Numerical Analysis of Orbital Perturbation Effects on Inclined Geosynchronous SAR

Transcription

1 snsors Articl Numricl Anlysis of Orbitl Prturbtion Effcts on Inclind Gosynchronous SA Xicho Dong 1, *, Chng Hu 1, Tng Long 1, nd Yunho Li 1 1 School of Informtion nd Elctronics, Bijing Institut of Tchnology, Bijing , Chin; cchchb@163.com C.H.; longtng@bit.du.cn T.L.; lyh @163.com Y.L. Bijing Ky Lbortory of Embddd l-tim Informtion Procssing Tchnology, Bijing , Chin * Corrspondnc: dongxicho@bit.du.cn; Tl.: ; Fx: Acdmic Editor: Json K. Lvy civd: 16 Mrch 016; Accptd: 30 August 016; Publishd: Sptmbr 016 Abstrct: Th gosynchronous syntic prtur rdr GEO SA is suscptibl to orbit prturbtions, lding to orbit drifts nd vritions. Th influncs bhv vry diffrntly from thos in low Erth orbit LEO SA. In this ppr, impcts of prturbtions on GEO SA orbitl lmnts r modlld bsd on prturbd dynmic qutions, nd n, focusing is nlyzd orticlly nd numriclly by using Systms Tool Kit STK softwr. Th ccurt GEO SA slnt rng historis cn b clcultd ccording to prturbd orbit positions in STK. Th prturbd slnt rng rrors r minly first nd scond drivtivs, lding to img drifts nd dfocusing. Simultions of point trgt imging r prformd to vlidt formntiond nlysis. In GEO SA with n inclintion of 53 nd n rgumnt of prig of 90, Dopplr prmtrs nd intgrtion tim r diffrnt nd dpndnt on gomtry configurtions. Thus, influncs r vrying t diffrnt orbit positions: t qutor, first-ordr phs rrors should b minly considrd; t prig nd pog, scond-ordr phs rrors should b minly considrd; t or positions, first-ordr nd scond-ordr xist simultnously. Kywords: gosynchronous SA; orbitl prturbtion; focusing 1. Introduction Gosynchronous syntic prtur rdr GEO SA [1] runs on n orbit hight of round 36,000 km, hs rvisit tim of lss thn 4 h nd covrg of mor thn 1000 km by 1000 km. cntly, GEO SA hs bcom hot topic, s it hs ovrwhlming dvntgs for monitoring rthquks nd or disstrs [ 4]. ltd rsrch is focusd on systm dsign nd optimiztion [5 8] nd dvlopmnt of ccurt imging lgorithms [9,10]. Or studis r dvotd to non-idl influncs, including tmosphric ffcts [11 14]. Howvr, orbitl prturbtions r rrly concrnd, though stllits r crtin to b impctd by prturbtions cusd by Erth s non-sphricl mss distribution, tmosphric drg, third body ttrction, solr rdition, tc. In spcborn SA, prturbtions will ld to orbit drifts nd vritions by influncing orbitl lmnts prmtrs rquird to uniquly idntify spcific orbit in clstil mchnics vrying with tim. Prturbtions imposd on stllits r rltd to orbit hight. For low Erth orbit LEO cs, J trm prturbtion kind of Erth s non-sphricl prturbtion nd tmosphric drg should b minly considrd. Comprtivly, min sourcs of prturbtions for gosynchronous orbit r J trm prturbtion nd third body ttrction. Th solr rdition should lso b considrd if r/mss rtio is lrg. Furrmor, in LEO SA, rdr img intgrtion tim is vry short t lvls of 1 s, nd thus, orbit cn b considrd to b frozn within intgrtion tim; whil in GEO SA, it bhvs vry diffrntly. Th prturbd orbit drifts mong dys or wks cn form sptil bslin btwn rpt pss GEO SA nd cn b Snsors 016, 16, 140; doi: /s

2 Snsors 016, 16, 140 of 4 mployd for rpt pss intrfromtry or diffrntil intrfromtry [15,16]. In comprison, prturbd orbit vritions within on dy or GEO SA prtur tim svrl hundrds or vn thousnds of sconds cn ld to chnging of slnt rng historis nd rsult in dgrdtion nd vn filur of focusing [17,18]. Kou t l. [17] studis prturbd orbit vritions ffcts on focusing nd intrfromtry of circulr SA CSA. Howvr, CSA is n innovtiv SA systm, which hs circulr ndir trck nd forms high rsolution imgs by cohrnt intgrtion of whol dy s orbit priod cquisitions. Thrfor, its signl modls nd corrsponding conclusions cnnot b usd dirctly in inclind GEO SA, whos intgrtion tim is comprtivly lss nd only t lvls of round svrl minuts. In [18], prturbtion ffcts r nlyzd for GEO SA in circulr gosynchronous orbit, nd pproximt prturbd qutions r doptd in nlyzing rrors of Dopplr prmtrs nd focusing prformnc. Th fourth-ordr Dopplr prmtrs r doptd [19]. Thrfor, for mor gnrl css of GEO SA, which is running in llipticl or nr-circulr orbit, pproximtion mthod is not vry ccurt. In ctulity, it is complx nd difficult to driv nlyticl solutions of vrious prturbtions influncs on orbitl lmnts nd slnt rng historis bcus ir ffcts r coupld with ch or. In ordr to void possibl nd grt rrors inducd by pproximtion, numricl pprochs r rcommndd, in which AGI Systm Tool Kit STK Anlyticl Grphics, Inc., Exton, PA, USA softwr is mployd to gnrt highly ccurt prturbtions nd prturbd slnt rng historis. This ppr studis prturbtion influncs on orbitl lmnts, slnt rng historis nd focusing prformnc of GEO SA bsd on ccurt numricl simultion using STK. Howvr, ory of prturbtion influncs on orbitl lmnts is lso dmonstrtd nlyticlly. Thn, corrsponding vritions of slnt rng historis r drivd bsd on Tylor xpnsion of slnt rngs. Th nlyss r bsd on prturbd motion qutions nd STK. Th lws of vritions nd corrsponding influncs r dducd. As nlyticl solution for prturbtion dynmic qutions nd ir compositd influncs on slnt rng r difficult to obtin, numricl pproch bsd on STK is doptd to construct SA signls. Th slnt rng historis with nd without ffcts of prturbtions r gnrtd. Finlly, focusing nd vlutions r prformd to vlidt nlyss bsd on STK simultions.. Prturbtions nd Thir Influncs on GEO SA Orbitl Elmnts Stllits r influncd by Erth s non-sphricl mss distribution, tmosphric drg, third body ttrction, solr rdition nd ors. Th stllit orbit drifts inducd by s prturbtions cn b dpictd by ir influncs on orbitl lmnts bsd on corrsponding prturbtion motion qutions. In this sction, ll kinds of prturbtions in GEO SA r discussd nlyticlly, long with diffrncs from thos in LEO SA. Thn, xprssions, priods nd ordrs of mgnitud of prturbtions influncs on orbit lmnts r prsntd. Ths will b usd s bridg to nlyzing prturbtions influncs on slnt rng historis in nxt sction..1. Prturbtion Motion Equtions For GEO SA, prturbtions will produc cclrtions, which r composd of rdil r, norml z nd trnsvrs θ componnts following xis ordr of θ z nd r, rspctivly. Th rdil componnt is long dirction pointing from gocntric cntr to stllit; norml componnt is prpndiculr to orbit pln; nd trnsvrs componnt lis within orbit pln nd follows right-hnd rul. Th imging gomtry of GEO SA is shown in Figur 1.

3 Snsors 016, 16, of 4 Snsors 016, 16, of 6 z z r O O f y i x N Figur Figur 1. Gomtry 1. Gomtry of of gosynchronous GEO GEO SA SA imging. imging. O-xyz O-xyz is is Erth-cntrd inrtil inrtil coordint coordint systm. systm. O - O-rθz r θ z is is stllit stllit locl locl coordint systm. systm. Equtions Equtions of stllit of stllit motion motion hv hv to b to rprsntd b rprsntd by prturbd by prturbd qutions. qutions. According According to to orbitl orbitl prturbtion prturbtion dynmic dynmic ory, ory, orbitl orbitl lmnts lmnts influncd influncd by by prturbtions prturbtions cn cn b b obtind. obtind. Howvr, Howvr, Lgrngin Lgrngin prturbd prturbd qution qution cn cn b b only only usd usd for for prturbtion, prturbtion, which which cn cn b prsntd b prsntd by by potntil potntil function function i.., i.., Erth s Erth s non-sphricl non-sphricl prturbtion prturbtion nd nd third third body body prturbtion. prturbtion. Considring Considring non-consrvtiv non-consrvtiv prturbtion prturbtion forcs, forcs, such s such solrs rdition solr rdition nd tmosphric nd tmosphric drg, nodrg, potntil no potntil functions functions xist for us; xist rfor, us; rfor, Lgrngin prturbd Lgrngin qution prturbd of motion qution cnnot of motion b dirctly cnnot usd b indirctly such usd cs. Th in such Gussin cs. prturbd Th Gussin qution prturbd of motion qution cn b of mployd motion for cn ny b typ mployd of prturbtion. for ny typ Th of tmporl prturbtion. drivtivs Th of tmporl orbitl drivtivs lmntsof hv bn orbitl xprssd lmnts nlyticlly hv bn [0]. xprssd Th dtild nlyticlly lgbrs [0]. cnth b found dtild in [0]. lgbrs Th diffrntil cn b found qutions in [0]. r: Th diffrntil qutions r: [ ] d = p p sin f d p p sin f 1 d f µ 1 r 1 df 1 + cos f 1 r cos f θ cos f 1 cos f [ d = p sin f d p sin f cos f cos f d f µ 1 + cos f r + + cos f + ] cos f r 1 + cos f 3 θ 3 df 1 cos f 1cos f di = p cos f + ω d f diµ 1 p + cos cos f f 3 z 3 3 z 3 dω df 1 cos f = p 1 sin f + ω d f µ sin i d p sin cos f f 3 z 4 [ ] 3 z 4 dω df sin i 1 cos f = p cos f d f µ 1 + cos f + cos f sin f r cos f 3 θ 1 sin f + ω tn i 1 + cos f 3 z 5 d p cos f cos f sin f 1 sin f whr p is ltus r 3 3 z 5 df rctum nd p = 1 1cos f, µ is gocntric grvittionl 1cos f tn i constnt.,, i, f, ω 1cos f r orbitl lmnts nd rprsnt smi-mjor xis, ccntricity, inclintion, tru nomly nd rgumnt of prig, rspctivly. p 1, Th d/df is isgocntric diffrntil grvittionl oprtor. constnt.,, whr p is ltus rctum nd i, f, r orbitl lmnts nd rprsnt smi-mjor xis, ccntricity, inclintion, tru nomly nd rgumnt of prig, rspctivly. Th d/df is diffrntil oprtor.

