PDF Evaluation of the Integer Ambiguity Residuals

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1 PDF Evlutio o th Itgr Ambiguity Rsiduls S. Vrhg d P.J.G. Tuiss Dprtmt o Mthmticl Godsy d Positioig Dlt Uivrsity o Tchology, Thijsswg, 69 JA Dlt, Th Nthrlds E-mil: S.Vrhg@GEO.TUDlt.l Abstrct. A prmtr stimtio thory is icomplt i o rigorous msurs r vilbl or vlidtig th prmtr solutio. Sic th clssicl thory o lir stimtio dos ot pply to th itgr GPS modl, rigorous vlidtio is ot possibl wh us is md o th clssicl rsults. As with th clssicl thory, irst stp or big bl to vlidt th itgr GPS modl is to mk us o th rsiduls d thir probbilistic proprtis. Th rsiduls qutiy th icosistcy btw dt d modl, whil thir probbilistic proprtis c b usd to msur th sigiicc o th icosistcy. I this cotributio w will prst d vlut th joit probbility dsity uctio (PDF o th multivrit itgr GPS crrir phs mbiguity rsiduls. Sic th rsiduls d thir proprtis dpd o th itgr stimtio pricipl usd, w will prst th PDF o th mbiguity rsiduls or th whol clss o dmissibl itgr stimtors. This icluds th stimtio pricipls o itgr roudig, itgr bootstrppig d itgr lstsqurs. I ordr to gt bttr udrstdig o th vrious turs o th joit PDF o th mbiguity rsiduls w will us stp-by-stp costructio idd by grphicl ms. Although th rsults pply or y dimsio, th o-dimsiol cs d th two-dimsiol cs r highlightd. Kywords. GPS, itgr mbiguity rsiduls, prmtr distributios Itroductio Ay GPS modl o obsrvtio qutios tht icluds crrir phs dt o two or mor rcivrs c b prmtrid i o-itgrs d itgrs. Th o-itgrs rr to th bsli compots d dditiol ukows lik tmosphric dlys. Th itgr prmtrs rr to th ukow cycl mbiguitis o doubl-dircd crrir phs dt. Wh th itgrss o ths prmtrs is xplicitily tk ito ccout i th prmtr stimtio procss, w spk o crrir phs mbiguity rsolutio. It c b pplid to grt vrity o GPS modls tht r usd i pplictios lik survyig, vigtio, d gophysics. A ovrviw o GPS modls c b oud i txtbooks lik Hom-Wllho t l. (997, Lick (995, Prkiso d Spilkr (996, Strg d Borr (997, Tuiss d Klusbrg (998. Prmtr stimtio provids th stimts o th ukow prmtrs, togthr with th corrspodig vric mtrics. I th clssicl thory o lir stimtio, th vric mtrics provid suicit iormtio o th prcisio o th stimtd prmtrs. Th rso is tht lir modl pplid to ormlly (Gussi distributd dt, provids lir stimtors tht r lso ormlly distributd, d th pkdss o th multivrit orml distributio is compltly cpturd by th vric mtrix. Uortutly, this rltivly simpl pproch cot b pplid i cs itgr prmtrs r ivolvd i th stimtio procss, sic th itgr stimtors do ot hv Gussi distributio, v i th modl is lir d th dt r ormlly distributd. Istd o th vric mtrics, w thror hv to us th prmtr distributio itsl i ordr obti th pproprit msurs tht c b usd to vlidt th itgr prmtr solutio. For tht purpos, th probbility dsity uctio (PDF o th mbiguity rsiduls c b usd, sic th rsiduls qutiy th icosistcy btw dt d modl, whil th PDF dscribs thir probbilistic proprtis, which r msur or th sigiicc o th icosistcy. Our gol is to vlut th PDF o th mbiguity rsiduls or th whol clss o dmissibl itgr stimtors, sic th rsiduls d thir probbilistic proprtis dpd o th stimtio pricipl tht is usd.

