LAAS Ranging Error Overbound for Non-zero Mean and Non-gaussian Multipath Error Distributions

Transcription

1 LAAS Ragig Error Overboud for o-zero Mea ad o-gauia Multipath Error Ditributio Irfa Sayim ad Bori Perva Illioi Ititute of Techology, Chicago, Illioi BIOGRAPHY Irfa Sayim received a B.S. degree from Marmara Uiverity (1990), Itabul, Turey, M.E. (1996) ad Ph.D. (2003) from Illioi Ititute of Techology i Mechaical ad Aeropace Egieerig. Curretly, he i worig o the avigatio Itegrity of Local Area Augmeted GPS for aircraft preciio approach ad ladig. Bori Perva received a B.S. from the Uiverity of otre Dame (1986), M.S. from the Califoria Ititute of Techology (1987), ad Ph.D. from Staford Uiverity (1996), all i Aeropace Egieerig. From 1987 to 1990, he wa a Sytem Egieer at Hughe Space ad Commuicatio Group. Dr. Perva wa a Reearch Aociate at Staford from 1996 to 1998, ervig a project leader for GPS Local Area Augmetatio Sytem (LAAS) reearch ad developmet. He wa the 1996 recipiet of the RTCA William E. Jaco Award ad the 1999 M. Barry Carlto Award from the IEEE Aeropace ad Electroic Sytem Society. Curretly, Dr. Perva i Aitat Profeor of Mechaical ad Aeropace Egieerig at the Illioi Ititute of Techology i Chicago. ABSTRACT Providig ufficiet avigatio itegrity i oe of the mot critical elemet i the developmet Local Area Augmetatio Sytem (LAAS) architecture ad remai a uolved ey techical problem. For example, i LAAS, the avigatio itegrity i eured through the computatio of the aircraft poitio error boud (i.e., Protectio Level, VPL/LPL) i which implicitly zeromea, ormally ditributed fault-free error model are aumed. However, it i udertood that true ragig error caot eaily be decribed i thi way, epecially i the preece of groud reflectio multipath. Therefore, to eure avigatio itegrity of LAAS, the true ragig error mut be bouded with zero-mea ormal ditributio. I thi regard, a major ragig error ource, groud reflectio multipath, i pecifically targeted i thi reearch ad ew theoretical approache are defied to boud ad accout for it effect. ITRODUCTIO The Local Area Augmetatio Sytem (LAAS) i a differetial atellite-baed avigatio ytem architecture deiged to provide avigatio ervice for civil aircraft uer durig preciio approach ad ladig. I the deig of architecture, oe of the mot challegig ad remaiig uolved techical problem i to provide ufficiet itegrity-related broadcat iformatio (i.e., peudorage broadcat correctio error tadard deviatio, gd ) to aircraft. The itegrity-related iformatio i tralated to itegrity ri aemet through the computatio of Vertical ad Lateral Protectio Level (termed VPL ad LPL, repectively) to determie whether or ot LAAS i available to upport preciio approach ad ladig. Thee level are the poitio boud that ca be eured with a acceptable level of itegrity ri. The precribed algorithm for the geeratio of the Protectio Level aume a zero-mea ormally ditributed fault-free error model for the broadcat peudorage correctio. The tadard deviatio of correctio error i preumed by the aircraft to be equal to the broadcat value of for each atellite. I reality, while uch a error ditributio model appear to be reaoably coitet with referece receiverrelated ragig error due to receiver oie ad diffue gd 1

2 multipath, remaiig error uch a groud reflectio multipath ad ytematic referece receiver/atea error may be o-gauia, o-zero-mea, or both. Therefore, LAAS mut tae pecial care whe geeratig gd to guaratee itegrity ri at deired level of probability. For ormally ditributed error ource uch a receiver-related oie ad diffue multipath, tadard deviatio ca be etimated uig experimetal data aloe. I thi cae, however, it i till eceary to accout for the additioal itegrity ri icurred by tatitical ucertaity (due to fiite ample ize) i the owledge of referece receiver error tadard deviatio ad error correlatio betwee multiple referece receiver. I prior wor [3], a detailed methodology wa propoed for the defiitio of miimum acceptable iflatio parameter for the ample tadard deviatio. Furthermore, i order for uch a empirical proce to be applied, it i eceary to defie a proper method to collect data ito bi prior to igma etimatio. While large bi ize are deired to maximize ample ize (to limit required iflatio factor), bi ize i ultimately cotraied by the eed for patial tatioary of all data withi the bi (i.e., all error data withi a bi mut have the ame uderlyig ditributio). Oe cadidate approach toward the reolutio of thi critical tradeoff i a adaptive etimatio techique ow a Expadig Bi method (EB-method) [9]. Bia-type error ource, uch a ytematic referece receiver/atea error mut be accouted for by a appropriate method of boudig a o-zero mea ditributio with a zero mea ormal ditributio. Sice LAAS Groud Facility (LGF) mea error value will ot be broadcat to the aircraft by the LGF, the effect of a o-zero mea mut be accouted for by further iflatio of. However, to eure tight poitio domai gd (VPL ad LPL) overboud