4 Snsors 016, 16, of 4.. Prturbtions in GEO SA Th vrious prturbtions hv diffrnt influncs on stllits t diffrnt orbit hights [1]. For low Erth orbit stllits, J trm ccounts for bout on prcnt of two-body cntriptl forc. Th tmosphric drg prturbtion dcrss s stllit orbit hight incrss, nd it is round ordrs of of J trm for most of LEO stllits nr 160,000 km orbit hights. Whn orbit hight bcoms highr, its influncs vnish. Th third body ttrction nd solr rdition prturbtions r rltivly smll nd r round ordrs of 10 4 ~10 5 of J trm. In comprison, influncs of prturbtion on high orbit stllits r vry diffrnt. Th J trm will dcrs s orbit hight incrss; whil third body ttrction incrss it will rriv t sm lvl s J trm t gosynchronous orbit. Th solr rdition prturbtion rmins lmost unchngd. Furrmor, for ny orbit hight, or prturbtions, such s tidl prturbtion, r svrl ordrs smllr thn J trm nd, thus, cn b ignord. Th influncs of prturbtion on GEO SA r simultd using STK. Th initil orbitl lmnts of rfrnc GEO SA hv smi-mjor xis of 4,164. km, ccntricity of 0.07, inclintion of 53, right scnsion of scnding nod AAN of 110 nd rgumnt of prig of 70. Th 10 dys orbit drifts cusd by ch prturbtion lon r producd nd prsntd in Tbl 1. Th ordrs of ch prturbtion s influnc r consistnt with formntiond nlysis. For GEO SA, only Erth s non-sphricl mss distribution, third body ttrction nd solr rdition should b considrd. Tbl 1. GEO SA orbit drifts ftr 10 dys whn considring ch prturbtion lon. Erth s Non-Sphricl Mss Distribution J Trm Ordrs of Third Body Attrction Solr dition Ors Tidl Prturbtion km km 95.9 km 34. km km 7.8 km Totl Furrmor, influncs hv obvious priodicity. According to influncs on GEO SA orbitl lmnts, prturbtions cn b ctgorizd into sculr componnt nd priodicl componnt. Th sculr componnt cn cus long-trm nd continuous drifts, whil priodicl componnt will produc priodicl chngs, including short priodicl trm nd long priodicl trm. Th short priodicl trm hs smllr priod thn stllit orbit priod, whil long priodicl trm hs much grtr on. Th priodicl prturbtions will cus oscilltions round crtin position. For GEO SA, short priod is within 4 h nd is usully 1 h or 4 h. Th long priod cn rch up to hlf month, hlf yr or vn svrl yrs..3. Erth s Non-Sphricl Mss Distribution In idl two-body movmnt Kplrin motion, stllit nd Erth r considrd s two prticls, nd grvity is clcultd btwn m. Howvr, Erth is not rgulr sphr, nd its mss distribution is unvn, so Erth s grvity on stllits cnnot b considrd s point-to-point form in rl css. If Erth is considrd s n irrgulr llipsoid, which is composd of n prticls pproximtly, grvity on stllit P outsid Erth is sum of grvitis on P from n prticls of Erth. As grvity is vctor, summtion is complx. Thrfor, gnrlly, potntil function V of Erth s grvittionl fild is firstly clcultd, nd n, grvity is producd bsd on potntil function V. According to clstil mchnics, sris xpnsion of potntil function V is sphricl hrmonics sris nd cn b prsntd s []: Vr, ϕ, λ = µ r n=0 E n n r C nk cos kλ + S nk sin kλp nk sin ϕ 6 k=0

5 Snsors 016, 16, of 4 whr r, ϕ, λ is gocntric rng, ltitud nd longitud. E is Erth qutoril rdius. C nk nd S nk is sphricl hrmonic cofficints. n nd k r ordrs. P nk x is ssocitd Lgndr function, i..,: P nk x = 1 x k/ d k dx k P n x 7 whil P n x is Lgndr polynomils: P n x = 1 d n n n n! dx n x 1 8 Msurmnts nd clcultions of s sphricl hrmonic cofficints r undr rsrch. Vrious modls of Erth grvittionl fild r constructd for pproximtion to Erth grvitis. Currntly, ltst modl is Erth Grvittionl Modl 008 [3] EGM008, by Ntionl Gosptil-Intllignc Agncy, NGA, which hs ordrs up to 159 nd sptil rsolution of 9 km. In modls, lowr ordr trms hv mor importnt rol in modls of Erth grvittionl fild, spcilly for scond-ordr of zonl hrmonics, i.., J trm. Actully, tssrl hrmonics will induc orbit prcssion nd longitud drifts. Howvr, y r long priodicl trms nd will not influnc focusing. Whn only J trm is considrd, Erth cn b simplifid s n llipsoid with som obltnss. For simplifiction for nlyzing mchnicl modl, only J trm is considrd for influncing orbitl lmnts in this sction. Howvr, or highr trms hv similr impcts nd nlyzing mthods. Whn only J trm is considrd, prturbtion trm in potntil function V of Erth s grvittionl fild cn b xprssd s: V prt r, ϕ = µ E r 3 J 3sin ϕ 1 9 whr J is Erth obltnss cofficint. xprssd s: = grd Vprt r, ϕ = iz Th cclrtions cusd by prturbtions cn b [ ir ] iθ iz r r f r sin f +ω i V prt r, ϕ 10 whr i r, i θ, r unit vctors of thr orthogonl dirctions in stllit locl coordint systm. Considring sin lw in sphricl tringl: sin ϕ = sin i sin f + ω 11 w cn obtin rdil, norml nd trnsvrs componnts of J prturbtion cclrtions: r = 3µ J r 4 1 3sin i sin f + ω 1 θ = 3µ J r 4 sin i sin f + ω 13 z = 3µ J r 4 sini sin f + ω 14

6 Snsors 016, 16, of 6 Snsors 016, 16, of 4 Substituting Equtions 1 14 into Equtions 1 5, w cn obtin vrition of orbitl lmnts undr influncs of J prturbtion s Appndix A for dtild dvitions: Substituting Equtions 1 14 into Equtions 1 5, w cn obtin vrition of orbitl lmnts undr influncs d 3 of J prturbtion s Appndix A for dtild dvitions: J d = 3 J [ ] 1cos f sin f sin isin u 1 cos f 3sin f df 15 p 1 d f p cos f sin f + sin i sin u 1 + cos f 3 sin f 15 di 3 J sini 1 cos f sin u cos u 16 df di = 3 p J sin i d f p 1 + cos f sin u cos u 16 d 3 J cos f 1cos f d = 3 [ sin f sin i sin u cos f 3sin f J d f p 1 + cos f sin f + sin isin u cos f 3sin f + + cos f ] 17 df p 1cos f cos f d 3 dω = 3 J cosi 1 cos f sin u 18 df Jp cos i d f p 1 + cos f sin u 18 d 3 J dω d f = 3 1cos f df p J 1 + cos f [ p ] 1 1+ cos f sin sin f +cos f cos f +sin f f sin f sin i sin u 19 sin isin u 19 cos f 1+ cos f sin u cos i cos f 1 cos f whr u = f + ω. sin ucos i Infrrd from Equtions 15 19, ch orbitl lmnt hs short priodicl trm with priod whr u f. of on GEO SA orbit T undr influncs of J trm. Howvr, r xists no long priodicl Infrrd from Equtions 15 19, ch orbitl lmnt hs short priodicl trm with trm. In comprison, sculr trm will giv sculr dcrsing in AAN nd sculr priod of on GEO SA orbit T undr influncs of J trm. Howvr, r xists no long incrsing in rgumnt of prig. Th spcific lw of GEO SA orbitl lmnts vrition priodicl trm. In comprison, sculr trm will giv sculr dcrsing in AAN is summrizd in Tbl. Th dsh indicts no influnc t ll. nd sculr incrsing in rgumnt of prig. Th spcific lw of GEO SA orbitl lmnts Tbl vrition. Lwis ofsummrizd GEO SAin orbitl Tbl lmnts. Th dsh vritions indicts undr no influnc influncs t ll. of J trm Tbl. Lw of GEO SA orbitl lmnts vritions undr influncs of J trm Orbitl Elmnts Short Priod Long Priod Sculr Orbitl Elmnts Short Priod Long Priod Sculr Smi-mjor xis T - Smi-mjor xis T - - Eccntricity T - Eccntricity T - - Inclintion T - Inclintion T - - AAN T Argumnt of AAN prig T T - Argumnt of prig T - Not: T is priod of GEO SA; is incrs; is dcrs. Not: T is priod of GEO SA; is incrs; is dcrs. Th Th prturbtion prturbtion motion motion qutions qutions cn cn b b nlyticlly solvd solvdby by cnonicl cnonicltrnsformtion, but but solving solving procss procss is complictd is complictd nd nd ccurcy ccurcy limitd. limitd. In this In sction, this sction, numricl numricl intgrl pproch intgrl pproch is doptd is doptd to solvto solv qutions qutions using using MATLAB. MATLAB. Th Th orbitl orbitl lmnts lmnts vritions vritions r prsntd r prsntd in Figur in Figur, which, which shows shows consistncy consistncy with with formntiond orticl orticl nlysis. nlysis. Smi-mjor xis km Smi-mjor xis b Eccntricity c Inclintion Figur. Cont.

7 Snsors 016, 16, of 4 Snsors 016, 16, of 6 d AAN Argumnt of prig Figur. Curvs of short priodicl trms of GEO SA orbitl lmnts vritions undr Figur. Curvs of short priodicl trms of GEO SA orbitl lmnts vritions undr influncs of J trm. influncs of J trm..4. Attrction of Sun nd Moon.4. Attrction of Sun nd Moon Th third body ttrction prturbtion is minly cusd by ttrction of Sun nd Th third body ttrction prturbtion is minly cusd by ttrction of Sun nd Moon. Moon. Espcilly for GEO SA, s prturbtion ffcts should b considrd. Whn orbit Espcilly for GEO SA, s prturbtion ffcts should b considrd. Whn orbit hight riss hight riss bov 50,000 km, influncs of ttrction of Sun nd Moon will xcd bov 50,000 km, influncs of ttrction of Sun nd Moon will xcd J trm. J trm. Th cclrtion cusd by Moon ttrction cn b xprssd s [] p. 65: Th cclrtion cusd by Moon ttrction cn b xprssd s [] p. 65: µ M rm M r M r = M r m mr r M µ Mr M 0 r mr mr 3 M M M M is Moon's grvittionl constnt, r whr µ M is Moon's grvittionl constnt, M is rdil vctor of Moon in Erth cntrd inrtil coordint systm ECI. r r M is rdil vctor of Moon in Erth cntrd inrtil coordint systm ECI. r is rdilis vctor rdil of vctor stllit of in ECI. stllit m is in ECI. stllit m mss. is Th stllit cclrtion mss. Th cusd cclrtion by Sun cusd ttrction by hs Sun ttrction similr form hs s similr Eqution form 0. s Eqution 0. Th prturbtion cclrtions by by Sun Sun nd nd Moon Moon cn lso cn b lso dcomposd b dcomposd into into rdil, rdil, norml norml nd nd trnsvrs trnsvrs componnts componnts nd nnd b n substitutd b substitutd into into prturbtion prturbtion motion Equtions motion Equtions 1 5, obtining 1 5, obtining nlyticl vrition nlyticl forms. vrition As priod forms. of As Erth priod rvolution of round Erth rvolution Sun is on round yr nd Sun priod is on ofyr Moon nd rvolution priod of round Moon Erth rvolution is on month, round GEO Erth SAis orbitl on month, lmnts vritions GEO SA undr orbitl influncs lmnts of vritions third body undr prturbtions influncs of Sun ndthird Moon body ttrction prturbtions hv Sun longnd priod Moon trm ofttrction hlf month hv nd hlf long yr. priod If only trm of hlf ttrction month of nd Sun hlf nd yr. Moon If only is considrd, ttrction of lws of Sun nd prturbd Moon vritions is considrd, r summrizd lws inof Tbl 3. prturbd vritions r summrizd in Tbl 3. Tbl 3. Lw of GEO SA orbitl lmnts vritions undr influncs of third body Tbl prturbtions. 3. Lw of GEO SA orbitl lmnts vritions undr influncs of third body prturbtions. Orbitl Elmnts Short Priod Long Priod Sculr Orbitl Elmnts Smi-mjor xisshort Priod T 0.5 months/0.5 Long Priod yrs - Sculr Smi-mjor xis Eccntricity T T months/0.5 yrs - Eccntricity Inclintion T 0.5 months/0.5 yrs T 0.5 months/0.5 yrs AAN T 0.5 months/0.5 yrs Inclintion Argumnt of prig T T months/0.5 yrs yrs AAN T 0.5 months/0.5 yrs.5. Argumnt Solr dition of prig Prssur T 0.5 months/0.5 yrs Light illumintd on objct s surfc will produc prssur, which is nmd light prssur..5. Solr dition Prssur Bsids visibl light, lctromgntic wvs lso hv light prssur intrction. Th light prssur Light will illumintd impos prturbtion objct s forcs surfc on will stllit produc inprssur, spc nd which willis cus nmd orbit light drifts prssur. nd vritions. Bsids visibl Thislight, kind of lctromgntic prturbtion iswvs clldlso light hv prssur light prturbtion prssur intrction. which Th solr light rdition prssur prturbtion will impos contributs prturbtion most. forcs Thon cclrtions stllit cusd in spc by nd solr will rdition cus orbit prturbtion drifts nd is vritions. xprssdthis s: kind of prturbtion is clld light prssur prturbtion in which solr rdition prturbtion contributs most. Th cclrtions cusd by solr rdition prturbtion is xprssd s: 3

8 Snsors 016, 16, of 4 = Cr A m Kφ c 1 ˆr 1 r whr C r is rdition prssur cofficint, which is gnrlly st s btwn on nd two. C r is dfind s 1 + ε, whr ε is rflctivity. Whn incidnt powr is prfctly bsorbd, ε = 0 nd C r = 1; whn rflction is prfctly diffus, C r = 1.44; whn rflction is prfctly spculr, ε = 1 nd C r =. A/m is stllit r-mss rtio; K is illumintion fctor i.., on or zro dpnding on whr stllit is in sunlight or not; φ is light prssur cting t on stronomicl unit from Sun constnt; c is light vlocity; r is distnc btwn stllit nd Sun; nd ˆr is corrsponding unit vctor. Whn ignoring ccntricity of Erth rvolution nd considring no shdow rgion sunlight incidnt ngl is zro, rdil, norml nd trnsvrs componnts of solr rdition prturbtion cclrtions cn b xprssd s: r = C r A m Kφ c cos f cos l + sin f cos i sin i θ = C r A m Kφ c cos f cos l sin f cos i sin i 3 z = C r A m Kφ c sin f sin i 4 whr l = θ + ω + Ω, f is tru nomly of Sun in ECI nd i is inclintion of Sun in ECI. Substituting Equtions 4 into prturbtion motion Equtions 1 5, w cn obtin lw of GEO SA orbitl lmnts vritions undr influncs of solr rdition prturbtion. As f hs long priod of on yr, orbitl lmnts vritions hv n idnticl long priod. Th lw of vritions is listd in Tbl 4. Tbl 4. Lw of GEO SA orbitl lmnts vritions undr influncs of solr rdition prturbtion. Orbitl Elmnts Short Priod Long Priod Sculr Smi-mjor xis T 1 yr - Eccntricity T 1 yr - Inclintion T 1 yr AAN T 1 yr Argumnt of prig T 1 yr -.6. Influncs on Orbitl Elmnts Th compound influncs of vrious prturbtions on orbitl lmnts nd corrsponding priodicl chnging lws r summrizd in Tbl 5. Th short priodicl vritions r within on orbit priod nd cn influnc GEO SA slnt rng historis. Th long priodicl nd sculr vritions will influnc covrg nd loctions. Tbl 5. Summry of priodicl chnging lws of prturbd GEO SA orbitl lmnts vritions SP: short priodicl trm, LP: long priodicl trm, S: sculr trm. Orbitl Elmnts Erth s Non-Sphricl Mss Distribution Attrction of Moon nd Sun Solr dition Prssur Smi-mjor xis SP SP + LP SP + LP Eccntricity SP SP + LP + S SP + LP Inclintion SP SP + LP + S SP + LP + S AAN SP + S SP + LP + S SP + LP + S Argumnt of prig SP + S SP + LP + S SP + LP

9 Snsors 016, 16, of 6 Snsors 016, 16, of 4 Th formntiond nlyss of prturbtions influncs on GEO SA r givn bsd on Th dynmic formntiond qutions sprtly. nlyss ofhowvr, prturbtions in rl css, influncs prturbtions GEO SA impct r givn GEO bsd SA on simultnously, dynmic qutions nd thus, sprtly. orbitl Howvr, chnging inruls rl css, undr prturbtions totl prturbtions impctcnnot GEO SA b simultnously, prsntd in nlyticl nd thus, forms. orbitl Instd, chnging influncs ruls undr r clcultd totl prturbtions numriclly. cnnot STK provids b prsntd two in numricl nlyticl modls, forms. Instd, i.., high-prcision influncs r orbit clcultd propgtor numriclly. HPOP STK nd provids long-trm two numricl orbit modls, prdictor i.., LOP. high-prcision Th HPOP orbit cn propgtor ccurtly HPOP clcult nd orbit long-trm undr orbit prdictor comprhnsiv LOP. Th prturbtion HPOP cn nvironmnt ccurtly clcult nd is suitbl orbit for undr short priod comprhnsiv nd highly prturbtion ccurt orbit nvironmnt clcultion. Th LOP cn clcult vrgd influncs of prturbtions nd cn rduc grtly nd is suitbl for short priod nd highly ccurt orbit clcultion. Th LOP cn clcult clcultion tim undr som prcision prcondition. vrgd influncs of prturbtions nd cn rduc grtly clcultion tim undr som LOP is suitbl for orbit clcultion in much longr tim scls, such s svrl months or prcision prcondition. svrl yrs. Th chnging ruls of GEO SA orbit within fiv yrs r clcultd using LOP LOP is suitbl for orbit clcultion in much longr tim scls, such s svrl months or in STK. Firstly, w will us STK to build scn whr GEO SA stllit is ddd. Th orbit svrl yrs. Th chnging ruls of GEO SA orbit within fiv yrs r clcultd using LOP in prmtrs of GEO SA will b input s ttributs of stllit. Th initil orbit prmtrs of STK. Firstly, w will us STK to build scn whr GEO SA stllit is ddd. Th orbit prmtrs GEO SA r s follows: smi-mjor xis is 4,164. km; ccntricity is 0.07; orbit of GEO SA will b input s ttributs of stllit. Th initil orbit prmtrs of GEO SA inclintion is 53 ; AAN is 110 ; nd rgumnt of prig is 70. Thn, simultion r s follows: smi-mjor xis is 4,164. km; ccntricity is 0.07; orbit inclintion is 53 ; prmtrs in AAN is 110 Tbl r filld into STK for simulting ; nd rgumnt of prig is 70 prturbtions. Hr, svrl ssumptions r. Thn, simultion prmtrs in Tbl r filld md: 1 ccntricity of Erth s rvolution is not considrd; shdow rgion of into STK for simulting prturbtions. Hr, svrl ssumptions r md: 1 ccntricity of Erth is not considrd; 3 sunlight is incidnt t ngl of 0. In or words, cs of Erth s rvolution is not considrd; shdow rgion of Erth is not considrd; 3 mximum r-mss rtio is considrd sunlight is incidnt t ngl of 0 in simultions; whil in ctul css, r-mss rtio. In or words, cs of mximum r-mss rtio is could b vrint nd smllr. considrd Simultion in rsults simultions; r s whil shown in in ctul Figur css, 3. Th orbitl r-mss lmnts rtio could undr b vrint influncs nd smllr. of compound Simultion prturbtions rsults r hv s shown long in nd Figur short 3. Th priodicl orbitl vritions. lmnts undr All of orbitl influncs lmnts, of compound but smi-mjor prturbtions xis, hv hv long sculr nd short vritions. priodicl Undr vritions. influncs All of of orbitl prturbtions, lmnts, but chnging smi-mjor priod xis, of hv smi-mjor sculrxis vritions. is sm Undr s tht of influncs longitud of prturbtions, drifts nd GEO chnging SA priod orbitl ofpriod smi-mjor vritions xis round is sm.7 yrs. s thtin offigur longitud 3f, initil drifts nd orbitl GEO priod SA of orbitl GEO SA priod is vritions idnticl round to Erth.7 yrs. rottion In Figur nd is 3f, 86,164 initil s. Whn orbitl influncd priod ofby GEO prturbtions, SA is idnticl to GEO SA Erth rottion orbitl priod nd is 86,164 incrss s. Whn nd bcoms influncd longr by prturbtions, thn Erth rottion. GEO SAMnwhil, orbitl priod incrss longitud nd of bcoms scnding longrnod thndcrss Erthnd rottion. bhvs Mnwhil, s nodl rgrssion. longitud of Aftr scnding 1.38 yrs, nod GEO dcrss SA nd orbitl bhvs priod s nd nodl longitud rgrssion. of scnding Aftr 1.38 yrs, nod bhv GEOoppositly. SA orbitl Finlly, priodthis ndwill longitud ld to of scnding rciprocting nodmotion bhvround oppositly. crtin Finlly, longitud this will nd ld tovrition rciprocting rng is motion round round 50, which crtin is longitud function of nd initil vrition longitud rngrltiv is round to 50, stbl whichlongitud is function points, of but initil obtind longitud through rltiv tostk stbl simultion longitud hr. points, Th mximum but obtind vlu through pprs t STK simultion momnt whn hr. Th mximum stllit is vlu snt pprs into orbit. t momnt whn stllit is snt into orbit. Inclintion Smi-mjor xis b Eccntricity c Inclintion Figur 3. Cont.

10 Snsors 016, 16, of 4 Snsors 016, 16, of 6 d AAN Argumnt of prig f Orbit priod nd longitud of scnding nod Figur 3. Curvs of prturbd GEO SA orbitl lmnts vritions within fiv yrs. Figur 3. Curvs of prturbd GEO SA orbitl lmnts vritions within fiv yrs. As short priodicl prturbtions will influnc focusing, mgnituds of prturbd orbitl As lmnts short priodicl r simultd prturbtions bsd on will influnc HPOP modl focusing, in STK. Th mgnituds simultion of prturbd prmtrs orbitl nd lmnts options r r similr simultd to thos bsd in on LOP, s HPOP shown modl in Tbl in STK. 6. Th Th simultion simultion strt prmtrs tim is 4:00 nd.m. options on 1 r My similr 013 to nd thos in nd LOP, tim s is shown 4:00.m. in Tbl on 6 6. My Th 013. simultion Aftr strt simultion, tim is w 4:00 will.m. us on 1 MATLAB My 013 nd connctor ndto tim intrfc is 4:00with.m. on STK. 6 My Th 013. prturbd Aftrorbit simultion, lmnts wftr will us prst MATLAB tim intrvl connctor cn b to intrfc output with in MATLAB. STK. Th prturbd Thus, orbit vrition lmnts rngs ftr of prst orbitl tim intrvl lmnts cn b cusd output by in MATLAB. short Thus, priodicl vrition prturbtions rngscn of b orbitl rtrivd, lmnts nd cusd rsults by r listd short in priodicl Tbl 7. prturbtions cn b rtrivd, nd rsults r listd in Tbl 7. Tbl 6. STK prmtrs nd options for prturbtions simultion. Tbl 6. STK prmtrs nd options for prturbtions simultion. Prmtrs Vlus Prmtrs Vlus Prmtrs Simultion strt tim 0:00 Vlus.m. 1 July 007 Simultion Prmtrs nd tim 1:00 p.m. 11 July Vlus 01 Simultion Coordint strt timsystm 0:00.m. 1J000 July 007 Erth Simultion grvity nd modl tim EGM :00 p.m July 01 Coordint Solr systm rdition modl J000 Sphricl Surfc Erth grvity rflctivity modl EGM Solr rdition Ar-mss modl rtio Sphricl m /kg Th third Surfc body rflctivity ttrction Th Sun/Moon 1.3 Ar-mss rtio m /kg Th third body ttrction Th Sun/Moon Tbl 7. ngs of orbitl lmnts vritions cusd by simultd short priodicl Tbl prturbtions. 7. ngs of orbitl lmnts vritions cusd by simultd short priodicl prturbtions. Erth s Non-Sphricl Attrction of Moon Orbitl Elmnts Solr dition Prssur Ors Totl Erth s Mss Non-Sphricl Distribution Attrctionnd of Moon Sun Orbitl Elmnts Solr dition Prssur Ors Totl Smi-mjor xis m Mss Distribution 308 nd Sun Smi-mjor Eccntricity xis m Inclintion Eccntricity Inclintion AAN Argumnt AAN of prig Argumnt of prig Influncs of Prturbtions on GEO SA Focusing 3. Influncs of Prturbtions on GEO SA Focusing GEO SA focusing is impctd by slnt rng rrors within intgrtion tim. Th slnt rng GEO rrors SA r focusing cusd is by impctd vritions by of slnt prturbd rng rrors GEO within SA orbitl intgrtion lmnts, tim. consisting Th slnt of rng short rrors priodicl, r cusd bylong vritions priodicl of nd prturbd sculr componnts. GEO SA orbitl Bcus lmnts, long consisting priodicl of nd short sculr priodicl, orbit drifts long will priodicl impct nd chrctristics sculr componnts. of covrg Bcus nd loctions, long priodicl y will nd not sculr influnc orbit focusing drifts will nd, impct thus, r chrctristics not discussd of in this covrg sction. nd In summry, loctions, this ysction will not crris influnc out focusing modlling nd, thus, of r slnt notrngs discussd nd inn this sction. nlyzs In summry, slnt rng this sction vritions crris whn out ch modlling orbitl oflmnt slnt rngs hs nd smll n vrition nlyzs of, which slnt rng is rltd vritions to short whn priodicl ch orbitl prturbtions. lmnt hs smll vrition of, which is rltd to short priodicl prturbtions GEO SA Slnt ng Modl 3.1. GEO SA Slnt ng Modl In Erth-cntrd Erth-fixd coordint ECEF systm, considring tht tru nomly t In prtur Erth-cntrd momnt Erth-fixd t0 is f0, coordint coordints ECEF of systm, point considring trgt bing tht focusd tru r: nomly t prtur cntr momnt t 0 is f 0, coordints of point trgt bing focusd r: rt f0 xt f0, yt f0, zt f0 5 r t f 0 = x t f 0, y t f 0, z t f 0 5 whr:

11 Snsors 016, 16, of 4 whr: x t = [cos f + ω cos βcos α sin f + ω sin βcos icos α sin βsin isin α] y t = [cos f + ω sin βcos α + sin f + ω cos βcos icos α + cos βsin isin α] z t = [sin f + ω sin icos α cos isin α] 6 whr is Erth rdius, α is gocntric ngl nd β quls Ω Ω G nd is rltd to longitud. Ω G is Grnwich sidrl hour ngl nd quls Ω G0 + ω t, whr Ω G0 is initil Grnwich sidrl hour ngl nd ω is Erth rottion ngulr vlocity. f is tru nomly; ω is rgumnt of ltitud; i is orbit inclintion. Th coordints of GEO SA cn b xprssd s: r s f = x s f, y s f, z s f 7 whr: x s f = cos f + ω cos β r sin f + ω sin βcos i r y s f = cos f + ω sin β r + sin f + ω cos βcos i r z s f = sin f + ω sin i r 8 Thrfor, slnt rng cn b clcultd s: [ ] [ ] [ = x s f x t f 0 + y s f y t f 0 + z s f z t f 0 ] 9 Th simplifiction of Eqution 9 cn b xprssd s: = r + rcos α 0 30 whr is slnt rng nd r is gocntric distnc of GEO SA nd quls 1 / 1 + cos f. α 0 is gocntric ngl of prtur cntr. Infrrd from Eqution 30, is minly dpndnt on smi-mjor xis nd ccntricity. Whn substituting tru nomly f into Eqution 30, with rspct to f cn b considrd s rng historis of GEO SA. Assuming rng-dopplr lgorithm is doptd, loction of trgt will b influncd by, s wll. Thrfor, loction will lso b dpndnt on smi-mjor xis nd ccntricity. Furrmor, loction will lso b ffctd by loction of stllit nd bm pointing. If loction of stllit chngs whil bm pointing rmins unchngd, loction rrors will b inducd in trgt loction. Conclusivly, slnt rng history will b influncd by smi-mjor xis nd ccntricity, whil loction of trgt will b influncd by ll of orbit lmnts. In ordr to nlyz influncs of orbitl lmnts vritions on focusing, GEO SA slnt rngs should b firstly Tylor xpndd for rtriving Dopplr prmtrs. Thus, drivtivs of slnt rngs will ffct focusing qulity. Th nlyticl forms of ch ordr of drivtiv of slnt rng r obtind. Thn, influncs of orbitl lmnts vritions on focusing r nlyzd. Th slnt rng in Eqution 30 is Tylor xpndd to third-ordrs t prtur cntr, nd w hv: 0 + d dt t t t=t0! d dt t t t=t0 3! d3 dt 3 t t t=t0 whr t 0 is prtur cntr momnt nd t is slow tim.

12 Snsors 016, 16, of 4 Th tim drivtivs of slnt rng cn b clcultd s rfr to Appndix A for dtild drivtions: d µ dt = Ṙ = P sin f [r cos α 0 ] 3 d dt =.. = µ Ṙ + cos f µ P r cos α 0 cos f 33 d 3 =... dt 3 = cos α 0 µ 3 P 1 r sin f µ. µ 3 P 1 r 3 At prtur cntr whr t = t 0 nd f = f 0, w hv: d dt d 3 t=t0 µ = 3 dt 3 P sin f 0 = t=t0. 3.Ṙ Ṙ + cos f + P. µ sin f + 3 sin f cos f + cos f r µ P sin f 0 cos θ L µ 1/ sin f 0 cos θ L r 0 1/ + d t=t0 = µ sin f 0 cos θ L + cos f 0 cos α 0 cos f 0 dt r cos f 0 r 0 0 µ cos f 0 r cos α 0 r 0 r 0 0 µ 3 P sin f 0 = r 0 0 = µ cos f 0 cos θ L r 0 r cos θ L sin f 0 cos θ L cos f cos f cos θ L r 0 0 cos f 0 cos α r cos f 0 cos α 0 [ 0 1 r0 3 cos θ ] L cos f cos θ L cos f 0 0 cos α 0 r r cos f 0 cos α 0 µ 3 P sin f 0 cos θ L 1+3 cos f 0 r 0 µ 3/ sin f 0 cos θ L r 0 7/ In clculting Equtions 35 37, cos α 0 is projction of Erth rdius to gocntric rng of GEO SA. Thrfor, w hv rltionship of cos θ L = r 0 cos α 0 / 0 whr θ L is down-look ngl. Substituting Equtions into Eqution 31, w hv: µ 1/ sin f 0 cosθ L r 0 1/ t t 0 + µ cos f 0 cosθ L r 0 t t 0 µ 3/ sin f 0 cosθ L 6r 0 7/ t t 0 3 = 0 + C 1 t t 0 + C t t 0 + C 3 t t whr C i cn b xprssd s: C 1 = µ 1/ sin f 0 cosθ L r 0 1/ C = µ cos f 0 cosθ L r 0 C 3 = µ 3/ sin f 0 cosθ L 6r 0 7/. In ordr to vlidt slnt rng modl, w will chck phs rrors inducd by Tylor xpnsion to pproximt idl slnt rng s bov discussion. Th simultion prmtrs of imging vlidtion r listd in Tbl 8. It should b notd tht for highly inclind GEO SA with inclintion of 53, intgrtion tim of round 100 s cn chiv modrt rsolution of 0 m. 39

13 0 In ordr to vlidt slnt rng modl, w will chck phs rrors inducd by Tylor xpnsion to pproximt idl slnt rng s bov discussion. Th simultion prmtrs of imging vlidtion r listd in Tbl 8. It should b notd tht for highly inclind GEO SA with inclintion of 53, intgrtion tim of round 100 s cn chiv Snsors 016, 16, of 4 modrt rsolution of 0 m. Tbl 8. Simultion prmtrs of of imging vlidtion xprimnt. Prmtrs Vlus Unit Prmtrs VlusUnit Prmtrs Vlus Unit Prmtrs Vlus Unit Smi-mjor xis 4,164. km Down-look ngl 4.65 Smi-mjor xis 4,164. km Down-look ngl 4.65 Eccntricity Antnn dimtr 4 m Eccntricity Antnn dimtr 4 m Inclintion 53 Inclintion 53 Wvlngth 0.4 m Wvlngth 0.4 m AAN AAN PF PF 00 Hz 00 Hz Argumnt ofargumnt prig of prig Intgrtion Intgrtion tim tim 100 s 100 s Mn nomly Mn nomly 0 0 Puls Puls width width 0 us 0 us m m Signl Signl bndwidth 18 MHz18 MHz Smpling rt rt 0 MHz0 MHz Assuming GEO SA trnsmitting nd rciving slnt rngs r idnticl to, phs rrors Assuming whn using GEOTylor SAxpnsion trnsmitting r nd shown rciving in Figur slnt 4. Infrrd rngs r from idnticl Figur 4, to, phs phs rrors whn using Tylorxpnsion 4 rrors r t ordrs of 10 t prig r nd shown pog. in Figur Th mximum 4. Infrrdphs from rror Figur t 4, qutor phs is rrors r 0.3 t. Thus, ordrs of phs 10 4 πrrors t prig r ll nd blow pog. orticl Th mximum thrshold phs of rror 4. Th t pproximtion qutor is 0.3π. Thus, cn stisfy phs nlysis rrors r of ll influncs blow of orticl orbitl thrshold lmnts of vritions π/4. Th on pproximtion drivtivs cn of stisfy slnt nlysis rng. of influncs of orbitl lmnts vritions on drivtivs of slnt rng. Prig b Equtor c Apog Figur 4. Phs rrors of Tylor xpnsion pproximtion to idl GEO SA slnt rng. Figur 4. Phs rrors of Tylor xpnsion pproximtion to idl GEO SA slnt rng. 3.. Influncs of Orbitl Elmnts on Slnt ng 3.. Influncs of Orbitl Elmnts on Slnt ng As formntiond nlysis, only smi-mjor xis nd ccntricity should b considrd As formntiond in nlyzing influncs nlysis, on only GEO smi-mjor SA slnt xis rngs. ndthrfor, ccntricity in this should sction, bonly considrd in rrors nlyzing of influncs slnt rng on drivtivs GEOC1, SA C slnt nd C3 rngs. r drivd Thrfor, whn in this vritions sction, of only smi-mjor rrors of slnt xis nd rng drivtivs ccntricity Cxist. 1, C nd C 3 r drivd whn vritions of smi-mjor xis nd ccntricity xist Influncs of Smi-Mjor Axis Whn smi-mjor xis hs smll vrition of, ll ordrs of slnt rng cofficint cn b pproximtd by first drivtiv: C i dc i 40 d whr C i cn b rfrrd to Eqution 38 nd r 0 = 1 / 1 + cos f 0. It cn b trnsformd into s Appndix A for dtild dvitions: C 1 dc 1 d = 1 C1 41 C dc d = C 4 C 3 dc 3 d = 7 1 C3 43

14 dc1 1 C1 C 1 d 41 dc C C d 4 Snsors 016, 16, of 4 dc3 7 C3 C 3 d 1 43 According to Figur 3, hs long priodicl vrition with mximum rng of 5 km. According to Figur 3, hs long priodicl vrition with mximum rng of 5 km. Thrfor, ll ordrs of slnt rng cofficint cn b producd s mximum vlu of = 5 km, Thrfor, ll ordrs of slnt rng cofficint cn b producd s mximum vlu of nd syntic prtur tim is 100 s. Th corrsponding rsults r shown in Figur 5. Th phs = 5 km, nd syntic prtur tim is 100 s. Th corrsponding rsults r shown in rrors r lmost sm for ll of down-look ngls. Thrfor, rsults in Figur 5 tk Figur 5. Th phs rrors r lmost cs of down-look ngl of 4.65 sm for ll of down-look ngls. Thrfor, rsults. From Figur 5, third-ordr phs rrors r blow π in Figur 5 tk cs of down-look ngl of From Figur 5, third-ordr phs rrors nd cn b ignord. 4 At prig nd pog, minly lds to scond-ordr phs rrors. r blow 4 10 nd cn b ignord. At prig nd pog, minly lds to At scond-ordr qutor, phs minly rrors. lds At to qutor, first-ordr phs rrors. At or orbit positions, cn produc minly lds to first-ordr phs rrors. At both or orbit first-ordr positions, nd scond-ordr cn produc both phs rrors. first-ordr nd scond-ordr phs rrors. First-ordr phs rrors b Scond-ordr phs rrors c Third-ordr phs rrors Figur 5. Phs rrors of on orbit long with tru nomly cusd by vritions of Figur 5. Phs rrors of on orbit long with tru nomly cusd by vritions of smi-mjor smi-mjor xis. xis Influncs of Eccntricity 3... Influncs of Eccntricity Whn ccntricity hs smll vrition of, ll ordrs of slnt rng cofficint cn b Whn pproximtd ccntricity by s Appndix hs smll A for vrition dtild of, dvitions: ll ordrs of slnt rng cofficint cn b pproximtd by s Appndix A for dtild dvitions: dc1 1 1 dr C1 C1 44 d r d C 1 dc 1 1 d = C 1 dc 1 1 r dr 44 dr d C C 45 d r d C dc 1 d dc = C 3 1 7r dr dr 45 d C3 C3 46 d r d C 3 dc 3 1 Th vritions of ll ordrs of dslnt = rng C 3 cofficint 7 r dr 46 dcusd by r t lvls of , 10 6 Th vritions nd of 10 ll ordrs. According of slnt to Figur rng cofficint 3b, hs cusd sculr by vrition, r t nd lvls nnul of 10 3, 10 1 nd According to Figur 3b, hs sculr vrition, nd nnul dvition is round Thrfor, in css of = nd syntic prtur tim of 100 s, ll ordrs phs rrors cusd by r clcultd nd prsntd in Figur 6. Th phs rrors r lso lmost sm for ll of down-look ngls. Thrfor, rsults in Figur 6 tk cs of down-look ngl of From Figur 6, third-ordr phs rrors r blow 0.03π nd cn b ignord. At prig nd pog, minly lds to scond-ordr phs rrors. At qutor, minly lds to first-ordr phs rrors. At or orbit positions, cn produc both first-ordr nd scond-ordr phs rrors. Conclusivly, smi-mjor xis nd ccntricity vritions cusd by prturbtions cn rsult in rrors of first-ordr nd scond-ordr of slnt rng nd, thus, corrsponding first-ordr nd scond-ordr phs rrors, lding to focusing dgrdtion. Th third-ordr phs rrors or bov will not ffct focusing qulity.

15 rrors r lso lmost sm for ll of down-look ngls. Thrfor, rsults in Figur 6 tk cs of down-look ngl of From Figur 6, third-ordr phs rrors r blow 0.03 nd cn b ignord. At prig nd pog, minly lds to scond-ordr phs rrors. At qutor, minly lds to first-ordr Snsors phs 016, rrors. 16, 140 At or orbit positions, cn produc both first-ordr nd scond-ordr 15 of 4 phs rrors. First-ordr phs rrors b Scond-ordr phs rrors c Third-ordr phs rrors Figur 6. Phs rrors of on orbit long with tru nomly cusd by vritions of Figur 6. Phs rrors of on orbit long with tru nomly cusd by vritions of ccntricity. ccntricity Influncs of Orbitl Elmnts Vritions on Focusing Conclusivly, smi-mjor xis nd ccntricity vritions cusd by prturbtions cn rsult According rrors to focusing of first-ordr ory, ll nd ordrs scond-ordr of phs rrors of xcpt slnt rng constnt nd, phs thus, could ffct corrsponding focusing. first-ordr Th first-ordr nd scond-ordr will only ld phs to rrors, img drifts, lding whil to focusing scond-ordr dgrdtion. will chng Th third-ordr frquncyphs modultion rrors or rt bov f dr will nd not lds ffct to focusing dgrdtion, qulity. including brodning nd dclintion of min lob, long with sidlobs rising. Th third-ordr will rsult in symmtricl 3.3. Influncs sidlobs of Orbitl nd Elmnts brodning Vritions on min Focusing lob. Th According point trgts to t focusing prigory, nd qutor ll ordrs rof slctd phs rrors for vlidting xcpt constnt influncs phs of orbitl could lmnts ffct vritions focusing. on Th focusing. first-ordr Th will focusing only ld lgorithms to img mploying drifts, whil sris scond-ordr rvrsion will [4] chng will b usd. frquncy In simultion, modultion rt initil fdr nd idl lds twoto body motion focusing nd dgrdtion, slnt rngs including r firstly brodning obtind using nd dclintion STK. Thn of prturbd min lob, slnt long rngs with ftr nsidlobs incrmnt rising. of Th for third-ordr smi-mjor will xis rsult ndin for symmtricl ccntricitysidlobs r lso producd nd brodning using STK. min Th idl lob. two body slnt rngs will b usd s rfrnc to mtch Th point prturbd trgts slnt t prig rngs ndqutor focus. Th r simultion slctd for prmtrs vlidting of influncs imging vlidtion of orbitl r lmnts rfrrdvritions to Tbl 8. on According focusing. toth focusing simultion lgorithms in Tbl 7, mploying vritions of sris smi-mjor rvrsion [4] xiswill nd b ccntricity usd. In cusd simultion, by prturbtions initil will idl btwo 588body m ndmotion nd 4. So inslnt imging rngs r vlidtion, firstly obtind nd using r ststk. s 5300 Thn m ndprturbd slnt, rspctivly. rngs ftr n incrmnt of for smi-mjor xis As nd shown for in Figur ccntricity 7 c, it cnr blso wllproducd focusd inusing rng STK. nd Th zimuth idl two tbody prig slnt if no rngs vritions will r b ddd usd s inrfrnc orbitl to lmnts, mtch nd prturbd pk slnt sidlob rngs rtios nd focus. PSL Th cnsimultion chiv prmtrs idl lvlof of 13. imging db. As shown vlidtion in Figur r rfrrd 7d f, whn to Tbl n incrmnt 8. According of 5300 to msimultion is ddd toin Tbl smi-mjor 7, vritions xis i.., itof riss from smi-mjor 4,164. 4,169.5 xis nd km, ccntricity it cnnotcusd b wllby focusd prturbtions in zimuth will b 588 t prig. m nd Th 1.7 sidlobs So riss in sriously, imging nd vlidtion, zimuth PSL nd is dtriortd r st s 5300 to 11.0 m nd db. 1.7 Th 10focusing 4, rspctivly. in rng is not ffctd. Mntim, As shown trgt in Figur rmins 7 c, t it cn scn b cntr, wll focusd nd noin symmtricl rng nd sidlobs zimuth xist. t prig This suggsts if no tht vritions onlyr rsults ddd in in scond-ordr orbitl lmnts, of nd slnt rng pk without sidlob rtios first-ordr PSL nd cn scond-ordr chiv vritions. idl lvl As of shown 13. db. in Figur As shown 7g i, in whn Figur n7d f, incrmnt whn of n incrmnt is ddd of 5300 to m is ccntricity ddd to i.., itsmi-mjor riss from , xis i.., it riss it cnnot from 4,164. 4,169.5 b focusd t ll in km, it zimuth cnnot b t prig. wll focusd Th min lob zimuth hs bn t ovrwhlmd prig. Th by sidlobs sidlobs, riss sriously, but trgt nd is stillzimuth in scn PSL cntr, is dtriortd nd no symmtricl to 11.0 sidlobs db. Th xist focusing ir. in Th rng rng is focusing not ffctd. is good. Mntim, This suggsts trgt similr rmins vrition t s tht scn for, cntr, nd nd only no scond-ordr symmtricl slnt sidlobs rng xist. is chngd. This suggsts tht only rsults in scond-ordr of slnt rng For without focusing first-ordr t qutor, nd scond-ordr rsults r shown vritions. in Figur As shown 7j l. It in cn Figur b wll 7g i, focusd whn in n rng incrmnt nd zimuth of whn is ddd no orbitl to lmnts ccntricity vritions i.., it xist. riss Whn from , vritions it cnnot nd b r considrd, focusd t ll only in loction zimuth of t prig. point trgt Th min is chngd lob hs without bn ovrwhlmd ny influnc by on focusing. sidlobs, Thus, but it trgt is still in scn cntr, nd no symmtricl sidlobs xist ir. Th rng focusing cn b concludd tht only nd chng first-ordr slnt rng of GEO SA t qutor. Th focusing qulity is not impctd, nd thus, prturbd imging rsults r not shown hr. In conclusion, GEO SA slnt rng is minly rltd to smi-mjor xis nd ccntricity. Th vritions of s two itms will rsult in rrors of first-ordr nd scond-ordr slnt rngs, whil influncs of third-ordr nd bov could b ignord. At diffrnt orbit positions, influncs hv diffrnt bhviors. At qutor, first-ordr phs rrors should b minly considrd; t prig nd pog, scond-ordr phs rrors should b minly considrd; t or positions, first-ordr nd scond-ordr phs rrors xist simultnously.

16 Amplitud db ccntricity. Th vritions of s two itms will rsult in rrors of first-ordr nd scond-ordr slnt rngs, whil influncs of third-ordr nd bov could b ignord. At diffrnt orbit positions, influncs hv diffrnt bhviors. At qutor, first-ordr phs rrors should b minly considrd; t prig nd pog, scond-ordr phs rrors should016, b 16, minly considrd; t or positions, first-ordr nd scond-ordr phs 16 rrors Snsors 140 of 4 xist simultnously. b Azimuth profil idl, prig c ng profil idl, prig Amplitud db Contour idl, prig ng smpls d Contour, prig Azimuth profil, prig f ng profil, prig g Contour, prig h Azimuth profil, prig i ng profil, prig j Contour idl, qutor k Azimuth profil idl, qutor l ng profil idl, qutor Snsors 016, 16, of 6 Figur 7. Prturbd imging rsults of simultd point trgts in GEO SA whn considring Figur 7. Prturbd imging rsults of simultd point trgts in GEO SA whn considring vritions of smi-mjor xis nd ccntricity. vritions of smi-mjor xis nd ccntricity. 4. STK Simultion nd Vrifiction 4. STK Simultion nd Vrifiction Th prturbd GEO SA slnt rng history nd ccurt signl modl cnnot b obtind Th prturbd GEO SA slnt rng history nd ccurt signl modl cnnot b obtind nlyticlly from GEO SA gomtry nd prturbtion dynmic qutions dirctly. For nlyticlly from GEO SA gomtry nd prturbtion dynmic qutions dirctly. For scond bst, prturbd signls cn b gnrtd by using dducd vritions of slnt scond bst, prturbd signls cn b gnrtd by using dducd vritions of slnt rngs. rngs. This is usful in summrizing chnging lws bsd on rror propgtion ory, but This is usful in summrizing chnging lws bsd on rror propgtion ory, but not vry not vry convincing in nlyzing influncs of prturbtions on GEO SA ccurtly. Instd, convincing in nlyzing influncs of prturbtions on GEO SA ccurtly. Instd, numricl numricl pproch is good ltrntiv in simulting influnc dirctly. In this sction, pproch is good ltrntiv in simulting influnc dirctly. In this sction, HPOP in STK HPOP in STK is usd to simult prturbd GEO SA slnt rng historis nd n to gnrt is usd to simult prturbd GEO SA slnt rng historis nd n to gnrt chos. chos. Th sris rvrsion lgorithm is doptd for focusing. Th simultion prmtrs r sm s in Tbl 8. Th simultion rsults r prsntd in Figur 8. Figur 8 givs drifts of minimum slnt rng quivlnt to rng t prtur cntr. Th drivtions from initil position incrs continully within 30 dys nd ccumult up to 65 km t prig in comprison with 34 km t qutor. Ignoring constnt componnt in slnt rng, prturbd slnt rng historis within 100-s prtur tim r prsntd in Figur 8b. Th vritions r chnging for

17 Snsors 016, 16, of 4 Th sris rvrsion lgorithm is doptd for focusing. Th simultion prmtrs r sm s in Tbl 8. Th simultion rsults r prsntd in Figur 8. Figur 8 givs drifts of minimum slnt rng quivlnt to rng t prtur cntr. Th drivtions from initil position incrs continully within 30 dys nd ccumult up to 65 km t prig in comprison with 34 km t qutor. Ignoring constnt componnt in slnt rng, prturbd slnt rng historis within 100-s prtur tim r prsntd in Figur 8b. Th vritions r chnging for 1 5 dys from initil position. Th phs rrors cusd by prturbtions within intgrtion tim of 100 s cn chiv up to 0.14π, s shown in Figur 8c. If prtur incrss, phs rrors lso dtriort Snsors 016, nd 16, 140 vn hv to b considrd whn chivd up to crtin xtnt. 18 of 6 slnt rng incrmnt b Slnt rng history c Phs rrors d fdc vritions fdr vritions f fdrr vritions Figur 8. Vritions of slnt rng history nd Dopplr cofficint Dopplr cntroid: fdc, Figur 8. Vritions of slnt rng history nd Dopplr cofficint Dopplr cntroid: f dc, frquncy modultion rt: fdr nd third-ordr Dopplr rt: fdrr of prturbd GEO SA within frquncy modultion rt: f dr nd third-ordr Dopplr rt: f drr of prturbd GEO SA within on prtur t prig. on prtur t prig. Figur 8d f shows vritions of first-ordr to third-ordr Dopplr cofficint, which will influnc focusing. Th first-ordr Dopplr rt cofficint is Dopplr cntroid f dc ; scond-ordr is Dopplr modultion rt f dr ; nd third-ordr is scond drivtivs of Dopplr history f drr. At prig, mximum incrmnt of f drr ftr 30 dys is up to 10 7 Hz/s within intgrtion tim of 100 s. Corrspondingly, ccumultd phs rror within 100 s is only π, which is fr lss thn thrshold of π/4 in SA ory. Thus, influnc of f drr cn b ignord. In comprison, vritions of f dc r non-zro nd will induc img drifts; vritions of f dr would rsult in ccumultd qudrtic rrors of bov π/4, which will cus dfocusing. Conclusivly, vritions Contour of 1 f dy b Azimuth profil 1 dy c ng profil 1 dy dc nd f dr should b considrd. Figur 9 shows prturbd imging rsults of point trgts t prig ftr 1, nd 6 dys prturbtions. Th prturbtion cn produc phs rrors within on prtur nd is crtin to dgrd focusing. Th zimuth PSL of imging rsult ftr on dy s Figur 9 c will drop to 1.8 db, whil rng focusing is good. As for imging rsults ftr two dys in Figur 9d f, sidlobs ris pprntly; zimuth PSL drops to 10.7 db; nd point trgt dvits from scn cntr. This mns tht r xist obvious first-ordr nd scond-ordr phs rrors. As for imging rsults ftr six dys in Figur 9g i, r is srious dfocusing in zimuth, nd d Contour dy Azimuth profil dy f ng profil dy

18 Snsors 016, 16, 140 d fdc vritions 18 of 4 fdr vritions f fdrr vritions Figur 8. Vritions of slnt rng history nd Dopplr cofficint Dopplr cntroid: fdc, drift is lso srious. In comprison, focusing in rng is not impctd. Th vlutions of frquncy modultion rt: fdr nd third-ordr Dopplr rt: fdrr of prturbd GEO SA within imging on rsults of prturbd GEO SA point trgts t prig r listd in Tbl 9. prtur t prig. Contour 1 dy Snsors 016, 16, 140 c ng profil 1 dy Azimuth profil dy f ng profil dy h Azimuth profil 6 dy i ng profil 6 dy 19 of 6 Azimuth smpls d Contour dy b Azimuth profil 1 dy ng smpls g Contour 6 dy Figur 9. Imging rsults of prturbd GEO SA point trgts t prig ftr 1, nd 6 dys Figur 9. Imging rsults of prturbd GEO SA point trgts t prig ftr 1, nd 6 prturbtions. dys prturbtions. Tbl 9. Evlutions of prturbd GEO SA point trgts t prig. Tbl 9. Evlutions of prturbd GEO SA point trgts t prig. PSL db ISL db Drift m ng Azimuth ng AzimuthAzimuth PSL db ISL db Idl two body motion ng Azimuth ng 8.0 Azimuth 1 dy s prturbtion dys prturbtion Idl two body motion 6 dys prturbtion Drift m Azimuth 0 1 dy s prturbtion dys prturbtion Conclusions 6 dys prturbtion Prturbtion is min rror sourc influncing GEO SA focusing. It will cus vritions of GEO SA orbit lmnts, mong which chngs of smi-mjor xis nd ccntricity cn rsult 5. Conclusions in vrying slnt rng historis. Th studis bout prturbtion influncs on orbitl lmnts nd r crrid out nlyticlly bsditon cus prturbd motion of Prturbtion is slnt minrng rrorhistoris sourc influncing GEO SA focusing. will vritions qutions nd Tylor xpnsion pproximtion. Th chnging lws of prturbd orbitl GEO SA orbit lmnts, mong which chngs of smi-mjor xis nd ccntricity cn rsult in lmnts nd slnt rng vritions r dducd, long with corrsponding influncs. Th vrying slnt rng historis. Th studis bout prturbtion influncs on orbitl lmnts vritions of prturbd GEO SA slnt rng will induc first-ordr nd scond-ordr nd slnt rng historis r crrid out nlyticlly bsd on prturbd motion qutions rrors within intgrtion tim. Thus, ccumultd linr nd qudrtic phs rrors will nd Tylor xpnsion pproximtion. Th chnging lws of prturbd orbitl lmnts nd dtriort focusing qulity. slnt rng vritions r dducd, long with corrsponding influncs. Th vritions of Th focusing prformnc is nlyzd bsd on numricl pproch using STK, which is doptd to construct SA signls with nd without ffcts of prturbtions. Thn, imging rsults r vlutd nd comprd. Th simultions hv good consistncy with formntiond orticl nlyss tht first-ordr nd scond-ordr phs rrors should b considrd, whil cubic nd highr-ordr phs rrors will not impct imging nd, thus, cn b ignord. For GEO SA with llipticl orbit, inclintion of 53 nd rgumnt

19 Snsors 016, 16, of 4 prturbd GEO SA slnt rng will induc first-ordr nd scond-ordr rrors within intgrtion tim. Thus, ccumultd linr nd qudrtic phs rrors will dtriort focusing qulity. Th focusing prformnc is nlyzd bsd on numricl pproch using STK, which is doptd to construct SA signls with nd without ffcts of prturbtions. Thn, imging rsults r vlutd nd comprd. Th simultions hv good consistncy with formntiond orticl nlyss tht first-ordr nd scond-ordr phs rrors should b considrd, whil cubic nd highr-ordr phs rrors will not impct imging nd, thus, cn b ignord. For GEO SA with llipticl orbit, inclintion of 53 nd rgumnt of prig of 90, influncs r dpndnt on gomtry configurtions s Dopplr prmtrs nd intgrtion tim r diffrnt. Thrfor, t diffrnt orbit positions, influncs hv diffrnt bhviors. At qutor, first-ordr phs rrors should b minly considrd; t prig nd pog, scond-ordr phs rrors should b minly considrd; t or positions, first-ordr nd scond-ordr xist simultnously. Though s conclusions r drivd from spcific rfrnc GEO SA orbit chosn, numricl pproch cn b gnrlizd to GEO SA pplictions. In oprtion, prturbtions influncs will ccumult during mission lif, nd thus, prformnc will dtriort. Whn prturbtion rrors bcom intolrbl, it is rcommndd to compnst m through ccurt msurmnts. Actully, orbit mintnnc is lso possibl compnstion ltrntiv. It is dirct wy to impos crtin forc, which countrcts prturbtion forcs. Corrspondingly, prturbd motion qutions cn b modifid by dding such forc. Howvr, in spc missions, on limiting fctor is ful consumption for orbit mintnnc. If ful runs out, spc missions will b dgrdd or vn fil. Thus, though continuous orbit mintnnc would b ncssry for compnsting influncs of orbit oscilltions on focusing, dmnds for ful will b hug nd byond tolrnc. sultntly, continuous mintnnc is not option. Furrmor, GEO SA hs hug pltform nd lrg ntnn, which r mployd for compnsting hug slnt rng loss. Thrfor, if continuous orbit mintnnc is mployd, it is difficult to stbiliz pltform within focusing tim. Th focusing prformnc cn b dgrdd sriously in this cs. In summry, orbit mintnnc is only prfrrd whn ccumultd orbit drifts cusd by long-trm priodicl nd sculr prturbtions r chivd up to crtin lvl whn obsrvtion pln is impctd. If so, oprtion lif tim cn b prolongd. For improving GEO SA focusing influncd by prturbtions, it is rcommndd to mploy ccurt orbit msurmnts or som signl procssing mthods, such s phs grdint utofocus PGA lgorithm. Acknowldgmnts: This work ws supportd by Ntionl Nturl Scinc Foundtion of Chin Grnt No , No , No ; Chin Postdoctorl Scinc Foundtion 015M570941; Chng Jing Scholrs Progrm T011. Author Contributions: Xicho Dong nd Chng Hu concivd nd dsignd mthods; Xicho Dong nd Yunho Li prformd simultion; Chng Hu nd Tng Long nlyzd dt; Xicho Dong wrot ppr. Conflicts of Intrst: Th uthors dclr no conflict of intrst. Appndix A Th diffrntil qutions of orbitl lmnts vritions undr influncs of J prturbtion cn b obtind by substituting Equtions 1 14 into Equtions 1 5. Considring rltionship btwn ltus rctum p nd rdium r, which is: r = p 1 + cos f, A1

20 Snsors 016, 16, of 4 Equtions cn b drivd by following lgbr: d d f = p µ d d f = p 3µ J µ r 4 = p µ p 3µ J 1 r 4 = 3 J p1 p 4 r cos f = 3 J p1 1 + cos f sin f 1+ cos f sin i sin f + ω 1+ cos f sin i sin f + ω [ sin f 1 3sin i sin u [ cos f sin i sin u sin f 3 sin f sin i sin u +sin i sin u + cos f sin i sin u = 3 J p1 1 + cos f [ sin f + sin i sin u 1 + cos f 3 sin f sin f 3µ J 1+ cos f r 4 + +cos f + cos f 1+ cos f 3 = 3 J p 4 1 p r 4 1+ cos f 3 1 3sin i sin f + ω 3µ J sin i sin f + ω r 4 sin f 1 3sin i sin f + ω 1+ cos f sin i sin f + ω 1+ [ cos f 3 ] sin f 1 + cos f 1 3sin i sin f + ω + +cos f + cos f + + cos f + cos f sin i sin f + ω [ ] = 3 J sin f 1 3sin i sin u + cos f 3 cos f sin i sin u 1 + cos f p + + cos f + cos f [ sin i sin u sin f + sin f cos f + + cos f + cos f 3sin f 3 sin f cos f sin i sin u = 3 J 1 + cos f p [ ] = 3 J sin f 1 + cos f cos f p sin [ i sin u + cos f + cos f 1 + cos f 3sin f 1 + cos f ] = 3 J 1 + cos f sin f + sin i sin u cos f 3sin f + +cos f p 1+ cos f A3 di d f = p µ = 3 J p 4 p r 4 cos f +ω 3µ J 1+ cos f 3 sini sin f + ω r 4 cos f +ω sini sin f + ω 1+ cos f 3 = 3 J p cos f + ω 1 + cos f sini sin f + ω = 3 J sini p 1 + cos f sin u cos u ] ] ] ] A A4 dω d f = p µ 1 sin i sin f +ω 3µ J 1+ cos f 3 sini sin f + ω r 4 = 3 J p sin f + ω p r 4 1+ cos f 3 sin i sini sin f + ω = 3 J p = 3 J cos i p 1 + cos f sin u 1 + cos f sin u sin i cos i sin i sin u A5

21 Snsors 016, 16, of 4 dω d f = p µ cos f 3µ J 1+ cos f r 4 + cos f sin f 3µ J 1+ cos f 3 + tn 1 sin f +ω 3µ J i 1+ cos f 3 = p 3µ J 1 µ r 4 1+ cos f 3 = 3 J p 4 1 p r 4 1+ cos f 3 = 3 J 1 + cos f p = 3 J p 1 + cos f 1 3sin i sin f + ω r 4 sin i sin f + ω r 4 sini sin f + ω cos f 1+ cos f + cos f sin f + tn 1 i cos f 1+ cos f + cos f sin f 1 3sin i sin f + ω sin i sin f + ω sin f + ω sini sin f + ω 1 3sin i sin f + ω sin i sin f + ω sin f + ω sini sin f + ω + tn 1 i cos f 1+ cos f 3sin i sin u + cos f sin f sin i sin u + cos sin i i sin i cos isin u [ 1+ cos f sin f +cos f +sin f cos f 1+ cos f sin ucos i cos f 1+ cos f sin i sin u whr u = f + ω. Dtils of drivtions of Equtions 3 34 r s follows. Th slnt rng cn b xprssd s: ] A6 = r + r cos α 0 = + r r cos α 0 A7 Tking prtil diffrntil oprtions for Eqution A7, w cn obtin: Ṙ = ṙ r cos α 0 + rṙ = ṙ r cos α 0. A8 Th drivtivs of r cn b clcultd s:. r = 1 sin f 1+ cos f = r α. sin f p = µ p sin f,. α A9.. r = α. µ p cos f = µ p cos f µp r = µ r cos f. A10 In dvitions,. α is stllit ngulr vlocity. W will lso us rltionships p = 1 nd r. α = µp [5]; Eqution A8 cn b trnsformd s: d µ dt = Ṙ = p sin f r cos α 0. A11 Similrly, w will tk scond drivtivs t both sids of Eqution A7, nd w gt:..ṙ.. + = r r cos α 0 + ṙ A1

22 Snsors 016, 16, 140 of 4 Thrfor, scond drivtiv of cn b xprssd s:.. µ r = cos f µ r cos α 0 + p sin f = = = µr µ r cos f r cos f cosα 0+ µ p sin f.. µ p cos f 1+ cos f + µ p sin f µ r cos f cos α 0 µ p cos f + µ r cos f cos α 0.. A13 Thn, w obtin: d dt =.. = µ + cos f µ p r cos α 0 cos f. A14 Similrly, w cn obtin third drivtiv of s Eqution 34. Hr, w will first us third drivtiv of r, which is:... r = µṙ cos f µ αsin. f r 3 r = µ r ṙcos f + r αsin. f 3 = µ µ r 3 p sin f cos f + µp r sin f = µsin f µ r 3 p 3 cos f + 1 A15 Whn considring smll vrition of, Equtions cn b drivd s: C 1 dc 1 d = µ 1/ dr 1/ sin f 0 cosθ 0 L d = 1 µ 1/ sin f 0 cosθ L r 1/ 0 dr 0 r 0 d = 1 C 1 1 /1+ cos f 0 r 0 = 1 C1 A16 C dc d = 1 µ cos f 0cosθ L r 0 r 1 dr 0 0 d = C dr 0 = C r 0 d C 3 dc 3 d = 1 7 µ3/ sin f 0 cosθ L r 7/ 0 r 1 dr 0 0 = 1 7 C3 d Similrly, whn considring smll vrition of, Equtions cn b drivd s: A17 A18 C 1 dc 1 d = µ 1/ dr 1/ sin f 0 cosθ 0 L d = µ 1/ r sin f 0 cosθ 1/ 0 d+dr 1/ 0 L = µ 1/ sin f 0 cosθ L = C 1 1 r 1 0 dr 0 d d r 1/ 0 d 1 r 3/ dr 0 0 d A19

23 Snsors 016, 16, of 4 C dc = µ cos f 0 cosθ L = µ cos f 0 cosθ L d dr 0 d r 0 d r 3 0 dr 0 d = C 1 r 0 dr 0 d C 3 dc 3 d = µ 3/ sin f 0 cosθ L 6 = µ 3/ sin f 0 cosθ L 6 = C r 0 dr 0 d dr 7/ 0 d r 7/ 0 d 7 r 9/ dr 0 0 d A0 A1 frncs 1. Tomiysu, K.; Pclli, J.L. Syntic prtur rdr imging from n inclind gosynchronous orbit. IEEE Trns. Gosci. mot Sns. 1983, GE-1, [Crossf]. Gurniri, A.M.; Broquts, A.; cchi, A.; occ, F.; uiz-odon, J. Advncd rdr gosynchronous obsrvtion systm: Argos. IEEE Gosci. mot Sns. Ltt. 015, 1, [Crossf] 3. Hobbs, S.; Gurniri, A.M.; Wdg, G.; Schulz, D. Gostr initil mission dsign. In Procdings of 014 IEEE Intrntionl Goscinc nd mot Snsing Symposium IGASS, Qubc City, QC, Cnd, July 014; pp Mdsn, S.N.; Chn, C.; Edlstin, W. dr options for globl rthquk monitoring. In Procdings of IGASS IEEE Intrntionl Goscinc nd mot Snsing Symposium, Toronto, ON, Cnd, 4 Jun 00; pp Hobbs, S.; Mitchll, C.; Fort, B.; Holly,.; Snpir, B.; Whittkr, P. Systm dsign for gosynchronous syntic prtur rdr missions. IEEE Trns. Gosci. mot Sns. 014, 5, [Crossf] 6. uiz-odon, J.; Broquts, A.; Mkhoul, E.; Monti Gurniri, A.; occ, F. Nrly zro inclintion gosynchronous sr mission nlysis with long intgrtion tim for rth obsrvtion. IEEE Trns. Gosci. mot Sns. 014, 5, [Crossf] 7. Hu, C.; Long, T.; Zng, T.; Liu, F.; Liu, Z. Th ccurt focusing nd rsolution nlysis mthod in gosynchronous sr. IEEE Trns. Gosci. mot Sns. 011, 49, [Crossf] 8. Pi, X.; Frmn, A.; Chpmn, B.; osn, P.; Li, Z. Imging ionosphric inhomognitis using spcborn syntic prtur rdr. J. Gophys. s. Spc Phys. 011, 116, [Crossf] 9. Hu, C.; Tin, Y.; Zng, T.; Long, T.; Dong, X. Adptiv Scondry ng Comprssion Algorithm in Gosynchronous SA. IEEE J. Sl. Top. Appl. Erth Obs. mot Sns. 016, 9, [Crossf] 10. Li, D.; Wu, M.; Sun, Z.; H, F.; Dong, Z. Modling nd procssing of two-dimnsionl sptil-vrint gosynchronous sr dt. IEEE J. Sl. Top. Appl. Erth Obs. mot Sns. 015, 8, [Crossf] 11. Bruno, D.; Hobbs, S.E. dr imging from gosynchronous orbit: Tmporl dcorrltion spcts. IEEE Trns. Gosci. mot Sns. 010, 48, [Crossf] 1. odon, J..; Broquts, A.; Gurniri, A.M.; occ, F. Gosynchronous SA focusing with tmosphric phs scrn rtrivl nd compnstion. IEEE Trns. Gosci. mot Sns. 013, 51, [Crossf] 13. Dong, X.; Hu, C.; Tin, W.; Tin, Y.; Long, T. Dsign of vlidtion xprimnt for nlysing impcts of bckground ionosphr on gosynchronous sr using GPS signls. Elctron. Ltt. 015, 51, [Crossf] 14. Tin, Y.; Hu, C.; Dong, X.; Zng, T.; Long, T.; Lin, K.; Zhng, X. Thorticl nlysis nd vrifiction of tim vrition of bckground ionosphr on gosynchronous sr imging. IEEE Gosci. mot Sns. Ltt. 015, 1, [Crossf] 15. Hu, C.; Li, X.; Long, T.; Go, Y. GEO SA intrfromtry: Thory nd fsibility study. In Procdings of 013 IET Intrntionl dr Confrnc, Xi n, Chin, April 013; pp Hu, C.; Li, Y.; Dong, X.; Long, T. Optiml dt cquisition nd hight rtrivl in rpt-trck gosynchronous sr intrfromtry. mot Sns. 015, 7, [Crossf] 17. Kou, L.-L.; Wng, X.-Q.; Xing, M.-S.; Chong, J.-S.; Zhu, M.-H. Effct of orbitl rrors on gosynchronous circulr syntic prtur rdr imging nd intrfromtric procssing. J. Zhjing Univ. Sci. C 011, 1, [Crossf]

24 Snsors 016, 16, of Jing, M.; Hu, W.; Ding, C.; Liu, G. Th ffcts of orbitl prturbtion on gosynchronous syntic prtur rdr imging. IEEE Gosci. mot Sns. Ltt. 015, 1, [Crossf] 19. Zho, B.; Qi, X.; Song, H.; Wng,.; Mo, Y.; Zhng, S. An ccurt rng modl bsd on fourth-ordr dopplr prmtrs for gosynchronous sr. IEEE Gosci. mot Sns. Ltt. 014, 11, [Crossf] 0. Gorg, W.; Collins, I. Th Foundtions of Clstil Mchnics; Pchrt Foundtion db Pchrt Publishing Hous: Tucson, AZ, USA, Dlgdo, M.. Modling Spc Environmnt; Univrsidd Politcnic d Mdrid: Mdrid, Spin, Montnbruck, O.; Gill, E. Stllit Orbits: Modls, Mthods nd Applictions; Springr-Vrlg Brlin: Brlin, Grmny, Pvlis, N.; Holms, S.; Knyon, S.; Fctor, J. Th dvlopmnt nd vlution of Erth Grvittionl Modl 008 EGM008. J. Gophys. s. 01, 117. [Crossf] 4. Hu, C.; Long, T.; Liu, Z.; Zng, T.; Tin, Y. An improvd frquncy domin focusing mthod in gosynchronous sr. IEEE Trns. Gosci. mot Sns. 014, 5, Curlndr, J.C.; Mcdonough,.N. Syntic Aprtur dr: Systms nd Signl Procssing; Wily: Nw York, NY, USA, by uthors; licns MDPI, Bsl, Switzrlnd. This rticl is n opn ccss rticl distributd undr trms nd conditions of Crtiv Commons Attribution CC-BY licns