Exploiting Satellite Motion in ARAIM: Measurement Error Model Refinement Using Experimental Data

Transcription

1 xplotg Satellte Moto ARAIM: Measuremet rror Model Refemet Usg xpermetal Data Matheu Joerger * ad ors Perva ** * he Uversty of Arzoa, ucso, AZ ** Illos Isttute of echology, Chcago, IL IOGRAPHIS Dr. Matheu Joerger obtaed a Master Mechatrocs from the Natoal Isttute of Appled Sceces Strasbourg, Frace,, ad a M.S. ad a Ph.D. Mechacal ad Aerospace geerg from the Illos Isttute of echology (II), ad 9 respectvely. He s the 9 recpet of the Isttute of Navgato (ION) radford Parso award, ad the 4 recpet of the ION arly Achevemet Award. He s curretly a research assstat professor at II, worg o mult-sesor tegrato, o seuetal fault-detecto for mult-costellato avgato systems, ad o safety of sese ad avod for umaed arcraft systems. Dr. ors Perva s a Professor of Mechacal ad Aerospace geerg at II, he coducts research o advaced avgato systems. Pror to jog the faculty at II, he was a spacecraft msso aalyst at Hughes Arcraft Compay (ow oeg) ad a postdoctoral research assocate at Staford Uversty. Prof. Perva receved hs.s. from the Uversty of Notre Dame, M.S. from the Calfora Isttute of echology, ad Ph.D. from Staford Uversty. He s a Assocate Fellow of the AIAA, a Fellow of the Isttute of Navgato (ION), ad dtor--chef of the ION joural NAIGAION. He was the recpet of the II Sgma X xcellece Uversty Research Award (, ), Ralph arett Mechacal ad Aerospace Dept. Outstadg eachg Award (9, ), Mechacal ad Aerospace Dept. xcellece Research Award (7), Uversty xcellece eachg Award (5), I Aerospace ad lectroc Systems Socety M. arry Carlto Award (999), RCA Wllam. Jacso Award (996), Guggehem Fellowshp (Caltech 987), ad Albert J. Zahm Prze Aeroautcs (Notre Dame 986). ASRAC I ths wor, a ew tme-seuetal postog ad fault detecto method s derved ad aalyzed for dualfreuecy, mult-costellato Advaced Recever Autoomous Itegrty Motorg (ARAIM). Ule covetoal sapshot ARAIM, the seuetal approach explots chages satellte geometry at the cost of slghtly hgher computato ad memory loads. From the perspectve of users o earth, the moto of ay gve GNSS satellte s small over short tme tervals. ut, the accumulated effect geometry varatos of redudat satelltes from multple GNSS costellatos ca be substatal. hs paper uatfes the potetal performace beeft brought by satellte moto to ARAIM. It specfcally addresses the followg research challeges: (a) defg ad expermetally valdatg raw GNSS code ad carrer error models over tme, cosstet wth establshed ARAIM assumptos, (b) desgg estmators ad fault-detectors capable of explotg satellte moto for postog, carrer phase cycle ambguty estmato, ad tegrty evaluato, ad (c) formulatg these processes a computatoally-effcet mplemetato. A modular algorthm s desged, oly reurg a mor augmetato of the sapshot arbore ARAIM multple hypothess soluto separato (MHSS) algorthm. Other modfcatos to eable tme-seuetal ARAIM clude addtoal groud segmet performace commtmets, ad the cluso of extra parameters the broadcast tegrty support message (ISM). Avalablty s aalyzed worldwde for arcraft precso approach avgato applcatos. Results show substatal performace mprovemets for seuetal ARAIM over sapshot ARAIM, ot oly to acheve localzer precso vertcal (LP) reuremets usg depleted GPS ad Galleo costellatos, but also to fulfll much more strget reuremets cludg a te-meter vertcal alert lmt. INRODUCION hs paper descrbes the desg, aalyss, ad evaluato of a ew tme-seuetal postog ad fault detecto method for Advaced Recever Autoomous Itegrty Motorg (ARAIM) usg dual-freuecy, multcostellato Global Navgato Satellte Systems (GNSS). he ew approach dffers from pror wor o sapshot (or stataeous) ARAIM algorthms [-3] that t also explots satellte moto, whch provdes observablty of costat measuremet bases [4]. hs

2 prcple s used ths wor to estmate floatg (real valued) carrer phase cycle ambgutes, thereby mprovg avgato accuracy ad tegrty. Wth the moderzato of GPS, the full deploymet of GLONASS, ad the emergece of Galleo ad edou, a greatly creased umber of redudat ragg sgals becomes avalable, whch has recetly draw a reewed terest RAIM. RAIM explots redudat GNSS measuremets to acheve self-cotaed fault detecto at the user recever [5, 6]. I partcular, RAIM ca help relax reuremets o groud-based tegrty motors. For example, researchers the uropea Uo ad the Uted States are vestgatg ARAIM for worldwde vertcal gudace of arcraft [-3]. Oe of the prmary tass ARAIM s to evaluate tegrty rs, or euvaletly, the protecto levels (PL), whch are probablstc bouds o postog errors. Itegrty rs s the probablty of udetected faults causg uacceptably large postog errors. Multple research efforts have recetly bee coducted to desg optmal estmators ad detectors that mmze the tegrty rs ARAIM, whle meetg specfed cotuty ad accuracy crtera [7-]. hese methods have bee employed the ARAIM Mlestoe Report [] to detfy the crcumstaces uder whch dualfreuecy GPS/Galleo could satsfy LP- reuremets globally. he localzer precso vertcal (LP) reuremets are set to support vertcal avgato durg approach operatos dow to -foot alttude above groud. Referece [] shows that worldwde coverage of LP- s achevable usg optmal sapshot ARAIM algorthms for a wde rage of omal measuremet error ad fault parameters. However, referece [] also pots out cases LP s ot achevable usg dual-freuecy GPS ad Galleo, for example, whe the omal costellatos are eve slghtly depleted e.g., by a sgle satellte each. I addto, the alert lmt (.e., the lmt o acceptable postog errors, whch defes hazardous stuatos) s 35 meters for LP-, whch s much larger tha, for example, the Category II precso approach alert lmt reuremet of meters []. hus, gve that the methods used [7-] reach the best achevable performace, sapshot ARAIM algorthms caot provde global servce better tha LP-. I respose, ths wor explores the potetal beeft of ew tegrty motorg methods that explot satellte moto dual-freuecy mult-costellato ARAIM. Ule sapshot arbore ARAIM algorthms, carrer-smoothed code (CSC) measuremets from multple satelltes are combed at oe stat tme, posto estmates obtaed, for example, from a batch estmator (or fte-terval estmator []) are drectly derved from tme-seueces of raw measuremets. he raw batch s more computato ad memory expesve, but t gves the meas to explot satellte moto over short tme tervals. he upper boud o tme terval s lmted by the mmum msso durato for a ARAIMeupped arcraft. A worst-case scearo would be a stuato whch a arcraft s compelled to lad just after taeoff. he durato of such a msso cludg tax, taeoff, go-aroud, ad ladg s assumed to be o shorter tha -to-5 m. hs research bulds upo pror wor [3] a computatoally-expesve raw batch ARAIM algorthm was evaluated assumg prelmary error models over tme. I ths paper, we refe ad valdate these error models usg expermetal data, we mplemet a ew modular desg for the arbore batch ARAIM method (whch was brefly troduced [3] ad s refed here), ad we dscuss the eed for addtoal ARAIM groud segmet commtmets ad broadcast parameters the Itegrty Support Message (ISM). I the ext secto of ths paper, prevous research efforts are revewed, whch specfcally use geometrc dversty for carrer phase-based postog. hese pror refereces have commo that they rely o large chages geometry from a small umber of ragg sources. Coversely, ths paper, we explot the combato of small agular varatos from may space vehcles (S). o reforce ths pot, we derve a aalytcal expresso of the postog error varace for a llustratve arcraft avgato example usg a tmeseuece of carrer measuremets. he postog error varace s show to be versely proportoal to the accumulated agular varatos from vsble Ss, ad therefore decreases as the amout of geometry chage summed over redudat satelltes from multple GNSS creases. he thrd secto of ths paper addresses the ey challege of measuremet error modelg over tme. he ARAIM error models [-3] provde values of the stataeous CSC measuremet error stadard devatos due to satellte cloc ad orbt ephemers, troposphere, multpath ad recever ose. For each of these error sources, refereces [3, 4] descrbe parametrc models of the tme-correlato affectg raw code ad carrer measuremets. I ths paper, we valdate these error models over tme usg expermetal data. o acheve ths, GPS S cloc ad orbt ephemers errors are computed over a -moth perod by comparg broadcast ephemers (avalable o the Crustal Dyamcs Data Iformato System webste) to precse orbt ad cloc data from the Natoal Geospatal-Itellgece Agecy (NGA), whch s cosdered truth data. rror dstrbutos are establshed to determe realstcally