at the aircraft i the preece of a o-zero mea, the value of the required iflatio factor o i depedet o airbore error tatitic ( air gd ). Clearly prior iformatio of aircraft error tatitic will be uavailable at the LGF, o i thi paper we itroduce ew method for iflatio (due to o-zero mea), which reult i looer (i.e., more coervative) poitio domai boud, but do ot require ay pecific prior owledge of aircraft error tatitic. For o-gauia error ource, uch a groud reflectio multipath, tadard deviatio caot be etimated by the direct ue of data becaue empirically computed (ad iflated) value of are ot gd ufficiet to guaratee overboudig of the total LGF ragig error. Furthermore, it i impoible to rely o empirically cotructed ditributio (e.g., error data hitogram) aloe to defie the ature of the uderlyig error ditributio becaue little or o empirical data will exit i the tail (which are of greatet iteret i LAAS). Therefore, theoretical approache are ought i gd our wor to icorporate o-gauia error effect ito. I thi regard, groud reflectio multipath error, gd which i the larget cotributor ource to ragig error, i of particular iteret. Theoretical olutio are ivetigated for modelig groud reflectio multipath error igma. It i alo realized that for o-gauia error ditributio, poitio domai error boudig mut be pecifically addreed becaue a gauia boudig that hold i the rage domai (igle-atellite error ource) doe ot ecearily guaratee imilar boudig i the poitio domai (multiple-atellite error ource i weighted leat quare olutio). Therefore, poitio domai boudig mut be verified via multiple covolutio of o-gauia ditributio. A practical way i the itroduced to ytheize the empirical ad theoretical elemet decribed above to quatitatively etablih for LAAS. I particular, the gd approach tae for igma etablihmet i to compute a bet etimate of igma from data uig EB-method, ad the combie the reult with thoe obtaied by theoretical mea for groud reflectio multipath. It i clear that, however, that the theoretical model mut be coitet with the uderlyig phyical procee drivig the error. I order to avoid detailed validatio of phyical groud reflectio multipath model, we coider i thi wor two limit cae model ditributio for the overboudig of groud reflectio multipath. Thee are a wort-cae bia-type ditributio ad a wort-cae ymmetric, zero-mea, bimodal ditributio. The effectivee of ytheized igma reult i demotrated uig aalyi, imulatio ad experimetal data. BOUDIG COCEPT A overboud for aircraft poitio error i eeded whe the true ragig error ditributio are ot zero-mea gauia becaue, i LAAS, the aircraft aume that the broadcat igma i a tadard deviatio of a zero-mea gauia probability deity fuctio (PDF). It i the repoibility of the LGF to provide a tadard deviatio of a zero-mea gauia PDF that repreet the true correctio error ditributio. I reality, however, ituatio may exit i which o-gauia ad/or o zero-mea gauia ditributio mut be bouded. Boudig for ragig error i defied a a method by which a ufficiet zero-mea gauia ditributio i geerated to overboud the tail of true aircraft poitio error havig o-gauia or o-zero mea gauia PDF. To eure the validity of boudig i the poitio domai, method ivolve the covolutio of multiple ogauia PDF (i.e., um of o-gauia RV) ad the covolutio of the ame umber of gauia PDF. The reultig tail area of Cumulative Ditributio Fuctio (CDF) i evaluated to determie whether the gauia reult overboud the o-gauia at the probability level of iteret. The ratio of overboudig gauia tadard 2

3 deviatio to that of the o-gauia PDF parameter i called the Iflatio Factor. MEA BOUDIG FOR LAAS A metioed above, bia value i ragig error are either broadcat to the aircraft or defied a a part of Protectio Level equatio. Therefore, the exitece of uch biae mut be accouted for by igma overboudig. Overboudig o-zero mea gauia ditributio i eay i priciple, but difficultie arie whe the LGF ad aircraft are uaware of ditributio parameter corepodig to each other eror PDF. Thi difficulty limit LGF ability to ow whether or ot a meaigful protectio level i computed at aircraft prior it broadcat value. For example, broadcat igma of correctio error are geerated at the LGF, broadcat to the aircraft, combied with igma of the aircraft, ad the ued i Protectio Level equatio at aircraft for the fial avigatio itegrity ri aemet of preciio ladig. Both LGF ad aircraft error tatitic collectively mut be coidered i the mea boudig proce. To preciely accommodate for the exitece of a mea value i the correctio error, the LGF mut ow aircraft ragig error ditributio i order to geerate a appropriate overboudig value of. Ufortuately, the LGF gd obviouly caot ow aythig about the icomig aircraft error. The critical problem, the, i the boudig proce i to accout for o-zero LGF mea eror without prior owledge of each other eror ditributio parameter. It i alo oted that the mea value do ot ecearily alway have poitive value. I reality, however, it i ofte oly poible to defie a boud o the abolute value of the mea ragig error. Therefore, for a coervative boud, the mea value ca be aumed to be i the ame directio (abolute value). A decribed i [1] ad [4], the computatio of Protectio Level i baed o a Weighted Leat Square Etimatio, which i drive by the atellite geometry, ad ragig error covariace. I VPL, the igma are caled with projectio matrix elemet. Thee elemet chage with atellite geometry. Therefore, boudig i ot a pure fuctio of ditributio parameter. Satellite geometrie alo add further complexity i boudig proce. Let u recall the VPL equatio ad mae a imple modificatio by groupig the error ource i term of their aociatio with LGF ad aircraft, S z gd, air, VPL (1) where, md, M, ad. _ ff gd, gd, air, air, re, ow, the true protectio level Equatio (1) icludig biae ca be expreed a VPL i the followig form: act S z S VPL (2) act gd, air, z where, ad deote true value of igma ad mea decribig the true error PDF for the LGF ad aircraft. The aircraft poitio error boud i deired ad itegrity i maitaied if VPL VPL hold. Expaded form of thi act iequality ca be factored a follow, z S gd, air, We deire to defie Sz gd, Sz air, gd, air, S z ad gd, air, gd, gd, air, air, (3) to eure aircraft poitio error i bouded at deired level of probability of iteret (defied by value of ). Such a boudig i eured if we require that the followig iequalitie hold for each atellite,, gd, gd, air, air, (4a, 4b) where the Iflatio Factor ( ), a commo multiplier for both LGF ad aircraft igma, i Sz gd, air, Sz gd, air, It i eaily oberved that the Iflatio Factor ( ) i a oliear fuctio of the true igma ad mea value of both LGF ad aircraft. Sice iclude all tatitical parameter, the boudig of mea value become a difficult ta becaue the LGF i uaware of aircraft tatitic prior igma broadcat. Therefore olutio are ought that do ot deped o owledge of aircraft error ditributio parameter. Two uch olutio approache are examied here for accommodatio of the poible exitece of mea value: 1) a boudig i ought without ay pecific iformatio regardig LGF ad/or aircraft error PDF, 2) a direct accommodatio of mea value i ought with Alert Limit (i.e., VAL) bufferig. A boudig method for groud error i itroduced here to accout for lac of aircraft iformatio i the boudig proce. For example, let u recall boudig iequality, expad ad rewrite it i the vector form a follow, (5) (6) where, S S S... S z gd, z air, z gd, z air, ( ) S S S... S z gd, z air, z gd, z air, ( ) S Z gd, S Z air, S Z gd,... S Z air, ( ),,, ad i the maximum umber of atellite i view (ee by the LGF) at the time of igma broadcat. The bia term ca be bouded a (with it ecod orm) [5]. Applyig thi relatio to iequality (6) reult i the followig 3

4 expreio where we impoe a more coervative requiremet o. (7) A eve more coervative, but impler boud ca be defied a: (8) For example, let u ow expad thi iequality, S z gd, air, Sz gd, air, S (9) z gd, air, ad the etablih detailed geometry idepedet mea boudig model from (9) a dicued ext. Equatio (9) i the mot geeral form of boudig becaue it aume that the mea value exit ot oly at LGF but alo at aircraft. To date, the igma geeratio ad error proceig for aircraft ragig error i ot fialized ad it i yet uclear whether biae will exit i the aircraft error. I thi aalyi we coider all poible ceario, icludig aircraft biae a decribed i the followig cae. Cae-A: Mea value exit at the both LGF ad Aircraft. (i.e., gd, ad air, ). I thi cae, the followig expreio ca be writte for the LGF broadcat ad aircraft igma repectively,, gd, gd, gd,, air, air, air (10a, 10b) It i clear that iequality (9) i preerved by (10a. 10b) if it i applied to each atellite idividually. Cae-B: Mea value exit oly at the LGF (i.e., gd, ad air, ). I thi cae, the LGF ad aircraft igma ca be expreed a, (11a 11b), gd, gd, gd, air, air, Cae-C: Mea value exit oly at the Aircraft (i.e., gd, ad air, ). I additio to the above two ub-cae, the biae may exit oly at the aircraft. I thi cae the iflatio of the groud igma i ueceary. The reult, ummarized i Table 1 for Cae-A ad Cae-B are baed o rage domai boud for the LGF ad aircraft, repectively. A how, thee rage domai boud are geometry-idepedet ad ufficiet to eure a poitio error overboud. We ca thu accommodate rage error biae o idividual atellite imply by aumig a zero mea ad iflatig igma appropriately. The cot of uch practicality ad implicity i a additioal iflatio o igma of aircraft eve i the abece of aircraft mea value. I other word, eve if biae do ot exit at the aircraft we till eed to iflate aircraft igma by i order to eure a poitio domai error boud. However a lower iflatio may be poible i ome pecial cae. For example, i both Cae A ad Cae B, i aumed a maximum umber of atellite i view; however, eed oly be aociated with the umber of atellite whoe ditributio exhibit a mea value. Such a aumptio reduce the ueceary iflatio of LGF igma if ome atellite i view do ot have biae. However, eve i thi cae, the aircraft till