2 W will strt with th ormultio o th itgr GPS modl i sctio, whr lso th clss o dmissibl stimtors is did. I sctio 3 th PDF o th mbiguity rsiduls or th whol clss o dmissibl itgr stimtors is prstd. I prctic, th PDF tht is commoly usd is bsd o th icorrct ssumptio tht th itgr stimtor is dtrmiistic. It will b show how th rsultig PDF dirs rom th corrct o s prstd hr. Th vlutio o th PDFs is th subjct o sctio 4. W will ocus o th o-dimsiol d two-dimsiol cs. Thrby, w will lso look t rlistic GPS modl. I ordr to gt good udrstdig o th vrious turs o th joit PDF o th rsiduls, it is show how th PDF c b costructd stp-by-stp, idd by grphicl ms. Furthrmor, th scod momts o th mbiguity rsiduls with rltio to th prcisio o th GPS dt r show umriclly s wll s grphiclly. Th PDFs d scod momts will b comprd to th os tht r usully usd i prctic. Itgr mbiguity rsolutio Ay GPS obsrvtio modl c b prmtrid i itgrs d o-itgrs. This givs th ollowig systm o lir(id obsrvtio qutios: y = A + B b + (. mx mx x mxp px mx whr y is th GPS obsrvtio vctor o ordr m, d b r th ukow prmtr vctors o ordr d p rspctivly, d is th ois vctor. Th dt vctor y usully cosists o th obsrvdmius-computd DD phs d/or cod obsrvtios o o, two or thr rqucis d ccumultd ovr ll obsrvtio pochs. Th tris o th prmtr vctor wil th cosist o th ukow itgr crrir phs mbiguitis, which r xprssd i uits o cycls rthr th i uits o rg. Sic it is kow tht th tris r itgrs, Z. Th rmiig ukow prmtrs orm th tris o th vctor b. Ths prmtrs my b th ukow bsli compots d or istc tmosphric (ioosphric, troposphric dlys, which r ll rl-vlud, i.. b R p. I this cotributio w will rr to ths rl-vlud prmtrs s th bsli stimtor, lthough th vctor b my thus coti othr prmtrs th oly th bsli compots. Th clssicl lir stimtio thory c b pplid to modls tht coti rl-vlud prmtrs. Howvr, i th itgrss o th mbiguity prmtrs is tk ito ccout, w hv to ollow dirt pproch which icluds sprt stp or mbiguity rsolutio. Th complt stimtio procss will th cosist o thr stps (Tuiss 993. I th irst stp, th itgrss o th vctor is discrdd d 'lot' solutio is computd with stdrd lst-squrs djustmt. This rsults i rl-vlud stimts or d b d thir vric-covric (vc- mtrix:, b ˆ b ˆ ˆ b ˆ ˆ bˆ (. I th scod stp th itgr mbiguity stimt is computd rom th 'lot' mbiguity stimt â: = S (.3 ( whr S: R Z th mppig rom th - dimsiol spc o rl umbrs to th - dimsiol spc o itgrs. Th il stp is to us itgr mbiguity stimts to corrct th 'lot' stimt o b with b = bˆ (.4 ( b ˆ ˆ This solutio is rrrd to s th 'ixd' bsli solutio. Both qs.(.3 d (.4 dpd o th choic o th itgr stimtor. Dirt itgr stimtors r obtid or dirt choics o th mp S: R Z. This implis tht lso th probbility distributio o th stimtors dpds o th choic o th mp. I ordr to rriv t clss o itgr stimtors rom which to choos, w will strt with th mp S: R Z. Th spc o itgrs, Z, is o discrt tur, which implis tht th mp must b myto-o mp, d ot o-to-o. I othr words, dirt rl-vlud mbiguity vctors will b mppd to th sm itgr vctor. Thror, subst S R c b ssigd to ch itgr vctor Z : S = { x R = S x }, Z ( (.5 This subst S cotis ll rl-vlud 'lot' mbiguity vctors tht will b mppd to th sm itgr vctor, d it is clld th pull-i rgio o (Jokm 998, Tuiss 998. This implis tht = â S. Th itgr mbiguity stimtor c b xprssd s:

3 = s Z ( with s ( x = i x S (.6 othrwis whr w usd th idictor uctio s (x. Th itgr stimtor is compltly did by th pull-i rgio, so tht it is possibl to di clss o itgr stimtors by imposig vrious coditios o th pull-i rgios. Th clss o dmissibl itgr stimtors is did s ollows. Diitio. (Admissibl itgr stimtors. A itgr stimtor, = s Z (, is sid to b dmissibl wh its pull-i rgio, S = x R = S( x, Z, stisis ( i ( ii ( iii { } 7 Z It S S S = R It S = + S, Z =,, Z, whr 'It' dots th itrior o th subst. I Tuiss (998 th motivtio or this diitio is giv. Exmpls o itgr stimtors tht blog to th clss o dmissibl itgr stimtors r itgr roudig (R, itgr bootstrppig (B, d itgr lst-squrs (LS. I Tuiss (999 th corrspodig pull-i rgios r show. For rviw, s Tuiss (b. 3 Th Probbility Dsity Fuctio I Tuiss ( th joit d mrgil PDFs o both th itgr d o-itgr prmtrs wr dtrmid. I ordr to do so, it ws ssumd tht th 'lot' solutios r ormlly distributd. This implis tht th mrgil PDF o th 'lot' mbiguitis is giv s: ( x = dt xp (π { x } ˆ (3. It c b prov tht th joit distributio o â d is giv by:, ( x, = ( x s ( x, x R, Z (3. Th distributios o th 'lot' d 'ixd' mbiguitis c b rcovrd rom this PDF by summig ovr or itgrtig ovr x rspctivly. Th distributio o th itgr mbiguity stimtor is giv by probbility mss uctio (PMF, d ot by PDF. This PMF is qul to th itgrl o th PDF o th 'lot' mbiguity ovr th pull-i rgio S : ˆ, ( x, dx = ( x dx = P[ = (3.3 R S ] Our gol is to dtrmi th PDF o th mbiguity rsidul, which is did s: = (3.4 I prctic it is ot icorrctly ssumd tht th itgr stimts r dtrmiistic, which would imply tht th PDF o th rsiduls coicids with th PDF o th 'lot' mbiguitis. Istd, th PDF c b costructd oc th joit distributio o d is kow. This PDF c b obtid usig th trsormtio (s Tuiss, : I = I I (3.5 Sic, ( x is kow (q.(3., th ollowig c b obtid: (3.6, ( x, =, ( x +, = ( x + s ( x Th PDF o th mbiguity rsiduls th ollows rom summig ovr ll itgrs: ( x = ( x + s ( x, x R Z (3.7 Z, With q.(3. this illy givs: ( x = dt Z (π xp{ x + } s ( x (3.8 Not tht (x is oly ssitiv to th rctiol prt o bcus th summtio is ovr ll itgrs, so tht rplcig by -[] dos ot chg th rsult ([] ms tht is roudd to th rst itgr. Figur 3. shows ll stps rquird or th costructio o th PDF o th mbiguity rsiduls i th o-dimsiol (-D cs. Th PDF o â (top lt is plottd log th x-xis, th PMF o (top right log th -xis, d th joit PDF (top middl is plottd i th x-pl. Its costructio rom th mrgil PDF d PMF c b s s ollows. First th prts o th PDF o â r slicd out tht corrspod to ll pull-i rgios. For th - D cs, th pull-i rgios simply r itrvls with lgth, ctrd t th itgrs, S ={x R x- ½}. Ths slics r th trsltd log th -xis to th corrspodig itgrs.

