A New Three-Frequency, Geometry-Free, Technique for Ambiguity Resolution

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1 A New Three-Frequey, Geometry-Free, Tehique or Amiguity Reolutio Rold R. Hth, NvCom Tehology, I. BIOGRAPHY Rold R. Hth w oe o the ouder o NvCom Tehology, Joh Deere Compy, d i urretly the Diretor o Nvigtio Sytem. Prior to joiig NvCom, he worked GPS oultt with umer o ompie d govermet geie. Iluded mog thee were Lei, Hoeywell, Northrop, NASA d the FAA. He worked or 3 yer t Joh Hopki Applied Phyi Lortory, ive yer t Boeig d 3 yer t Mgvox. Ro reet work h oetrted o highury pplitio o GPS. He i the uthor o my urvey d vigtio pper d h over doze ptet ivolvig GPS proeig tehique. Ro reeived B.S. degree i 196 i Mth d Phyi rom Settle Pii College. He h erved i umer o poitio with the Ititute o Nvigtio (ION iludig Chir o the Stellite Diviio d i Preidet. Ro w the 1994 reipiet o the Stellite Diviio Kepler Awrd d i 000 reeived the Thom L. Thurlow wrd rom the ION. He w eleted Fellow o the ION i 000. ABSTRACT A ew tehique or miguity reolutio t log dite i deried. It ue the ode d rrier meuremet o three requeie i uuul wy. Speiilly, it ue vergig method to rrive t urte miguity-reolved d rertio-orreted meuremet tht lrgely overome the didvtge o the loe pig etwee the L d the L5 requeie. O oure, it work etter whe the eod d third requeie re rther prt, uh the Glileo L5 d E6 requeie. The tehique i uique i tht there i o requiremet to reolve the miguitie o the udmetl L1, L d L5 rrier phe meuremet. Ited wide-le, ut oiy, rertio-orreted rrier phe meuremet i ormed rom two o the three widele rrier phe dieree ormed rom the dieree o pir o the udmetl phe meuremet. Thee dieree re miguity reolved uig ioopherimthig ode meuremet d re the omied ito rertio-orreted ompoite meuremet. While thi wide-le ompoite i quite oiy, it e moothed with rertio-orreted, ompoite meuremet with muh lower oie. The miguitie o thi low-oie ompoite meuremet re ot required ie it i imply ued to mooth the oie i the wide-le rertio-orreted ompoite. By ot requirig the teppig rom oe miguity-reolved rrier phe meuremet to other, it i lrgely immue to lok dieree t the dieret requeie whih otge the tepped pproh. I dditio, eue the iitil miguity reolutio i doe with wide-le omitio, the reliility o the miguity reolutio i rout d reltively ieitive to the preee o mll ode-rrier ie. The geometry-ree pproh o idividully reolvig the miguitie remove the tropopheri rertio rom the miguity reolutio prolem. Thu, the il moothed, rertio-orreted ompoite meuremet i ieitive to oth ioopheri d tropopheri rertio eet. Though the moothig proe my require ome miute to reh the optiml ury level, the reult hould igiitly exted the rge over whih RTK reult e otied without requirig the modelig o the ioophere. INTRODUCTION The moothig o the ode meuremet with the rrier phe meuremet i routiely doe i mot GPS or GNSS reeiver. I igle-requey reeiver thi moothig i ormlly limited to time ott o oe to two miute to void iig the moothed meuremet with ompoet o the divergig ioopheri eet. Whe two or more requeie re ville, the ode meuremet e moothed with lier omitio o rrier phe meuremet whih mth the ioopheri rertio eet. Thi llow igiitly loger vergig time ott. O prtiulr iteret, the wide-le rrier phe meuremet ormed y diereig the idividul rrier phe meuremet e ued to mooth requey weighted ompoite ode meuremet whih mthe the ioopheri rertio eet o the wide-le rrier phe