eed to iflate it ow igma by a factor of. Summary of Mea Boudig Model. A mea boudig model i propoed. The model coit of two ub-cae, correpodig to ceario where the biae exit at the LGF ad/or at the aircraft. It i how that thi model coervatively boud biae; therefore it i appropriate from the perpective of itegrity. The igificace of the Model i the boud doe ot require ay pecific prior owledge of aircraft error tatitic. The ummary i how i Table 1. Table 1 Summary of Mea Boudig Cae A LGF pr _ gd, gd, gd, Aircraft air, air, air, Cae B pr _ gd, gd, gd, air, air, GROUD REFLECTIO MULTIPATH Multipath i a well-ow caue of igal tracig error ot oly for GPS but alo for all type of Radio Frequecy (RF) applicatio. It i caued by udeired reflected/diffracted igal from variou urface i the ear viciity of the atea. Thee urface ca be the groud or other object uch a a buildig, a movig vehicle, or a tree. I geeral, due to the lowly varyig ature of uch evirometal factor, it i uliely that the effect of multipath error ditributio ca be quatified by experimetal mea aloe. Collectig error data ample over may day (for example, for oe year over all four eao) may ot reliably decribe all variatio of the error. There are two reao for thi: 1) ample ize are limited ad ca be highly correlated over adjacet day; 4

5 therefore, iformatio about ditributio tail caot be eaily extracted from a limited et of ample, ad 2) for practical LGF iitializatio purpoe it i impoible to rely o a log term error collectio prior to ytem iitializatio. Groud reflectio multipath alo ca be reitat to filterig. For example, atellite at the ame elevatio but with differet agular rate will have differet multipath error becaue for atellite with higher agular rate, carrier aided moothig of peudorage meauremet will for the mot part mitigate multipath. Thi pheomeo ca be explaied by the oie atteuatio performace of the moothig filter, which i a fuctio of filter badwidth. The peritecy of groud reflectio multipath error after the carrier-moothig filter ca be how eaily. For example, multipath error may be characterized by it Doppler rate a ( 4h )E co E. Where, h i the atea height, i the GPS wavelegth of 19 cm, ad E ad E are elevatio ad elevatio rate, repectively. The badwidth of the carrier moothig filter loop i / Hz or / rad/, where i the moothig filter time cotat. For LAAS, ecod, o the badwidth of the filter loop i rad/. If the filter badwidth i higher tha the multipath Doppler rate the the moothig filter i uable to atteuate the multipath error. [7]. A example of oie atteuatio veru atea height i performed by imulatio the GPS cotellatio of 24 atelite at O Hare Iteratioal Airport, Chicago. The reult of thi imulatio how that a atea height of > 5 meter i required for the moothig filter to reliably atteuate error caued by the groud reflectio multipath. For compario, the LAAS Tet Prototype atea height are approximately 2 meter, o mot groud reflectio multipath will ot atteuated by the filter. It i oted that a dicrete object reflectio multipath error ca be modeled i a imilar maer but the delay are uually loger becaue of the larger reflectio ditace betwee atea ad reflectio urface (relative to groud reflectio). Therefore, the Doppler rate for uch multipath aturally i high. Secodly, a importat characteritic of dicrete object reflectio multipath i the duratio of reflected igal. The atellite poitio ad reflectio urface orietatio may produce a pecific geometry that cotribute multipath for a loger duratio of reflected igal but at a fater Doppler rate. I cotrat, ome geometrie may cotribute horter duratio of multipath ad lower Doppler rate. I either cae, the filter moothig ca uccefully average out thee type of error. The groud reflectio multipath, ulie thee cae, cotai both diadvatage: loger duratio of multipath ad very low Doppler rate. I thi reearch, therefore, theoretical approache are emphaized oly for groud reflectio multipath error. Two cadidate ditributio model are propoed for the etablihmet ad iflatio of correctio error tadard deviatio to accout for the effect of groud reflectio multipath. I thi aalyi, a Multipath Limitig Atea (MLA) implemetatio i aumed at the LGF. The MLA coit of a top( high zeith )atea for tracig high elevatio atellite ad a bottom atea (Dipole) for tracig low elevatio atellite. The primary driver i the MLA deig i to limit groud multipath error. The elevatio cutoff agle betwee two atea i called the traitio agle ad i about 35 degree. To addre groud reflectio multipath, we begi with the followig defiitio: e i the ragig error due to groud reflectio multipath, i the amplitude of reflected igal relative to direct, i the phae of reflected igal relative to direct, d i the GPS receiver Delay Loc Loop (DLL) half correlator pacig (e.g.,. chip or m), ad hi E i the multipath delay. eglectig ecod order ad higher term i (ice for the MLA), a implified error model for MLA groud multipath wa defied i referece [2] a follow: e mi[,d] co. I the groud reflectio multipath error model, the firt term, mi[,d], i aociated with atea height, atellite poitio, ad receiver correlator pacig. Sice thee value are ow for ay give ite ad atellite poitio, they may be defied a a cotat value c mi[ hi E,d]. Dividig groud reflectio multipath by c reult i the followig ormalized groud reflectio multipath (MP) error ê e c co. I priciple, MP i a determiitic error ource. However, it i impoible to fully characterize the phyical/patial characteritic ad temporal variatio i the local RF eviromet. Therefore, it i treated a a radom error i thi aalyi. I the MP error model, there are two variable: 1) relative igal tregth,, ad 2) relative phae,. Thee two quatitie will be treated a radom variable. Cadidate model for their ditributio are decribed below. The goal i to defie a wort-cae, realitic ditributio for thee quatitie for ue i the overboudig aalyi. Relative Phae Variatio: A uiform phae ditributio ( varie radomly betwee ad ) i a widely ued model ditributio i RF applicatio. Such a model i alo coitet with the aumptio that the phae varie over widely paced day betwee ad due to mall chage i reflectio urface height ad reflectivity for a give atellite elevatio. Thu, i firt treated a a ~ U,. The uiformly ditributed radom variable: correpodig PDF i: f ( ) u( ) u( ), where u(.) i the uit tep fuctio. A ecod model for the phae, amely, a cotat value of phae (wort-cae ceario), i alo coidered. I maig uch a aumptio, the effect of phae variatio i directly elimiated ice a cotat value of phae i paed through the coie fuctio. I thi cae, for example, the relative 5

6 phae ditributio ca be expreed by f ( ) ( ). Where i the Dirac Delta fuctio. Sice we have defied cadidate PDF for relative phae variatio, ow, a ew radom variable, z co, ca ow be itroduced. With a uiform phae variatio, ~ U,, the correpodig PDF of z become, u( ) u(z z (z) ) z f. Thi PDF i ymmetric, bimodal, ad ha igularitie at z. Bi-modality ad igularity are importat characteritic that lead difficultie i boudig proce ice the poitio domai boudig coditio require multiple covolutio of uch PDF. Therefore, pecial care i tae whe uig thi o-gauia ragig error ditributio for boudig. With a cotat phae, the PDF become f z (z) z (z ). I thi cae, the relative igal tregth ( ) ditributio will be the oly effective radom ource i the fial PDF (a the PDF of a product of two idepedet radom variable, ad z ). Relative Sigal Stregth: A wort-cae cotat value, b, i aumed for reflectio amplitude,. The PDF i expreed a f ( ) ( b ). A upper limit for thi cotat value ca be defied from the performace of MLA ued i LTP a plotted i the upper trace of Figure 1 a a fuctio of atellite elevatio. O-GAUSSIA AD O-ZERO MEA MULTIPATH ERROR DISTRIBUTIOS The MP error i a product of ad z. Sice two ditributio are defied for radom variable z ad a igle ditributio i defied for, it i poible to geerate two model ditributio for MP error a product of ad z. Thee model repreet two limit cae ditributio of groud reflectio multipath error: 1) a wort-cae ymmetric zero-mea o-gauia ditributio (i.e., a bi-modal ditributio) ad 2) a mea value (i.e., a bia-offet). Additioal le coervative model for groud reflectio multipath were explored i [8], but the uefule of thee model i cotiget o their experimetal validatio. Model-1: Cotat value of ad Uiformly ditributed. With thi model, the MP error will be imply a product of a cotat value of relative igal tregth ( b, ee Figure 1 for b value) ad the radom variable, z. The reultig PDF of MP error i f (x) [u(x b) u(x b )]. Direct aalytical ê b x / b covolutio of thee PDF yield itegral that are ot tractable i cloed form. The characteritic fuctio (Fourier Traform) of the Model 1 PDF ca be readily how to be a Beel Fuctio of the firt id. However, the ivere Fourier Traform of product (equivalet to covolutio i the rage domai) of Beel Fuctio i ot readily acceible i cloed form. Furthermore, direct umerical covolutio of PDF i alo difficult ice the PDF fuctio ha igularitie at b, requirig impractically fie dicretizatio for accurate reult. To circumvet thee difficultie, we itroduce a coervative approximatio for thi PDF a: f ê (x) [ (x b) (x b )]. The aociated CDF of MP error i the, F ê (x) u(x) u(x b) u(x ). We firt coider the hypothetical limitig cae of a covolutio of a large umber of Idepedet, Idetically Ditributed (IID) ource. I thi cae, the Cetral Limit Theorem defie the eceary boudig value of igma for the actual PDF b / ad for the coervative ê model b. Ufortuately, while thi reult hold ê whe i very large, it i ot ufficiet for fiite value of ( 12). I thi cae, further iflatio of the gauia will be eceary to eure CDF overboudig. For a give atellite, we coider the MP error, ê, i ditributed accordig to the model ê ~ [ (x ) (x )]. Sice for thi ditributio, the maximum poible MP error reultig from a liear combiatio of the ource i ê max max ê.... tot i Equality (larget e max ) i the boud above occur whe all are the ame (i.e., IID ource). I thi cae, o iflatio of i required whe. ( ) ice the error will ever exceed. (which i the tot limit criterio for poitio domai tail overboudig et by the defiitio of Category 1 LAAS VPL H1 i [1]). I geeral, a iflatio/deflatio factor of /. may be ued i the rage domai to eure that the poitio domai MP error ever exceed.. For example, for tot, a iflatio factor of. i implied. However, a zero probability of exceedig. i clearly ot tot eceary. We oly require that the gauia boud the actual MP error i the CDF ee above.. A tot ufficiet iflatio factor of. i obtaied from direct covolutio of (IID) ource. Thi reult i ufficiet for all up to (ad coervative for ). We mut alo explicitly coider the effect of the additio of other cotributig gauia ource (due to diffue MP ad receiver oie). Clearly, the additio of uch error i ot a iue i the followig limitig cae: Gauia ource are very mall ( ): gau ê Model-1 domiate, o a iflatio factor of. i ufficiet. Gauia ource are very large ( ): gau ê Model-1 error are egligible by compario (Model-1 iflatio factor i irrelevat). 6

7 I cae where gauia ad Model-1 error are of comparable ize, the reult are agai evaluated by direct covolutio. I all cae, the iflatio factor of. i ee to be ufficiet to eure overboudig. Fially, the groud reflectio multipath igma ca be writte a:. bc (12) e Model-2: Cotat value of ad. With thi model, the MP error will be imply a product of a cotat wort-cae value of relative igal tregth ( b, ee upper trace of Figure 1) ad a cotat wortcae value of radom variable, z. The ormalized multipath error PDF for thi cae ca be expreed a f ê (x) (x b) ad the aociated CDF a, F ê u(x) u(x b). With thi PDF, the groud reflectio multipath error i imply a bia value uch a wort directio ad maximum value (i.e., ormalized groud reflectio multipath error b ): ê bc (13) e A previouly dicued i Model 1, we alo explicitly coider the effect of the additio of other cotributig gauia ource (due to diffue MP ad receiver oie). Clearly, the additio of uch error reult o-zero mea gauia ditributio. The exitece of a o-zero mea i the correctio error ca reult i uacceptable itegrity ri. Therefore, the mea value mut either be pecifically accommodated via igma etablihmet or it mut be how to have a egligible effect o itegrity. Both of thee two alterative are purued i followig ectio of paper. Size of bc. The lower trace of Figure 1 how the ize of product bc veru atellite elevatio for three differet atea height. The upper trace how the relative igal tregth, b. The relative igal tregth value are obtaied from [6] where they are uggeted a miimum requiremet for MLA atea for LAAS. bc (m) b (db) Elevatio (deg) Figure 1 Size of Product bc Veru Satellite Elevatio Agle for Three Differet Atea Height GAUSSIA ERROR SOURCES I LAAS, it i realized that uig data aloe i ot ufficiet to etablih broadcat igma, particularly if the data commiioig i limited to a igle day (or a few day). The baic reao for thi i that, with limited data, the ditributio tail caot be determied empirically. Simply treatig the data a gauia i coitet with certai compoet error ource, uch a diffue multipath ad receiver thermal oie. However, uch treatmet i ot ufficiet by itelf, becaue it i ow that o-gauia ource (groud reflectio multipath) are alo preet. Therefore, while the overboudig olutio hould clearly be baed o oberved data, it mut alo be augmeted by theoretical boudig of groud-reflected multipath. I the dicuio below, we briefly review the cadidate data proceig methodology of referece [9] ad how how multipath boud decribed i thi paper ca be icorporated to geerate the overall broadcat groud igma. However, data proceig approache other tha thoe decribed i [9] are equally applicable withi the empirical/theoretical fuio methodology decribed here. Serial Correlatio ad otatioarity. For a give data et (e.g., oe day worth of commiioig data), the effect of thee two propertie o the igma etimate have a reciprocal relatio a bi ize chage. The adaptive bi method uggeted i [9] avoid thi difficult tradeoff by chooig the bi ize at each elevatio (E) uch that the computed tadard deviatio (iflated for ucertaity due to the fiite ample ize withi the bi) i maximized. The empirical reult for a give referece receiver m ad elevatio E i deoted a. m,e Correlatio Betwee Referece Receiver. I the LAAS Protectio Level computatio, it i implicitly aumed that ragig error are ucorrelated acro groud referece receiver. I reality, however, it i poible that ome meaurable correlatio exit. Furthermore, eve if a egligibly mall correlatio coefficiet i computed from a fiite ample et, the tatitical ucertaity i the etimate mut alo be accouted for. Such ucertaity i leeed, a oe would aturally expect, a the ample ize ued to etimate correlatio coefficiet icreae. To accommodate the effect of correlatio, detailed empirical methodologie are preeted i [3] ad [9] to defie a factor by which c igma hould be icreaed:. m,e Seaoal Variatio. LGF multipath ragig error ca chage over time due to evirometal factor. For practical reao, however, it i clearly deirable to commiio a cadidate LGF uig a hort time pa of data (e.g., over few day). Oe potetial olutio i to ue archived LAAS Tet Prototype (LTP) data. I referece [9] a method i decribed for iflatio due to low eaoal variatio baed o: 1) a ormalizatio of log-term LTP error data with omial igma computed o a igle day, m m,e m 7

8 ad 2) igma iflatio by a factor derived from the m maximum oberved eaoal variatio betwee ormalized igma. The reult for a give referece ct c receiver m ad elevatio E i. The ue of m,e LTP data i uch a maer may be ufficiet util ufficiet ite pecific data i collected. GEERATIO OF BROADCAST SIGMA Sythei of Broadcat Sigma. Becaue either empirical error data or theoretical approache aloe are adequate, the fial broadcat will be a reult of both. I. Sigma Etimated From Data. Gauia (or early gauia) error ource ca be quatified uig data. However, a oted above, the gauia igma etimate mut iclude iflatio to accommodate the followig effect: -Sample tadard deviatio ucertaitie ( ). gd -Correlatio betwee receiver ( ). -Log-term temporal variatio ( ). Cadidate method for thee purpoe are decribed i detail i [9] ad briefly ummarized i the previou ectio. II. Sigma Geerated From Theoretical Aalye ): Overboudig gauia ditributio mut ( MP, E be defied to accommodate the o-gauia ad/or o-zero mea effect due to groud reflectio multipath. Thi ubject wa detailed earlier i thi paper. (Additioal related material may alo be foud i referece [8].) I thi ectio we apply thee methodologie to LTP empirical data. I. Sigma Etimated From Data. For a arbitrarily elected atellite, divergece-free code-miucarrier error trace for three LTP referece receiver (RR) are plotted i Figure 2. Alo how i the plot are the igma trace (olid curve) computed uig the adaptive bi method of [9]. Thee trace implicitly accout for data otatioarity ad iflatio due to tatitical ucertaity (). It i oberved that the igma reult for RR1 ad RR2 are geerally larger tha that for RR3. We hould alo ote that the wort igma value are obtaied ear the traitio elevatio agle (vertical dahed lie) betwee the High Zeith Atea (which trac high elevatio atellite degree) ad the MLA (a dipole atea that trac low elevatio atellite, degree). m E Sigma (m) Error ad Sigma (m) Sample Idex Figure 2 Adaptive Bi Sigma Reult for Each RR Figure 3 Compoite Empirical Sigma Reult II. Sigma Geerated From Theoretical Aalye Cadidate Broadcat Sigma for Model-1. Here we ytheize the empirical (compoite) gauia igma reult from Figure 3 with the gauia overboud groud reflectio multipath error igma for Model 1 from Equatio (12). The idividual reult are plotted i the upper ad lower trace of Figure 4 repectively. A wa how earlier, the gauia (empirical) ad gauia-multipath-overboud igma may be combied via root-um-quare to geerate the fial reult. Multipath (m) Sigma (m) MLA HZA MLA Cut-off Agle Sample Idex MP comp comp Cut-off Agle Sample Idex Figure 4 Multipath Model 1 RR1 RR2 RR3 8

9 Sigma (m) Sigma (m) Multipath (m) Sigma (m) gd Sample Idex Figure 5 Combied Empirical ad Model 1 Reult comp Sample Idex Figure 6 Multipath Model 2 gd Sample Idex Figure 7 Combied Empirical ad Model 2 Reult Cadidate Broadcat Sigma for Model-2. Here the groud reflectio multipath error i treated a wort-cae bia (Model-2). I Figure 6, the upper trace i agai the iflated empirical reult from Figure 3, ad the lower trace repreet the bia cotributio to the LGF boudig formula i Table 1. Thee two lower trace correpod to differig umber of atellite i view at the LGF (the miimum,, ad the maximum, ). The empirical ad theoretical compoet plotted i Figure 6 are combied uig the LGF mea boudig formula i Table 1 to geerate the overall boudig igma i Figure 7. B3 C3 It i clear that the reult here are more coervative tha the Model 1 cae. Thi i a expected reult becaue groud reflectio multipath i treated a a wort cae bia, with wort cae phae, i Model 2, wherea i Model 1 credit i tae for radome i the phae. LAAS Availability Cocer. I thee example/reult, avigatio itegrity i eured for the etablihed. However, the igma reult exceed C3 gd ad B3 pecificatio. Thi mea that LAAS availability may be affected. Sice both the empirical ad theoretical cotributio are relatively imilar i magitude, reductio i either compoet will have beeficial effect with regard to availability. I thi regard, wor i ogoig to refie both the empirical procee of [9] ad the boud developed i thi paper to achieve tighter igma boud. egligibility of Mea Effect withi the VAL. The exitece of a o-zero mea value i the correctio error ca potetially reult i uacceptable itegrity ri. Therefore, either the mea value mut be pecifically accouted for (e.g., uig oe the method defied above) or it mut be how to have a egligible effect o itegrity. The ecod alterative wa previouly decribed i referece [8] ad i briefly reviewed here becaue of it direct relevace to the ubject at had. Give the computed VPL i defied i Equatio (1), ad the actual VPL give the exitece of mea error mea it i how i referece [8] that a geometry-idepedet upper boud for the actual VPL may be defied a follow: VPL 1 VPL. (14) act comp max. (15) gd, gd, I a practical iterpretatio of requiremet it i poible that the 10 m VAL requiremet i actually coervative for LAAS Category I operatio. For example, while a coervative threhold VAL of 10 m i ued for LAAS, a operatioal value of approximately 12 m may actually be ufficiet to eure itegrity. I thi regard, it i oly eceary to eure that the calig term 1 i /. or lower. Uig the mallet value of for Category I ( = for H 1 ) ad the larget umber md _ ff of atellite ( ), the maximum acceptable value of max i.. The reult for other value of are how i Figure 8. Recall that the umber of traced atellite i ow to the LGF, o the leat coervative value of ca alway be ued. Thu larger mea value ca be accommodated a the umber of atellite decreae. ote that thi i a ufficiet coditio to eure that the mea i egligible. Implemetatio may be achieved by verifyig that the rage-domai mea i mall eough i compario to the tadard deviatio a eceary to eure that the ufficiet coditio i atified. 9

10 Sice the coditio defied above i ufficiet but ot trictly eceary (i particular, it ca be coervative for good atellite geometrie), alterative are poible. For example, we may coider at the referece tatio (at ay give time) all ubet geometrie that the aircraft may ue to compute VPL. Firt aume the larget pecificatio-compliat value of tadard deviatio for air ad reidual (tropopheric ad ioopheric ucertaity) error, ad for all cadidate ubet geometrie compute both VPL ad VPL. The LGF H H verifie that i all cae where thee VPL are le tha 10 m the additio of the uaccouted for poitio domai mea term doe ot caue the actual boudig VPL to exceed 12 m. ote that by itelf, thi i ot truly a ufficiet coditio becaue the air ad reidual tadard deviatio value ued at the aircraft may be maller tha the maximum value aumed o the groud. If they are maller at the aircraft, it i poible that geometrie exit i which both VPL computed o the groud exceed 10 m (ad thu are ot verified) but at the aircraft are lightly le tha 10 m. To protect agait uch occurrece, the groud tatio would alo have to coider a rage of tadard deviatio value for air ad reidual below the larget value aumed above. Such a approach would probably be computatioally iteive, but i poible i priciple. Figure 8 Maximum Allowable Ratio of Mea/Sigma COCLUSIOS I thi paper, two groud reflectio multipath error ditributio model were ivetigated for overboudig of LGF broadcat correctio error. Model-1 i a wort cae, zero-mea, ymmetric bi-modal ditributio, ad Model-2 i a wort-cae bia. Model-2 i the mot coervative iterpretatio of groud reflectio multipath ad it doe ot require ay validatio with data give that pecified atea/receiver deig performace i achieved. Thi model i relevat to other ytematic biae a well. For both model, ufficiet mathematical boud o broadcat rage domai pr_gd were defied to eure error overboudig for the aircraft i the poitio domai. The coditio uder which groud bia error may be igored i LAAS were alo briefly reexamied ACKOWLEDGEMETS The author gratefully acowledge the Federal Aviatio Admiitratio for upportig thi reearch. However, the view expreed i thi paper belog to the author aloe ad do ot ecearily repreet the poitio of ay other orgaizatio or pero. REFERECES 1. RTCA (SC-159/WG-4), "Miimum Aviatio Sytem Performace Stadard for the Local Area Augmetatio Sytem (LAAS), RTCA/DO-245, RTCA Ic., Wahigto DC, 28 September Braach, M., "Multipath Effect," Global Poitioig Sytem: Theory ad Applicatio, Volume 1, AIAA Progre i Aeroautic ad Atroautic Vol. 163, Perva, B., ad Sayim, I., "Sigma Iflatio for the Local Area Augmetatio of GPS," IEEE Traactio o Aeropace ad Electroic Sytem, Volume 37, umber 4, October Specificatio: Performace Type Oe Local Area Augmetatio Sytem Groud Facility, U.S. Federal Aviatio Admiitratio, Wahigto, D.C., FAA-E-2937, 21 September Golub, H., Gee, ad Loa, F. V. Charle, Matrix Computatio, 2 d, Editio, The Joh Hopi Uiverity Pre, Marylad, Braf, R., Decriptio of the FAA Local Area Augmetatio Sytem of (LAAS), avigatio, Joural of The Ititute of The avigatio, Vol. 44, o. 4, Witer Ege, P., Local Area Augmetatio of GPS for the Preciio Approach of Aircraft, Proceedig of the IEEE, Vol. 87, o.1, Jauary Perva, P., Pulle, S., ad Sayim, I., "Sigma Etimatio, Iflatio ad Moitorig i the LAAS Groud Sytem," Proceedig of the 13 th Iteratioal Meetig of the Satellite Diviio of the Ititute of avigatio (IO GPS-2000), Salt Lae City, UT, September Sayim, I., Perva, B., Pulle, P., ad, Ege, P., "Experimetal ad Theoretical Reult o LAAS Sigma Overboud," Proceedig of the 15 th Iteratioal Meetig of the Satellite Diviio of the Ititute of avigatio (IO GPS-2002), Portlad, OR, September