4 Fig. 3.. Th costructio o (x rom â(x: PDF â(x (top lt; joit PDF â, (x, (top middl; PMF P[=] (top right; joit PDF, (x, (bottom middl; PDF (x (bottom right. Th joit PDF o d (bottom lt ollows rom othr trsltio o th slics, but ow log th x-xis, so tht thy r ll ctrd t th m vlu x =. Th PDF o (bottom right is illy obtid by summig ovr, i.. ll slics r gi trsltd log th -xis to th origi. Th distributio o th mbiguity rsiduls is clrly o-gussi, d it quls ro or ll vlus o x outsid th pull-i rgio. Tht implis tht th orm o th vctor o mbiguity rsiduls is lwys boudd, irrspctiv o th vlu tk by th 'lot' solutio. A vry importt implictio is tht th dirc btw th 'lot' d th 'ixd' bsli solutio is th lso boudd, s c b s i q.(.4. S lso Tuiss (. It c lso b s tht th PDF is symmtric roud th origi, but th shp dpds o th prcisio o 'lot' solutio. Mor xmpls r giv i th xt sctios. This ms tht th PDF is idpdt o th ukow itgr mbiguity vctor Z. I othr words, th m o th mbiguity rsidul quls ro, E{}=. Thus, th PDF is compltly kow oc th prcisio o th 'lot' solutio is kow d th choic o th itgr stimtor is md. 4 Evlutio o th PDF Th rsults o th prcdig sctio pply or y dimsio. Howvr, or vlutio w ocus o th o-dimsiol d two-dimsiol cs. 4. O-dimsiol cs I th o-dimsiol cs w do ot d to distiguish btw th thr dmissibl stimtors, sic thy r ll idticl. I igur 3. stp-bystp grphicl costructio o th PDF o th rsiduls ws show. All probbility mss is loctd withi th pull-i itrvl (-½,½. Th distributio withi this itrvl dpds o th prcisio o th 'lot' mbiguitis. Figur 4. (lt pl shows th PDF o th rsiduls or dirt vlus o th stdrd dvitio, σ, o th mbiguitis. Also, th xtrms σ = d σ = r show. I th irst cs, th PDF bcoms impuls uctio, i th lttr cs th rsiduls hv uiorm distributio withi th pull-i itrvl. Th pkdss o th PDF clrly dpds o th prcisio.

5 σ = 4 σ = σ =..5 σ =..5.5 σ =.5 σ =.3.5 σ =.3 σ = Fig. 4.. Rsidul PDF or dirt vlus o th stdrd dvitio σ: corrct PDFs (lt; d PDFs s usd i prctic (right. Th right pl o igur 4. shows th PDF o th 'lot' mbiguitis or th sm stdrd dvitios. I prctic, ot th rdomss o th 'ixd' solutio is igord, implyig tht: ( x : ( x =. This distributio uctio hs iiit tils. Not tht wh th prcisio is high, or xmpl σ =. cycls, tht th PDFs bcom lmost idticl. O th othr hd, wh th prcisio is low, or xmpl σ =.3 cycls, th PDFs r vry dirt. I o would you us th ls PDF by ssumig tht th 'ixd' solutio is dtrmiistic, tht my ld to ls hypothsis tsts o th itgrss o th mbiguitis. Kowldg o th PDF lso llows us to driv xprssios or th scod momts, i.. th vric d covric, o th mbiguity rsiduls. Th vric c b computd with: σ σ = x ( x dx (4.. /.5 x S σ Fig. 4.. Vric o th rsiduls, σ, vrsus th vric o th 'lot' mbiguitis, σ, (-D. Figur 4. shows th vric o mbiguity rsiduls, σ, s uctio o th vric o th 'lot' mbiguitis, σ. As c b s th vric o th rsiduls is lwys smllr th / d it is lwys lowr or qul to th vric o th 'lot' mbiguitis. Agi this shows tht it is ot corrct to ssum tht th 'ixd' solutio is dtrmiistic, so tht σ = σ : 4. Two-dimsiol cs I th two-dimsiol cs th PDFs o th dirt dmissibl stimtors my b quit dirt, spcilly