3 achevable values of the User Rage Accuracy (URA) ad User Rage Rate Accuracy (URRA), whch are ey to tegrty rs evaluato. URA ad URRA are bouds o the satellte cloc ad orbt ephemers errors, ad o the rate of chage of these errors over tme, respectvely. Models to accout for tropospherc delay, multpath reflectos, ad recever ose are updated as well. he fourth secto of ths paper descrbes how these measuremet error models are used a batch arbore estmator ad detector. We defe a modular batch ARAIM mplemetato, whch oly reures mor modfcatos to covetoal sapshot ARAIM. he puts to the batch are CSC measuremets (same as sapshot), augmeted wth carrer measuremets, ad staced over few, freuet sample tmes wth the fxed batch terval (e.g., three sample tmes at 5 m tervals over m). Fault detecto s the performed usg a batch multple hypothess soluto separato (MHSS) algorthm. atch-mhss s smlar to sapshot ARAIM, except that a tme-seuece of measuremets s processed rather tha a sgle set of CSC data, ad that the omal bas due to sgal deformato (specfed [, ]) s treated dfferetly. he batch formulato rus seuetally usg a sldg-wdow mechasm. Kalma flter (KF) mplemetatos [5] wll be vestgated future wor to further reduce computato load. I the ffth secto of ths paper, we dscuss performace commtmets by the groud segmet,.e., by ether the costellato servce provder (CSP) or by the ar avgato servce provder (ANSP). I ARAIM, the groud commts to tegrty parameters, whch are broadcast the ISM to the arcraft they serve as bass for tegrty rs (or PL) evaluato [-3]. he tegrty parameter set would have to be exteded from cludg URA for sapshot ARAIM to URA-ad-URRA for batch ARAIM. URRA specfcatos are readly descrbed for GPS [6], but ot fully defed yet. Also, the broadcast ISM must be augmeted to clude formato o URRA (oly few bts are eeded to actvate the tme-seuetal ARAIM module). he sxth secto of ths paper aalyzes batch ARAIM. A performace assessmet s preseted for arcraft approach applcatos usg ARAIM wth dual-freuecy GPS ad Galleo satellte measuremets. We uatfy the mpact of S moto by comparg the tegrty performace assumg froze (.e., tme-varat) versus ufroze satellte geometres. Performace sestvty to URRA ad to batch terval legth s also evaluated. It suggests that, based o curretly-acheved URRA values, oly three samples over a te-mute batch perod are eeded to effcetly explot satellte moto. I addto, worldwde avalablty maps show that batch ARAIM ca effectvely augmet sapshot ARAIM order to meet LP- reuremets eve wth depleted costellatos. Uder codtos descrbed the paper, batch ARAIM ca satsfy much more strget reuremets, cludg a meter vertcal alert lmt. A FIRS ANALYSIS OF H IMPAC OF GOMRIC DIRSIY ON IAS SIMAION hs wor ams at explotg chages satellte geometry to obta fast ad accurate estmates of carrer phase cycle ambgutes [4]. Implemetatos of ths prcple for arcraft precso approach ad ladg clude the pseudo-satellte ( pseudolte )-based Itegrty eaco Ladg System (ILS) the early 99 s [7-9], a augmetato of GPS usg low earth orbtg (LO) GlobalStar satelltes [], ad a GPS augmetato system usg Irdum LO satelltes [4]. I each of the above refereces, greatly mproved postog performace was acheved by explotg the fast relatve agular moto betwee user recever ad the few ragg sources (pseudoltes or LO Ss) vew durg short (fve-to-te mute log) msso duratos. I cotrast, ths wor, we oly cosder measuremets from medum earth orbtg (MO) GNSS satelltes, whch are slowly movg from the perspectve of a user ear the surface of the earth. It s the multplcty of ragg sources from several GNSS costellatos that s exploted here to acheve a sgfcat accumulated geometry chage. o reforce ths dea, we aalyze a llustratve example of a two-step carrer phase-based arcraft posto estmato process represeted Fgure. hs ualtatve exercse ams at demostratg that the accumulated geometrc varatos form multple Ss ca be a drvg mechasm for carrer phase postog. levato Agle levato Agle ertcal Posto x tme ertcal Dsplacemet x tme Fg. Arcraft Postog xample to Illustrate the Impact of S Moto o stmato rror Reducto

