Experimental and Theoretical Results on the LAAS Sigma Overbound

Transcription

1 xperiental and Theoretical Reult on the LAAS Siga Overbound Irfan Sayi *, Bori Pervan *, Sa Pullen and Per nge * Illinoi Intitute of Technology, Chicago, IL Stanford Univerity, Stanford, CA BIOGRAPHY Irfan Sayi received a B.S. degree fro Marara Univerity (1990), Itanbul, Turkey and M fro Illinoi Intitute of Technology (1996), in Mechanical and Aeropace ngineering. Currently, he i a Ph.D. candidate at Illinoi Intitute of Technology and working on Local Area Augentation of GPS navigation integrity for aircraft preciion approach and landing. Bori Pervan received a B.S. fro the Univerity of Notre Dae (1986), M.S. fro the California Intitute of Technology (1987), and Ph.D. fro Stanford Univerity (1996), all in Aeropace ngineering. Fro 1987 to 1990, he wa a Syte ngineer at Hughe Space and Counication Group. Dr. Pervan wa a Reearch Aociate at Stanford fro 1996 to 1998, erving a proect leader for GPS Local Area Augentation Syte (LAAS) reearch and developent. He wa the 1996 recipient of the RTCA Willia. Jackon Award and the 1999 M. Barry Carlton Award fro the I Aeropace and lectronic Syte Society. Currently, Dr. Pervan i Aitant Profeor of Mechanical and Aeropace ngineering at the Illinoi Intitute of Technology in Chicago. ABSTRACT The Local Area Augentation Syte (LAAS) i the differential atellite navigation architecture tandard for civil aircraft preciion approach and landing. While the yte proie great practical benefit, a nuber of key technical challenge have been encountered in the definition of the architecture. Perhap chief aong thee ha been the need to enure copliance with tringent requireent for navigation integrity. In thi context, thi paper define a practical way to decribe and quantitatively etablih LAAS correction error broadcat iga for final integrity rik aeent at aircraft. The ethod involve a yntheized olution of both data-baed analyi for gauian (or nearly gauian) error ource and a theoretical bound for non-gauian error ource uch a ground reflection ultipath. In addition, thi paper cover the largely unreolved iue (binning, etc.) concerning iga quantification by direct ue of data. INTRODUCTION Local Area Augentation Syte (LAAS) integrity rik i quantified at the aircraft via the coputation of Vertical and Lateral Protection Level (tered VPL and LPL, repectively). The precribed algorith for the generation of thee protection level iplicitly aue zero-ean, norally ditributed fault-free error ditribution for the broadcat peudorange correction. While the aued error odel ay be conitent with the effect of theral noie and diffue ultipath, it i undertood that reaining error uch a ground reflection ultipath and yteatic reference receiver/antenna error are not necearily reliably odeled by zero-ean noral ditribution. Therefore, to enure that the coputed value of VPL and LPL at the aircraft are eaningful and that integrity rik i properly anaged, pecial care ut be taken by the LAAS Ground Facility (LGF) in the etablihent of the broadcat peudorange correction error tandard deviation ( ). In thi paper, we addre aor reaining unreolved iue concerning the etablihent of. Thee include the definition of a ufficient proce by which epirical error data ay be proceed to enure patially tationarity of error, quantification and copenation for the effect of eaonal variation of error, and a ethodology to account for potential non- Gauian error ource. For norally ditributed error uch a receiver theral noie and diffue ultipath, tandard deviation can be etiated uing experiental data alone. In thi cae, however, it i till neceary to account for the additional integrity rik incurred by tatitical uncertainty (due to finite aple ize) in the knowledge of reference receiver

2 error tandard deviation and error correlation between ultiple reference receiver. In thi regard, a detailed ethodology ha been developed for the definition of iniu acceptable inflation paraeter for the aple tandard deviation [1]. (The inflation paraeter are function of the nuber of aple available and the aple correlation coefficient.) However, in order for uch an epirical proce to be applied, it i firt neceary to define a proper ethod to collect data into patial bin prior to iga etiation. While large bin ize are deired to axiize aple ize (to liit required inflation factor), bin ize i ultiately contrained by the need for patial tationarity of all data within the bin (i.e., all error data within a bin ut have the ae underlying ditribution). The quantitative reolution of thi critical tradeoff, which i conceptually illutrated in Figure-1, i a aor ubect of the work decribed in thi paper. the tail (which are of greatet interet in LAAS). Therefore, theoretical approache are ephaized in our work to incorporate ground reflection ultipath effect into. The ultiate goal of thi work i to define a ufficient ethodology for the etablihent of the LAAS broadcat. Neither theoretical approache nor epirical error data alone are adequate in thi regard. The final broadcat peudorange iga will be a reult of both eleent. In thi paper, we introduce a practical way to yntheize the epirical and theoretical eleent to quantitatively etablih for LAAS. RANGING RROR CHARACTRISTICS In general, the peudorange error ha three iportant characteritic. Thee are: Required Inflation Factor on pr gnd all Size of Bin Figure-1 Sketch of rror Variation within Bin large Spatial Variation of rror Ditribution Within Bin I -Repeatability II -Serial Correlation III -Nontationarity A iple illutration of data howing thee characteritic i ketched in Figure-2. In thi paper, we ephaize the quantification and accoodation of thee characteritic in LAAS broadcat iga etablihent rather than their caue and itigation. The effect of eaonal variation in peudorange correction error (in particular ultipath) ut alo be accounted for in the broadcat. However, it i clearly ipractical to collect a full-year pan of data (prior to coiioning) for each LGF to account for uch effect. Therefore, archived error data collected at the LAAS Tet Prototype (LTP) facility at the W. J. Hughe FAA Technical Center i ued to define a baeline LGF odel for eaonal variation in. The oberved LTP teporal variation i ued to define a coon tandard inflation factor for ue in the etablihent of in future LGF intallation until ufficient ite-pecific data i collected. Becaue ground ultipath error i not necearily norally ditributed, epirically coputed (and inflated) value of are not ufficient to guarantee overbounding of the total LGF ranging error. Furtherore, it i ipoible to rely on epirically contructed ditribution (e.g., error data hitogra) alone to define the nature of the underlying error ditribution becaue little or no epirical data will exit in DAY # N DAY # 2 DAY # 1 e e e ak ak ak III III II ax II ax II Figure-2 Sketch of rror Characteritic ax I I ak ak ak

3 I- Repeatability (Day-to-Day Correlation): It i well known that the ranging error, in general, i repeatable (or correlated) day-to-day. The repeatability characteritic i ainly caued by ultipath error and it can be eaily oberved with a tationary (fixed) antenna when the environental condition are contant. Becaue of the repeatability property, calibration of error i poible in principle. For exaple, ue of a prior day error data to correct error on the current day ay reduce ranging error ize. However, the fundaental proble regarding tatitical decription of the (reidual) error ditribution and iga bounding reain unolved. Another raification of error reparability i that iga cannot be eaily etablihed by eneble of data over any day. There are two baic reaon for thi: 1) data enebled over any day will exhibit ignificant correlation effect between day (i.e., aple are not independent), 2) the iga etablihent proce ut be reaonably hort for practical LGF initialization. Therefore, the approach taken in thi work i to generate iga fro data collected over a ingle (coiioning) day and then inflate the reult to account for long-ter eaonal variation of the error oberved at the LTP ite (where everal year worth of archived data are available). II- Serial Correlation: One of the ot ignificant characteritic of the oberved ranging error i erial correlation between recorded aple of data. Thi correlation effectively liit the nuber of independent aple that can be aued in the calculation of inflation factor that account for tatitical uncertainty in the etiated iga. In general, the nuber of independent aple for coputing inflation factor will be a function of the ize of the bin and the correlation tie of the data within it. III- Nontationarity: A nontationary proce i defined a proce in which tatitical paraeter of ditribution do not tay contant in tie. levation dependency of ultipath delay and GPS antenna gain pattern are a coon ource of nontationarity in oberved ranging error. With the antenna ued for LAAS, thee effect are reduced, but they are till preent and ignificant near the cut-off angle between the and the (High Zenith Antenna). In addition, nontationarity ay alo exit due to aziuthal variation (e.g., dicrete reflector or diffractor) in the antenna. XPANDING BIN (B) CONCPT The xpanding Bin (B) ethod i a new approach for data-baed iga etablihent that iultaneouly anage the effect of nontationarity and erial correlation of oberved error data. Traditional approache toward data-baed iga etablihent rely on fixed bin width, which are elected a priori with intent to both iniize the effect of ixing of error data derived fro ditribution and axiize the nuber of aple within each bin. In practice, the appropriatene of prior bin ize election i difficult to quantitatively validate and i therefore often udged via ad hoc inpection of the data. In contrat, the B ethod i an adaptive chee which autoatically elect the bin width at a given tie (or elevation) for each atellite eparately. In practice, the reult i achieved by conidering not only a ingle bin width but all poible bin width at the given tie/elevation. The inherent tradeoff in bin ize election reulting fro the iultaneou preence of nontationary and erial correlation i gracefully controlled by electing the wort-cae inflated iga a repreentative of given tie/elevation ( ). The B ethod i ipleented eparately for each atellite by the following ean: Firt, a core bin BI i defined to provide a iniu allowable independent aple ize for iga etiation. Thi i perfored by uing the entire data et fro the atellite pa to copute the error correlation tie, which i in turn tranlated into the tie between independent aple. With thi reult the core bin ize BI i et. Second, a axiu-ize bin BM i defined in order to provide a wide range for etiating any candidate iga. In principle, BM can be elected to include the entire data et. A iple ketch of inner (core) and outer bin, BI and BM repectively, i hown in Figure-3 for an arbitrary tie/elevation. The atheatical decription of the proce for a repreentative iga at tie/elevation can be expreed a: { } = ax (1), where, i the bin width index ranging fro BI to BM. After iga i elected for a given tie/elevation uing equation (1), the entire proce repeated at next data epoch until the end of the data et i reached.

4 Bin Center ( ) Figure-3 Sketch of B Concept COMPUTATION OF SIGMA Given the conceptual introduction of B ethod in previou ection, the echanization of the proce can now be decribed in greater detail. To aid in thi decription, a flow chart of iga coputation proce i hown in Figure 4. flow chart of Figure-4. Firt the aple autocorrelation function (ACF) for the elected data et i coputed, and a reulting correlation tie etiate i extracted fro the ACF by auing a firt order Gau-Markov proce [5]. Siultaneouly, the aple variance are coputed fro the elected data et. The nuber of independent aple in the binned data i coputed by iply dividing the nuber of recorded aple in the elected data et by twice the etiated correlation tie. Baed on the nuber of available independent aple, an inflation factor i generated to account for tatitical uncertainty the coputed aple tandard deviation. The detail of the coputation of thee inflation factor fro an integrity rik perpective are provided in reference [1]. In thi paper, however, inflation factor are directly generated by the ue of a 99.9% confidence interval. The aociated in factor i plotted a a function of the nuber of independent aple in Figure-5. It i clear that a all nuber of independent aple require high inflation factor on iga to cover etiation uncertainty. For each bin ize, the coputed aple tandard deviation i inflated and the reult i tored. When all candidate bin ize are proceed, the upper bound inflated iga i elected a follow:, { } = ax{ α (n ) ˆ } = ax (2),,,,,,, where α n ) i the inflation factor given that,(,, n,, independent aple are available for reference receiver, tie epoch, and bin width index ; ˆ, i the coputed tandard deviation of data at epoch and bin width index. DATA DATA BIN (,) n, τ, ACF α, ˆ,,1,, ax α Figure-4 Flow Chart for Siga Coputation ax { }, n Figure-5 Inflation for Uncertainty on Coputed Siga a a Function of Independent Saple Size A illutrated previouly (Figure-3), the coputation tart fro a predefined range of data ized with BI and follow by continuouly adding new eaureent in both tie direction of it. ach new increaed data et i paed through the coputation algorith hown in the BNCHMARK TST OF B MTHOD Two iga reult, obtained by two different ethod, are hown in Figure-6: 1) the iga hitory generated by the B ethod, and 2) the inflated iga generated by ue of

5 entire data et. It can be oberved, by viual inpection of Figure-6, that the B-generated iga appear to be a faithful repreentation of the variation of the error data itelf (which i alo hown on the plot), becaue iga v. tie profile i haped not only by low variation of error (erial correlation effect) but alo by the ize of aple tandard deviation (nontationarity) of error data. The flat line i the iga obtained by ue of entire data. Thi iga indicate that if we ix all the error ditribution within a ingle bin (without regard to nontationarity effect) the iga will not repreent error variation properly. rror & Siga Siga Of ntire Data Siga of B Figure-6 Siga Generation for Nontationary Proce Coparion of two iga in CDF (Cuulative Ditribution Function) ene are hown in Figure-7. In thi figure, the perforance of both ethod i copared againt a tandard noral ditribution by noralizing the actual error data by each of the two iga curve and then plotting their correponding CDF. overbound the error data noralized by the inflated iga of the entire data et. The baic reaon for thi fact i that the latter approach doe not account for nontationarity (i.e., ixing of data fro different error ditribution during the atellite pa). CORRLATION BTWN RCIVRS In the LAAS VPL coputation, it i iplicitly aued that ranging error are uncorrelated acro ground receiver. In fact, the exitence of any uch correlation i not trictly conitent with the VPL equation ince the for an individual reference receiver i alway divided by the nuber of receiver to account for the averaging of uncorrelated receiver eaureent. In reality, however, it i poible that oe eaurable correlation exit. Furtherore, even if a negligibly all correlation coefficient i coputed fro a finite aple et, the tatitical uncertainty in the etiate ut alo be accounted for. Such uncertainty i leened, a one would naturally expect, a the aple ize ued to etiate correlation coefficient increae. To accoodate the effect of correlation, we aue that the ground error tandard deviation for any given reference receiver i expreed by equation (2). The effect of correlation between receiver when averaging M reference receiver error can be odeled a an effective increae in, a follow: = β (3) c, where,, F(x) pirical CDF Data Noralized with Inflated Siga Of ntire Data β = 1+ M i 1 i = 1 ρ i ρ + ρ + ρ < ρ (4) Data noralized with Inflated iga of B Ν ( 0, 1) x Figure-7 Perforance Coparion of New Method It i clear fro the figure that for the B-noralized cae a ignificant argin exit with repect to an overbounding tandard noral ditribution. In contrat, the tandard noral ditribution i inufficient to where the total poitive correlation between receiver i + defined by, ρ = ( ρi ) M i 1 i = correlation by ρ = ( ρi < ) M i 1 i = 0, and the total negative 0. Note that any inflation of iga due to negative correlation i irrelevant ince the initial (iplicit)

6 auption of uncorrelated receiver error will already reult in over inflation in thi cae. The tatitical relationhip between eaured (aple) correlation ( r ) and the true correlation coefficient correlation ( ρ ) i given in [3]: 1 ( ) ρ ( ) ( ) r ~ ln, ln (5) 1 Ν 2 1 r 2 1 ρ n 3 where n i nuber of independent aple ued to copute r. The 99.9% confidence contour for ρ are plotted in Figure-8 veru r and n. r n Figure-8 Contour of 99.9% Confidence Correlation a a Function of Meaured Correlation and Independent Saple TMPORAL VARIATION OF RROR In thi ection, the proce of quantification and accoodation of teporal variation i detailed uing a repreentative exaple. The procedure i baed on the relative axiu variation between average iga acro eaon. The goal, a entioned earlier, i to etablih iga fro a liited duration of LGF coiioning data (one day) and cale by a factor ( γ ), derived fro long-ter archived LTP data, to account for teporal variation: = γ (6) ct, ρ c The long-ter teporal variation factor ( γ ) i obtained uing a one-year pan of LTP data, with four eaonal apling of two week per eaon,. ach day of archived LTP data conit of error eaureent fro three LAAS Integrated Multipath Liiting Antenna. A atellite (PRN#2, arbitrarily choen in thi exaple), for which we previouly etablihed iga value uing the B ethod on the initial day worth of data, i ued to generate teporal variation effect. All of the ranging error on the ubequent day of data for thi atellite are firt noralized by initial day B iga value. (Thi i done o that the teporal variation relative to the initial day B reult can be directly oberved.) Then the tandard deviation of each noralized error data et coputed. The noralized error tandard deviation are then grouped into four averaged eaonal apling. The reaon for eaonal grouping and averaging iga eaonally i that we are intereted in characterizing the effect of long-ter, lowly-varying effect due to the weather-related environental change. Finally, the average tandard deviation of each eaon i orted fro iniu to axiu, and the ratio between axiu and iniu average iga i elected a teporal variation cale factor ( γ ) a hown: ax(,eaon ) γ =, (7) in( ),eaon where, =, and,eaon [ ],w int er,pring,fall,uer,w int er,, pring,, fall, and, uer are average of noralized error tandard deviation for the winter, pring, fall, and uer eaon, repectively. The eaonal variation inflation factor reult for the particular atellite conidered here (PRN#2) i γ 1 = It i iportant to note that in thi work only a ingle atellite/ingle receiver cae i conidered o far. A ore detailed analyi (for ultiple atellite and receiver) ut be conducted to define a generalized eaonal variation inflation factor uitable for ue in the LGF iga etablihent proce. GROUND RFLCTION MULTIPATH Due to the lowly-varying nature of ground-reflection ultipath, it i unlikely that it effect on iga can be quantified by experiental ean alone. Therefore, in pat work [5], a nuber of candidate theoretical approache were defined for the etablihent and inflation of iga in thi regard. A relatively conervative exaple odel, Uniforly Ditributed Relative Phae and Contant Reflection Strength, i elected fro thee candidate a a repreentative exaple. For thi odel, it i hown in reference [5] that ground reflection ultipath can be bounded by a zero ean gauian ditribution with tandard deviation ( D / U) in[ hin,d] (8) MP 2

7 Where, D / U i the aplitude of reflected ignal relative to direct, h i antenna height, i elevation angle, d i the half correlator pacing (e.g., 0.05 chip = 15). In Figure-9, thi exaple odel, equation (8), i plotted in the lower trace by uing the value of relative ignal trength [6] given in the upper plot and LTP receiver characteritic. D/ U h = Becaue neither theoretical approache nor epirical error data alone are adequate, the final broadcat peudorange iga will be a reult of both eleent. The candidate iga etablihent proce i defined here: 1- Ue the B ethod to generate the axiu obtainable iga value fro data. (The B approach iplicitly incorporate nontationarity effect and inflation for aple tandard deviation etiation uncertainty.), { } = ax{ α (n ) ˆ } = ax (9),,,,,,, MP Cut-Off Angle 2-Account for correlation effect between reference receiver, = β (10) c,, levation (degree) Figure-9 Multipath Siga v. Satellite levation Angle While the ground reflection ultipath reulting fro the elected odel i not gauian, it i hown in reference [5] that if the bounding value of ultipath iga in equation (8) i cobined via root-u-quare with iga fro other gauian error ource, gauian overbounding i preerved. Thi i an iportant iue relevant to the iga ynthei dicuion given in the next ection. SYNTHSIS OF BROADCAST SIGMA To accoodate all contributing, independent error ource, etablihent of broadcat iga ut include: 1-Siga tiated Fro Data: To accoodate gauian (or nearly gauian) error ource, A very liited (one day) data analyi ut be done for every new intallation and iga ut include: - Inflation due to aple tandard deviation uncertaintie ( α ) - Accoodate the effect of nontationarity ( ixing ) of error data within bin. - Inflation due to correlation between receiver ( β ). Seaonal Data Analyi: - Inflation due to long-ter teporal variation ( γ ). 2-Theory/Analyi: To accoodate ultipath (nongauian) error ource, A theoretical odel ut be defined (e.g., ground reflection ultipath odel fro reference [5]) and cobined (RSS) with etiated iga fro data. 3-Account for long-ter teporal (eaonal) error variation, = γ (11) ct, c 4- Generate the copoite iga fro data, M 1 ct ( ), 2 cop = M, = (12) M 5-Cobine with the theoretical ultipath iga bound, cop 2 2 ( ) + = M, MP (13) AN ILLUSTRATIV XAMPL RSULT The following data pecification are ued in the exaple analyi that follow: Site: FAATC/LAAS Tet Prototype BI: 1000 Recorded Saple BM: 5000 Recorded Saple Conf. Interval for Inflation: 99.9% Nuber of Reference Receiver: 3 Satellite: PRN#2 levation Mak: 5 degree Cut-Off Angle of : 35 deg C/N O Mak: 40 db-hz. Soothing Tie Contant: 100 ec. Raw Data Saple Rate: 2 Hz Tie of Data Record: February 2000 Firt, iga are etiated by direct ue of data with the B ethod. ach iga trace i then plotted (olid

8 curve) in Figure-10 for RR1 (reference receiver #1), RR2, and RR3, fro top to botto repectively. For coparion, the actual error data i alo plotted. It i oberved that the iga trace for RR1 and RR2 are generally larger than that for RR3. We hould alo note that the wort iga value are obtained near by the cutoff angle (vertical dahed line) of and alo that a ignificant contribution of error to (Dipole) iga i clearly oberved. that each reference receiver noralized error i conervatively overbounded by a tandard noral CDF. F(x) pirical CDF Ν( 0,1 ) RR2 1 RR1 RR1 RR3 rror and Siga () 2 3 Saple Index Figure-10 Siga of B Method for each RR RR2 RR3 x Figure-12 CDF Overbound of B iga Correlation effect between receiver are applied to iga of B and then plotted in Figure-13. For thi exaple, the copoite iga of three reference receiver i inflated a a function of the nuber independent aple within the data, a defined in equation (4). The upper curve how copoite iga that i generated after each reference receiver iga i independently inflated for correlation uing the value lited in Table-1. The iga trace of Figure-10 are plotted in the upper plot of Figure-11 for ore direct coparion of relative perforance between reference receiver. The botto plot within the figure how the copoite (average) iga of three RR. The copoite broadcat iga of three-reference receiver i about 10 c except near cutoff angle region. Siga RR1 Reference Receiver tiated Siga RR2 RR3 r / ρ = 1 = 2 = 3 β = / / = / / = / / Table-1 Correlation Value In Table-1, the eaured ( r ) and 99.9% confidence ( ρ ) value of correlation between reference receiver are given. Thee value are converted to correlation inflation factor ( β ) lited in the lat colun. Siga Copoite Siga Siga Copoite Siga Fro B Method Saple Index Figure-11 B Siga (Individual and Copoite) The B-iga-noralized error ditribution (CDF) for the three receiver are plotted in Figure-12. It i clear Saple Index Figure-13 B Siga Inflated by Correlation ffect In the Figure-14, the effect of teporal variation i applied to iga (after correlation effect have been applied). For the tie being, we have available only the

9 teporal variation inflation factor reult for RR1. Therefore, we aue that the teporal inflation factor for the other two receiver are the ae ( γ1 = γ 2 = γ 3 = ). The upper curve i the final iga trace obtained fro data-baed etiation. Figure-16 Final Siga Reult v. Specification In thi exaple, it i clear that etablihed exceed C3 and B3 pecification. However, it i poible that ay be reduced by one or ore of the following ean: 1. Refined calibration of code-carrier phae center offet for the LGF data ued in thi work. Siga Figure-14 Data Baed Siga (Siga of B Inflated by Correlation and Teporal Variation ffect) In Figure-15, the overbounding ground reflection ultipath iga, plotted in Figure-9, i cobined with data-baed iga of Figure-14. Siga Siga Figure-15 Final Siga (Data Baed Siga cobined with Multipath Siga) The final copoite broadcat iga reult,, i plotted in Figure-16 and copared with LGF C3 and B3 broadcat iga pecification [2]. Saple Index Saple Index B3 C3 Saple Index 2. Cut-off angle (tranition angle between and Dipole) can be varied to oe degree to reduce iga peak. 3. The iga perforance of one or two receiver ay be acceptable without the aid of the reaining receiver, which ay have higher iga. (For thi exaple, RR3 ha ignificantly lower error than the other two receiver near the peak at 35 deg). Therefore, RR aking at certain elevation where iga i large ay prevent unacceptable copoite iga reult. 4. It ay be poible to ue aller inflation factor baed on the entire data et, given that data noralized by B iga i overbounded by tandard noral CDF. An exaple of the application of ite 4 i dicued in ore detail below. The approach i otivated by the oberved reult that error data noralized by B-iga which are not inflated for tatitical uncertainty are neverthele till overbounded by a tandard noral ditribution. In Figure-17, the previou final iga reult are odified by caling uninflated B-iga by the (tatitical uncertainty) inflation factor derived fro the nuber of independent aple within the entire et (atellite pa) of data. Siga for thi cae i defined a,,all,all { ˆ } = α ( n )ax (14),, where α ( n, all ) i the inflation factor given that independent aple ize n, all i available for a bin width correponding to the entire data et. Obviouly, the entire data et ha ore independent aple than any ubet bin, o the effect of inflation due to tatitical uncertainty will be uch aller in thi cae. Therefore, the final iga reult will be reduced even if the other effect (correlation and teporal variation) reain unchanged. It i clear that the new reult are ignificantly iproved uch that the C3 pecification i nearly atified, with exception at the iga peak near the cut-off angle region. However, becaue the ipetu for thi odified approach to inflation i derived priarily fro epirical

10 obervation, rather than fro theoretical arguent, additional work i required to validate the applicability of thee reult. Siga B3 ACKNOWLDGMNTS At the FAA Technical Center, the aitance and contructive coent of John Warburton i greatly appreciated. The author gratefully acknowledge the Federal Aviation Adinitration for upporting thi reearch. However, the view expreed in thi paper belong to the author alone and do not necearily repreent the poition of any other organization or peron. RFRNCS Figure-17 Final Siga Reult v. Specification CONCLUSIONS C3 Saple Index In thi paper, a new adaptive bin election approach, known a the xpanding Bin (B) Method, i defined and applied to nontationary and autocorrelated ranging error data i propoed for etablihent of LGF broadcat iga. The reult in thi paper how that: Uing the B ethod, the upper-bound iga trace (a a function of tie/elevation during the atellite pa) i directly extracted fro the available data. The B-ethod iplicitly account for nontationarity and inflation for tatitical uncertainty. rror data noralized by B-iga are conervatively overbounded by a tandard noral CDF. Abrupt variation in iga acro bin boundarie, which exit in the fixed-bin approache, are naturally eliinated uing the B approach. An analyi of long-ter (eaonal) error variation i perfored. The axiu, relative, noralized teporal variation oberved uing archived LTP data wa elected a a iga cale factor. The approach i uggeted for ue a a baeline for other LGF ite until they have collected ufficient data. [1] B. Pervan, and I. Sayi, "Siga Inflation for the Local Area Augentation of GPS," I Tranaction on Aeropace and lectronic Syte, Volue 37, Nuber 4, October 2001 [2] RTCA (SC-159/WG-4), "Miniu Aviation Syte Perforance Standard for the Local Area Augentation Syte (LAAS), RTCA/DO-245, RTCA Inc., Wahington DC, 28 Septeber [3] Juliu S. Bendat and Allan G. Pierol, Rando Data, Analyi and Meaureent Procedure, 2 nd, dition, John Willey & Son, 1986 [4] Arthur Gelb, Applied Optial tiation, The M.I.T. Pre, 1999 [5] B. Pervan, S. Pullen, and I. Sayi, "Siga tiation, Inflation and Monitoring in the LAAS Ground Syte," Proceeding of the 13 th International Meeting of the Satellite Diviion of the Intitute of Navigation (ION GPS-2000), Salt Lake City, UT, Septeber 2000 [6] R. Braff, Decription of the FAA Local Area Augentation Syte (LAAS), Journal of the Intitute of Navigation, Vol. 44, No. 4, winter In addition, the effect of correlation between receiver and prior theoretical reult for (non-gauian) ground reflection ultipath error are directly addreed. Finally, a detailed candidate ethodology for LGF iga etablihent i propoed in thi paper.