i thr is high corrltio btw th 'lot' mbiguitis. Thror, w hv lookd t th rsultig PDFs or vc-mtrix tht would b obtid by choosig rlistic GPS modl. I this xmpl tht ms tht w us th cod d phs obsrvtios o th L d L rqucy with udircd stdrd dvitios o.3 m d.3 m or cod d phs obsrvtios rspctivly. Th rsultig vc-mtrix o th DD mbiguitis rds: â = ( Th rsidul PDF is costructd i similr wy s show or th -D cs. This is show i igur 4.3 i cs o roudig: th probbility mss o â i ll pull-i rgios, which r squrs i this cs (top, is ddd to th mss i th pull-i rgio ctrd t

6 th origi (middl. Th rsult is th PDF o th rsiduls show i th bottom pl ( drk r corrspods to lrg probbility mss. Figur 4.4 shows th PDFs o th 'lot' mbiguitis (top, d o th mbiguity rsiduls obtid with th dirt stimtors. I igur 4.5 th PDFs o th corrspodig 'lot' mbiguitis d th rsiduls tr Z-trsormtio r show. This Z-trsormtio is usd to dcorrlt th vcmtrix (s.g. Tuiss, 998: T T = Z Z, ˆ Z (4.3 ˆ = I our xmpl this rsults i: ˆ =, Z = ( I w irst look t th rsults obtid or th vcmtrix giv i (4. w s tht th distributio clrly dpds o th itgr stimtor tht is usd, d tht it my b cosidrbly dirt rom th PDF o â. Firstly, th PDF o â is ot boudd, whrs th rsidul PDFs r boudd by th corrspodig pull-i rgio. Scodly, th rsidul PDFs i this xmpl r multimodl. This is du to th ct tht th PDF o â is vry logtd d its orittio is ot log th x -, x - or x =x -xis. This implis tht thr is lso high probbility mss loctd r wy rom th origi. So, this coirms tht th 'ixd' mbiguitis my ot lwys b cosidrd dtrmiistic. Th PDF o th rsidul or itgr LS, LS, is show i th bottom pl o igur 4.4. Th pull-i rgio o th itgr LS stimtor ollows vry much th shp o th PDF o â d is thus vry logtd. Thror, it is hrd to visully dtct y dircs btw th PDFs o â d LS, lthough thr r dircs spcilly r th boudris o th pull-i rgio d o cours outsid th pull-i rgio. Howvr, this will b sir to s or th rsiduls o th Z-trsormd mbiguitis. Not tht oly or th itgr LS stimtor th ollowig is tru: T T = ˆ = Z ( = Z (4.5, LS LS LS LS so tht T ( x = ( Z x (4.6, LS LS Fig Th costructio o (x rom â(x i cs o roudig (-D: PDF â(x (top; summtio ovr ll pull-i rgios (middl; PDF (x (bottom. Thror, th shp o th PDF o th Z- trsormd rsiduls is similr to th PDF o th utrsormd rsiduls, oly tht it is trsormd lik th pull-i rgio. Figur 4.5 shows tht spcilly or th bootstrppd d itgr LS stimtor th shp o th rsidul PDF 'its' th shp o th pull-i rgio quit wll. Du to th dcorrltio, th rsultig PDFs r ow uimodl. Howvr, th shp r th boudris o th pull-i rgios is clrly dirt rom th shp o th PDF o â. For clrity this is show by th blck cotour lis which r plottd t vlu o.6. Not tht ths cotour lis r ot closd, which ms tht biss i crti dirctios will b hrdr to dtct th biss i othr dirctios, dpdig o th corrltio btw th 'lot' mbiguitis.

7 Fig Rsidul PDFs (-D: PDF â(x (top; PDF (x or roudig ( d rom top; PDF (x or bootstrppig (3 rd rom top; PDF (x or itgr LS (bottom. Fig Rsidul PDFs tr Z-trsormtio (-D: PDF â(x (top; PDF (x or roudig ( d rom top; PDF (x or bootstrppig (3 rd rom top; PDF (x or itgr LS (bottom.