4 Let ad C respectvely be the umber of satelltes vew ad the umber of costellatos used for avgato. I a frst step, carrer phase measuremets dffereced over tmes t ad t from 3 C satelltes are used to determe the three dmesoal arcraft dsplacemet ad the C chages recever cloc bases over the tme terval t t t. he vertcal dsplacemet x estmated usg these tme-dffereced carrer measuremets s uambguous ad accurate, but the absolute arcraft posto at t, oted x for the vertcal coordate, s uow. I a secod step, the cotrbuto of the remag * Ss over t s exploted, * 3 C. Appedx A shows that the resultg estmato error varace o x ca be expressed as: * s s * * () s the varace of the oosphere-free tmedfferetal carrer measuremet, adjusted for x s the elevato agle for S at some tal tme t s the elevato agle for S at a later tme t uato () descrbes two fudametal mechasms by whch ca be reduced. Frst, f agular varatos betwee ad amout to zero for all Ss, the carrer phase sgals aloe provde o absolute postog formato. ut, wth S moto, the larger the agular varato s, the smaller becomes []. Secod, the larger the umber of vsble Ss * s, the smaller becomes. Further results Appedx A suggest that low-elevato satellte moto cotrbutes more to vertcal postog tha hgh-elevato S moto does. I mult-costellato ARAIM, eve though - varatos may be small over short tme perods, the umber * of cotrbutg terms the deomator of uato () ca be large eough to provde sgfcat reducto. For a coarse umercal example, f we assume hypothetcal values of,, 3deg for,..., *, t m, * 5m (accoutg for the x -adjustmet, recever ose, multpath, ad resdual tropospherc errors), ad a rate of chage elevato agle.5deg/ s, we fd 4.7m. hs value of, obtaed eve wthout CSC measuremets, ca be suffcet to meet a 35 m alert lmt. hs s because multpled by a tegrty 7 multpler of 5.3 (correspodg to a tegrty rs reuremet) s 5 m, sgfcatly smaller tha 35 m. he remader of the paper ams at evaluatg postog tegrty for more realstc estmato processes, measuremet error models, mult-gnss S geometres, ad tegrty reuremets. oth sapshot ad batch estmato are represeted Fgure. Let be the umber of satelltes vew, ad the umber of samples, collected from flter tato at tme-epoch to curret tme. For sapshot estmato, raw code measuremets for S (as,..., dcated by the left superscrpt) are smoothed usg carrer data sampled at tmes to (rght,..., subscrpts). Carrer-smoothed code measuremets are output by Hatch flters (HF) at tme for satelltes,...,. ARAIM error models specfy steady-state CSC stadard devatos. HF tato tme s therefore assumed to be far eough the past to reach steady-state. CSC measuremets are the used, for example, a weghted least suares (WLS) estmator [] ad a MHSS detector [-3], to determe user posto xˆ SNAP ad protecto levels at oe stat tme. I parallel, the modular desg of batch ARAIM s represeted Fgure. hree addtoal parameters are troduced. Let be the batch perod,.e., the fte tme terval over whch measuremets are processed: determes the amout of chage S geometry (e.g., we wll use = m the ext sectos of the paper). Also, let S be the samplg terval,.e., the tme betwee raw samples wth the batch (e.g., S =.5 s). Let S be the batch samplg terval wth the batch perod. ecause ths modular batch process, ose averagg s performed separate HF s, S ca be selected much larger tha the raw measuremet samplg perod S ( S >> S ). C

5 raw code ad carrer raw code ad carrer I Fgure, CSC data measuremets sample tervals are augmeted wth carrer ad staced at freuet batch (omally 5m ) over the S batch terval (omally S m ). S s selected much larger tha the raw measuremet samplg perod S =.5s ( S >> S ). hs compact tmeseuece of few measuremets s processed a WLS estmator to smultaeously estmate user posto xˆ ACH at tme ad floatg cycle ambgutes, ad a MHSS detector smlar to the sapshot algorthm. hs CSC-based batch ARAIM method oly reures a mor augmetato of the exstg sapshot ARAIM algorthms. It explots two ey estmato prcples code ose averagg usg CSC from HFs also used sapshot ARAIM S moto usg few carrer measuremets ot used sapshot ARAIM hs modular desg s also much more computato ad memory effcet tha the raw batch [3], whch had to process more tha a thousad raw code ad carrer sgals collected at S =.5s samplg perods. Prelmary evaluato [3] suggests that the performace of ths ew batch mplemetato (wth 5m, m ) s oly slghtly poorer tha S that of a raw batch (wth S =.5 s,,...,,...,,...,,...,,...,,...,,...,,..., Hatch Flter Hatch Flter Hatch Flter S, S, Hatch Flter CSC at tme S, S, S m ). Sapshot WLS stmator MHSS Detector atch WLS stmator MHSS Detector CSC ad raw carrer at freuet tmes (e.g., 3 tmes over m) S,, S,, xˆsnap xˆach posto estmates posto (ad cycle ambguty) estmates Fg. Overvew of Sapshot ARAIM stmato (top) vs. atch ARAIM (bottom) he frst ey challege ths wor s that the ARAIM measuremet error models [] were oly establshed for CSC code, usg large data collecto campags [, 3]. However, evaluatg the batch ARAIM performace reures raw measuremet error models for ad,...,. he ext secto addresses ths ssue.,..., DRIAION OF RAW MASURMN RROR MODLS OR IM hs secto ams at dervg statstcal error models over tme for raw measuremets used batch ARAIM. hs wor provdes refemet ad expermetal valdato of pror research o measuremet error tme correlato [3, 4]. It also ams at fdg error models that are cosstet wth the well-establshed CSC models expermetally valdated [, 3] ad used ARAIM [-3]. he learzed oospherc-error-free code ad carrer phase measuremet euatos for satellte at tme respectvely are: e x () e x,, MP,, RN,, b,,, MP,, RN,, b, e s the 3 le-of-sght vector a local referece frame (e.g., North-ast-Dow or ND) for satellte at tme x s the 3 user posto vector ND, s the recever cloc offset s the tropospherc error s the S cloc ad orbt ephemers error, b, s a omal bas o prmarly due to code sgal deformato (SD) b, s a omal bas o, such that b, << b, (carrer s uaffected by SD), cluded to accout for small mea values the pared overboudg process [4]. s the carrer phase cycle ambguty (costat over tme) MP,, ad MP,, respectvely are code ad carrer errors due to multpath RN,, ad RN,, respectvely are code ad carrer recever ose terms

6 he followg subsectos descrbe models for each dvdual source of error. Satellte cloc ad orbt ephemers error he satellte cloc ad orbt ephemers error euato for satellte at tme s: b g t t RS, b g, (3) s a uow, costat bas s a uow, costat gradet t s the frst tme-epoch of the batch terval t represets ay tme durg the batch terval, such that, s the last epoch of the batch terval resdual errors ot captured by the bas plus RS, he frst term ramp model stataeous ucertaty ARAIM assumptos,.e., b uato (3) captures the,. It s set accordg to b s assumed zero-mea ormally dstrbuted wth varace URA. We use the otato: b ~ N(, URA ) ad we assume: URA = m []. hs stadard devato represets URA values that are expected to be achevable wth the ext decade, whe GPS satelltes older tha loc IIF are decommssoed. It ca easly be show that a tme-varat bas b affectg a seuece of raw code measuremets put to a Hatch flter causes the same error b the resultg output CSC. ut, the ARAIM error model does ot specfy whether omal S cloc ad orbt errors ca be assumed costat over ay partcular tme terval. o accout for temporal error varatos, the secod term uato (3) s added: t s a ramp over tme ( t t ) wth a uow but costat gradet g, accoutg for lear devatos from the tal value. Referece [4] gves evdece that perodc varatos of orbt errors are o the order the MO GNSS orbtal perod, whch supports the use of a smple lear model over te mutes. It also ctes refereces [5-7] to establsh a dstrbuto o the rate of chage of orbt/cloc errors: 4 ~ N(,(4.7 m/s) ). g o further valdate uato (3), a expermetal data aalyss s carred out, whch completes the wor [3]. Smlar to pror wor [, 6, 8], precse GPS satellte orbt ad cloc estmates from the Natoal Geospatal- Itellgece Agecy (NGA) are cosdered truth referece data, ad are compared to broadcast GPS ephemerdes archved by the Crustal Dyamcs Data Iformato System (CDDIS). o complemet the wor [, 6, 8] that addressed stataeous errors, the focus ths wor s o the valdato of error model characterstcs over tme. I ths aalyss, a set of data from Jauary 4, 5 to September 9, 5 s processed, for eght loc IIF satelltes (PRNs, 3, 6, 9, 5, 6, 7, ad 3). he basplus-ramp model uato (3) s ft to truth-musbroadcast data. ecause NGA orbt data s oly provded every fve mutes, the ft terval s selected to be 3 mutes log. hs s larger tha the example -mute batch perod assumed the ext sectos of ths paper. It wll provde coservatve results because resdual fttg errors are larger over 3 m tha they would be over m. ased o ths data, emprcal parameter dstrbutos for b, g ad are establshed. hese are the RS, bouded the cumulatve dstrbuto fucto (CDF)- sese [4, 9] usg Gaussa dstrbutos. he stadard devatos of the over-boudg Gaussa fuctos are gve able for the cloc error cotrbuto, ad for the three-dmesoal orbt errors. Orbt errors are expressed a local-level, satellte-fxed referece frame, terms of the -trac, cross-trac ad radal compoets [8]. ecause GPS satelltes are at alttudes of about, m, user recevers ear the surface of the earth are affected by ragg errors that are mostly due to the radal orbt ad cloc compoets. Fortuately, orbt radal ad cloc errors are sgfcatly smaller tha orbt -trac ad cross-trac compoets, as show able. o get a coservatve estmate of the error parameter varace for b whle tag to accout the worstcase geometry betwee S ad user recever ear earth surface, we use the followg euato from [8]: (4), CLK, R.4, I, C.4 s a multpler accoutg for the worst-case projecto of o-radal orbt error compoets;,,clk,r,,i ad,c, respectvely, are the varaces of the cloc, orbt radal, -trac ad cross-trac error compoets cotrbutg to b. he same ca be doe for the varace G of g. uato (4) s cosstet wth euato (A-) [6] (although provdg a slghtly looser boud because dfferet S elevato mas agles are cosdered).

7 xpermetal data aalyss able shows that 4.6m ad.8 m/s, whch are both G much smaller tha the assumptos metoed above. I the upcomg avalablty aalyss, we wll assume 4 m ad 4.7 m/s. URA G URRA I addto, the dstrbuto of resdual errors are RS, plotted for the orbt radal ad orbt -trac compoets Fgs. 3 ad 4, respectvely. s obtaed by RS, removg the best ft bas-plus-ramp model from the truth-mus-broadcast data. he fgures show that RS, s ot eglgble as compared to other error sources affectg carrer measuremets. As could be expected from pror aalyses [, 6, 8], radal error compoets are much larger tha -trac errors. We caot determe what parts of these resduals are caused by errors NGA truth data. We therefore attrbute the etre resdual errors to S cloc ad orbt ephemers errors, ad accout for them the avalablty aalyss, assumg: RS, ~ N, (.56 m) (5) the stadard devato of.56 m was obtaed usg the same formula as uato (4) but appled to (values lsted the rghtmost colum of able ). RS, Future wor wll clude a detaled aalyss of the tmecharacterstcs of whch, for ow, s modeled as RS, whte ose for samples tae at tervals S, several mutes apart. (We wll assume a S 5m the sxth secto later ths paper.) As a alteratve to uato (3), Appedx evaluates a uadratc model for S cloc ad orbt ephemers errors. hs uadratc model halves the stadard devato of resdual errors as compared to the lear model. RS, ut, t would reure addtoal CSP or ANSP commtmets o the User Rage Accelerato Accuracy (URAA), ad extra parameters the broadcast ISM. he uadratc model s ot employed ths paper, but may be further vestgated future wor. ropospherc Delay he troposphere error model s also modfed as compared to [3]. he tropospherc delay for S at tme s modeled as:, c, ZD, (6) able. Lear rror Model Parameter Over- oudg Stadard Devatos Cloc compoet Orbt radal Orbt -trac Orbt cross-trac Overall (usg (5)) Folded Cumulatve Dstrbuto Fucto Folded Cumulatve Dstrbuto Fucto Stadard devato of boudg Gaussa fucto for b for g for RS.7 m. -4 m/s.4 m.3 m. -4 m/s.7 m. m m/s.84 m.46 m.4-4 m/s.7 m.6 m.8-4 m/s.56 m sample CDF boudg Gaussa: O =.7 m Resdual Orbt-Radal rror Compoet (m) Fg. 3 Folded CDF (mprcal ad oudg Gaussa) for the Orbt-Radal Compoet of the Resdual rror sample CDF overboudg Gaussa: O =.835 m Resdual Orbt-I-rac rror Compoet (m) Fg. 4 Folded CDF (mprcal ad oudg Gaussa) for the Orbt-I rac Compoet of the Resdual rror

8 s the zeth tropospherc delay for S at h h ZD, tme s the tropospherc scale heght ( h = 7m) s the arcraft heght at tme c, s the tropospherc zeth-to-slat mappg c coeffcet:,.. (s( )) s the elevato agle for S at tme Pror wor [3] modeled the tropospherc delay tmecorrelato assumg that tropospherc refractvty was uow, but costat over [3]. I ths paper, we o loger mae ths assumpto, the resultg error model s much more coservatve. We use: ~ N(, ),. m [] (7) ZD, ZD ZD Smlar, he tme-ucorrelated CSC recever ose stadard devato assumed ARAIM [] uts of meters, s gve by: c.5.43exp( / 6.9) () RN,, IF Raw code ad carrer stadard devatos are the expressed as [3]: RN,, 9. 8 RN,,, RN,,. RN,, () he oly source of error left uaddressed uato () s, whch s dealt wth the ext secto. It s b SD, worth otg that the robustess of the error models ca be esured by selectg a large batch samplg terval. he correlato tme costats for multpath error MP,, ad MP,,, tropospherc delay,, ad resdual S cloc ad orbt ephemers errors ca RS, be roughly approxmated, as log as they are sgfcatly smaller tha S. vdece of ths wll be establshed future wor. Recever Nose ad Multpath rror We use the recever ose ad multpath error model derved [3], based o [, 4, -4, 9]. mecorrelated raw code ad carrer measuremet errors due to multpath reflectos are modeled as frst order Gauss Marov Processes (GMP) wth tme costat, wth 8s : MP MP ~ N (, ), ~ N (, ) MP,, MP, MP,, MP, MP,,. 5 MP,,, MP,,. 5 MP,, (8) he ARAIM elevato-depedet model for MP gve [] for GPS s, uts of meters: c.3.53exp( /) (9) MP,, IF c IF s the oosphere-free measuremet combato multpler: f L, L5 c IF 4 4 ( fl fl5)/( fl fl5) f respectvely are L ad L5 freueces s the elevato agle degrees for S at tme, AIRORN ACH SIMAOR AND DCOR DSIGN I ths secto, CSC ad raw code ad carrer measuremet error models derved above are corporated a batch estmator ad a batch-mhss RAIM faultdetecto algorthm that eables tegrty rs evaluato. hs batch mplemetato s formulated a computatoally effcet process llustrated Fgure ad referred to as batch ARAIM. atch Measuremet uato For each S, CSC ad carrer measuremets at p / S sample tmes, from tmes to cremets, are respectvely staced p vectors S ρ ad φ. hese vector are the arraged a p batch measuremet vector ( s umber of vsble Ss): φ ρ φ φ ρ ρ () hs vector ca be expressed terms of state varables, ad of measuremet error vectors as: φ G ρ G H N p H H RR RR u η s RR v v RNM, RNM, b b (3)

9 u s a 5 vector of postos ad GPS ad Galleo recever cloc offsets at all tmes (assumg a dual-costellato GNSS) η s a vector of cycle ambgutes: s RR s a ( ) vector of costat error states. s a a b matrx of zeros ab ector u s costructed usg the followg euatos: u u u, x u (4) GPS, GAL, x s the three-dmesoal arcraft posto vector at tme, GPS, ad GAL, are the recever cloc offsets for GPS ad Galleo respectvely, assumg that the tmeoffset betwee the two costellatos s uow. he vector of costat cycle ambgutes s defed as: he vector of error states s gve by: η (5) RR s b g (6) b ad g are the vectors of costat cloc ad orbt ephemers bases ad gradets for all satelltes costructed followg the exact same patter as η uato (5). rror states s RR are cluded the state vector, ot because ther estmated values are of partcular terest, but because state augmetato s a practcal way to corporate measuremet error dyamcs. Pror owledge o state varables u s captured a dagoal state formato matrx (verse of covarace matrx) wth dagoal elemets: (5 p) G hs a-pror owledge of state estmate errors ca drectly be corporated the estmator by addg up formato matrces as descrbed Secto.. of []. Or, t ca be cluded by measuremet vector augmetato,.e., assumg pseudo-measuremets o s as [4, 3]. RR he batch geometry matrx G ad state coeffcet matrces H ad H are ot explctly expressed here N RR to lmt the legth of the paper. Other batch realzatos cludg error states for dfferet applcatos ca be foud [4, 3]. ector v v ] accouts for radom [ RNM, RNM, troposphere, recever ose, ad multpath errors affectg carrer ad code sgals. Its covarace matrx s oted. Matrx s bult as follows. Frst, for the troposphere cotrbuto, matrx s parttoed four p p detcal blocs made of dagoal matrces capturg the fact that the troposphere detcally affects code ad carrer. he, the troposphere covarace s added to that of recever ose ad multpath errors. he multpath ad recever ose covarace cotrbuto s a fully populated matrx that accouts for multpath tmecorrelato, for HF smoothg tme-correlato, ad for CSC-to-carrer correlato. A detaled descrpto of how to costruct s gve [3]. he p vector of omal bases due to code sgal deformato b for all Ss at all tmes s arraged the same way ρ s uato (). ector b s arraged a smlar fasho, but taes much smaller values because carrer phase measuremets are ot affected by sgal deformato. (Note that for carrer sgals, bases that are costat over the short batch terval wll be absorbed to the floatg cycle ambguty states.) Next we cosder the batch measuremet uato (3), wth a addtve p batch fault vector f, whch wll have to be detected. he resultg batch observato euato ca be rewrtte a stadard form as: z H x v atch stmator Desg z Hx v f (7) s the batch measuremet vector s the batch observato matrx, s the batch state vector s the batch measuremet error vector: v ~ N([ b b ], ) he batch weghted least-suares (WLS) estmate for the state of terest (e.g., for the vertcal posto coordate, whch s of prmary terest arcraft approach avgato) at the curret tme s defed as: ˆ s z (8) x s s the p vector of batch WLS coeffcets. (he same otatos, wth addtoal detals, are used

10 [3]). ˆx uato (8) s obtaed usg all avalable measuremets, ad s also referred to as full-set soluto. he full-set estmate error s oted : x ˆx, x s the true value of the state of terest. s such that: ~ s s (9) N b sf, b s the mpact o state estmato of b ad b. he ARAIM error model [-] does ot specfy whether these bases are costat over tme, but t assumes that the elemets of b ad b ca all be bouded by a maxmum value oted b om: b om=.75 m []. We further assume that ragg errors due to these bases ca be dfferet for all satelltes. It follows that b ca be bouded by: operator, ad: p b s A b () om deotes the elemet-wse absolute value.5 I A p p p p p I p () a factor of.5 s used to capture the fact that carrer measuremets are ot mpacted by sgal deformato, but may be mpacted by small o-gaussa but bouded errors. atch Detector Desg A multple-hypothess soluto separato (MHSS) batch RAIM method [,, 4, 5, 3, 33] s adopted for detecto of f. Let h be the umber of fault hypotheses that eed to be motored agast (refer to [3, 33] for detals o how to determe h ). A set of mutually exclusve, exhaustve hypotheses H, for,..., h, s cosdered. Uder H, a umber of measuremets are smultaeously mpacted by the fault. he fault-free subset soluto, whch excludes these measuremets, s wrtte as: xˆ s z, s s the p vector of the subset soluto s batch WLS coeffcets wth zeros for elemets correspodg to the faulted measuremets [3, 33]. Uder of xˆ s such that: b H, the estmato error ~ N, s s () b A p s b. (3) om he batch MHSS test statstcs are the defed as: ˆ, for,..., h, (4) xˆ x s ormally dstrbuted wth varace [5, 3, 33]: (5) Itegrty ad Cotuty Rs valuato he tegrty rs, or probablty of hazardous msleadg formato P HMI, s evaluated usg the followg upperboud [3]: P HMI P h P H P H H P H (6) s the alert lmt (AL) that defes hazardous stuatos: [], the vertcal AL s =35m H occurrece P H s the pror probablty of s the fault-free hypothess H H for,..., h are the fault hypotheses correspodg to faults o subset measuremet (cludg sgle-satellte, mult-satellte, ad costellato faults []) Uder fault-free hypothess H, the detecto threshold s set based o a allocated cotuty rs reuremet C RQ (specfed []) to lmt the probablty of false alarms. ca be defed as: CRQ ( hph ) Q (7) the fucto Q {} s the verse tal probablty dstrbuto of the two-taled stadard ormal dstrbuto. It s worth otg that ths batch mplemetato s compatble wth estmator optmzato methods [, 7, 9], whch are specfcally desged to mmze tegrty rs whle meetg cotuty reuremets. hese methods wll be vestgated for batch ARAIM future wor.

11 GROUND SGMN AND INGRIY SUPPOR MSSAG AUGMNAION O NAL ACH ARAIM hs secto dscusses the mplcatos of usg batch ARAIM as a modular complemet to sapshot ARAIM. Just as the User Rage Accuracy (URA) s a prmary tegrty parameter for sapshot ARAIM [-3], the User Rage Rate Accuracy (URRA) s ey to batch ARAIM. Frst, the foudatoal defto of a fault, whch sapshot ARAIM s defed terms of the URA, must be refed whe batch ARAIM s mplemeted to address URRA. A straghtforward modfcato of [34] gves the followg defto: A Sgal Space (SIS) fault state s sad to exst o satellte costellato j whe the magtude of the stataeous SIS ragg error s greater tha f, URA,, j at the worst user locato, or whe the magtude of the stataeous SIS rage rate error over a predefed tme perod (e.g., m) s greater tha f, URRA,, j at the worst user locato. hs fault defto ca the be used as [34] to defe pror probabltes of satellte faults P sat,, j ad of costellato faults URA,, j, P sat, j P,. he tegrty parameters cost j,, ad P cost, j, ad the omal bas boud b om uato () are eeded at the arcraft to evaluate avgato tegrty. he groud segmet,.e., the costellato servce provder (CSP) ad/or the ar avgato servce provder (ANSP) provde uatfable assurace o the valdty of these parameters. I batch ARAIM, a ffth parameter eeds to be cluded: the URRA, oted. URRA,, j URRA specfcatos are readly descrbed for GPS [6], but they are ot fully defed yet. ( A future verso of ths SPS PS may establsh a stadard [6].) Referece [6] otes that a 6-sgma upper boud o the SPS SIS stataeous User Rage Rate rror (URR) s. m/s over ay 3-secod terval. hs umber correspodg to a.3 m/s -sgma value s a order of magtude larger tha our observatos able. hs s because the value s specfed over 3-secod perod, ad probably accouts for hgh-freuecy S cloc varatos. As such, t s ot applcable over our perod of terest of - 5m. Assumg a.3 m/s -sgma URRA over m yelds urealstcally large user rage errors (much larger tha specfed URAs). hus, for the performace aalyss the ext secto, we use the URRA value of.4 m/s establshed prevous sectos. Fally, cosstetly wth the ARAIM archtecture defed [], the tegrty support message (ISM) ca be employed as a meas to modfy the URA ad URRA values assumed at the arcraft. hs facltates the tegrato of ewly-deployed costellatos ad satelltes, whch are ot expected to acheve low URA s early phases of operato, but ca mprove as the GNSS matures, ad degrade aga as t gets older. Itegrty parameters are ot expected to chage freuetly, but should stay costat over moths. Multple alteratves are cosdered to broadcast URRA values, depedg o how may data bts ca be allocated to ths parameter the ISM: URRA values ca be broadcast for each dvdual satellte or, URRA values ca be broadcast for sets of satelltes, for example, a sgle boudg URRA could be trasmtted per satellte loc, or per costellato or, a batch ARAIM valdty flag could be corporated the ISM, dcatg that preset, stored URRA are valdated by the groud. hese preset values could be uploaded mothly as part of the arcraft database. If these practcal ssues are overcome, batch ARAIM ca acheve substatal reductos tegrty rs as compared to sapshot ARAIM. xamples of such mprovemets are uatfed the ext secto. ACH ARAIM INGRIY AND AAILAILIY PRFORMANC ANALYSIS hs secto presets a tegrty ad avalablty performace aalyss of batch ARAIM as compared to sapshot ARAIM. Sestvty of avgato tegrty rs to batch terval legth ad URRA value s uatfed, ad global avalablty maps are gve. Sestvty Aalyss A method was developed [3] to aalytcal establsh the codtos of euvalecy betwee sapshot ad batch ARAIM. hese aalytcal relatoshps were used to esure that raw measuremet error models parameters that are ot expermetally set were cosstet wth ARAIM assumptos [, ]. uvalecy was obtaed by artfcally freezg the smulated satellte geometry, ad by adeuately selectg [3]. I Fgure 5, we freeze ad ufreeze the S geometry to emphasze the mpact of explotg S moto batch ARAIM. he fgure shows the tegrty rs P HMI uato (6) evaluated over 4 hours, at a example

12 Chcago locato (5.5 deg N, -8. deg ), assumg dual-freuecy measuremets from GPS ad Galleo. he omal batch terval s m, ad the omal URRA s URRA.4m/s. Other parameter values that are ot drectly relevat ths subsecto ad that are ept costat clude: S 3 s, a example tegrty rs reuremet reuremet 7 I RQ.98 [], a cotuty rs 6 C RQ 3.9 [], a vertcal alert lmt m, ad pror probabltes of satellte ad costellato faults, both oted respectvely are P 5 ad sat P H euato (6), P 8. cost Fgure 5 shows that for froze S geometres, batch ARAIM performs smlar to sapshot ARAIM. It s actually slghtly better tha sapshot ARAIM because the batch terval s sgfcatly loger tha the Hatch flter (HF) traset respose tme, ad over, the process gets some observablty o the error states s RR uato (3). P HMI decreases substatally whe ufreezg the S geometry, especally at the P HMI peas whch are a prmary cause for uavalablty. I the remader of the paper, we wll o loger artfcally freeze geometry. Fgure 6 llustrates the sestvty of P HMI to. It shows that usg batch ARAIM wth 5m already provdes sgfcat PHMI -reducto as compared to sapshot ARAIM. Further reducto s obtaed whe creasg to m ad 5 m. Larger -values are ot cosdered because of the operatoal costrats o mmum that are metoed Itroducto. I Fgure 7, we vestgate the sestvty of P HMI to URRA, for the omal value = m. Just as URA s ey sapshot ARAIM [], low URRA (labeled URRA ) values are strumetal achevg low P HMI batch ARAIM. Whe the expermetally-valdated omal URRA (.4m/s ) s creased to URRA URRA.4m/s, the the beeft of usg S moto batch ARAIM becomes eglgble, ad the correspodg PHMI -curve wth damod marers Fg. 7 matched that of batch ARAIM for the froze geometry case Fg. 5. Avalablty Maps hs sub-secto evaluates the global avalablty of LP- avgato reuremets to support localzer precso vertcal arcraft approach operatos dow to feet above the groud, usg ARAIM wth dual-freuecy measuremets from GPS ad Galleo. Nomal smulato parameters are defed [], ad clude: Itegrty Rs a fve degree satellte elevato mas ARAIM CSC measuremet error model parameters euatos () to (), ad () avgato reuremets: 6 C RQ 3.9, 35 m pror probabltes of faults: P 5, P 4 sat cost 7 I RQ.98, reduced-batch perod ad samplg terval: m, 5 m S omal costellatos, comprsg 4 GPS satelltes ad 4 Galleo Ss [35] reured P HMI CSC: Sapshot ARAIM atch ARAIM (FROZN geometry) atch ARAIM (UNFROZN) me (hr) Fg. 5 Itegrty Rs oud Obtaed Usg Sapshot s. atch ARAIM: all parameters detcal to Fg. 8 ( = 6s, R = 3s). Itegrty Rs reured P HMI CSC: Sapshot ARAIM atch ARAIM ( =3s) atch ARAIM ( =6s) atch ARAIM ( =9s) me (hr) Fg. 6 Itegrty Rs oud Obtaed Usg Sapshot s. atch ARAIM: all parameters detcal to Fg. 8 ( = 6s, R = 3s).

13 Itegrty Rs reured P HMI CSC: Sapshot ARAIM atch ARAIM ( URRA =.4m/s) atch ARAIM ( URRA =.4m/s) 45 N 45 S 35 W 9 W 45 W me (hr) Fg. 7 Itegrty Rs oud Obtaed Usg Sapshot s. atch ARAIM: all parameters detcal to Fg. 8 ( = 6s, R = 3s). Fg. 9 Avalablty Map for atch ARAIM usg depleted costellatos, Pcost = -4, l = 35 m (coverage of 99.5% avalablty s 9.%). hese parameters are modfed below to evaluate performace sestvty. I order to accout for potetal satellte outages, depleted costellatos of 4- GPS satelltes ad 4- Galleo Ss [35] are cosdered as well. Addtoal reuremets, cludg ffectve Motor hreshold (M) ad fault-free accuracy reuremets [], are cluded the smulato but ot dscussed ths paper as they oly have a mor mpact o overall avalablty. Fgures 8 ad 9 dsplay avalablty maps for a deg deg lattude-logtude grd of locatos, for depleted 4- GPS ad 4- Galleo costellatos, for satellte geometres smulated at regular 5 mute tervals over a 4 hour perod. Avalablty s computed at each locato as the fracto of tme the PHMI -boud meets I RQ. 45 N 45 S W 9 W 45 W Fg. 8 Avalablty Map for Sapshot ARAIM usg depleted costellatos, Pcost = -4, l = 35 m (coverage of 99.5% avalablty s 6.5%). I the fgures, avalablty s color-coded: whte color correspods to a value of %, blac represets 8%. Costat avalablty cotours are also dsplayed. he gray areas Fg. 8 dcate that sapshot ARAIM s clearly outperformed by batch ARAIM Fg. 9. he worldwde avalablty metrc gve the fgure captos s the weghted coverage of 99.5% avalablty: coverage s defed as the percetage of grd pot locatos exceedg 99.5% avalablty. he coverage computato s weghted at each locato by the cose of the locato s lattude, because grd pot locatos ear the euator represet larger areas tha ear the poles. Fgures 8 ad 9 show that the coverage of 99.5% avalablty creases from 6.5 % for sapshot ARAIM, to 9.% for batch ARAIM, assumg = 35 m. hs result was obtaed usg the basele sapshot ARAIM algorthm descrbed [-3]. Referece [] also provdes a optmzed estmator, whch, whe evaluated uder the same assumptos as Fg. 8, provdes coverage of 99.5% avalablty of 8%. herefore, eve usg a optmzed estmator, coverage usg sapshot ARAIM s 9% lower tha usg batch ARAIM. Furthermore, the optmal estmator approach [, 7, 9] s compatble wth batch ARAIM, so that avalablty Fg. 9 ca be further mproved. hs wll be vestgated future wor. Fgures ad evaluate the potetal of batch ARAIM to meet reuremets that are more strget tha LP, cludg a tght alert lmt: = m. I ths case, oe ey assumpto to acheve hgh avalablty s that the pror probablty of costellato-wde faults must be reduced to P 8. hs assumpto, also cost cosdered ARAIM (for horzotal ARAIM or H- ARAIM []), may be accomplshed usg the

14 gudeles [36]. hese gudeles explot addtoal formato from the ANSP groud segmet. I Fgs. ad, avalablty maps assume omal 4-satellte GPS ad Galleo costellatos. he same color code as Fgs. 8 ad 9 s used. Aga, batch ARAIM Fg. provdes a dramatc mprovemet as compared to sapshot ARAIM Fg.. Worldwde coverage of 99.5% avalablty, ad of 95% avalablty (gve paretheses), for the above cofguratos, for omal ad depleted costellatos, are lsted able. able uatfes the global performace mprovemet brought by batch ARAIM as compared to sapshot ARAIM. As metoed throughout the paper, ths mprovemet comes at the cost of hgher computato ad memory loads, of addtoal groud segmet commtmets o URRA, ad of few extra parameters the ISM. It must also be oted that, for arcraft avgato stadards that are more strget tha LP, ad that clude = m, addtoal reuremets are typcally volved, e.g., o the commucato l betwee ANSP groud segmet ad arcraft. Such cosderatos are beyod the scope of ths paper. able suggests that a meter alert lmt may be reachable usg omal costellatos, but for ow, oly wth low coverage. he coverage metrc ca be decevg (e.g., avalablty averaged over all locatos Fg. s 98%). Stll, addtoal research s eeded to mprove coverage, for example usg optmzed estmators. CONCLUSION I ths paper, a ew ARAIM tegrty motorg method was devsed, whch explots the moto of satelltes from multple GNSS. Raw measuremet error models over tme were establshed. hese models were the corporated batch-type estmato ad solutoseparato fault-detecto processes. he mpact of satellte moto o batch ARAIM was aalyzed as a fuctos of batch perod, ad the uatfed globally comparso wth covetoal sapshot ARAIM. he proposed batch ARAIM mplemetato s slghtly more computato ad memory expesve tha sapshot ARAIM. ut, t ca provde sgfcat performace mprovemets both whe amg to acheve LP reuremets usg depleted costellatos, ad whe tryg to meet a much more strget te-meter alert lmt. he ext step of ths research s to vestgate ways to further mprove avalablty whe tryg to acheve - meter alert lmts, for example by employg o-leastsuares estmators specfcally desged to mmze tegrty rs. 45 N 45 S W 9 W 45 W Fg. Avalablty Map for Sapshot ARAIM usg omal costellatos, Pcost = -8, l = m, (coverage of 99.5% avalablty s %). 45 N S W 9 W 45 W Fg. Avalablty Map for atch ARAIM, usg omal costellatos, Pcost = -8, l = m, =9s, (coverage of 99.5% avalablty s 43%). able. Coverage of 99.5% Avalablty, ad Coverage of 95% Avalablty ( paretheses) omal costellatos, P cost = -4, =35 m, =6s depleted costellatos, P cost = -4, =35 m, =6s omal costellatos, P cost = -8, = m, =9s ACKNOWLDGMNS Sapshot ARAIM 94% (%) 63% (98%) % (%) atch ARAIM % (%) 9% (%) 43% (86%) he authors would le to tha the Federal Avato Admstrato (FAA) for ther support of ths research.

15 However, the opos ths paper are our ow ad do ot ecessarly represet those of ay other perso or orgazato. APPNDIX A: IMPAC OF S MOION ON POSIIONING RROR ARIANC DRIAION he aalyss descrbed ths appedx ad llustrated Fgure s a exteso of the Statc Surveyg problem gve []. I ths case, the measuremet vectors φ ad φ clude dfferetal carrer phase sgals (at tmes ad ) from satelltes, for,...,. Smplfyg the problem to be oe-dmesoal alog the vertcal drecto a local referece frame at the arcraft locato, φ ad φ are wrtte by: We defe: φ φ g x φ g ( x x ) η v η v * φ gx g x η v* hese measuremets are expressed matrx form as: ad φ φ * g g I x v I η v* φ, φ x η, s g, s * * * g s s s the vertcal posto to be estmated (dsplayed Fgure ) detty matrx I s a carrer phase cycle ambguty for S he carrer phase measuremet ose vector [ v ] * v * s assumed zero-mea ormally dstrbuted wth covarace matrx I *. he WLS estmate error varace for x s gve by: g g g I * (A.) I I g I s a a b matrx of zeros ab Usg a popular matrx verso formula, becomes: (A.3) g g g g ( g g )( g g ) whch, usg the above deftos of g ad g ca be wrtte as: s s (A.4) hs result s commeted upo Secto A Frst Aalyss of the mpact of Geometrc Dversty o as stmato. uato (A.4) ca be further aalyzed cosderg small agular varatos over a ftesmally small tme terval δ t. ecause ths llustratve example, becomes ftely large whe, let us assume that we have pror owledge o x wth varace, pror. he reasog s pursued the formato form because formato (verse of varace) ca drectly be added up. We are terested determg the cotrbuto δ of the moto of S over δ t (other Ss are assumed statc) to the a-posteror, post estmate formato : he formato δ ca be wrtte as: δ, post, pror δ x -. s s( δ ) (A.5) Usg a frst order aylor seres approxmato for small agular varatos about of the secod term the sum, uato (A.5) becomes: cos( ) δt (A.6) δ

16 3 C cos( ) δt, * Low elevato satelltes have a zeth agle approachg 9. uato (A.6) shows that the formato cotrbuto δ s larger for these satelltes tha for hgh elevato Ss, both because of the s -term, ad because s larger at low elevatos. APPNDIX : QUADRAIC RROR MODL he satellte cloc ad orbt ephemers error euato for satellte at tme s: t t t t RS, b g, s a uow, costat uadratc coeffcet hs uadratc error model s processed usg e moths of truth-broadcast data processed as descrbed Secto Dervato of Raw Measuremet rror Models Over me. he resultg overboudg Gaussa stadard devatos are lsted able 3. he resdual error devato the rghtmost colum s reduced from 5.7 cm for the lear model (see able ) to 3. cm for the uadratc model. ut, as metoed the text, mplemetg the uadratc error model would reure addtoal commtmets o URAA, ad extra parameters broadcast the ISM. able 3. Quadratc rror Model Parameter Over- oudg Stadard Devatos Stadard devato of boudg Gaussa fucto for b for g for for RS CLK.7 m.5-4 m/s m/s.4 m R.3 m.3-4 m/s m/s. m I.4 m m/s. -7 m/s.6 m C. m m/s.4-7 m/s.53 m O.6 m m/s.3-7 m/s.3 m CLK : cloc compoet; R : orbt radal; I : -trac; C : cross-trac O: overall value (usg uato (5)) RFRNCS [] Worg Group C. ARAIM echcal Subgroup. Iterm Report Issue.. echcal report, U-US Cooperato o Satellte Navgato,. [] Worg Group C. ARAIM echcal Subgroup. Mlestoe. Report. echcal report, U-US Cooperato o Satellte Navgato, 4. [3] lach, J.,. Walter, P. ge, S. Waller, F. A. Feradez, R. Dellago, R. Ioades, I. F. Heradez,. elabbas, A. Spletter, M. Rppl, Crtcal lemets for a Mult-Costellato Advaced RAIM, NAIGAION, ol. 6, No., 3, pp [4] Remod,., Pseudo-Kematc GPS Results Usg the Ambguty Fucto Method, Proceedgs of the 46th Aual Meetg of the Isttute of Navgato. Atlatc Cty, NJ. 4-49, NJ, 99. [5] Lee, Y. C., Aalyss of Rage ad Posto Comparso Methods as a Meas to Provde GPS Itegrty the User Recever, Proc. of the 4d Aual Meetg of he Isttute of Navgato, Seattle, WA, 986, pp. -4. [6] Parso,. W., ad Axelrad, P., Autoomous GPS Itegrty Motorg Usg the Pseudorage Resdual, NAIGAION, ol. 35, No., 988, pp [7] lach, J., Walter,., ad ge, P., Optmal Postog for Advaced RAIM. NAIGAION, Joural of he Isttute of Navgato, ol. 6, No. 4, 3, pp [8] lach, J., Walter,., ge, P., Results o the Optmal Detecto Statstc for Itegrty Motorg, Proceedgs of the 3 Iteratoal echcal Meetg of he Isttute of Navgato, Sa Dego, Calfora, Jauary 3, pp [9] Joerger, M., Lagel, S., ad Perva,., Itegrty Rs Mmzato RAIM - Part - Optmal stmator Desg, submtted to Joural of Navgato of the Royal Isttute of Navgato, 4. [] Joerger, M., Lagel, S., ad Perva,., Itegrty Rs Mmzato RAIM - Part - Optmal Detector Desg. submtted to Joural of Navgato of the Royal Isttute of Navgato, 4. [] Rado echcal Commsso for Aeroautcs (RCA) Specal Commttee 59. Mmum Avato System Performace Stadards for the Local Area Augmetato System (LAAS). Documet RCA/DO-45, 4. [] Crassds, J., ad Jus, J., Optmal stmato of Dyamc Systems, Chapma & Hall/CRC, oca Rato, FL, 4, pp [3] Joerger, M., ad Perva,., Mult-Costellato ARAIM xplotg Satellte Geometry Chage,