2 meuremet. Ater uiiet moothig, the reultt moothed ode meuremet e ued to et the miguitie o the rrier phe dieree meuremet, umig, o oure, tht the ode meuremet ie re mll. Whe three or more requeie re ville, two idepedet rrier phe dieree meuremet e ormed d the miguitie reolved vi the mthed moothig proe jut deried. But hvig two miguity-reolved rrier phe dieree meuremet llow u to oti rertio orreted reult with the pproprite lier omitio. Uortutely, whe two o the rrier requeie re loe together, the lier omitio tht remove the ioopheri eet gretly mpliie the phe oie preet i the meuremet. Thi e overome y moothig the ompoite, rertio-orreted, miguityreolved, rrier phe meuremet with other rertio-orreted rrier phe meuremet whih i otruted to miimize the oie without regrd to the miguity reolutio prolem [1]. However, ome dditiol ury my e hieved y il reidul miguity reolutio. The detil o thi proe with exmple or GPS d or Glileo re deried elow. GENERAL CONSIDERATIONS There re ew geerl iue whih eed to e ddreed eore the proe deried i the itrodutory etio i detiled. Firt, or impliity, the equtio will e writte i there i o diereig o meuremet ro ite. I t, theoretilly, the proe e ued o igle-ite meuremet. However, there e igiit ode veru rrier ie i the trmiio rom the idividul tellite whih would prevet the proe rom workig o ite y ite i. But, give meuremet t kow ite roud the world, it my e poile to meure y ode or rrier ie d hrterize them utio o the gle to the reeiver ite reltive to the tellite ixed oordite. Suh lirtio proe would llow igle ite proeig. Without lirtio, the equtio e pplied diretly to the meuremet either diereed ro ite or the meuremet o give ite ter djutmet with orretio geerted t reeree ite. Seod, the equtio re writte i there i o diereig o meuremet ro tellite. The reeiver rot-ed ilter my rete, i eet, dieret lok reeree t the dieret reeived requeie. Thi rete i etwee the wide-le phe meuremet (the dieree o the reeree lok t the two requeie d the mthig requeyweighted ode meuremet ( weighted verge o the lok t the two requeie. I thi i i lrge it ould led to iorret miguity reolutio. Thi prolem e voided y utrtig the meuremet rom give tellite or rom verge ro ll tellite. Sie the tropophere et the meuremet y the me mout t eh requey, the miguity reolutio, d rertio orretio proee re trpret to tropopheri eet. Speiilly, the miguity reolutio proe d the ormig o the rertioorreted lier omitio will leve the tropopheri ompoet o the meuremet uhged. The dvtge o the geometry-ree pproh i tht the tropopheri idued rge error do ot dverely et the miguity reolutio proe. RESOLVING THE WIDE LANE AMBIGUITIES The irt tep i the proe o otiig low-oie rertio-orreted d miguity-reolved rrier phe meuremet i to orm t let two wide-le rrier phe dieree d to reolve the miguitie i thoe wide-le meuremet. To void the repetitio o the me equtio pplied to umer o dieret requeie, the proe i deried irt uig the ode d rrier phe meuremet t three geerl requeie leled, d. The ode meuremet, P d P, t the irt two o thee requeie re give y: P ρ + I (1 / P ρ + I ( / I thee two equtio, ρ i the geometri rge (iludig tropopheri rertio idued error d I i the ioopheri rge error utio o the ivere requey qured. I imilr hio we write the led rrier phe meuremet, d, utio o the rw phe meuremet, φ d φ : ( + N / ρ I / φ (3 ( + N / ρ I / φ (4 I thee two equtio N repreet the ukow yle miguity t the requey idited y the uript d i the peed o light. By omputig the requey-weighted verge o equtio (1 d ( we redue the multipth orruptio o the ode meuremet omewht d get

3 ew depedee o the ioopheri rertio error. Speiilly: P + P I ρ (5 P + + Thi requey-weighted ode meuremet mthe the ioopheri error o the wide-le rrier phe dieree meuremet. I ( φ φ λ + ( N N λ ρ + (6 The wvelegth deigted, λ -, i the wvelegth o the dieree requey. Diereig equtio (6 rom equtio (5 d dividig y the dieree wvelegth give diret meure o the wide-le miguity. P N ( N N φ φ (7 λ Sie the wide-le miguity doe ot hge log phe-lok i mitied y the reeiver trkig loop, thi vlue e verged over time (moothed to get ireigly urte miguity vlue. The moothig e doe with expdig verge ilter. 1, (, 1 N N + N, 1 N (8 (The iditor o the mout o moothig i dropped i uequet equtio. Ater moothig, thi vlue e plugged ito equtio (6 to give miguity reolved, wide-le rrier phe meuremet. I φ φ + [ N λ ] rd ρ + ( (9 The mout o moothig required to eure tht the orret wide-le miguity i determied i equtio (8 i utio o the wvelegth o the dieree requey (the loger, the etter. Tle 1 give the oited wvelegth o the dieree requeie or three dieret ytem hoie o the three requeie. The d requeie re hoe to orrepod to the L1 d L5 requeie o GPS, whih re ommo to the L1 d E5 requeie o Glileo. Three dieret hoie o the middle requey,, re oidered, the GPS L requey (whih i MHz ove the L5 requey, the Glileo E6 requey (whih i other MHz ove the L requey, d third requey other MHz ove the Glileo E6 requey. Mid. Freq λ - λ - λ - GPS L Glileo E GHz Tle 1: Dieree requey wvelegth REFRACTION CORRECTED WIDE LANE PHASE MEASUREMENTS The lt term i equtio (9 ove repreet the orruptio idued i the wide-le meuremet y the ioopheri rertio. Thi error eed to e removed. The irt tep i removig thi ioopheri error i to orm eod miguity reolved wide-le rrier phe meuremet rom the three ville primry phe meuremet. Oly two o the three poile wide-le dieree whih e ormed re idepedet. Thereore, it mke ee to orm the two wide-le dieree whih hve the widet le width, ie thoe will tke the let mout o vergig time to determie the miguity vlue. Thu, umig > > the wide le rrier phe repreeted y diereig the d meuremet hould e ormed ext. We do ot eed to go through the ove derivtio ll over gi. The pproprite equtio repreetig the miguity-reolved, wide-le rrier phe meuremet rom the d rrier phe meuremet i diretly logou to equtio (9 d i otied imply y replig the origil uript with the pproprite ew uript. I φ φ + N λ ρ + ( (10 Hvig determied the N - d N - miguitie, the N - miguity e otied imply y ummig the irt two. Thu, the third miguity-reolved wide-le rrier phe dieree i eily otied. Give two equtio whih how dieret depedee o the ioopheri rertio, it i reltively trightorwrd proe to geerte lier omitio o the two meuremet whih remove y ioopheri rertio error. Speiilly, equtio (10 i multiplied y / d the utrted rom equtio (9. ρ (11

4 Thi i the deired reult. Speiilly, equtio (11 give u rertio-orreted, miguity-reolved, rrier phe meuremet with o ioopheri rertio orruptio. THE NOISE PROBLEM Uortutely, the proe o ormig oth the wide le d the rertio orretio ue the oie i the meuremet to e uttilly mpliied. The oie mpliitio i idepedet o whih two o the three poile wide-le omitio re eleted or the rertio orretio proe. I t, umig tht it were poile to reolve the miguitie o the idividul rrier phe meuremet (deigte them, d repetively ee equtio (3 d (4, eh o the three poile pir o miguity reolved, wide-le meuremet whe rertio orreted i equtio (11 give rie to the me equivlet equtio. ( ( ( ( ( ( + + ρ (1 By pikig the requeie o L1 d L5, whih re ommo to GPS d Glileo, the d requeie repetively, we tudy the oie idued utio o the hoie o the middle requey,. Two dieret umptio regrdig the oie will e explored. Firt, we will ume tht there i oie equl to oe etimeter i eh o the phe meuremet whe led y their repetive wvelegth. The we will ume equl oie i eh o the phe meuremet (19 degree. Slig thee phe error y the wvelegth reult i oie o 1.0, /, d 1.34 etimeter repetively. Thee oie level re quite peimiti ut re iteded to ilude the eet o uttil multipth oie. The expeted oie i the rertio-orreted meuremet i give y the qure root o the um o the qure o the three oeiiet i equtio (1, weighted y the repetive oie. The oie reultig rom the three previouly oidered hoie o the middle requey re explored. Tle give the reultt oie or thee three hoie o the middle requey whe equl phe oie i oe m o eh. Mid. Freq. C1 C C3 RSS GPS L Glileo E GHz Tle : Ampliitio o oe m. oie Clerly, the Glileo E6 igl i uttilly etter th the GPS L igl or miimizig the oie o the rertio-orreted, miguity-reolved, wide-le rrier phe meuremet. Whe the oie i umed to e 19 degree o phe i eh o the primry phe meuremet, the dvtge o the Glileo E6 requey i eve more prooued. Thi i how i Tle 3 elow where the weighted oeiiet d RSS vlue re give. Mid. Freq wc1 wc wc3 RSS GPS L Glileo E GHz Tle 3: Ampliitio o 19 degree o phe oie The oie i thee rertio-orreted, miguityreolved, wide-le, ompoite meuremet rivl tht o the rw ode meuremet. However, the oie o the ompoite phe meuremet i till geerlly etter th rertio orreted ode meuremet would yield. Thi i prtiulrly true whe ie d oie vergig re oidered. CARRIER SMOOTHED CARRIER Jut the ode meuremet e (d uully re moothed uig rrier phe meuremet, the rertio-orreted, miguity-reolved, wide-le rrier phe meuremet geerted vi equtio (10 e moothed to uttilly redue the oie. To do the moothig we eed rertio-orreted rrier phe meuremet with miimum oie, ut the miguitie do ot eed to e reolved. Aumig oe etimeter o phe oie i eh o the primry led phe meuremet, the miimum-oie rertio-orreted omitio e determied rom three otrit. Speiilly, the vlue o the oeiiet,, d, ued to multiply the primry phe meuremet t the three requeie mut tiy: (13 / / + / 0 ( mi (15

5 The irt o thee three equtio eure the rge meuremet i ot led. The eod eure tht the ioopheri rertio error i eled, d the lt eure tht miimum oie mpliitio our. (I oequl oie i preet i the three phe meuremet, thi lt equtio eed to e modiied ppropritely. Solvig the irt equtio or give: 1 (16 Pluggig thi vlue or ito equtio (14 d olvig or give: ( (17 ( Uig oviou deiitio or thee requey utio, thi e impliied to: F F (18 Iertig the vlue o d, give i equtio (16 d (18 repetively, ito equtio (15 give the vlue v whih we wih to miimize. Ater impliitio we get: v F + F F F + F F (1 (1 (1 F + F + (19 Tkig the derivtive with repet to, ettig it to zero, d the olvig or the vlue o give: (1 F F (1 F + F F + F (0 The vlue o d ow e otied y k utitutio ito equtio (18 d (16. The vlue o,, d, otied whe the middle requey ume the three dieret vlue ued erlier, i give i Tle 4. The il olum give the etimted oie mpliitio o the equl oie primry meuremet. Thi how tht, i the primry miguitie ould e reolved, the GPS L requey tully reult i lightly lower rertio orreted oie th the Glileo E6 requey. Mid. Freq. RSS GPS L Glileo E GHz Tle 4: Coeiiet or rertio orretio with miimum oie Deiig the miimum-oie rertio-orreted vlue Μ, it vlue i omputed : + + (1 M Where,, d re the repetive rrier phe meuremet led y their wvelegth d with the whole-yle miguitie etimted (i.e. the miguitie my ot e orret. Both the vlue omputed rom equtio (11 d the vlue omputed rom equtio (1 oti meuremet o the rertio orreted rge. Thu, whe diereed, the vlue yielded will e utio o the multipth oie o the three requeie d ott i error ued y y iorret miguitie ued i equtio (1. Thu, O ( I thi oet dieree, O, i moothed i ireigverge ilter, it vlue will pproh the egtive vlue o the i error preet i equtio (1. Speiilly, the moothed oet i give y: M 1 S ( O S 1 + S 1 (3 where the vlue o iree y oe t eh meuremet epoh. Thi moothed i vlue the e dded k oto the vlue rom equtio (1 to give ireigly urte rertio-orreted rrier-phe meuremet without y i. Speiilly, moothed rertio-orreted meuremet,, i otied rom: + S (4 I the oie i the phe meuremet were rdom (white, the expeted oie i equtio (4 would deree rom the oie o equtio (11 or (1 ( give i the lt olum o Tle towrd the oie o equtio (1 ( give i the lt olum o Tle 4 i oe over the qure root o trjetory. However, the oie i domited y multipth eet, whih re ot white. The oie vergig deped upo the utoorreltio utio o the multipth d reeiver meuremet oie. The iitil poitive utoorreltio ue the oie vergig to e lower th idepedet oie; however, ter everl miute, the utoorreltio goe egtive whih ue the oie to verge out ter th idepedet oie. It hould lo e oted tht, ulike ode multipth eet, the rrier phe multipth h equl ditriutio o poitive d egtive error M

6 d hould verge to zero over time. The et reult i tht ter 15 to 30 miute o vergig oe ould expet the reidul oie to pproh ew etimeter. OPTIONAL RESOLUTION OF THE AVERAGE AMBIGUITY A idited previouly, oly two o the wide-le miguity-reolved rrier phe meuremet re idepedet. Thu, the third, wide-le miguity vlue e omputed rom the irt two. Thi leve oe degree o reedom i the whole-yle miguitie o the primry rrier phe meuremet. Speiilly, give the wide-le miguity vlue, i y oe o the wholeyle miguitie preet i the primry rrier phe meuremet e determied, the the other primry miguitie e omputed well. Aume, or exmple, the vlue o N ued i equtio (3 d uequetly i equtio (1 h ee etimted to e oe whole yle lrger th it true vlue. Sie the vlue o the wide le whole yle (N -N h ee determied orretly i equtio (7 ove, the vlue iged to N will lo e oe yle too lrge. I imilr hio, the vlue iged to N will e oe yle too lrge. Thu, otried y the orret wide-le miguity vlue, y etimtio error i oe o the primry miguity vlue will ue equl error i eh o the other primry miguity vlue, i.e. the verge vlue o the three primry miguity vlue will e o y oe. Thi hrteriti llow u to ompute the eet o improper etimte o whole-yle miguity o the rertio orreted vlue otied rom equtio (1 ove. I Tle 5 elow, the rertio orreted wvelegth, i.e. the rge error i equtio (1 reult o oe-yle etimtio error i the whole yle miguitie, i give. The otriutio to thi rge error rom eh requey i lo give uig the oeiiet o equtio (1 oud i Tle 4. Mid. req. λ λ λ λ GPS L Glileo E GHz Tle 5: Rertio orreted wvelegth I eh e, the eet o error o oe yle i the hoie o the primry miguitie reult i rertioorreted error itrodued ito equtio (1 o lmot 11 etimeter. Sie the rertio orretio i equtio (11 or (1 i idepedet o error i the primry miguitie d depedet oly o the wide-le miguitie, it i ler tht the dieree etwee the two, repreeted y the oet i equtio ( d (3, will hve error whih will el out the error i equtio (1. But thi me tht ter uiiet moothig the moothed oet vlue o equtio (3 will reh tedy tte vlue whih, whe divided y the rertio-orreted wvelegth (il olum o Tle 5 d rouded to the eret iteger, will give the umer o whole yle y whih to orret the primry whole yle miguity vlue. Tht i: N N + [ S / λ ] (5 rd N N + [ S / λ ] (6 rd N N + [ S / λ ] (7 Thu, ter uiiet moothig time, etimted t 15 to 30 miute, it hould e poile to orret the wholeyle miguity vlue ued i equtio (1 d tep to the il optiml ury repreettive o tht equtio idited i the il olum o Tle 4. A dditiol ptet h ee pplied or whih over the proe eeded to reolve the il rertio orreted miguity. ALTERNATE METHODS FOR COMPUTING THE NOISE OFFSET VALUE There re everl lterte me o geertig the oet vlue o equtio ( ove whih eed to e moothed to redue the oie d to qutize it to the eret whole yle. It tur out tht the three primry meuremet e omied to elimite the ioopheri rertio eet, levig oly the rge (d oie. Or the three meuremet e omied to elimite the rge levig the ioopheri eet (d oie. Tht leve oly oe degree o reedom to olve or other vrile. Tht idepedet omitio elimite oth the rge d the ioopheri rertio eet d leve oly peii omitio o the oie i the three meuremet. The prtiulr omitio o oie whih reult i tht whih i uorrelted with either the rge or the ioopheri eet. Thi me tht y meuremet omitio whih el out the eet o oth rge d ioophere will reult i led verio o thi me oie omitio. (Thi oie oet will e ied y y error i the miguity vlue. Sie the oet vlue deied i equtio ( i oe uh meuremet omitio tht el out oth rge d ioopheri eet, it i prtiulr ite o uh led oie omitio. There re, however, umer o other method (ll equivlet whih e ued to geerte the oet vlue. Oe ould ue dieret omitio o meuremet to olve or the ioopheri rertio eet d the dieree two o thee to geerte led oet vlue. Or oe ould ue dieret omitio o meuremet to geerte dieret rertio orreted meuremet o rge d the dieree y two o thee to geerte led oet vlue. Filly, oe ould i oe tep geerte rd

7 omitio o meuremet whih elimite oth the rge d ioopheri eet ( i doe elow. Adrew Simky, i Reeree, derie method o geertig ompoite meuremet rom three requeie whih h o rge depedee d o ioopheri rertio eet. Speiilly, it h oly oie d multipth depedee. It i thu other ite o led verio o the oet vlue give i equtio (. Hi equtio (dpted to my ottio i: O λ ( + λ ( + λ ( (8 I hve leled thi O ie it tur out it i imply led verio o the oet dieree give i equtio ( ove, whih i ueively moothed i equtio (3. Slig thi equtio it dieretly yield oeiiet vlue o pr with the oeiiet i equtio (1 ove. Speiilly, orm the vlue: O + + (9 Clerly, the rge ompoet i eled i eh pir o primry meuremet. The ioopheri ompoet re elled etwee term d the reidul i due etirely to oie d multipth plu the eet o y whole-yle miguity etimtio error. Simpliyig the equtio give: O ( ( + ( (30 Tle 6 give the oeiiet o equtio (30 i umeril orm whe L1, L5 d the middle requey,, i tht how i the irt olum. For uequet ue, the lt olum how the i i the vlue whih would reult rom poitive oe yle error i the three primry rrier phe whole yle miguitie leled λ m i the lt olum i Tle 5. Thi i imply meure o the vlue o the le tor whih i depedet o the prtiulr wy i whih the rge d ioopheri eet were elimited. The rge i elled ie the oeiiet um to zero. Mid. Freq. C C C λ m GPS L Glileo E GHz Tle 6: Coeiiet o Equtio (30 Two optio re ville t thi poit. Firt, equtio (30 e moothed i expdig vergig proe imilr to tht o equtio (3. Ater uiiet moothig the vlue overged to hould pproh multiple o the λ m vlue i the tle. Dividig y the λ m vlue d roudig to the eret whole umer will give the iteger y whih the primry whole yle miguitie eed to e orreted. A eod optio i etter, i oe wihe to ue the rertio orreted rrier-phe meuremet eore the moothig h ee ompleted. Speiilly, the oeiiet i equtio (30 (i.e. the vlue i eh row o Tle 6 e led y ott whih will ue the vlue o λ m to extly el the miguity idued error i equtio (1 repreeted y the il olum o Tle 5. Thee ie i Tle 5 were otied y iertig the oeiiet omputed i Tle 4 ito equtio (1 or give oe yle miguity error. The pproprite lig d the revied oeiiet re give i Tle 7. The λ m vlue or eh row (ot how i the egtive o the il olum o Tle 5. Mid. Freq. Sle C C C GPS L Glileo E GHz Tle 7: Reled Coeiiet Whe equtio (30 i multiplied y the le tor give i Tle 7, the modiied oeiiet o equtio (30 whih pper i Tle 7 orrepod preiely to the dieree etwee equtio (11 d (1. Thi i veriied y utrtig the oeiiet i Tle 4 rom the oeiiet i Tle. The dieree i the oeiiet vlue give i Tle 7, whih veriie tht, with the pproprite lig, the oet vlue o equtio (1 e otied vi lterte me. Sutitutig the oeiiet o Tle 7 ito equtio (31 give diret equtio whih e ued to ompute the oet, O. O C + C + C (31 It i pproprite to give t let oe more exmple o how oe might ompute lterte vlue o the led oet or uequet moothig. A tted ove, diereig y two rertio orreted meuremet o the rge hould el tht rge d leve led verio o the

8 oet o equtio (. I thi exmple we will ue the rertio-orreted rge equtio rom the irt d eod requeie d will utrt rom it the rertio-orreted rge equtio rom the eod d third requeie. Speiilly, rertio orretig equtio (3 d (4 to elimite the ioopheri eet give: ρ (3 The prllel equtio or rertio orreted rge rom the d requeie i: ρ (33 Sutrtig equtio (33 rom (3 give the led oet equtio: O ( + ( ( (34 The oeiiet d wvelegth (le re evluted or the three dieret middle requeie i Tle 8. Mid Freq. C C C λ m GPS L Glileo E GHz Tle 8: Coeiiet o Equtio (34 Like the prior ltertive thee oeiiet e reled to ue the eet o whole yle miguity error (i eh primry phe meuremet to extly el the depedee o the miguity idued error i equtio (1. Thi le tor i otied y dividig the vlue o the whole yle miguity i equtio (34, i.e. lt olum o Tle 8, ito the whole yle miguity i equtio 1, i.e. lt olum o Tle 5 (d hgig the ig. The reultt le tor d revied oeiiet re give i Tle 9. Mid. Freq. Sle C C C GPS L Glileo E GHZ Tle 9: Reled Coeiiet Tle 9 h extly the me reled oeiiet Tle 7 d the oet vlue, O, i otied gi y iertig thee oeiiet ito equtio (31. Thi how tht the me equtio or the oet vlue, O, reult rom multiple ltertive derivtio. The led verio o equtio (30 d equtio (34 re idetil. CONCLUSIONS A ew three-requey tehique or otiig geometry ree, rertio-orreted, miguity-reolved, rrierphe meuremet h ee deried. Firt, the miguitie o t let two wide le rrier-phe dieree re otied y vergig the orrepodig requey weighted ode meuremet. Thee two miguity-reolved meuremet re the omied ito ompoite rertio-orreted meuremet. The reultig ompoite i quite oiy due to the mpliitio o the multipth oie i the origil rrier-phe meuremet. However, thi oiy rertio-orreted rrier phe meuremet e moothed with other rertio-orreted rrier phe ompoite meuremet otruted to miimize the oie. Thi lter meuremet e otruted rom the primry rrier phe meuremet prior to reolvig their whole yle miguitie. By moothig the dieree i the two rertio-orreted meuremet, the oie e redued d the i i the low-oie meuremet (due to iorret miguitie e etimted d uequetly orreted. REFERENCE 1. Hth, R.R., US Ptet # 6,934,63. Method or uig three GPS requeie to reolve rrier-phe iteger miguitie, Aug. 3, Simky, A., Three the Chrm: Triple-Frequey Comitio i Future GNSS, Iide GNSS, My/Jue 006, pp