8 Although th PDFs o â d,ls hv th most similr shps, th igur shows tht th distributios r dirt rom ch othr. Agi th coclusio is tht igorig th rdomss o th 'ixd' solutio lds to ls ssumptios o th distributio d thus to ls tsts or itgrss o th prmtrs. Th scod momts o th mbiguity rsiduls i cs o roudig r computd with: σ dx dx (4.6 = xi x j ( xi, x i j i j j x S Th rsultig vc-mtrics r: = R.65.67, = R i j (4.7 Figur 4.6 shows th llipss tht corrspod to ths vc-mtrics o th rsiduls without d with Z-trsormtio. Also, th corrspodig vcmtrics o th 'lot' mbiguitis, ˆ d ˆ, r show. Ths r th scod momts o th rsiduls tht r usully usd i prctic. It c b s tht spcilly or th highly corrltd vcmtrix, th dvitio is lrg. 5 Coclusios I this cotributio it is show how th PDFs o mbiguity rsiduls c b costructd. This llows or rigorous tstig. Th xt stp will b to ormult hypothss d dtrmi tst sttistics ssitiv to ths hypothss or th -dimsiol cs or ll dmissibl itgr stimtors. Thrby w will hv to look t th si d th powr o th tst d th tst should b optiml. Currtly, th rdomss o th 'ixd' solutio is igord, which rsults i wrog rsidul PDFs, d wrog scod momts. This my ld to ls vlutio o th qulity o tstig th itgrss o th mbiguitis. Th kowldg o th corrct PDFs c b usd to ivstigt i which css th tst s usd i prctic givs suicit dscriptio, d i which css it will ld to ls coclusios Fig Ellipss corrspodig to th vc-mtrics o th 'lot' mbiguits (light gry d th rsiduls (blck: or ˆ (lt d ˆ (right. Rrcs Hom-Wllho, B., H. Lichtggr, d J. Collis (997. Globl Positioig Systm: Thory d Prctic. 4 th d., Sprigr, Brli Hidlbrg Nw York. Jokm, N.F. (998. Itgr Ambiguity Estimtio without th Rcivr-Stllit Gomtry. LGR Sris, o.8, Dlt Godtic Computig Ctr, Dlt. Lick, A. (995. GPS Stllit Survyig. d d., Joh Wily, Nw York. Prkiso, B. d J.J. Spilkr (ds (996. GPS: Thory d Applictios. Vols. d, AIAA, Wshigto, DC. Strg, G. d K. Borr (997. Lir Algbr, Godsy, d GPS. Wllsly-Cmbridg Prss. Tuiss, P.J.G. (993. Lst-squrs Estimtio o th Itgr GPS Ambiguitis. Ivitd lctur, Sctio IV Thory d Mthodology, IAG Grl Mtig. Bijig, August. Also i: LGR Sris, No.6, Dlt Godtic Computig Ctr, Dlt, pp Tuiss, P.J.G. (998. O th Itgr Norml Distributio o th GPS Ambiguitis. Artiic St 33(: Tuiss, P.J.G. (999. A Optimlity Proprty o th Itgr Lst-squrs Estimtor. J God 73: Tuiss, P.J.G. (. Itgr Estimtio i th Prsc o Biss. J God 75: Tuiss, P.J.G. (. Th Prmtr Distributios o th Itgr GPS Modl. J God 76: Tuiss, P.J.G. (b. Itgr Lst-Squrs. I: Proc. V Hoti-Mrussi Symposium. Mtr, Ju 7- (this volum. Tuiss, P.J.G., d A. Klusbrg (ds (998. GPS or Godsy. d d., Sprigr, Brli Hidlbrg Nw York.

Linear Algebra Existence of the determinant. Expansion according to a row.

Linear Algebra Existence of the determinant. Expansion according to a row. Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit