes> <>: Ioload_coecio, <0,> fo <o, yes> 3 <3>: Cood_ce oupu fomas, <,,3> fo <caesia,ellipsoid,boh> : CP_combie, <0,,> fo <code,phase,code-phase combiaio> : CP, <,> fo C o P used i sigle posiioig. 3 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

15 : WcWp code ad phase weigh facos, defaul <>: Ieaho, <0,> fo o,yes i - <->: Dopple_sig, <-,> <>: Ioload_sig <-,> <>: Iidesig, <-,> <>: Iopsig, <-,>, saic shows Iopsig mus be <>: Iclocksig, <-,> <>: Iclocksig, <-,> <>: Iclocksig3, <-,> 4 <4>: Ieo_a Ixyz > eaho used i asmiig ime coecio <>: Iaso Iluisa, <0,> asmiig ime oaio <o,yes>i <>: ReceivC, <0,> o,yes fo eceive clock coecio i <0>: Ku, fuhe u fo Ru-4 o o, <0,> fo o, yes i : Ambiguiy fixig seach seleco, 0,, fo o, yes, equivale i : Is_aea_c, 0, fo o,yes aea mass cee coecio 0 : Robus esimaio fo weigh egulaio, <0,> fo <o, yes> 50 : N_ cieium fo cacel N, defaul <0> 0 : Ifile, <0,,,3> fo <sequeial,kalma,adapive kalma,iellige> fileig 5 : Equivale mehod, 5 * 0 - zeo diffeecig mehod * - sigle diffeecig mehod * - double diffeecig mehod * 3 - iple diffeecig mehod * 4 - use defied diffeecig mehod 70 6 : I_C0 I_CS geopoeial odes, maximum : Iobi %d igs, igs_sf, 3 bob, 4 bob_sf : Isequeial %d 0,,3 fo epoch, sequecial, accumulae soluio : I_sf_ef %d, fomig equaio i ecsf o ecef coodiae sysems : I_ocood, fo fixed ewok o fee ewok 5 : Ipospocessig,,3,4,5 see ew983v.c 5 : Idck_obi 4,5,6,7 obi kow, deemie,, coecio 0 : Ispioi,,3,,,3,4,5,6,7 : Idaacodiio 0, fo o, yes o use he daa codiios. 0 : Iiiial_updae 0, fo o, yes o updae he iiial values of he ukows. EOF : ed of he ipu paamee file I above file, he pah ame of he IGS daa is give. The used saios ae give i a lis of seleced saios. The lis file of seleced saios is defied oe saio pe lie, wih saio id, 4 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

16 umbe, coie, ECEF coodiaes x,y,z, saio popey, ieal umbe of he saio. As soo as he fi chaace of he lie is a blak, he saio will be o icluded i he daa pocessig. I his way o add o educe a saio may be ealised easily by ediig he file. Of couse, he saio ad saellie elaed files, IGS ad boadcas obi daa files, ocea loadig paamees ad geopoeial files ae give. Daa of May s 00 is used fo oe day soluio fom 0:30:00 o 3:30:00. Saio coodiaes shall be solved fo oo. Obi deemiaio swihch is o, ECSF coodiae sysem mus be used. Boadcas obi daa is used. Equivale algoihm ad sequeial soluio is pefeed. No a pioi ifomaio is used. Ohe paamees ae sadad. 4.3 Sadad Debug Swiches i Ipu Paamee File The sadad debug swiches ae defied as follows defaul: * o be defied sadad debug swiches : Debug O_ipu <0,> :<ead daa, epi daa> : Debug O poe <0,> :<o, check> : Debug O oload <0,> :<o, check> : Debug O_oload_ide <0,> :<o, check> 3 : Debug O_igsda <0,,> :<all,o,check> i igsew0.c defied swich : Debug O_lagage <0,else>:<all,o> : Debug O_lagagef<0,else>:<all,o> : Debug O_igs <0,> :<all,o> used i _igs_d,_igs_h 3 : Debug O_boadcas<0,> : <all,check> i boad.c : Debug O_bobc <0,else>:<all,o> : Debug O_eph_check<0,else>:<all,o> : Debug O_boadcas_obi<0,else>:<all,o> : Debug O eph <0,else>:<all,o> : Debug O_ioo <0,> :<o,yes> pi ioo_esidual : Debug O_iexd <0,,else> :<all,pa,o> : Debug O_pepo <0,else> :<all,o> : Debug O_mai <0,>:<o,yes> gpsda, sigl : Debug O_ime0 ime swich IO_ome0 ou, else else : Debug O_iexh <0,else> :<all,o> swich i iex.c 5 : Debug O_efsa <0,,,3,4,5>:<all,...,o> i obi.c : Debug O_geob <0,else>:<all,o> : Debug O_geda <0,,else> :<all,pa,o> : Debug O_gedao <0,else> :<all,o> 5 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

17 : Debug O_oda_c <0,else> :<all,o> : Debug O_sdda <0,else> :<all,o> : Debug O_cycle <0,,else> :<all,pa,o> oupu 6 : Debug O_gpsda<-,0,,,3,4,5>:<all,o-,s-,d-,da,cyc,sigl> : Debug O_co <0,else> :<all,o> : Debug O_o <0,else>:<all,o> : Debug O_o <0,else>:<all,o> : Debug O_ela <0,else>:<all,o> : Debug O_Roa <0,else>:<all,o> 4 : Debug O_SL_eph <0,else>:<all,o> 4 : Debug O_ide<0,,>:<allide_day,alliepo_oa,o> 0 : Debug O_gcsls <0,,,3>:<all,pa,mii,o> i ad_coe.c 0 : Debug O_gcsls <0,,,3>:<all,pa,mii,o> 3 : Debug O_cls <0,,,3>:<all,pa,mii,o> 0 : Debug O_ls <0,,,3>:<all,pa,mii,o> : Debug O_ls <0,>:<all,o> 5 : Debug O_chol <0,,,,3>:<all,pa,mii,o> 5 : Debug O_gaussj <0,,,3>:<all,pa,mii,o> 7 : Debug O_eq <0,,,3,4,5,6,7>: i obs. equaio 3 : Debug O_om <0,,>:<all,pa,o> i omal equaio 0 : Debug O_model <0,,>:<all,pa,o> i model fucio 7 : Debug O level <0,,,3>:<all,pa,mii,o> i mai.c - : Debug O_ime ime swich ed i mai.c - : Debug O_Sop ime swich sop i mai.c 6 : Debug O_om_cp <0,,>:<all,pa,o> i om_dd_cp.c : Debug O_kalma <0,,>:<all,pa,o> i ykalma.c : Debug O_scee <0,>:<all,o> 6 : Debug O_siglep <0,,,3,4,5,6>:<all,pa,o> 6 : Debug O_velociy <0,,,3,4,5,6>:<all,pa,o> The debug swiches ae oly eeded fo he advaced uses o locae he eos ad o egulae he es ad iemidial oupu. 6 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

18 5. Muli-Fucioal GPS Daa Pocessig I his secio, he muli-fucialiies of he MFGsof will be oulied wih examples of he elaed ipu paamee files. They ae: sadad global ewok moioig wih dyamic obi deemiaio ad wih kiemaic/dyamic combied obi deemiaio as well as wih kow obi, egioal ewok moioig wih obi coecio ad wih kow obi, local ewok saic ad kiemaic posiioig, oboad LEO saellie kiemaic ad kiemaic/dyamic obi deemiaio. 5. Global Newok Moioig ad Dyamic Obi Deemiaio The ipu paamee file is exacly give i secio 3.3. Sigifica aleaive soluios may be obaied e.g. by exedig he daa legh up o hee days, adjusig he daa samplig ae, chagig daa combiaio mehods, fixig he ewok, selecig diffee efeece saios, o usig diffee a pioi ifomaio. The defiiios of he efeece saios have o be chaged by ediig he seleced saio file, ad ohe elaed paamee lies ae lised below: 00 5 : begiig dae Yea moh day 00 5 : DelaDay oal days, i : Begiig ime hou miue sec : Ed ime hou miue sec i : DelaT used samplig ae <4>: Dc, daa combiaio ype, <,,3,4,5,6,7> : CP_combie, <0,,> fo <code,phase,code-phase combiaio> : I_ocood, fo fixed ewok o fee ewok 0 : Ispioi,,3,,,3,4,5,6,7 5. Global Newok Moioig ad Kiemaic/Dyamic Obi Deemiaio The oly oe swich diffee fo such soluio compaed wih he compuaio oulied i secio 5., is ha he Idck_obi swich should be se o 6, i.e. 6 : Idck_obi 4,5,6,7 obi kow, deemie,, coecio The sigifica aleaive opios meioed i secio 5. above ae also applicable hee see 5.. The advaage of a kiemaic/dyamic combied obi deemiaio is ha he obi is fied by a dyamic model ad deemied geomeically. No dyamic foce models ae used, so he pocessig is fase ad easie. 7 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

19 5.3 Regioal Newok Moioig wih Obi Coecio The oly oe swich diffeece fo such soluio compaed wih he compuaio oulied i secio 5. is ha he Idck_obi swich should be se o 7, i.e. 7 : Idck_obi 4,5,6,7 obi kow, deemie,, coecio The seleced saio file is of couse diffee ad icludes oly ames of he egioal saios. The sigifica aleaive opios meioed i 5. ae also applicable. Hee he obi coecio model ae used o fi he daa so ha he pecisio of he egioal ewok moioig ca be modified. I is suggesed ha he IGS pecise obi daa shall be used i his case. 5.4 Local Newok Saic ad Kiemaic Posiioig Geeally speakig, he GPS daa file ames ae give i he ipu paamee file i case of local ewok daa pocessig. The IDaaFile swich has o be chaged o, i.e. : IDaaFile, <,,3> daafiles used <above, file, boh> Local ewok saic daa pocessig I he sadad ipu paamee file give i 5., hee ae wo daa files fom wo saios which have he saio popeies of ad. These wo lies ad he elaed wo coodiae lies shall be swiched o, i.e. cacel he * mak : Coodiaes xyz, sa_po /home/sf/xu/ew/popo/sylmh.0ox : RINEX daa file ame : Coodiaes xyz, sa_po /home/sf/xu/ew/popo/sylrh.0ox : RINEX daa file ame The efeece saio has he saio popey of, ad he ukow saic saio has he saio popey of. Theefoe he coodiaes give hee fo efeece saio shall be vey pecise. The appoximaed coodiaes of ukow saio shall be give. Because he daa ae measued o he May d, 00 wih samplig ae of 0. secods i he ime duaio fom 7:0:3 o 8:0:30, he elaed paamees have o be chaged. The IGS o boadcas obi files have o be chaged accodigly, i.e. */home/sf/xu/commo_daa/igs643.sp3 /home/sf/xu/commo_daa/igs644.sp3 o : IGSobi daa file ame : IGSobi daa file ame 8 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

20 */home/sf/xu/d/bdc0.0 : Boadcas ephemeides file ame */home/sf/xu/d/ifag0.0 : Boadcas ephemeides file ame /home/sf/xu/d/ifag0.0 : Boadcas ephemeides file ame /home/sf/xu/d/bdc0.0 : Boadcas ephemeides file ame ad 00 5 : begiig dae Yea moh day 00 5 : DelaDay oal days, i 7 3 : Begiig ime hou miue sec : Ed ime hou miue sec i : DelaT used samplig ae : DaaIeval,,3 fo >0sec, sec, 0.sec 3 : SKD, Saic/Kiemaic/Dyamic 0// / : Icoodiaes, coodiaes used 3 *,,3,4 - use xyz <give above, i Riex files, sigle poi, i saio file> * 3 fo kiemaic by sigel poi posiioig * 4 fo IGS ewok daa pocessig Local ewok kiemaic daa pocessig I he sadad ipu paamee file give i 5., hee ae wo daa files fom wo saios which have he saio popeies of ad 3. These wo lies ad he elaed wo coodiae lies shall be swiched o, i.e. cacel he * mak /home/sf/xu/ew/popo/sylrh.0ox : RINEX daa file ame : Coodiaes xyz, sa_po /home/sf/xu/ew/popo/sylrh.0ox : RINEX daa file ame : Coodiaes xyz, sa_po Ohe chages of he paamees ae he same as oulied above excep he lies of u module ad appoximae coodiaes: : SKD, Saic/Kiemaic/Dyamic 0// / 3 : Icoodiaes, coodiaes used 3 *,,3,4 - use xyz <give above, i Riex files, sigle poi, i saio file> * 3 fo kiemaic by sigel poi posiioig * 4 fo IGS ewok daa pocessig 9 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

21 5.5 Oboad Kiemaic/Dyamic Obi Deemiaio The ames of he LEO o boad GPS daa files ae give i a file called leoobifiles. Oe file pe lie ad he ame has maximal 80 chaaes. The daa files will be ead auomaically accodig o he ime duaio i which he daa will be pocessed. The example daa used hee ae hee days fom Apil 30h 00 o May d 004 ad he leoobifiles ae give below: /home/sf/xu/champ_daa/obi/00_8_.da /home/sf/xu/champ_daa/obi/00_9_0.da /home/sf/xu/champ_daa/obi/00_9_.da /home/sf/xu/champ_daa/obi/00_0_0.da /home/sf/xu/champ_daa/obi/00_0_.da /home/sf/xu/champ_daa/obi/00 0.da /home/sf/xu/champ_daa/obi/00.da May paamees ae o eleva fo his kid of daa pocessig. The o be chaged ipu paamees ae lised as follows: 4 : IDaaFile, <,,3,4> daafiles used <above, file, boh,exa> 0. : DelaT used samplig ae 0. 6 : Idck_obi 4,5,6,7 obi kow, deemie,, coecio 5.6 Ohe Fucios Fo easos oe eeds o kow he pecise coodiaes of he efeece saio of a local o egioal ewok. I his case he saio ca be combied io he IGS ewok fo a commo daa pocessig. To do so, oe jus eeds o lis he ieesed daa file ame i he ipu paamee file, ad ses he IDaaFile o 3, i.e. 3 : IDaaFile, <,,3> daafiles used <above, file, boh> Fo some special puposes of global o egioal ewok daa pocessig, such as oposphee o ioosphee soudig, he poblem may also be solved i a obi fixed o ewok fixed ways. The ewok fixig swich ad he obi fixig swich ca be used selecively. They ae: : I_ocood, fo fixed ewok o fee ewok 4 : Idck_obi 4,5,6,7 obi kow, deemie,, coecio 0 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

22 Ohe sigifica possibiliies ae lised follows: Swich o ad off he daa codiios Swich o ad off he a pioi ifomaio i diffee modulus Swich bewee epoch-block soluio, sequeial soluio ad oal soluio. Swich bewee all kids of daa pocessig algoihms ad dada combiaios Swich o o updae he iiial values of he ukows a kid of ieaio I case of obi fixed soluio, he poblem ca be solved i ECEF o ECSF coodiae sysems. 6. Sucue ad Diagam of MFGsof The mos impoa sofwae packages of MFGsof ae he physical models, algoihms ad ools. They ae lised below figues ae give i he appedicies. Physical Models:. Topospheic models. Ioospheic model 3. Relaiviy model 4. Eah ide model 5. Ocea loadig ide model 6. Saellie mass cee model 7. Sola adiaio model 8. Amospheic dag model 9. Geopoeial disubace 0. Tidal poeial disubace. Su, Moo, plaes obi models. Muli-bodies disubace 3. Dyamic obi fiig model Algoihms:. Equivale algoihm ---- udiffeeced algoihm ---- diffeecig algoihms. Diagoalisaio algoihm. Idepede paameeisaio --- opimal baselies --- daa codiios 3. Cycle slip deecio 4. Ambiguiy --- iiialisaio --- seach 5. Adjusme ad fileig --- Leas Squaes --- Block-wise elimiaio --- Sequeial --- A pioi ifomaio --- Kalma file --- iellige Kalma file 6. Diffeeiao/iegaos --- vaiace equaio Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

23 7. Diffeeial Dopple --- obi Tools:. Coodiaes asfomes. Time sysems asfomes 3. Su-Moo-plaes obi ceao 4. Boadcas obi asfome 5. Iepolaio ad iegaio 6. Maix ivese --- Gauss-Joda --- Cholesky 7. Helme asfomaio 8. Mappig fucios 9. Fligh sae compuaio 0. Specal aalysis mehods. Saisic aalysis. Gaphic epeseaio Daa pocessig diagam:. Pogam sa. Commo pa --- Read ipu paamee file --- Read ohe files give Read/ceae Su-Moo-plaes obis --- Compue Eah/ocea loadig ide Obi daa asfomaio Daa pe-pocessig if possible --- iiialisaio opimal baselies --- ambiguiy vaiaio equaio 3. Sequeial ime do loop --- ge eal ime daa --- models ad paamees acquisiio --- sigle poi posiioig velociy deemiaio --- ambiguiy check ad se --- modulus swich local e o boad global/egioal e 4. summay pa foecas ieaio aalysis modulus swich: modulus swich --- global/egioal e --- ambiguiy check --- models acquisiio --- obsevaio equaio Fi ad Keple/Jacoby maices --- ukow check wih daa codiios --- omal equaio --- exchage, elimiaio, accumulaio Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

24 --- adjusme/file diagoalisaio 7. Saegies ad Piciples Used The MFGsof is based o a ew heoeical backgoud ad is a ealisaio of seveal ew algoihms. The piciples will be oulied i his secio. As soo as hee exiss he efeeces, he heoies will be descibed biefly. 7. Equivale GPS Daa Pocessig Algoihm I GPS daa pocessig pacice, he commoly used mehods ae so-called zeo-diffeece udiffeeial, sigle-diffeece, double-diffeece ad iple-diffeece mehods Baue 994; Hofma-Wellehof e al. 997; Kig e al. 987; Leick 995; Remodi 984; Seebe 993; Sag ad Boe 997; Wag e al I is well-kow ha he obsevaio equaios of he diffeecig mehods ca be obaied by cayig ou a elaed liea asfomaio o he oigial equaios. As soo as he weigh maix is similaly asfomed accodig o he law of covaiace popagaio, all mehods ae equivale, heoeically. A heoeical poof of he equivalece bewee he u-diffeeial ad diffeeial mehods ca be foud i Schaffi ad Gafaed 986. A algeba poof is give i Xu 00 whee a uified GPS daa pocessig mehod based o equivalely elimiaed equaios is poposed. By selecig he elimiaed ukow veco as a veco of zeo, a veco of saellie clock eo, a veco of all clock eo, a veco of clock ad ambiguiy paamees, o a veco of use-defied ukows, he selecively elimiaed equivale obsevaio equaios ca be fomed, especively. The equaios ae equivale o he zeo-, sigle-, double-, iple-, o use-defied diffeeced equaios. The advaage of such a mehod is ha he obsevaioal veco emais he oigial oe ad he weigh maix keeps he u-coelaed diagoal fom. The deailed descipio ca be foud i Xu 003, 6.7 pp.07-8, 7.6 pp Fomaio of Equivale Obsevaio Equaios Fo he coveiece of eades, he mehod o fom a equivalely elimiaed equaio sysem is oulied hee. The heoy is give i Xu 00 i deail. I pacice, someimes oly oe goup of ukows is of iees; i is bee o elimiae he ohe goup of ukows called uisace paamees, fo example, because of hei size. I his case, usig he so-called equivalely elimiaed obsevaio equaio sysem could be vey beeficial Wag e al. 988; Xu ad Qia 986; Zhou 985. The uisace paamees ca be elimiaed diecly fom he obsevaio equaios isead of fom he omal equaios. The lieaised obsevaio equaio sysem ca be epeseed usig he maix: X V L, P A A X 3 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

25 whee L is he obsevaioal veco of dimesio m; A ad A ae coefficie maices of dimesio m ad m ; X ad X ae ukow vecos of dimesio - ad ; V is a esidual veco of dimesio m; ad m ae umbes of oal ukows ad obsevaios especively; P is a symmeic ad defiie weigh maix of dimesio m m. Fo u-coelaed obsevaio veco L, P is a diagoal maix. Leas squaes omal equaio of Eq. ca be fomed he by M M whee M X W, M X W T T A PA A PA M T T A PA A PA M Q Q M Q Q Q T T W A PL, W A PL M M M. 3 Elimiaig X ad X i Eq. especively, he elaed equivale omal equaios of X ad X ca be obaied. Combiig hem ogehe oe has he so-called diagoalised omal equaio Xu 003a M 0 0 X B, 4 M X B whee e.g. Cui e al. 98, Gohad 978 M, Q M Q, 5 M M MM M B W MM W M M M M M B W M MW, 6. 7 I is obvious ha Eq. ad Eq.4 ae wo equivale omal equaios. The soluios of he boh equaios ae ideical. The deivaio of Eq.4 fom Eq. is o a simple liea asfomaio ad is ievesible. The elaed equivale obsevaio equaios of he diagoal omal Eq.4 ca be wie Xu 003a U L D U L 0 0 X, D X P 0 0, 8 P whee U ad U ae esidual vecos which have he same popey as V i Eq., ad D I K A, D I J A, 9 whee I is a ideiy maix, ad Xu 003a, Wag e al. 988, Xu ad Qia 986, Zhou 985 T T J A M A P, K A M A P. 0 4 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

26 Maices J, K, I-J ad I-K ae idempoe, I-J T P ad I-K T P ae symmeic. These popeies ae used o deive Eq.8 fom Eq.4, ad of couse, ca be used o fom he omal equaio of Eq.8 o ge Eq.4. Agai, he deivaio of Eq.8 fom Eq. is o a simple liea asfomaio ad is ievesible. Fo easos, we defie he equivale cofaco maix Q e by M 0 Q 0 Q e. 0 M 0 Q Which has he same diagoal eleme blocks as he oigial cofaco maix Q ad guaaees ha he pecisio elaio bewee he ukows emais he same. Such a defiiio is ideed implicily used i he adiioal block-wise leas squaes adjusme cf. Xu 003 p7-3. A GPS daa pocessig algoihm ha uses he secod equaio of Eq.8 deoed by Eq.8_ is said o use he mehod of equivale obsevaio equaio selecively. Selecig X i he fis equaio of Eq.8 deoed by Eq.8_ as zeo veco, he algoihm is he same as he udiffeeced mehod. Selecig X as he saellie clock eo veco, as he veco coaiig all clock eos, as he veco coaiig all clock eos ad ambiguiies, o as he veco coaiig ay use defied paamees, he he algoihm is equivale o he sigle-diffeece mehod, he double-diffeece mehod, he iple-diffeece mehod, o he use defied elimiaig mehod, especively. The ukow X ca be compued sepaaely wih Eq.8_ if desied Xu 00. The equivalece popey is valid ude hee implici assumpios. The fis oe is ha he ideical obsevaio veco L is used. The secod is ha he paameeisaio of X is ideical. The hid is ha he veco X should be able o be elimiaed. Ohewise he equivalece does o hold Xu 00. The advaages of his mehod ae compaed wih u-diffeeial ad diffeeial mehods: The u-diffeeial ad diffeeial GPS daa pocessig ca be deal wih i a equivale ad uified way. The daa pocessig sceaios ca be seleced by a swich ad used i a combiaive way; The elimiaed paamees ca be also solved sepaaely wih he same algoihm; The weigh maix emais he oigial diagoal oe; The oigial obsevaios ae used; o diffeecig is equied. I is obvious ha he descibed algoihm meawhile has all he advaages of all u-diffeeial ad diffeeial GPS daa pocessig mehods. 7. Diagoalisaio Algoihm I he sequeial adjusme ad Kalma fileig, hee is a poblem which has o bee descibed exacly ad heoeically i lieaue befoe. Tha is how o give up he uisace paamee elaed ifomaio fom he pas. Fo a eal ime daa pocessig, i is specially impoa o keep he updaed poblem as small as possible. The so-called diagoalisaio algoihm is poposed iiially fo deivig a equivale ambiguiy seach cieio ad he used fo givig up he uisace paamee elaed ifomaio Xu 003a. Suppose he pas suveyig ifomaio is peseed i a omal equaio sysem 5 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

27 M M M X W, M X W ad he omal equaio of he pese epoch-block cosiss oly he paamee sub-veco X. The he Eq. ca be diagoalised by M 0 0 X B, 3 M X B so ha oe jus eeds o accumulae he X elaed pa i Eq.3 io he pese omal equaio ad he o solve he poblem. I such way he X is cosideed as uisace paamee sub-veco ad is give up duig he sequeial daa pocessig so ha he daa pocessig poblem will be able o be kep as small as possible. Followig gaphic shows he elaioship bewee ukow umbe ad ime fo a sequeial soluio of IGS ewok wih 47 saios ad fixed obi fo a ewok moioig poblem. Wihou diagoalisaio algoihm, he paamee umbe iceases o 300 wheeas wih diagoalisaio algoihm, he paamee umbe emais aoud 500. Wihou such a algoihm a exac ad effecive eal ime daa sequeial pocessig ad fileig is o ealisic. 7.3 Opimal Ambiguiy Seach Cieia The ambiguiy seach cieio is a coadic opic i ieaioal GPS eseach commuiy sice a few yeas. 6 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

28 Suppose GPS obsevaio equaio is Eq. ad is leas squaes omal equaio is Eq., whee X N N is he ambiguiy sub-veco ad X Y Y is he es ukow sub-veco. The leas squaes ambiguiy seach LSAS cieio Teuisse 995, Leick 995, Hofma-Wellehof e al. 997, Eule ad Ladau 99; Ha ad Rizos 997 is δ dn N N Iv Q N, N whee N 0 is he floa soluio of he ambiguiy sub-veco, dn N 0 N. The ambiguiy seach is a pocess o fid ou a veco N i he seachig aea so ha he value of δdn eaches he miimum. The so-called geeal ambiguiy seach cieio is deived i Xu 00a ad has a fom of T δ dx X X Iv Q X, X whee X Y N T, X 0 Y 0 N 0 T, dx X 0 X, idex 0 deoes he floa soluio. The seach is a pocess o fid ou a veco X icludes N i he seachig aea ad Y compued so ha he value of δdx eaches he miimum. The opimal popey of his cieio ca be foud i Xu 00a. Fo he equivale omal equaio 4, he elaed equivale cieio is T δ dx Y0 Y Iv Q Y0 Y + N 0 N Iv Q N0 δ dy + δ dn N, 6 whee idex is used o disiguish he equivale cieio fom he geeal oe give i Eq.5. I is woh meioig ha he well-kow ambiguiy fucio AF mehod is mahemaically icoec cf. Xu 003 pp ad he so-called leas squaes LS cieio is geeally o opimal cf. Xu 003 pp53-7. Fo ifomaio of he eades, a bief discussio is give below coceig he LS cieio. A excelle summay of he deivaio of he ambiguiy seach LS cieio is give i Vehage 004. I emiology of Teuisse 995 he obsevaio model is y Aa + Bb + e, 7 Q y ad he soluio of Eq.7 ca be obaied by he followig miimizaio poblem Qy mia, b y Aa Bb, a Z, b R. 8 Eq.8 ca be ohogoalised by y Aa Bb eˆ + aˆ a + bˆ a b. 9 Qy Qy Qy The explaaios of he symbols ca be foud i Vehage 004. The soluio of Eq.8 is he obaied by hee seps:. compue he floa soluio of Eq.7;. miimize he secod em deoed by δ a fo lae coveiece o he igh-had side of Eq.9 ad se he hid em deoed by δ b o zeo o ge he fixed ambiguiy; 3. compue he fixed coodiaes usig fixed ambiguiy. The key poblem is ha he δ b is o allowed o be se o zeo. The sigifica opposie agumes ae lised below: Qy 7 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

29 To se δ b o zeo o ay cosa ae equivale, howeve he δ b is eihe zeo o cosa fo all coodiae vecos compued fom all o be seached ambiguiy cadidae vecos. Theefoe o se δ b o zeo is ideed o omi he em δ b i Eq.9 i ohe wods, i he secod sep he effec of he em δ b has eve bee ake io accou i his way. To omi he em δ b i he miimisaio poblem Eq.9 is allowed oly if followig assumpio is ue, i.e. a miimum δ a will lead o a miimum δ b ad heefoe lead o a miimum of δ a + δ b. Howeve, heoy ad umeical examples show ha such a assumpio cao be geeally ue cf. Xu 00. The omissio of he em δ b i Eq.9 desoys he equivalece bewee Eq.8 ad Eq.9. The poblem of miimisaio of δ a + δ b is degaded o a poblem of miimisaio of δ a. I his way he obaied soluio cao be geeally he same as he soluio obaied diecly fom Eq.8. Noig opimaliy ud uiqueess popeies of he soluio of Eq.8, he soluio obaied by miimisig δ a cao geeally be he opimal oes of Eq.8. Ay idicaio obaied by such o opimal esuls may o be eally ue. To se δ b o zeo is he same as o se b as floa soluio he δ b is zeo. Howeve, b cao be he floa soluio. Poof: accodigly o he summaised mehod selec b floa coodiae soluio; hough he miimisaio of δ a oe ges he veco a, he oe ca compue b usig a; because a is o he floa ambiguiies, heefoe b is o he floa coodiae soluio. So he esul of b saes ha b does o equal floa coodiae soluio; his is i coflic wih he assumed saig value. To se δ b o a cosa is o allowed eihe. Poof: fo b floa coodiae soluio + ay cosa veco e.g. b 0 o b floa soluio + cosa veco of 000 km oe has δ b cosa, he followig miiδ a +δ b miiδ a ; he sough fo esuls ae he same by usig he summaised mehod, o mae he compuaio is saed fom give floa posiio o oe 000 km fahe away. Theefoe seig δ b cosa would be icoec. 7.4 Idepede Paameeisaio Mehod The paameeisaio poblem of he equivale GPS daa pocessig mehod is sudied i deails i Xu 004. As meioed i Secio 7., he equivalece popey is valid ude hee implici assumpios. The fis oe is ha he ideical obsevaio veco L is used. The secod is ha he paameeisaio of X is ideical. The hid is ha he veco X should be able o be elimiaed. Ohewise he equivalece does o hold Xu 00. The fis equivale codiio is ecessay fo he exacess of he equivalece because of he fac ha hough fomig diffeeces he u-paied daa will be cacelled ou i he diffeecig daa pocessig. The secod equivale codiio saes ha he paameeisaio of he udiffeeced ad diffeecig models should be he same. This may be iepeed as: he ak of he udiffeeced ad diffeecig poblem should be he same if he diffeecig is fomed by a full ak liea asfomaio. If oly he diffeecig equaios ae ake io accou, he he ak of he udiffeeced model should equal he ak of he diffeecig model plus he umbe of elimiaed idepede paamees. I is well-kow ha oe of he clock eo paamees is liealy coelaed wih he ohes. This may also be see i he poof of he equivale popey of he double diffeeces, whee he wo eceive clock eos of he baselie may o be sepaaed fom each ohe ad have o be asfomed o oe paamee ad he elimiaed Xu 00. This idicaes ha if i 8 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

30 udiffeeced model all clock eos ae modeled, he poblem will be sigula i.e. ak defec. Ideed, Wells e. al. oiced ealy i 987 ha he equivalece is valid if measues ae ake o avoid ak defec i he bias paameeisaio Wells e al Which clock eo has o be kep fixed is abiay. Because of he diffee qualiies of he saellie ad eceive clocks, a good choice is o fix a saellie clock eo he clock is called efeece clock. I pacice, he clock eo is a ukow, heefoe hee is o way o keep ha fixed excep o fix i o zeo. I such a case, he meaig of he ohe bias paamees will be chaged ad may epese he elaive eos bewee he ohe biases ad ha of he efeece Xu 003 pp The hid equivale codiio is impoa o esue a full aked paameeisaio of he o be elimiaed paamee veco X. The udiffeeced Eq. is solvable if he paamees X ad X ae o ove-paameeised. I case of sigle diffeeces, X icludes saellie clock eos ad is able o be elimiaed. Theefoe, o guaaee ha he udiffeeced model Eq. is o a sigula oe, X i Eq. mus be o ovepaameeised. I case of double diffeeces, X icludes all clock eos excep he efeece oe. Hee we oice ha he equivale obsevaio Eq.8_ deoes he secod equaio of Eq.8 is equivale o he double diffeecig obsevaio equaio ad Eq.4_ deoes he secod equaio of Eq.4 is he elaed omal equaio. I adiioal double diffeecig obsevaio equaio, he ambiguiy paamees ae epeseed by double diffeecig ambiguiies. Recall he equivale popey, he umbe o ak of o liealy coelaed ambiguiy paamees i X mus be equal o he umbe of he double diffeecig ambiguiies. I case of iple diffeeces, X icludes all clock eos ad ambiguiies. The fac ha X should able o be elimiaed leads agai o he coclusio ha he ambiguiies should be liealy idepede. The wo equivale liea equaios should have he same ak. Theefoe, if all clock eos excep he efeece oe ae modelled, he umbe of idepede udiffeeced ambiguiy paamees should be equal o he umbe of double diffeecig ambiguiies. Accodig o he defiiio of he double diffeecig ambiguiy ad a special example of obsevaio equaio, he mehod of idepede paameeisaio ad he so-called daa codiios as well as he exeded way of double diffeecig fomig ae deived Xu Sequeial Daa Dealig ad Geomeic Illusaio I case of sequeial daa dealig, as soo as he cofiguaio of he obseved saellies do o chage gealy ad he eceives wok ideally, he liea coelaios bewee he bias paamees emai he same. Wih he oaio of he eah ad chagig of he cofiguaios of he saellies, he idepede bias paamee se ad he o be kep fixed paamee se ae chagig oo. If he o be fixed paamees ae deemied befoehad, he of couse hey should be fixed o he eal values, o i ohe wods, hey should be elaxed fom fixig codiios. Theefoe oe may iclude all bias paamees i he model fis ad meawhile oice which paamees ae ove-paameeised oes, he modify he ecod of ove-paameeisaio hough accumulaio of he sequeial oes of he sequeial equaios. I his way he ove-paameeised biases ca be exacly ideified ad kep fixed if sequeial soluios ae eeded. The efeece clock paamee ad he ambiguiies of he efeece saio have o be kep fixed i ay way. A aificial chagig of he efeece clock ad efeece saio will make he sequeial equaios icosise o each ohe. The easo why he efeece paamees have o be fixed lies i he aue of disace measuemes, which cao povide he daum oigi cf. e.g. Wells e al. 987 p9. Suppose d is k he diec measueme of clock eos of saellie k ad eceive i, i.e. di δ i + δ k, o mae how may obsevaios wee made ad how he idices wee chaged, oe paamee i.e. efeece 9 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

31 clock is isepaable fom he ohes ad has o be fixed. The easo why he ambiguiies ad he clock biases ae paly liea coelaed ca be similaly aalysed. Suppose h is he diec k k measueme of clock eos of saellie k ad eceive i ad ambiguiy N, i.e. h i δ i + δk + Ni, he umbe of ove-paameeised biases is exacly he umbe of oal obseved saellies ad used eceives. This esues agai ha he paameeisaio mehod o fix he efeece clock ad oe ambiguiy of evey saellie as well as oe ambiguiy of he efeece saellie of evey saio is easoable. The case of combiaio of d ad h as code ad phase obsevaios will be discussed below Coelaio Aalysis i Case of Phase-Code Combiaio A phase-code combied obsevaio equaio ca be wie by cf. Xu 003 p3: V L A V L A A X wpp0 0 ad P 0 X 0 wcp0. 0 whee L ad L ae he obsevaioal vecos of phase scaled i legh ad code, especively; V ad V ae elaed esidual vecos; X ad X ae ukow vecos of ambiguiy ad ohes; A ad A ae elaed coefficie maices; P 0 is a symmeic ad defiie weigh maix; w p ad w c ae weigh facos of he phase ad code obsevaios. The phase, code ad phase-code omal equaios ca be fomed especively by: whee N N N N X X R R T M wp + wc AP0 A wp + wc N, M M T T M wp AP0 A wp N, T wp AP0 A wpn M M X, N X Rc, B M M X B,, T AP0 wpl + wcl w pr B + w R ad B T wp AP0 L w pr. Covaiace maix is deoed c c M M Q Q Q Iv, 3 M M Q Q whee Gohad 978; Cui e al. 98 Q M MM M, M M M M Q, 4 Q M MQ ad Q M M Q. 30 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

32 i.e. Q Q Q wp + wc N wpnn N, w N wp wp + wc NN N p ad 5 N N wp + wc N wpnn N. So he coelaio coefficie C ij is a fucio of w p ad w c, i.e. C f w, w, 6 ij p c whee idices i ad j ae ukow idices i X ad X, especively. Fo w c 0 oly phase is used, X ad X ae paly liea coelaed ad w c w p X ad X ae ucoelaed hee exiss idices ij, so ha C f w, w 0 ad C f w, w w 0. 7 ij p c ij p c p Fo phase ad code combiaio, w c 0.0w p ca be seleced, so ha C f w, w 0.0w. 8 ij p c p Eqs.6, 7 ad 8 idicae ha fo he coelaed ukow pais ij he coelaio siuaio may o chage much by combiig he code o he phase because of he lowe weigh of he code elaed o he phase. A umeical es cofimed his coclusio. 7.5 Diffeeial Soluio of he Vaiace Equaio We deoe he age ad age ae fucio geeally by ρ ; hei paial deivaives wih espec o he obi sae veco ae give i Xu pp ad have he foms of ρ ρ ρ,, o. 9 &v X Theefoe, he obi paamee elaed pas i he lieaised GPS obsevaio equaios ae he T X ρ, &v ρ T y, o y, 30, &v y X y whee,,,, X T T X y X 0 Y y X 0 Y. y X, Y 0 X, Y ae he sae veco of saellie ad he paamee veco of he foce models, ad idex 0 deoes he elaed iiial vecos of ime 0. y is he oal ukow veco of he obi deemiaio poblem, he elaed coecio veco is y y y0, ad X 0 is he coecio veco of he iiial sae veco. The paial deivaives of X wih espec o y is called asiio maix which has he dimesio of 6 6+, whee is he dimesio of veco Y. The equaio of moio of he saellie is cf. Xu 003 pp.54 Eq Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

33 3 0 0 X X F X &. 3 The paial deivaives of he equaio of moio of he saellie wih espec o he veco y ae * + y F y X X F y F y X &, 3 whee he supescip * deoes he paial deivaives of F wih espec o he explici paamee veco y i F, ad B A E f m f m E X F D &, * G Y f m y F C, 33 whee E is a ideiy maix; he paial deivaives ae discussed ad deived i Xu 003, pp I is oable ha he foce paamees ae o fucios of. Theefoe he ode of he diffeeiaios i Eq.3 ca be exchaged. Deoig asiio maix by Φ, 0, he Eq.3 us ou o be,, 0 0 C D d d + Φ Φ. 34 Eq.34 is called diffeeial equaios of he asiio maix o vaiaioal equaios cf., e.g., Moebuck ad Gill 000. Deoig,,, Φ &, 35 a aleae expessio of Eq.34 ca be obaied by subsiuig Eqs.35 ad 33 io Eq.34,,, G d d B A d d The iiial value maix is iiial sae veco does o deped o foce paamees: E , Φ. 37 Tha is, i he GPS obsevaio equaios, he asiio maix has o be obaied by solvig iiial value poblem of diffeeial equaio 34 o 36. The poblem is adiioally solved by iegaio. Howeve, by caefully sudyig he iegao, oe may oice ha by iegaio fom 0 o seveal fucio values a he pis bewee 0 o have o be compued. This is o he same as he obi iegaio, i which he igh fucio is he fucio of kow foces. Hee he igh fucio icludes he ukow fucio Φ, 0. So wihou appoximaio, he iegao cao be applied diecly fo solvig he vaiaio equaio. This fac leads o a ew developme of a diffeeiaio mehod o solve he vaiaio equaio ad is give fis ime i Xu 003 pp Fo coveiece of eades, he algoihm is oulied below. Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

34 33 Eq.36 is a maix diffeeial equaio sysem of size Because A ad B ae 3 3 maices, he diffeeial equaios ae idepede fom colum o colum. Tha is, we eed jus o discuss he soluios of he equaios of oe colum. Fo colum j, he Eqs.36 ad 37 ae,,3, i G d d B A d d ij k kj ik kj ik ij, 38,,3, i j i ij ij ij δ δ &, j k if j k if kj 0 δ, whee idex ij deoes he elaed eleme of he maix. Fo ime ieval [ 0, ] ad diffeeiaio sep h 0 /m, oe has 0 + h,,,m ad,,3, + + i h d d ij ij ij ij,,,3,, + i h d d ij ij ij ij ij. 39 The Eq.38 us ou o be,,3,, i h ij ij ij ij ij &.,,3, i G h B A h ij k kj kj ik kj ik ij ij ij, 40 whee,,, m. Fo i,, 3 ad he sequeial umbe, hee ae hee equaios ad hee ukows of ime + ; so ha he iiial values poblem has a se of uique soluios sequeially. Eq.40 ca be ewie as R R R h B h E j j j, 4 whee j j j j j j j j j G G G h B h E A h E R R R. 4 Fo,, m, above equaios ae solvable. I is oable ha he hee maices + + h B h E A h E h B h E,, ae idepede fom he colum umbe j. The soluios of Eq.4 ae vecos j j j ad,...,, m j j j & & &, 43 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

35 whee he velociy veco ca be compued usig defiiio of Eq.39. Solvig he equaios of all colum j, he soluios of he iiial value poblem of Eqs.36 ad 37 ca be obaied. The eeded esuls ae elaed o ime, which ca be obaied by aveagig he values of + ad -. This diffeeial algoihm fo solvig he vaiaio equaio ca be used as iiiao of he iegaio mehod fo solvig he same poblem o ca be used o eplace he iegaio mehod ad o solve he vaiaio equaio diecly. 7.6 Adjusme Models of he Sola Radiaio ad Amospheic Dag Based o he commoly used sola adiaio ad amospheic dag models of he obi heoy, a so-called disubace coodiae sysem is ioduced ad he o be adjused models ae poposed. A simulaio sudy show ha i is vey advaageous o descibe he models i he disubace coodiae sysem. The adiioal models o be adjused ae paly o well fomulaed ad he bias paamees ae paly ove-paameeised. The ew models ows much bee abiliy o absob he u-modelled eos wih fewe paamees. The ew models ae he implemeed i a sofwae ad show excelle pefomace Ioducio of he models Sola adiaio pessue model Sola adiaio pessue is foce acig o he saellie s suface caused by he suligh. The adiaio foce ca be epeseed as Seebe 993 S su su fsola mγ Ps C m ad 44 su su whee γ is he shadow faco, P s is he lumiosiy of he Su, C is he suface efleciviy, S/m is he suface o mass aio of he saellie, ad su ae he geoceic vecos of he saellie ad he Su, he elaed disaces ae ad su, is he ideiy veco poied fom he Su o he saellie. Usually, P s has he value of Newo/mee, C has values fom o, is fo he complee absopio of he suligh, ad fo alumiium, C.95. Because of he complex shape of he saellie, he use of cosa efleciviy ad homogeous lumiosiy of he Su, as well as he exisece of idiec sola adiaio efleced fom he Eah s suface, he model of Eq.44 is o accuae eough fo pecise puposes ad will be used as a fis ode appoximaio. A adjusme model fo fiig sola adiaio effecs ca be epeseed as cf. Beule e al. 994; Rohache & Meva 996 f sola foce f sola a + a a 3 a a a 3 a a a cosu. 45 si u 34 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

36 Whee 9 paamees ae used o model he sola adiaio foce eo fo evey saellie. Tha is, oe bias ad wo peiodic coefficies ae used i evey coodiae diecio o model ad absob he u-modelled eos. Amospheic dag model Amospheic dag is he disubace foce acig o he saellie s suface caused by he ai. Ai dag foce ca be epeseed as Seebe 993, Liu ad Zhao 979 C S f d m & & & & ai dag σ ai p ad p 46 m & & whee S is he coss secio o effecive aea of he saellie, C d is he dag faco, m is he mass of he saellie, & ad & ai ae he geoceic velociy vecos of he saellie ad he amosphee, ad σ is he desiy of he amosphee, p is he amospheic dag ideiy veco of he & & ai. Usually, S has a value of / 4 of he oue suface aea of he saellie, ad C d has labou values of. ±0.. The velociy veco of he amosphee ca be modelled by y & ai k ω kω x, 47 0 whee ω is he agula velociy veco of he Eah s oaio, ad ω ω, k is he amospheic oaio faco, ad x, y, z T. Fo he lowe laye of he amosphee, k, i.e., he lowe laye of he amosphee, is cosideed oaig wih he Eah. Fo he highe laye, k., because he highe ioosphee is acceleaed by he Eah s mageic field. A adjusme model fo fiig he amospheic dag foce model ca be epeseed as simila wih he Eq. cf. Xu 003 p38 Eq..85 f ai dag f dag b + b b 3 b b b 3 b b b ai cosv. 48 si v Whee v ω + β, ω is he agle of peigee of he saellie, β is he agle bewee adial ad ageial diecios of he saellie obi. Because β is o a cosa fo ay ellipse obi, cosu ad siu ae idepede fom cosν ad siν. Howeve, he six bias paamees ae pai-wise liealy coelaed. Oly hee of he biases ca be deemied. I his way he wo models ae paly isepaable. Fuhe poblem is ha he models ae vey empiical oes which se he mai disubace fequecy as ha of he moio of he saellie The disubace coodiae sysem ad eo models Sola adiaio foce veco is poied fom he Su o he saellie. If he shadow faco is compued exacly, he lumiosiy of he Su is a cosa, ad he suface efleciviy of he 35 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

37 saellie is a cosa, he he legh of he sola foce veco ca be cosideed as a cosa, because ad su su su su, 49 + su su ± su su ± su su m su ±... m su m 3 0 5, 50 whee ad su ae he geoceic disaces of he saellie ad he Su, especively. Ay bias eo i P s, C ad S / m may cause a model eo of α f sola, whee α is a paamee. So he α f sola ca be cosideed as mai eo model of he sola adiaio. Because he aio of he geoceic disaces of he saellie ad he Su is so small so ha he diecio ad disace chages of he Su-saellie veco ae egligible. Wih he moio of he Su, he sola adiaio foce veco chages is diecio wih he ime i he ECSF Eah-Ceed-Space-Fixed coodiae sysem ca. degee pe day. Such a effec ca oly be cosideed as a small dif, o a peiodical chagig fo he obi deemiaio. To model such a effec i ECSF sysem oe eeds hee bias paamees i hee coodiae axes, ad hee dif ems isead of a few peiodical paamees. I is obvious ha o model such a effec i he diecio of, jus oe paamee α is eeded. Ad we see ha he adiioal sola adiaio adjusme model may o able o fi he mai eo model well i ode o absob he effec. Theefoe, i is vey advaageous o defie a so-called disubace coodiae sysem as follows: he oigi is he geo-cee, he hee axes ae defied by edial veco of he saellie, he Su-saellie ideiy veco ad p he amospheic dag ideiy veco. These hee axes ae always i he mai disubace diecios of he idiec sola adiaio efleced fom he Eah s suface, diec sola adiaio ad amospheic dag, especively. This coodiae sysem is o a Caesia oe ad he axes ae o ohogoal o each ohe. The paamees i idividual axes ae maily used o model he elaed disubace effecs, ad meawhile o absob he emaied eo of ohe u-modelled effecs. I he amospheic dag model Eq.3, he velociy veco of he amosphee is always pepedicula o he z-axis of he ECSF coodiaes ad he saellie velociy veco is always i he ageial diecio of he obi. The vaiaio of he em & & deoed by g is domiaed by he diecio chages of he velociy vecos of he saellie ad he amosphee. Ay bias eo i S effecive aea of he saellie, C d dag faco ad σ desiy of he amosphee may cause a model eo of µ f dag, whee µ is a paamee. So he µ f dag ca be cosideed as mai eo model of he u-modelled amospheic dag. Fo simplificaio ou discussio, we coside he velociies of he saellie ad amosphee ae cosas, ad call he saellie posiios wih maxz ad maxz he highes ad lowes pois, especively. Saellie a he lowes poi, he wo velociy vecos ae i he same diecio ad heefoe he g eaches he miimum. A he ascedig ode he wo vecos have he maximum agle of icliaio i ad he g eaches he maximum. The g eaches he miimum agai a he highes poi ad eaches he maximum agai a he descedig ode, ad a he ed eaches he miimum a he lowes poi. I is obvious, besides he cosa pa, g has a domia peiodical compoe of cosf ad sif. Whee f is he ue aomaly of he saellie. So we see ha he adiioal amospheic dag adjusme model may o able o fi he mai eo model well i ode o absob he effec. ai 36 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

38 7.6.3 Numeical simulaio ad models aalysis Numeical simulaed daa of he sola adiaio eo model eo model α f sola ad he amospheic dag µ f dag ae caied ou based o a simulaed GPS saellie obi ad sola obi daa. Kepleia elemes fo GPS saellie ae seleced: semi-majo axis a 600km, icliaio i 55, ascedig ode Ω abiay, agle of peigee ω abiay, ecceiciy e , ue aomaly f 0 π. The obial peiod T is assumed as hous ad Kepleia elemes fo he Su ae seleced: semi-majo axis a AU asoomical uis, icliaio i 3, ecceiciy e 0.067, ue aomaly f f πt/yea i hous, f 0 is a abiay cosa bewee 0 π. Theoeically speakig, i is obvious ha he eos compued fom above wo eo models ca be compleely absobed by he wo model paamees α ad µ. The he dada i hee axes of he ECSF coodiae sysem ae specal aalysed o fid ou he fequecy popeies. FFT fas Fouie asfomaio mehod ae used. The heoeical aalysis abou he popeies of he mai eo models ae vey well show i he FFT esuls. Based o umeous aalysis of he daa, coclusios ae obaied. The sola adiaio pessue eo model ca be epeseed aleaively by a b α f sola a b, 5 a3 b3 whee b-ems ae vey small. The amospheic dag eo model ca be epeseed aleaively by µ [ a + bϕ ω cos f + cϕ3ω cos3 f + dϕ ω cos f ]p, 5 whee f dag sikω if coskω 0 ϕ k ω, k,, 3 53 if coskω 0 coskω whee ω is he agle of peigee ad f is he ue aomaly of he saellie, a, b, c ad d ae model paamees o be deemied. Accodig o he simulaio, a-em ad b-em ae he mos sigifica ems. The amou of d is jus abou % of he amou of c, ad he amou of c is abou % of ha of b Summay ad Coclusios I is advaageous o epese he sola adiaio pessue ad amospheic dag eo models i he so-called disubace coodiae sysem. The mai eo models ae cosideed heoeically o be able o be peseed by models α f sola ad µ f dag especively. The simulaio sudy show ha he adiioal models ae vey empiical oes ad ae o able o be used eve fo 37 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

39 fiig he mai eos, ad he peiodical ems ae eihe o easoable oes o wih wog fequecies. Two aleaive adjusme models epeseed by paamees i hee axes of he ECSF sysem ae poposed based o iesive aalysis. The paamees used ae fewe ad wihou poblem of ove-paameeisaio. Fo he emaiig eo model, he modulaio shall be based o a esidual aalysis isead of empiical seig. The poposed models ae implemeed i sofwae ad show good pefomace. 7.7 Iellige Kalma Fileig Techique Based o he followig ioducio of Kalma file ad he file usig velociy ifomaio, he eeds of he iellige Kalma fileig will be discussed ad he algoihm will be oulied Ioducio of Kalma File The piciple of he classical Kalma file ca be summaised as below Yag e al. 999: The lieaised obsevaio equaio sysem ca be epeseed by V i Li Ai X i, i P, 54 whee L : obsevaioal veco of dimesio m, A : coefficie maix of dimesio m, X : ukow veco of dimesio, V : esidual veco of dimesio m, : umbe of ukows, m : umbe of obsevaios, i : sequeial idex, i,,3,, ad P i : weigh maix of idex i. Suppose sysem equaios ae kow ad ca be peseed as whee U i X i Fi, i X i, i, 3,K, 55 F : asiio maix of dimesio, ad U : esidual veco of dimesio. U ad V ae u-coelaed ad have zeo expecaios. Usig he covaiace popagaio law, oe has fom Eq.55 T X i Fi, i Q X i Fi, i QU. 56 Q + The omal Eq.54 ca be fomed as M X B. 57 i i i Fo he iiial sep o epoch, i.e., i, Eq.54 has he soluio ude he leas squaes piciple ~ i Qi Bi, whee Qi M i X, Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

40 ad hee oe will assume ~ Q i Q i, 59 whee X ~ i ad Q ~ i ae called esimaed values. Usig he esimaed values ad asiio maix, oe ca pedic he ukow values ad covaiace maix of he ex epoch say i : ~ X F X ad 60 whee i i, i i ~ Q + T F i i, i Qi Fi, i QU, 6 X i ad Q ae called pediced values veco ad maix. The esimaed values of his i epoch ca be calculaed by ~ X X + K L A X, 6 i i i i i i i i ~ Q E KA Q ad 63 T T i i i i i V K Q A A Q A + Q, 64 V i whee K is he gai maix, Q P. Fo he ex sequeial sep i, he pediced values have o be compued by usig Eqs.60 ad 6, ad he esimaed values ca be compued by usig Eqs.6 ad 63. Such a ieaive pocess is called Kalma fileig. I classical Kalma fileig, i is assumed ha fo he poblem of Eq.54 hee exiss a sysem asiio maix F i,i i Eq.55 ad he cofaco Q U. Theefoe, he esimaed values i he Kalma file pocess ae depede o F i,i ad Q U. The asiio maix shall be based o segheed physical models, ad he cofaco shall be well-kow o easoably give. If he sysem descipio is accuae eough, of couse Kalma fileig will lead o a moe pecise soluio. Howeve, if he sysem is o sufficiely well-kow, he esuls of Kalma file will someimes o covege o he ue values divegece. Fuhemoe, a kiemaic pocess is geeally difficul o be pecisely epeseed by heoeical sysem equaios. Howeve, fo a dyamic pocess like o-boad GPS fo saellie o saellie ackig o obi deemiaio he sysem equaio ca be well-fomulaed by a obial equaio of moio. Aohe poblem of Kalma fileig is he sog depedecy of he give iiial values. May sudies have bee made i his aea o ovecome he above-meioed shoages. The developme of he heoy leads o a so-called adapive obus Kalma file i which Eq.6 ca be peseed by Yag e al. 00 ~ T T X i X i + QX Ai Ai QX Ai V i i + α Q L A X. 65 i Whee α is he adapive faco o adapive diagoal maix ad Q V is he obus weigh maix. Robus weigh ca be used o adjus he weigh of obsevaios accodig o he esiduals ad he adapive maix ca be used o adjus how much he pas ifomaio shall be ake io accou i he fileig. i i 39 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

41 7.7. Kalma Fileig Usig Velociy Ifomaio As applied i KSGSof Xu e al. 998, velociy ifomaio fom he diffeeial Dopple ca be used o descibe he sysem which is eeded i Kalma fileig. Whehe he eceive is movig o esig, he diffeeial Dopple icludes ifomaio abou he moio sae of he eceive. Theefoe, usig velociy ifomaio as a sysem descipio should be sigificaly bee ha ay empiical model. The piciple of Kalma fileig usig velociy ifomaio ca be oulied as follows Xu 003: Fo he iiial o pediced veco Z, he omal equaio of he phase obsevaio equaio ca be fomed by X M z Z Bz, Z, 66 N whee M z is he omal maix, ad B z is he veco o he igh side of he equaio. These ae fomed by usig iiial veco Z ; Z icludes sub-veco X coodiaes ad N ambiguiies ad ohes. The esimaed soluio of Eq.66 is he ~ ~ ~ Z Q B, Q M. 67 z z z z The omal equaio of he diffeeial Dopple obsevaio equaio cf. Xu 003, Eq.9.50 pp.97, oly he velociy veco is ukow ca be fomed by M & X &, 68 x B x & whee X & is he velociy veco of he eceive; i is also used as a idex o deoe he elaed omal maix ad veco o he igh side of he equaio. The soluio of Eq.68 is he X &. 69 Qx& Bx&, Qx& M x& Thus fo he ex epoch, deoed as k, he pediced veco us ou o be ~ Z k Z k + Z& k, 70 whee is he ime ieval of he epoch k ad k, ad X& k Z& k. 7 0 Equaio 70 idicaes ha he diffeeial Dopple has o be used i Eq.69 as obsevaios, because he velociy is cosideed a aveage oe hee. The elaed covaiace maix of he pediced veco is he ~ Q k Q z The weigh maix is z k + z k Q 0 x& P k Q. 73 z The omal Eq.66 of epoch k is 40 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

42 M z k Z k Bz k, 74 ad he Kalma file soluio of Eq.74 is he ~ ~ Z ~ k Qz k Bz k, Qz k M z k + Pz k. 75 I is oable ha he omal equaio of Eq.74 mus be compued usig he pediced veco Z k of Eq.70. Repeaig he seps fom Eq.68 o 75 fo he fuhe epoch is a pocess of Kalma fileig usig velociy ifomaio. The algoihm oulied above is suiable boh fo he kiemaic ad saic daa pocessig. This is ue especially fo saic daa pocessig, because he saio has o bee exacly assumed as fixed as descibed by Eq.68; such a algoihm will modify he popey of he sog depedecy o he iiial value of he Kalma file. The fomig of omal Eq.68 is a ieaive pocess, i.e., he velociy ifomaio has o be used fo fomig he equaio. Equaio 68 epeses a ealisic sysem descipio The Theoeical Poblem vs. Pacical Requieme Kalma file has bee boadly used i eal ime o sequeial daa pocessig, howeve almos all of hem ae case sudies i avigaio aea. The easo why he o-had Kalma fileig echiques cao be geeally used i he daily GPS avigaio ad moioig is a heoeical oe. The GPS obsevaio equaio descibes ha he posiio ukows will be deemied hough measuemes, wheeas he sysem equaio says ha, well, we do o kow he posiio ukows, bu we do kow he posiio chagig egulaio. Ad his is o ue i mos cases. If oe cosides he coodiae chage as a moio, he oly he moio ude he acig of he cosevaive foce will obey he moio law of Newo. As soo as hee exiss o cosevaive ad o egligible ukow disubace, he moio will be vey had o be descibed by a equaio give by pesos. Fo example, a peso dives a ca alog a sai see wihou ay affic lamp ad ohe affics wih a cosa velociy. The moio ca be descibed easily ad he accumulaed pas ifomaio epeses he smooh moio ad ca be used o esimae he moio i he fuue. Howeve, ow, he ma sees he ame of a side see ad emembes ha oe of his old fieds lives i he ea; he he ma decided o make a visi o his fied ad slowed dow ad soped he ca quickly ad deived his ca i backwad diecio a few mees ad soped, he ued o he igh. I such a case, he moio is o decided by ay equaio, bu by he idea of he dive. I is ealy impossible o y o use some equaios o descibe he faasy of he people. The aemp o y o foge he pas ifomaio adapive i such a case is a coec measue, howeve afe he eve, he fileig pocedue will sa agai o use some kids of empiic equaios o descibe he moio i he fuue; ad his cao be geeally wok well. To use he velociy ifomaio o descibe he sysem equaio of he moio is a gea sep fowad i he heoeical developme of he Kalma fileig echique. This echique was developed by a coopeaio bewee auho ad D. Yag duig Yag s gues say i GFZ i 997 Xu 003. The GPS phase obsevable came always fom a ieal Dopple measueme, o mae he Dopple is oupued fo use o o. So ha as soo as hee is o cycle slip happeed, he diffeeial Dopple ca always be deived fom he phase measuemes afewad, ad he diffeeial Dopple descibed a aveage moio of eceive duig he daa samplig epochs. Tha is he sysem equaio i such way is deemied meawhile by he addiioal measuemes. Fo a o acceleaed moio such a descipio would be good eough, 4 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

43 howeve, i case of high kiemaic moio, he acceleaio o eve highe ode acceleaio have o be ake io accou. We do o have a few uivesal equaios which could be used o descibe ay kid of moio, so he Kalma fileig echique cao be used geeally i ou avigaio ad moioig pacice. Howeve, if we by he way of suveyig, meawhile use he possibiliies of GPS o ohe addiioal sesos o measue ad deemie he sysem equaio, he we have desiged a uivesal Kalma file fo GPS daa pocessig ad we call such a file as a iellige Kalma file. The moe ad moe measueme ae ow made i a eal ime mae o moio he eal ime moio. The developme of a iellige Kalma file is a uge desie of he avigaio ad posiioig pacice especially fo he applicaio of GPS ad Galileo sysems Algoihm of Iellige Kalma File The adapive obus Kalma file possess aleady a kid of iellige. I ca adjus he weigh auomaically due o he eal esiduals of he obsevaio equaios called obus. I ca decide if o o ad how o adap he pas accumulaed sysem ifomaio called adapive. Howeve, he sysem equaios ae assumed kow ad fixed, ad o mae how he moio is, he sysem equaios will be always used. This is physically o coec ad heefoe may o be valid uivesally. Kalma file usig velociy ifomaio i GPS applicaio makes i possible fis ime o deemie he sysem equaio isead of o defie he sysem equaio empiically. Eve his is jus a appoximaio o a aveage moio duig he samplig epochs, howeve, his is a fis ode appoximaio ad could be well fi o he o acceleaed moio. The key poi is ha i his way he sysem equaio could be deemied ad updaed auomaically. Fo a geeal moio, especially he high kiemaic ad high dyamic disubed moio, he sysem descibed by a aveage moio could be o eough. Theefoe, a iellige Kalma fileig is o be able o adjus he weigh obus ad o adap accodigly he pas ifomaio adapive, ad o deemie he sysem equaio geeally iellige. The aveage velociy is a kid of smoohed velociy. Wihi he wo samplig epochs, he aveage moio is o pecise eough fo a acceleaed moio. Howeve, he acceleaio deived fom he wo adjace aveage moio may have high qualiy because of he smoohig. So we may use he pas ifomaio of he aveage velociy o deive he acceleaio o eve highe ode oes. The poblem is ha i his case he sysem equaio will be o a liea oe. This will be o a poblem fo pedicig he ex values of he ukows, howeve a poblem fo pedicig he ex covaiace maix of he ukows. The mehod of sysem equaio deemiaio by usig GPS fo geeal avigaio ad posiioig poblem has o be caefully sudied ad he cocep of such a iellige fileig has o be woked ou. How he saisic ifomaio will be popagaed ad he pecisio ca be esimaed exacly have o be sudied. I geeal, he so-called iellige Kalma fileig is a o liea pocess. The pecisio esimaio is impoa because i is he cieio fo he obus weigh ad adapaio as well as he judge of goodess of he sysem deemiaio. A well desiged MFGsof which povide sable ad homogeous soluios wihou usig empiic ad paly u-physical a pioi ifomaio equies a caeful sudy of such a iellige Kalma file fo is applicaios. 4 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

44 7.8 Fomig Idepede Baselies ad Exeded Double Diffeecig I ode o exed he way of double diffeecig fomig, we have o discuss fis how o fom a idepede ad opimal baselie ewok. I is well-kow ha fo a ewok wih saios hee ae - idepede baselies. A idepede baselie ewok ca be saed i wods: all saios ae coeced hough hese baselies, ad he shoes way fom oe saio o ay ohe saio hough hese baselies is uique. Geeally speakig, a shoe baselie leads o a bee commo view of he saellies. Theefoe he baselie should be fomed so ha he legh of he baselie falls as sho as possible. Fo a ewok, a opimal choice should be ha he summaio of weighed leghs of all idepede baselies should be miimum. This is exacly a mahemaic poblem called miimum spaig ee cf. e.g. Wag e al Thee exiss algoihms ad sofwae o solve his miimum spaig ee poblem. Theefoe we jus show a example hee. A IGS ewok wih ca. 00 saios ad he elaed opimal ad idepede baselie ee is show i Fig. The aveage legh of he baselies is ca. 300 km. The maximum disace is ca km. Fig.: Opimal idepede baselie ewok I he adiioal double diffeecig model, he u-paied GPS obsevaios of evey desiged baselie have o be omied because of he equieme of diffeecig fomig. The example i Xu 004 show ha, if he diffeecig is o limied by baselie desig, may obsevaios have o o be cacelled ou by diffeecig fomig. A opimal way of double diffeecig fomig should be based o a opimal baselie desig o fom he diffeecig fis, he, wihou limiaio of he baselie desig, o check fo he u-paied obsevaios i ode o fom possible diffeecig. This measue is useful fo isig he daa usig ae by diffeecig mehod ad fo a bee fulfilme of he fis equivale codiio he same daa should be used. 43 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

45 7.9 Sadad Models ad Algoihms The sadad models ad algoihms used ae summaised ad heoeically caefully veified i Xu 003 accodig o he expeiece of KSGSof ad lae iesive sudy. Fo coveiece of he eades, he elaed coes i he lieaue ae lised as follows. Models: opospheic models pp.5-57; ioospheic model pp.39-50; elaiviy model pp.57-6; Eah ide model pp.6-65; ocea loadig ide model pp.68-7; saellie mass cee model pp.78-8; sola adiaio model pp.3-34; amospheic dag model pp ; geopoeial disubace pp.3-6; idal poeial disubace pp.8-30; muli-bodies disubace pp.37; dyamic obi fiig model pp Algoihms: equivale algoihm pp.3-34, 07-7,93; udiffeeced algoihm pp.00, 09, 9-9; diffeecig algoihms pp.0-07; diagoalisaio algoihm pp. 7; idepede paameeisaio see above 7.4, Xu 004; opimal baselies ad daa codiios see above 7.4, Xu 004; cycle slip deecio pp.5-53; ambiguiy iiialisaio see souce code; ambiguiy seach cieio pp.53-7; leas squaes adjusme pp.9-; blockwise elimiaio pp.7-3; sequeial leas squaes pp.3; a pioi ifomaio 44-47; Kalma file pp.35-43; iellige Kalma file see above 7.7; diffeeial soluio of vaiace equaio pp.56-57; obi iegaos pp.57-59; diffeeial Dopple ad velociy deemiaio pp.98-99, 53, 95, sigle poi posiioig pp Tools: coodiaes asfomes pp.3-3; ime sysems asfomes pp.3-5; he Su-Mooplaes obi ceao pp.40-43; boadcas obi asfome pp.8-30; iepolaio ad iegaio pp.30, 57; Gauss-Joda ad Cholesky ivesios; Helme asfomaio pp.6-7; mappig fucios pp.47-49, 55-56; fligh sae compuaio pp.08-09; specal aalysis mehods Xu 99; saisic aalysis; gaphic epeseaio; 8. Numeical Examples 8. Daa ad Saio Newok Fo es u of he MFGsof, he IGS GPS daa of 47 saios measued o May s 00, ad saic/kiemaic GPS daa as well as CHAMP o-boad GPS daa ae used. A gaphic of he used IGS saio ewok is give below Fig.. 44 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

46 Fig.: Used IGS Saio Newok The umeical example of a IGS ewok wih 47 saios ad oe day s obsevaios show ha 87.9% of all daa is used i double diffeecig fomig based o he opimal baselie desig, wheeas 99.% of all daa is used i he exeded mehod of double diffeecig fomig. 8. Numeical Examples ad Ieal Tes Numeous examples ae compued usig MFGsof o have boad ieal ess. They ae lised below ad oulied biefly. 8.. Soluios i ECEF ad ECSF coodiae sysems Fo he global ewok moioig poblem wih fixed obi, he soluios ae compued i ECEF ad ECSF coodiae sysems, especively. The soluios ae he same excep egligible diffeeces. This idicaes ha he coodiae ad vaiables asfomaio as well as he coecios ae well doe i he wo sysems. 8.. Equivale Popeies Fo he global ewok moioig poblem wih fixed obi, he soluios ae compued wih equivale algoihm, u-diffeeial algoihm ad double diffeecig algoihm, especively. The soluios ae geeally he same excep egligible diffeeces. These esuls esue ha he equivale heoy hold umeically oo. 45 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

47 8..3 Daa Codiio ad Idepede Paameeisaio Fo he global ewok moioig poblem wih fixed obi, he soluios ae compued wih ad wihou he daa codiio ad idepede paameeisaio, especively. The ivese pocess ae sable wih daa codiio ad idepede paameeisaio ad he esuls ae homogeous. Wheeas wihou he codiio ad idepede paameeisaio, he ivese pocess is vey isable ad qui ofe fails by ivesio of he omal maix ad he esuls ae ihomogeeous ad obviously icoec Phase ad Phase-Code Soluios Fo he global ewok moioig poblem wih fixed obi, he soluios ae compued fo phase oly soluio ad phase-code combied soluio, wih ad wihou daa codiio, especively. The soluios show ha he phase-code combiaio does o chage much he sigulaiy poblem ad cofimed he heoeical aalysis meioed i Secio Wih he daa codiio ad idepede paameeisaio, he soluios ae a lile bi diffee i a ode of ca. --%. This may be explaied as he cosequece of he lowe weigh of he code Velociy Deemiaio Kiemaic ad saic esuls of a local ewok ae o give hee due o he easos. Velociy deemiaio of a saic saio is give i Fig.3. The mea value ad sadad deviaio ae boh less ha cm/secod. Fig.3: Kiemaic velociy of a saic saio blue-v x, gee-v y, ed-v z, uis: m/s 46 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

48 LEO o-boad GPS deemied velociy is give i Fig.4 wihou exeal compaiso. Fig.4: LEO saellie o-boad GPS deemied velociy blue-v x, gee-v y, ed-v z, uis: m/s 8..6 Regioal Newok Moioig wih Obi Coecio Solvig a egioal a pa of a global ewok moioig poblem wih obi coecio is ealised by usig MFGsof. The soluio pocess is sable ad he esuls ae homogeous. Howeve, he esuls ae obviously depeded o he model used fo he obi coecio. How o model he obi coecio of a egioal ewok poblem has o be fuhe sudied ad he expeiece has o be added o he sofwae Global Newok Moioig wih Kiemaic/Dyamic Obi Deemiaio Solvig a global ewok moioig poblem wih kiemaic/dyamic obi deemiaio is ealised by usig MFGsof. The obi is deemied by a geomeic way ad fied wih a dyamic model. Geeally speakig, he obi soluios ae smoohe ha ha of he kiemaic oes. The daa pocessig is simple wihou he complicaed dyamic foce model ad iegaio. The esuls ae o compaed exeally Global Newok Moioig wih Dyamic Obi Deemiaio Fo he global ewok moioig poblem wih dyamic obi deemiaio, he fucio is ealised i MFGsof. The vaiace equaio ae solved usig a diffeeial mehod oulied i Secio 7.5. The soluio show he poblems of he adjusme model of he sola adiaio. A sudy is give i Secio 7.6 ad he model is implemeed i he MFGsof ad show good pefomace. The soluio is o exeal compaed. 47 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

49 The sofwae is able o u i a sequeial way, a epoch-block wise way ad a oal pos pocessig way. The epoch-block wise soluio is eeded fo fileig Time Cosumig The global ewok moioig poblem wih fixed obi ca be solve i hee ways. The ime cosumig is lised hee. Fo a epoch-block soluio, po hou oe soluio, i ook miues. Fo daa of oe day i ook.5 -- hous, ha is ca. secods fo po epoch daa pocessig. Fo a sequeial soluio, po hou oe soluio, i ook 0 5 miues. Fo saa of oe day i ook 3 hous, ha is ca. 3 secods po epoch daa pocessig. Fo a oal soluio po day oe soluio, i ook 4 5 hous, ha is ca. 6 secods po epoch daa pocessig. The ime cosumig esuls have oly elaive sese of compaiso. I depeds o he compue ad he use umbe of he compue ec. The daa is ecoded befoe ioducig he daa codiio ad implemeig he idepede paameeisaio algoihm. Checkig fo he daa codiio ad paameeisaio ae ime cosumig vey iesive. 8.3 Exeal Compaisos Exeal compaisos ae mos ieesed opic ad impoa ask fo a ew sofwae. The basis fo a meaigful compaiso is ha he adjusme ad fileig piciples used shall be equivale, he daa used shall be ealy he same, he paameeisaio mehods shall be equivale, ad if he a pioi ifomaio is used he hey shall be similaly used. The a pioi ifomaio such as coodiae pecisio, may be obaied by aalysis of a log em sable soluios obaied wihou a pioi ifomaio. Howeve as kow, mos u-diffeeial GPS scieific sofwae ae dealig wih a sigula equaio because of he paameeisaio poblem, heefoe wihou a pioi ifomaio he sofwae ae o able o poduce sable soluios i pacice. So he chai bewee soluios ad he used a pioi ifomaio is o closed. Befoe he sofwae ca sably solve he poblem wihou a pioi ifomaio, oe may geeally do o able o have he a pioi ifomaio fom he soluios. I u, wihou a pioi ifomaio he sofwae ae o able o solve he poblem sably o obai homogeous soluios. I his aspec, MFGsof is he fis sofwae which is able o povide he sable soluios wihou a pioi ifomaio ad iu is able o ge he a pioi ifomaio hough a log em daa pocessig. Because of diffee paameeisaio mehods, he paamees have paly also diffee physical meaigs. The clock bias is he bias of he clock elaed o he efeece oe ad icludes he elaed ambiguiy bias. Fo saellie clock, he ambiguiy bias is depeded o he selecio of he elaed efeece saio of evey saellie, ad fo eceive clock, he ambiguiy bias is depeded o he selecio of he efeece saellie of evey saio. All paamees of he efeece saio ae cosideed as eo-fee o coecio-fee i MFGsof due o he aue of he disace measueme of he GPS. Due o above easos a successful exeal compaiso is o possible up o ow. The umeical esuls cofim above discussio. Ideed, abiay esuls ca be poduced by usig diffee empiical ifomaio. Theefoe he a pioi ifomaio mus be a easoable oes. 48 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

50 9. Summay A muli-fucioal GPS sofwae based o he equivale daa pocessig algoihm is developed. The sofwae, equipped wih a algoihm of idepede paameeisaio, may povide a sable soluio ad homogeous esuls wihou ay a pioi ifomaio. The fucioal abiliies of he sofwae eached fom he local, o he egioal ad global ewok applicaios ad is able o pocess he daa wih fixed obi, obi coecio, kiemaic/dyamic combied ad dyamic obi deemiaio i eal ime ad pos pocessig modules. The pecisio of he esuls, compaed wih he commo epoed oes, ae sill o saisfacoy. The sofwae has o u wih much moe umeical daa o lea he ielligece ad o modify is qualiy of fucioaliy. This sofwae suely ca be a good basis fo a excelle GPS sofwae. Ad suely is a good simulaio sofwae fo he Galileo sysem sudy o simulae he daa ad he o solve he avigaio ad posiioig poblem of he Galileo applicaios. Ad suely will be a good basis fo a GPS ad Galileo combied sofwae. 0. Ackowledgemes Gaefully ackowledged ae he suppo fom Pof. D. D. Ch. Reigbe. Wihou his ecouageme he developme wok may eve each his saus. Scieis Abbas Kha of KMS Demak is haked fo may discussios coceig he ocea loadig compuaio ad fo his kidly povidig us he ocea loadig compuig ouies. Pof. D. JiaCheg Li ad D. ZhegTao Wag of Uivesiy Wuha ae haked fo hei cowok as scieiss i GFZ fom 00 o 003. They ae maily esposible fo he eseach ad sofwae developme of he foce models of dyamic obi deemiaio ad he umeical iegaio ools as well as he iegaio ad exesive umeical ess of his sofwae. May valuable ad iesive discussios have bee held wihi he eam. Specially woh meioig is ha hough hei wok a misake i he heoeical descipio of he vaiaio equaio is coveed. They also poied ou ha hee ae may iiial values cao be compued by solvig he vaiaio equaio usig iegaio mehod as adiioally did. This leads o he ew developme of he algoihm of diffeeial soluio of he vaiaio equaio. They eviewed also pas of he exbook GPS Theoy, Algoihms ad Applicaios. Some of he fomulae of he iegaio mehods ae also coeced ad checked hough umeical ess. They coibued gealy fo his eseach ad developme wih hei excelle wok. The heoeical sudy of he sofwae developme is caied ou i 00 ad a leas leads o a deailed descipio of he dyamic obi deemiaio heoy by usig GPS. Afe oe ad a half yeas wok, he pogam developme is fiished a he middle of 003, howeve, due o he paameeisaio poblem, which has eve bee caefully sudied befoe, he sofwae us isable ad he esuls ae ihomogeeous. Ude he exeme pessue of failig o be able o poduce esuls wih he sofwae, he ime which I go o eseach ad fially leads o a udesadig ad solvig of he poblem a he middle of 004 mus be ackowledged. Wihou udesadig ad solvig he paameeisaio poblem, his maual will be eve bo. This sudy was caied ou ude he ga of he Gema Fedeal Miisy of Educaio ad Reseach BMBF No. 50EP Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

51 . Refeeces Abamowiz M, Segu IA 965 Hadbook of mahemaical fucios. Dove Publicaios, Ic., New Yok Adese OB 994 M,, ad S,, ocea ide models fo he Noh Alaic Ocea ad adjace seas fom ERS- alimey, Space a he sevice of ou eviome. I: Poceedigs of he secod ERS- symposium, Hambug, 4 Ocobe 993, Vol.., Jauay 994, Noodwijk, pp Balmio G, Schama E, Seeuw N 996 Compaibiliy of fis-ode cicula obi peubaios heoies: Cosequeces fo coss-ack icliaio fucios. J Geodesy 70,9: Beule G 996 GPS saellie obis. I Kleusbeg A, Teuisse PJG eds GPS fo geodesy. Spige-Velag, Beli Beule G 996 The GPS as a ool i global geodyamics. I: Kleusbeg A, Teuisse PJG eds GPS fo geodesy. Spige-Velag, Beli Bow R.G. ad P.Y.C. Hwag 99 Ioducio of Radom Sigals ad Applied Kalma File, Secod Ediio, Joh Wiley & Sos, Ic., New Yok Baue M. 994 Vemessug ud Oug mi Saellie, Wichma Velag, Kasluhe, i Gema. Cao M.E., Lachapelle G., Szames M., Hebe J., Keih J. ad Jokes S. 997 DGPS Kiemaic Caie Phase Sigal Simulaio Aalysis fo Pecise Velociy ad Posiio Deemiaio, Poceedigs of ION NTM 97, Saa Moica, CA. Cui X., Yu Z., Tao B. ad Liu, D. 98 Adjusme i Suveyig, Suveyig Publishig House, Pekig, i Chiese Davis P, Rabiowiz P 984 Mehods of umeical iegaio, d Ed. Academic Pess, INC Eule H.J., ad Ladau H. 99 Fas GPS ambiguiy esoluio o-he-fly fo eal-ime applicaios, Poceedigs of 6 h I. Geod. Symp. o saellie Posiioig, Columbus, Ohio, 7-0. Goad C., Remodi B. 984 Iiial Relaive Posiioig Resuls Usig he Global Posiioig Sysem, Bullei Geodesique, 58, Gohad E. 978 Eifuehug i die Ausgleichugsechug, Hebe Wichma Velag Kalsuhe. Feahesoe W, Deih M, Kiby J 998 Saegies fo he accuae deemiaio of ohomeic heighs fom GPS. Suvey ev 3467:78 96 Ha S. ad Rizos C. 995 O-The-Fly Ambiguiy Resoluio fo Log Rage GPS Kiemaic Posiioig, IAG Symposia 5, edied by Beule, Hei, Melboue ad Seebe. Ha S. ad Rizos, C. 997 Compaig GPS Ambiguiy Resoluio Techiques, GPS Wold, Oc. 997, pp54-6. Hofma-Wellehof B., Licheegge H. ad Collis, J. 997 GPS Theoy ad Pacice, Spige-Velag, Wie Hosee G. H. 987 Hadbook of Digial Sigal Pocessig, Egieeig Applicaios, Academic Pess, INC. Kig R.W., Mases E.G. Rizos C., Solz, A. ad Collis J. 987 Suveyig wih Global Posiioig Sysem FERD, Duemmle Velag, Bo. Leick A. 995 GPS Saellie Suveyig, New Yok: Joh Wiley & Sos. Mohamed AH, Schwaz KP 999 Adapive Kalma fileig fo INS/GPS. J Geodesy 73:93 03 Pakiso B.W., Spilke J.J. Eds 996 Global Posiioig Sysem: Theoy ad Applicaios, Volume I, II, Pogess i Asoauics ad Aeoauics, Volume 63 Reigbe C, Koeig R 995 O he accuacy of IGS coodiae soluio. I: Poceedigs of he Fis Tukish Ieaioal Symposium o Defomaio Isabul-94, Isabul, Sep. 5 9, 995, pp Reigbe C, Schwize P, Lueh H 996 CHAMP a challegig mii-saellie payload fo geoscieific eseach ad applicaio. Ese Geodaeische Woche, Suga, 7.-. Okobe 996, 4 p Remodi B. 984 Usig he Global Posiioig Sysem GPS phase obsevable fo elaive geodesy: modellig, pocessig, ad esuls, Uivesiy of Texas a Ausi, Cee fo Space Reseach. Rizos C, Ha S, Che HY 000 Regioal-scale muliple efeece saios fo caie phase-based GPS posiioig: A coecio geeaio algoihm. Eah Plaes Space 50: Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

52 Rohache M, Meva L 996 Beese GPS Sofwae Vesio 4.0. Asoomical Isiue of Uivesiy of Be Rohache M, Schae S 995 GPS-Ausweeechike. Schifeeihe des Deusche Veeis fü Vemessugswese, Bd. 8, pp 07 Schafi B. ad Gafaed E. 986 Geeaig classes of equivale liea models by uisace paamee elimiaio, Applicaios o GPS obsevaios, Mauscipa Geodeica :6-7 Schaffi B 99 Geeaig obusified Kalma files fo he iegaio of GPS ad INS. Techical Repo, No. 5, Isiue of Geodesy, Uivesiy of Suga Schaffi B 995 O some aleaive o Kalma fileig. I: Saso F ed Geodeic heoy oday. Spige-Velag, Beli, pp Schwize P, Kag Z, Reigbe C 995 GPS saellie-o-saellie ackig fo TOPEX/Poseido pecise obi deemiaio ad gaviy field model impoveme. J Geody 0:55 66 Seebe G. 993 Saelliegeodäsie, Wale de Guye, i Gema Sjoebeg L.E. 998 O he esimaio of GPS phase ambiguiies by iple fequecy phase ad code daa, ZfV, 39985, pp. 6-63, Suga, 998. Sjoebeg L.E. 999 Ubiased vs biased esimaio of GPS phase ambiguiies fom dual-fequecy code ad phase obsevables, Joual of Geodesy : 8-4 Sag G. ad Boe K. 997 Liea Algeba, Geodesy, ad GPS, Wellesley-Cambidge Pess Teuisse P.J.G. 995 The leas-squaes ambiguiy decoelaio adjusme: a mehod fo fas GPS iege ambiguiy esimaio, Joual of Geodesy, 70, Vehage S. 004 Iege ambiguiy validaio: a ope poblem? GPS Soluios 004 8:36-43 Wag G., Che Z., Che, W. ad Xu G. 988 The Piciple of he GPS Pecise Posiioig Sysem, Suveyig Publishig House, Pekig, i Chiese Wag LX, Fag ZD, Zhag MY, Li GB, Gu LK, Zhog TD, Yag XA, She DP, Luo ZH, Xiao BQ, Chai H, Li DX 977 Mahemaic hadbook. Educaioal Pess, Pekig, ISBN Wells D., Lidloh W., Schaffi B., Gafaed E. 987 GPS Desig: Udiffeeced Caie Bea Phase Obsevaios ad he Fudameal Diffeecig Theoem, Uivesiy of New Buswick Xu G., Qia Z. 986 The Applicaio of Block Elimiaio Adjusme Mehod fo Pocessig of he VLBI Daa, Cusal Defomaio ad Eahquake, Vol.6, No.4, i Chiese Xu G., Hehl K., Agema D. 994 GPS Sofwae Developme fo Use i Aeogavimey: Saegy, Realizaio, ad Fis Resuls, Poceedigs of ION GPS-94, p Xu G., Basos L., Timme L. 997 GPS Kiemaic Posiioig i AGMASCO Campaigs Saegic Goals ad Numeical Resuls, Poceedigs of ION GPS-97 Cofeece i Kasas Ciy USA, p73-83 Xu G., Schwize P., Reigbe Ch. 998 KSGSof Kiemaic/Saic GPS Sofwae --- Sofwae Use Maual, Scieific Techical Repo 9/998, GeoFoschugsZeum Posdam Xu G. 00 GPS daa pocessig wih equivale obsevaio equaios, GPS Soluios, Vol. 6, No. -, 6:8-33 Xu G. 00a A geeal cieio of iege ambiguiy seach, Joual of GPS, Vol. No.: -3 Xu G. 003a A diagoalisaio algoihm ad is applicaio i ambiguiy seach, Joual of GPS, Vol. No.:37-43 Xu G. 003 GPS Theoy, Algoihms ad Applicaios, Spige Velag, Heidelbeg Yag Y 997 Robus Kalma file fo dyamic sysems. Joual of Zhegzhou Isiue of Suveyig ad Mappig 4:79 84 Yag Y 999 Robus esimaio of geodeic daum asfomaio. J Geodesy 73:68 74 Yag Y, He H, Xu GC 00 Adapively obus fileig fo kiemaic geodeic posiioig. J Geodesy 75:09 6 Zhou J. 985 O he Jie faco, Aca Geodaeica e Geophysica, No.5, i Chiese 5 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

53 Zhou J., Huag J., Yag Y. ad Ou, J. 997 Robus leas squaes mehod, Publishig House of Huazhog Uivesiy of Sciece ad Techology, Wuha, i Chiese 5 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

54 . Appedixes. Appedix : Diagams of he Sofwae Algoihms diagam Diffeecig algoihms Udiffeeced algoihm Equivale algoihm Vaiace equaio Diagoalisaio algoihm Diffeeiao/ iegaos Obi Cycle slip deecio Algoihms Diffeeial Dopple Opimal baselies Idepede paameeisaio Ambiguiy Iiialisaio Daa codiios Adjusme ad fileig Seach Leas Squaes Block-wise elimiaio Sequeial A pioi ifomaio Kalma file Iellige Kalma file 53 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

55 Pogam diagam Pogam sa Commo pa ead ipu paamee file ad ohe files give ead/ceae Su-Moo-plaes obis/ Eah/ocea loadig ide compue Eah/ocea loadig ide obi daa asfomaio daa pe-pocessig if possible iiialisaio opimal baselies ambiguiy vaiaio equaio Sequeial ime loop ge eal ime daa models ad paamees acquisiio sigle poi posiioig velociy deemiaio ambiguiy check ad se modulus swich local e o boad global/egioal e ambiguiy check models acquisiio obsevaio equaio Fi ad Keple/Jacoby maices ukow check wih daa codiio omal equaio exchage,elimiaio,accumulaio adjusme/file diagoalisaio Summay pa focas ieaio aalysis 54 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

56 Models diagam Topospheic models Sola adiaio model Ioospheic model Amospheic dag model Relaiviy model Eah ide /ocea loadig ide Physical Models Geopoeial disubace Tidal disubace Dyamic obi fiig model Su, Moo, plaes disubaces Saellie mass cee coecio model Muli-bodies disubace 55 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

57 Tools diagam Coodiaes asfome Gaphic epeseaio Time sysems asfome Saisic aalysis Su-Moo-plaes obi ceao Boadcas obi asfome Tools Specal aalysis Fligh sae compuaio Iepolaio ad iegaio Mappig fucios Maix ivese Helme asfomaio Gauss-Joda Sholesky 56 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

58 . Appedix : Lis of Fucios of he Sofwae Complee fucios used i his sofwae ae lised bellow: models.c --- sofwae package of models double ela_iviy --- elaiviy models double io_mappig --- ioospheic mappig fucios double o_mappig --- opospheic mappig fucios double io_model --- boadcas iospheic model double opospheemodel --- opospheic models ad hei combiaio double iellis --- Niels opospheic model double NHMF --- mappig fucio i Niels model double NWMF --- mappig fucio i Niels model double yioo --- Vioo opospheic model double davis --- Davis opospheic model double saasa --- Saasamoie opospheic model double hopfield --- Hopfield opospheic model double o_saasa --- modified Saasamoie opospheic model double o_hopfield --- modified Hopfield opospheic model ools.c --- sofwae package of ools double disace --- disace compuig fucio void helme_ --- Helme asfomaio double zeihal --- zeih disace ad azimuh compuig fucio double aa --- acage fucio void xyzphila --- coodiaes asfomaio bewee x,y,z ad phi,lambda,h void oaior --- oaio maix bewee global ad local coodiae sysems void oaio --- oaioal fucio double zeiha --- zeih disace ad azimuh compuig fucio void TRANF --- coodiaes asfomaio bewee x,y,z ad geodeic phi,lambda,h double lagagef --- floa Lagage iepolaig fucio double lagage --- floa Lagage iepolaig fucio wih iege sep void lagage_cha --- Lagage fucio fo Foa call 57 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

59 double lie_iegae --- umeical iegaio wih d ode liea fiig model double lie_iepo --- umeical iepolaio wih d ode liea model double lie_iepo --- liea iepolaig fucio void oopolyom --- soluio of he d ode polyomial void oo3polyom --- soluio of he 3 d ode polyomial void oo4polyom --- soluio of he 4 h ode polyomial cood_ime.c --- sofwae package of ime ad coodiae sysems void _hms --- asfom fucio of ime fom floa o hou, mi, secod void oa --- oae a veco void oam --- oae a maix void Roaiom --- he pole moio oaio maix void Roaios --- he Eah oaio maix void Roaio --- he uaio maix void Roaio --- he uaio maix fo Foa call void Roaiop --- he pecessio oaio maix i RoaioT oaio maix es fucio double MJD --- compue JD fom yea, moh ad day double JD --- compue Julia dae double JD compue he Julia dae fom JD000.0 double JD_GPST --- JD ad GPST asfomaio double TDT_GPST --- TDT ad GPST asfomaio void d_gps --- TDT ad GPST asfomaio called by Foa double UTC_GPST --- UTC ad GPST asfome double Huc --- compue UTC fom GPST double Hu --- compue UT fom GPST void YMDHNW_JD --- compue yea, moh, day, hou, day_of_week, GPS_week, day_of_yea fom JD void GPSwdow --- fom yea, moh, day, hou compue GPS_day_of_week, week, day_of_yea i DayofYea --- compue day_of_yea fom yea, moh, day, hou i able --- leap secods able help.c --- help sofwae package void DdE_e --- chage DdE o e fo ead daa void Gblak --- chag G o blak 58 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

60 i ssx --- fid a sig i a sig void blak_ --- chage special chaaces o blak void blak_ chage special chaaces o blak i Fix --- oudig fucio double dfix --- double oudig fucio log lfix --- log oudig fucio i ifix --- i oudig fucio void fileam --- aach a umbe o file ame void check_balace --- check balace i c code void em_comme --- emove he comme lies void cou --- cou fo ifo i c code void seach_ex --- seach fo ex i c files i idex --- fid idex of a sig i a sig log cou_lie --- cou fo lies i c code i cp --- copy fom file o file i lowe_f --- chage file ame o lowe case i uppe_f --- chage file ame o uppe case void lowe_s --- chage sig o lowe case void uppe_s --- chage sig o uppe case i gelie --- ead a lie fom a file cha lowe_c --- chage a chaace o lowe case cha uppe_c --- chage a chaace o uppe case void addsos --- aach 0 o sig i sc --- fid a chaace i a sig void _gid_kms --- ead kms foma gid daa void w_gid_kms --- wie kms foma gid daa void w_gid_sf --- wie sufe foma gid daa void D_e --- chage D o e i a daa lie chol_.c --- sofwae package of Cholesky decomposiio void cp_alow --- copy a lowe case maix void cp_av --- copy a veco void ii_av --- iiial a veco void checke --- ideiy maix check void oaliv --- oal ivese 59 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

61 void checkoal --- check oal ivese maix void checkivdecompose --- check ivese decomposiio void checkdecompose --- check decomposiio void ivdecompose --- check decompose void decompose --- decompose void pm_low --- wie lowe case maix void pm_d --- wie d maix void pv_dow --- wie a floa veco void pv_dowlog --- wie a log veco void pv_dowi --- wie a iege veco void pv_d_vd --- wie a floa veco void pv_d_vlog --- wie a log veco void pv_d_vi --- wie a iege veco void pv_vd --- wie wo floa vecos void p3v_ --- wie hee floa vecos void pv_vlog --- wie wo log vecos void pv_vi --- wie wo i vecos adjus.c --- sofwae package of adjusme double ls --- leas squaes algoihm i chol_iv --- cholesky ivese fucio i gauss_joda --- gauss-joda ivese fucio void pm_low_cl_v --- wie a maix i lowe case wih ifo void m_low_cl_v --- ead a maix i lowe case wih ifo void pm_d_cl_v --- wie d maix wih ifo void m_d_cl_v --- ead d maix wih ifo void pv_v --- wie vecos wih ifo void v_v --- ead wo vecos wih ifo void psv_v --- pi a vaiable wih ifo double sv_v --- ead a vaiable wih ifo void sv_v --- ead a vaiable wih a sig void pv_d_v --- wie a veco void v_d_v --- ead a veco 60 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

62 idkep.c --- sofwae package of idal compuaio ec void SMP_eph_Iday --- compue oe day Su Moo plaes eph void SMP_eph --- compue a epoch Su Moo plaes eph void plae_eph0 --- compue plaes eph double omegapf --- omega ad f void SM_eph --- Su Moo eph ceao void SM_eph0 --- Su Moo eph ceao0 void keple_el --- keple elemes ad xyz asfome void keple_el_ --- keple elemes asfome fo Foa call double Keple_equaio --- keple equaio void ide_ --- iepolae he ide fo void ide_iday --- compue day idal effecs void ide --- ide ceao void iepo_oa --- iepolaio ad oaio i SL_eph --- ead Su Moo eph void Roaio_M --- compue 4 oaioal maices _ipu.c --- sofwae package fo ead files ad paamees void _oload --- ead ocea loadig paamees void _poe --- ead poeial model paamees void po_s --- special opeaio o a sig void _seleced_sa --- ead seleced saio file void check_sa_da --- check saio daa file void _igssie da --- ead igs saio ifo void ge_sie_ecceiciy --- ead saio ecceiciy file void ge_aea_phasecee --- ead aea phase cee daa void ge_sie_eceive --- ead eceive ype file void ge_sie_dome_ifo --- ead saio dome file void _saio_da --- ead saio ifo void ge_sa_id_code --- ead saio id code void ge_plae_id --- ead plae ifo of saios void blak_4id --- special opeaio o a sig void _sada --- ead saellie elaed ifo void _pola --- ead pola moio file void _ipu --- ead ipu paamee file 6 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

63 i _s --- sig opeaio void ime_debug --- ime debug fucio void w_coda --- ead codiio daa i idexx --- seach fo a chaace void DOP --- DOP compue fucio void w_ifo --- wie ifo i oveead --- ove ead fucio double woif --- combie wo iege o a floa mai.c --- sofwae package used i mai pogam void sm_ef_ --- compue he Sum, Moo coodiaes i ECEF sysem void se_said_idex --- se saellie ideifie veco void oae_sf_ef --- coodiaes asfomaio bewee ECSF ad ECEF coodiae sysems void V_oae_sf_ef --- velociy asfomaio bewee ECSF ad ECEF sysems void oae_a_ef_sf --- asfomaio of 3*3 maix fom ECEF o ECSF sysem void oam_ --- muliplicaio of asfomaio maix o he igh void oam_l --- muliplicaio of asfomaio maix o he lef void oae_sf_ef --- void oae_sf_ef fo Foa call void oae_maices_ --- ge he oaio maices of ime void oae_maices_ --- void oae_maices_ fo Foa call void smp_sf_ --- ge Sum, Moo, plaes daa i ECSF of ime void smp_sf_ --- void smp_sf_ fo Foa call void check_t_ --- idal coecio check accodig o he saio popey void ou_t --- oupu idal coecios void sa_xyz --- pepae saio umbe ad coodiaes fo idal compuaio void oload_ --- ge ocea loadig ide coecio of ime void pola_xy_ --- iepolae pola moio void pola_xy_ --- void pola_xy_ fo Foa call oload_ide.c --- sofwae package of ocea ide loadig effec Foa oigiao: Kha void oload --- call Foa o C ouie o compue oload effec void oload_sa_ --- Su vesio of oload C call Foa void oload_sa --- Vampie vesio of oload C call Foa void oload_sa_c --- subouie aslaed fom Foa void ease_c --- subouie aslaed fom Foa oigiao: Wezel 6 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

64 void doodso_c --- subouie aslaed fom Foa, se Doodso coefficies double das_a_c --- subouie aslaed fom Foa double as_a_c --- subouie aslaed fom Foa void ime_sei_c --- subouie aslaed fom Foa void emuc_c --- subouie aslaed fom Foa void ejul_c --- subouie aslaed fom Foa boad.c --- sofwae package of boadcas obi void oae_ob --- oae IGS ad boadcas obi o ECSF sysem void check_ob --- saisic aalysis of he wo obis void boadcas --- boadcas obi compuaio fom give paamees void bobc --- boadcas obi compuaio fom give paamees i fidisa --- fid ou he saellie ideifie void boadcas_obi --- boadcas obi compuaio fom give paamees double Keple_equaio --- solvig Keple equaio void _eph --- ead eph daa void eph_check --- eph daa check obi.c --- sofwae package of obi i efsa --- decide he efeece saio i geob_ --- ge obi daa of ime void geob_ --- i geob_ fo Foa call i geob --- ge obi daa due o sa id ad ime void d_leo_ob --- ead LEO obi daa void ge_leoob_ --- ge LEO obi daa of ime void igs --- combie IGS daa ad ead void igsda --- ge IGS daa void _igs_h --- ead IGS file heade pa void _igs_d --- ead IGS file daa pa pe67.c --- sofwae package of GPS daa pe-pocessig void gedaa_ --- ge GPS daa of oe epoch void ope heade --- ope GPS daa files ad ead he heades void _iexheade_oly --- ead he heade pa of all files void pepo_gps_da_0 --- pe-pocessig GPS daa 63 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

65 void pepo --- pe-pocessig void _iexdaa --- ead Riex daa void _iexheade --- ead Riex file heade void d_sa --- delee some daa due o sa id void ead_afile --- ead so-called a file void id_obs --- se obsevaio id void id_ij --- iege asfome void iealobs_ype --- se ieal obsevaio ype void d_da --- delee he daa saed o o iege epoch sp_pv.c --- sofwae package of sigle poi posiioig void siglep_c --- sigle poi posiioig void aso --- asmissio ad oaio coecio void aso --- void aso fo Foa call void eaho --- eah oaio coecio double disace --- disace fucio void velociy --- velociy fucio ambi.c --- sofwae package of ambiguiy void sepaaenxgoup --- sepaae N ad X goups void deeminaea --- deemie N aea void gen_ew --- ge N ew void mim0_ --- miimum of wo m0 void ambifixc --- ambiguiy fixig void seach --- ambiguiy seach ambi_io04.c --- sofwae package of ambiguiy-ioosphee equaio void ambi_io --- ambi-ioo equaio void shif_add --- shif-ad-add opeaio of maix void shif_add_ --- shif ad add opeaio void d_low --- d ad lowe case void exchageij_ --- exchage wo ow ad colum void exchageij_ --- exchage wo ow ad colum void exchageij_ --- exchage wo ow ad colum void ambiopef --- ambi, op, ef decisio 64 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

66 void i_ambi --- ieal fucio void ou_aambi --- oupu i picue void aambi --- aalysis ambiguiy void seach_ --- seach void VWSblak --- sig opeaio aohe.c --- sofwae package of aohe ouies void es8 --- es fucio void heoy_ambi --- heoeical ambiguiy void sa_view --- oupu viewed saellie i picue void diagoal_q --- Q maix diagoalisaio void cp_combie --- code-phase combiaio void exchage_mx --- exchage ukow ode of he omal maix void b_eli_mx --- elimiaio of ukows void ou_i_veco --- oupu iege veco void ou_d_veco --- oupu floa veco void ou_ig_veco --- oupu wih codiio double weigh_obus --- obus weigh fucio double weigh_zdis --- weigh fucio of zeih disace void ou_sa_d --- oupu d of saios i j_x_id --- iege opeaio i j_x_id4 --- iege opeaio void fom_om_eq --- fom omal equaio void fom_om_eq --- fo Foa call void accum_om_eq --- accumulaio of omal equaios dyamic_ob.c --- sofwae package of dyamic obi void s_vaiaio_eq --- solvig he vaiaio equaio void ge_fi_maix --- ge Fi maix void ge_fi_maix9 --- ge Fi maix void a_f_geo --- so-called a maix of foces void ai_dag --- amospheic dag model void ai_dag --- fo Foa call void ai_dag --- fo Foa call double ai_desiy --- ai desiy fucio 65 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

67 void sola_adiaio --- sola adiaio model void sola_adiaio --- fo Foa call void g_maix_c --- ge so-called g maix void sola_adiaio --- fo Foa call double shadowf --- shadow fucio void s_m_aio_ --- Suface ad mass aio fucio double mi --- miimum fucio void mc_coecio_ef --- GPS saellie mass cee coecio void masscee --- mass cee fucio void b_ai_dag --- b maix of ai dag fo Foa call void a_sola_adiaio --- a maix of sola adiaio fo Foa call void f_body --- -body foce void a_f_body --- a maix of -body foce double bea_ --- bea fucio double bea_ --- bea fucio i koecke_dela --- koecke-dela fucio i gauss_joda3 --- gauss-joda ivese void R_weigh_f --- weigh fucio xyz.c --- sofwae package of addiioal fucios void m33m maicies muliplicaio void ddodkep6 --- Jacoby maix of Keple elemes ad xyz_xyzdo void ge_f_veco --- ge foce veco void uge_kua0 --- Ruge_Kua iegaio iiial void adams_cowell_bea --- Adams-Cowell bea fucio void uge_kua --- Ruge_Kua iegaio void uge_kua --- Ruge_Kua iegaio void adams_cowell --- Adams-Cowell iegaio fucio void heoy_baselie --- heoeical baselies void pm_d_v --- oupu maix i d void miimum_ee3 --- miimum ee poblem void apioi --- a pioi fucio void eli_fx --- elimiaio of efeece coodiaes void ou_i4_veco --- oupu veco wih codiio void b_eli_efclock --- elimiae efeece clock 66 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam

68 double ls8 --- ls fucio modified vesio i chol_iv8 --- Cholesky ivesio modified vesio i gauss_joda8 --- Gauss-Joda ivese modified vesio void dpdxgrefsf --- paial diffeeiaios of obi fiig model ec. sofwae package kalma.c --- Kalma fileig fucios daa_codiios.c --- daa codiios mfgsof.c --- mai pogam of MFGsof mai.c --- mai pogam void posiio --- LEO kiemaic/dyamic obi deemiaio void posiio --- local ewok saic/kiemaic posiioig void posiio3 --- egioal ewok moioig wih obi coecio void posiio4 --- global ewok moioig wih kiemaic/dyamic obi deemiaio void posiio5 --- global ewok moioig wih dyamic obi deemiaio 67 Scieific Techical Repo STR 04/7 GeoFoschugsZeum Posdam