Availability Benefit of Future Dual Frequency GPS Avionics under Strong Ionospheric Scintillation

Transcription

1 Avalablty Beneft of Future Dual Frequency GPS Avoncs under Strong Ionospherc Scntllaton Jwon Seo, Todd Walter, and Per Enge Stanford Unversty BIOGRAPHY Jwon Seo s a Ph.D. canddate n aeronautcs and astronautcs at Stanford Unversty. He receved hs B.S. n mechancal engneerng (dvson of aerospace engneerng) from KAIST (Korea Advanced Insttute of Scence and Technology) and receved M.S. degrees n aeronautcs/astronautcs and electrcal engneerng from Stanford. Hs current research focuses on arcraft navgaton usng GPS and WAAS under severe onospherc scntllaton of the equatoral regon. He was a recpent of the Samsung Lee Kun Hee Graduate Fellowshp for fve years. Todd Walter receved hs B.S. degree n physcs from Rensselaer Polytechnc Insttute, Troy, NY, and hs Ph.D. degree from Stanford Unversty, Stanford, CA, n 993. He s currently a Senor Research Engneer n the Department of Aeronautcs and Astronautcs, Stanford Unversty. He has long been actve n the development of the Wde Area Augmentaton System and ts nternatonal counterparts. Hs early work ncluded some of the frst operatonal prototypng and development of many key operatonal algorthms and standards n use on these systems. Hs current research focuses on the future use of the two cvl frequences avalable through modernzed GPS and new satellte navgaton systems beng mplemented around the world. Dr. Walter s a Fellow of the Insttute of Navgaton (ION). He was a corecpent of the 00 Early Achevement Award from the ION and wnner of the 009 Thurlow Award. Per Enge receved hs Ph.D. degree n electrcal engneerng from the Unversty of Illnos, Urbana Champagn. He s the Klener-Perkns, Mayfeld, Sequoa Captal Professor n the School of Engneerng, Stanford Unversty, Stanford, CA. He s also Drector of the GPS Research Laboratory, whch poneers satellte-based navgaton systems for avaton and martme use. Two of these systems are n wdespread use today. The frst uses medum frequency beacons to broadcast dfferental GPS correctons to some. mllon, mostly marne, users around the globe. The second uses geostatonary satelltes to broadcast dfferental correctons and realtme error bounds to GPS users n North Amerca. Ths latter system came on lne for avaton n the Unted States n July 003, and smlar systems are beng developed n Europe, Japan, and Inda. Prof. Enge s a Fellow of the Insttute of Navgaton and the IEEE, and a member of the Natonal Academy of Engneerng. He has receved the Kepler, Thurlow, and Burka Awards for hs work. ABSTRACT A Global Postonng System (GPS) recever may lose carrer trackng lock to the GPS sgnal under deep and frequent fades due to onospherc scntllaton. The frequent loss of lock observed durng a strong scntllaton perod from the past solar maxmum can sgnfcantly reduce GPS avaton avalablty. However, the frequency dversty (L and L frequences) from the future GPS s expected to mtgate scntllaton mpact on GPS avaton. In order to assess ts mtgaton effectveness, we propose a way to generate correlated fadng processes based on our defnton of a correlaton coeffcent between fadng channels. Usng the correlated fadng process model that we propose, navgaton avalablty of Localzer Performance wth Vertcal gudance (LPV)-00 durng severe scntllaton s parametrcally studed. We then present results showng that hgh navgaton avalablty s attanable f a recever reacqures the lost channel wthn or seconds. Based on ths result, we propose a new performance requrement for the future dual frequency GPS avaton recever performance standards to guarantee hgh navgaton avalablty durng strong scntllaton. - September 009, Savannah, GA

2 INTRODUCTION The Global Postonng System (GPS) wth ntegrty nformaton from the Wde Area Augmentaton System (WAAS) [] s currently used n the Unted States to gude the approach of arcraft down to 00 ft above the runway. Ths approach procedure, also known as Localzer Performance wth Vertcal gudance (LPV)-00 [], can be expanded globally by future GPS wth augmentatons that provdes dual frequency cvlan codes n the avaton band (L and L frequences) [3, 4]. The onospherc delay can be drectly measured by future dual frequency GPS avoncs, and hgher-order onosphere errors are not problematc for LPV-00 []. However, deep and frequent GPS sgnal fades due to onospherc scntllaton [6] rases a concern about the operatonal avalablty of LPV-00 n the equatoral area durng solar maxma. Deep and frequent sgnal fadng observed durng the past solar maxmum (00) led to frequent loss of the carrer trackng lock of GPS recevers, whch can sgnfcantly reduce navgaton avalablty dependng on a recever s reacquston capablty [7]. However, the frequency dversty from L and L frequences from future GPS satelltes can mtgate scntllaton mpact on GPS avaton avalablty. For example, f a recever tracks both L and L frequences from the same satellte, t can rely on one frequency when t brefly loses the other frequency. Hence, the correlaton of sgnal fades across the two frequences s mportant to understand n order to assess the effectveness of mtgaton from the frequency dversty. If sgnal fades of two frequences are hghly correlated,.e., f the probablty of the smultaneous fadng of L and L channels of the same satellte s hgh, the actual beneft from frequency dversty would be lmted. A recent study reported very hgh correlaton coeffcents (exceedng 0.7) among sgnal ntenstes of L and L channels based on early GPS data collected at Thule, Greenland durng [8]. However, the conventonal defnton of sample correlaton coeffcent between two tme seres of sgnal ntensty used n the study evaluates the smlarty of the trends of two tme seres. It does not descrbe how often both frequences are lost smultaneously. Hence, the conventonal correlaton coeffcent s not a good metrc for evaluatng scntllaton mpact on GPS avaton. Snce there s no approprate metrc to measure the probablty of smultaneous fadng of two channels, ths paper suggests a new correlaton coeffcent from the perspectve of GPS avaton. Although strong scntllaton data from the L sgnal are not yet avalable, correlated L and L fadng processes wth arbtrary correlaton coeffcents can be generated by the method proposed n ths paper. Usng ths method, the avalablty of LPV-00 durng strong scntllaton can be parametrcally studed wth any correlaton levels between frequences, even before L scntllaton data can be collected durng the next solar maxmum (around 03). A new performance requrement for GPS avaton recevers s proposed based on the avalablty analyss results. DEEP GPS SIGNAL FADES AND CORRELATION COEFFICIENTS Electron densty rregulartes n the onosphere can cause ampltude fadng and phase jtter of transonospherc rado waves, whch s referred to as onospherc scntllaton [6]. Most of the Carrer-to-Nose-densty rato (C/N 0 ) fluctuatons due to scntllaton n Fgure (bottom) are not problematc for GPS avaton whle the recever mantans ts carrer trackng lock. However, the deep sgnal fadng (marked n red) causng loss of lock s of real concern. Our prmary nterest s not how the GPS sgnal fluctuates, but when a recever loses lock due to deep fadng. Fgure. Comparson of 0 Hz C/N 0 outputs of nomnal (top) and scntllaton (bottom) cases. Future GPS satelltes wll provde cvlan codes at dual frequences (L and L frequences) for avaton applcatons. When a recever loses lock on the L channel due to deep fadng, t can stll track the L channel to provde range measurements unless L and L channels are lost smultaneously. Hence, the correlaton of sgnal fades between L and L frequences should be well understood to assess mtgaton effectveness from frequency dversty. A hgh correlaton of sgnal fades n ths paper means that the sgnal fades of dfferent frequences occur smultaneously wth a hgh probablty. However, the conventonal defnton of the sample correlaton coeffcent of Equaton () does not provde any nformaton wth respect to how often the deep fades of two channels occur smultaneously. - September 009, Savannah, GA

3 ( x x)( y y) ˆ ρ = () ( x x) ( y y) Fgure llustrates a weakness of ths defnton of the sample correlaton coeffcent. The x and y of Equaton () are the C/N 0 sample ponts n green and blue n Fgure. The x and y are the mean values. In the top plot of Fgure, only one channel (blue) experences deep fadng. Hence, the recever would stll be able to track the other channel (green) when the other channel (blue) s brefly lost due to deep fadng. The bottom plot of Fgure represents a more challengng scenaro. In ths case, two channels sometmes fade smultaneously (red crcles); as a result, the recever does not have a backup channel durng these outage perods. by the large number of samples out of deep fades, whch are not of our nterest. Fgure clearly demonstrates the weakness of the conventonal defnton of sample correlaton coeffcent for the study of scntllaton mpact on GPS avaton. We suggest and valdate an alternatve defnton of a correlaton coeffcent n the next secton. SUGGESTION FOR A BETTER CORRELATION COEFFICIENT In order to suggest a better correlaton coeffcent of sgnal fades for ths study, we analyzed statstcs of sgnal fades durng a strong scntllaton perod of the prevous solar maxmum. The data were collected at Ascenson Island for nne days n 00 and the worst 4 mn observaton was used for ths study. Detaled nformaton about the data set s n [9]. The emprcal Probablty Densty Functon (PDF) of the tme between deep fades s shown n Fgure 3, based on the defnton of tme between deep fades of Fgure. An nterestng observaton s that the dstrbuton of the tme between deep fades approxmately follows an Exponental dstrbuton wth a mean of 9.7 s. Fgure. Weakness of the conventonal defnton of a sample correlaton coeffcent. Obvously, the bottom plot s more problematc for GPS navgaton, but t has a lower sample correlaton coeffcent (0.3) than the other case (0.4) when the correlaton coeffcent of Equaton () s used. Ths s because the defnton of Equaton () merely evaluates the smlarty of the trends of two tme seres. The green tme seres of Fgure (top) follows the general trend of the blue tme seres. Consequently, the correlaton coeffcent of the top fgure s hgh. However, the green tme seres of Fgure (bottom) sometmes goes up whle the blue tme seres goes down and vce versa (red arrows). Ths opposte trend sgnfcantly reduces ts sample correlaton coeffcent. Agan, we are nterested n how often deep fades of two frequences occur smultaneously. However, deep fades are very bref and sample ponts durng deep fades are few n number, so they have lttle nfluence on the sample correlaton coeffcent obtaned by Equaton (), whch s domnated Fgure 3. Emprcal probablty densty functon of tme between deep fades durng strong scntllaton. An mportant mplcaton of the exponentally-dstrbuted tme between deep fades s that the correspondng countng process of the fadng tself s a Posson process [0]. Fgure 4 llustrates ths relatonshp. The tme between deep fades corresponds to an nter-arrval tme of an arrval process. Snce the nter-arrval tme (or the tme between deep fades) follows the Exponental dstrbuton wth a mean of 9.7; the countng process of the number of arrvals (or number of deep fadngs) s the Posson process of rate / September 009, Savannah, GA

4 Fgure 4. Modelng a fadng process as a Posson process. The Posson process has many nterestng propertes. For nstance, the number of arrvals durng an nterval t, that s N(t) n Fgure 4, of a Posson process of rate λ s a Posson random varable wth a mean and varance λ t. If two ndependent Posson processes of rates λ and λ are combned together, the resultng process s stll a Posson process of added rate λ + λ [0]. Usng ths property, we can generate two correlated Posson processes from three ndependent Posson processes (Fgure ). Fgure. Generaton of two correlated Posson processes from three ndependent Posson processes. In Fgure, the common Posson process, C(t), of rate λ s combned wth two ndependent Posson processes, X (t) and X (t), of rates λl λ and λl λ, respectvely. The result s two correlated Posson Processes, L (t) and L (t), of rates λ L and λ L, respectvely. Snce the covarance between L (t) and L (t) s the same as the varance of the common Posson process, C(t) (See the Appendx for proof), the correlaton coeffcent between L (t) and L (t) s obtaned as follows. The physcal nterpretaton of λ t s the expected number of arrvals of a Posson process of rate λ durng an nterval t. Hence, when a fadng process s modeled as a Posson process, λ t s nterpreted as the expected number of deep fadngs durng an nterval t. Snce the common Posson process C(t) n Fgure models the smultaneous fadng of L and L channels, the correlaton coeffcent of Equaton () can be nterpreted as follows. Therefore, the correlaton coeffcent defned n Equaton (3) descrbes how often smultaneous fadng of two frequences occurs. We propose ths defnton of a correlaton coeffcent as an approprate metrc to measure the correlaton level between fadng channels. Ths defnton of a correlaton coeffcent s mathematcally derved from the statstcal observaton and makes good physcal sense. Furthermore, Fgure llustrates a way to generate two correlated fadng processes wth an arbtrary correlaton coeffcent. The rate of L fadng (number of L deep fadngs per unt tme) was obtaned from the real scntllaton data ( λ L = ), but the rate of L fadng s 9.7 not yet known. Currently, only one GPS satellte s transmttng an L testng sgnal [, ]. Accordng to [3], the ampltude scntllaton ndex (S 4 ndex) of the L frequency (.76 GHz) s expected to be greater than the S 4 ndex of the L frequency (.7 GHz).. f L S 4, L = S 4, L =. S 4, L f (4) L However, Equaton (4) does not relate the fadng rates of L and L. The fadng rate refers to the number of deep fades or the number of losses of lock of the channel per unt tme. Although the sgnal fluctuatons of L could be worse than the L sgnal fluctuatons, the hgher power and the better code structure of the L sgnal [4] would reduce the chance of loss of lock of the L trackng channel. Hence, we may assume that the L fadng rate s - September 009, Savannah, GA

5 smlar to the L fadng rate, whch should be valdated durng the next solar maxmum (around 03). After assumng that λ L s equal to λ L, we can obtan λ for an arbtrary correlaton coeffcent ρ from Equaton (). ρ λ = ρ λl λl = () 9.7 Once λ s obtaned for a gven correlaton coeffcent, three ndependent Posson processes, X (t), C(t), X (t), of rates λl λ, λ, λl λ, respectvely, can be generated. Fnally, the correlated fadng processes, L (t) and L (t), wth the gven correlaton coeffcent ρ are obtaned by combnng these three Posson processes (Fgure ). Usng ths method, the navgaton avalablty of LPV-00 under strong scntllaton s parametrcally studed n the followng secton. AVAILABILITY BENEFIT FROM DUAL FREQUENCY GPS AVIONICS Avalablty Smulaton Usng Correlated Fadng Processes A prevous study evaluated the GPS avaton avalablty durng an observed strong scntllaton perod [7]. The study consdered shortened carrer smoothng tme due to frequent fades and presented avalablty results as a functon of a recever s reacquston tme. However, the expected beneft from uncorrelated L/L channels was not consdered. Ths prevous study assumed 00% correlaton between L and L channels, whch means that whenever an L channel was lost due to deep fadng, the L channel was assumed to be lost as well. It s a very conservatve assumpton. We assume a m User Range Accuracy (URA), the onofree dual frequency code nose and multpath model based on the WAAS Mnmum Operatonal Performance Standards (MOPS) [], the troposphere model from the WAAS MOPS, and the same satellte constellaton observed durng the strong scntllaton perod n [7]. The avalablty of ths paper s also obtaned for a sngle user at Ascenson Island where the scntllaton data were collected. However, the avalablty smulaton of ths paper does not rely on the fadng tme seres of the collected scntllaton data. The collected data s L only and there s no L strong scntllaton data avalable. For ths reason, we generate correlated L/L fadng processes wth arbtrary correlaton coeffcents and parametrcally evaluate the expected beneft from frequency dversty. Two parameters are consdered for the avalablty smulaton of ths paper. These are the correlaton coeffcent between L and L channels and the reacquston tme of a recever. The reacquston tme s defned as a tme to reestablsh the lock after loss of lock and rentroduce the lost channel nto the poston calculaton. If a recever reacqures the lost channel quckly, the satellte outage duraton s reduced; consequently, scntllaton mpact s also reduced. The correlated fadng processes generated by the method of the prevous secton smulate tme nformaton of nstances of deep fades for all channels (L/L channels of eght satelltes n vew n our case). The recever s conservatvely assumed to lose ts carrer lock wth 00% probablty at each deep fadng. The severe scntllaton scenaro for the avalablty smulaton of ths paper s much worse than the strong scntllaton data collected at Ascenson Island durng the prevous solar maxmum. In the real data, 7 satelltes out of 8 were fadng for some fracton of 4 mn. In the severe scntllaton scenaro that we generated, L/L channels of all 8 satelltes are fadng at the maxmum rate for the whole tme perod. There s one more ssue to consder for the avalablty analyss. A GPS recever has to track both frequences to drectly measure the onospherc delay. If a recever brefly loses one frequency after deep fadng, the onospherc delay cannot be drectly measured. Hence, we need a way to bound ths unknown onospherc delay error durng bref outage perods. Three dfferent ways to bound onospherc delay errors are dscussed n the followng secton, and the operatonal avalablty of LPV-00 s obtaned for each case. Avalablty under the Most Conservatve Ionospherc Delay Estmaton The most conservatve approach to bound the onospherc delay error s that a recever does not trust range measurements from a satellte f the recever lost ether the L or L channel of the satellte. If ths approach s appled, hgher correlaton between L and L channels gves better avalablty. The hgher correlaton of ths paper means that fadng of L and L channels occurs smultaneously wth hgh probablty. In ths most conservatve approach, a recever does not use range measurements for poston calculatons unless dual frequency measurements are avalable. Hence, f a recever loses one frequency followed by the other, the total duraton of satellte loss would be extended. Therefore, t s better to lose both frequences together nstead of losng one frequency after the other, f the fadng rates of two frequency channels are fxed. - September 009, Savannah, GA

6 Ths ntutve explanaton s valdated by the avalablty smulaton result n Fgure 6. The 3 m Vertcal Alert Lmt (VAL) and the 40 m Horzontal Alert Lmt (HAL) for LPV-00 are consdered for ths analyss. Snce the approach to bound the onospherc delay error s so conservatve, even a short reacquston tme of s and 00% correlaton between L and L channels provdes only about 9% avalablty. Fgure 7. Avalablty of LPV-00 wth the most optmstc onospherc delay estmaton. Avalablty under the Sharpest Ionospherc Gradent Observed Fgure 6. Avalablty of LPV-00 wth the most conservatve onospherc delay estmaton. Avalablty under the Most Optmstc Ionospherc Delay Estmaton Now we consder the most optmstc approach for boundng the onospherc delay error. Snce fadng duraton s very bref about 0. s (9%) at 0 db-hz [9] the onospherc delay error may not grow sgnfcantly durng ths short perod. Hence, a recever may use the most recent onospherc delay measurement durng the bref perod of sngle frequency loss. In ths case, the recever does not lose a satellte unless both frequency channels are lost smultaneously. The avalablty of LPV-00 usng ths approach s presented n Fgure 7. A recever fully reles on the most recent onospherc delay measurement and the range measurement from one frequency when t brefly loses the other frequency. As expected, lower correlaton between two frequences provdes better avalablty. In ths case, a s reacquston tme of a recever gves more than 99.% avalablty f a 0% correlaton between L/L channels s assumed. Although the actual correlaton level between L/L channels should be valdated durng the next solar maxmum, Fgure 7 parametrcally llustrates the expected avalablty levels dependng on a correlaton coeffcent and a recever s reacquston tme under a severe scntllaton scenaro. Two extreme approaches for boundng the onospherc delay error were dscussed. The conservatve approach s too conservatve to provde hgh navgaton avalablty. The optmstc approach gves hgh avalablty, but t may be too optmstc to guarantee navgaton ntegrty. In equatoral area, scntllaton often accompanes depletons, regons of steep onospherc gradent. Thus, sudden large changes n onospherc delay are expected to occur smultaneously wth strong scntllaton. In order to guarantee the ntegrty, the sharpest observed onospherc gradent and the worst-case geometry s assumed. Accordng to [6], 4 mm/km was the largest observed gradent n slant onospherc delay n the Unted States. Based on ths sharpest onospherc gradent, the worst-case combnaton of geometres of Fgure 8 s consdered. If a satellte and a plane move n the same drecton and f the sharpest onospherc gradent moves towards the plane, the onospherc range error can grow wth the rate of 0. m/s. Fgure 8. Maxmum temporal range error under the sharpest onospherc gradent. - September 009, Savannah, GA

7 Ths maxmum temporal range error can be bounded by a new VPL equaton. VPL New = VPL Nomnal + S vert, at (6) where a s the growng rate of the onospherc range error, and t s the tme after losng the range measurement of one frequency of the satellte. (Refer to Page J- of [] for the defntons of VPL Nomnal and S matrx.) Fgure 9 graphcally llustrates ths approach to bound the maxmum temporal range error. Once a recever loses one frequency of the satellte, the a t term of the new VPL ncreases wth the rate of a = 0. m/s to bound the error. If both frequences are lost, the satellte s excluded from calculatng poston solutons. If the recever retracks both frequences, the new VPL value drops to the nomnal VPL value, snce the onospherc delay can be drectly measured agan. Note that ths approach s stll very conservatve because t assumes the sharpest onospherc gradent under the worst-case combnaton of geometres (Fgure 8) for all satelltes n vew. Fgure 0. Avalablty of LPV-00 wth the new VPL boundng the maxmum temporal range error (0. m/s). Fgure 9. Vertcal Protecton Level (VPL) calculaton under sngle frequency loss. Usng ths approach, the avalablty of LPV-00 s obtaned n Fgure 0. Comparng to the most optmstc case of Fgure 7, there s not much dfference n avalabltes f a recever reacqures the lost channel wthn 3 s. In order to evaluate the senstvty of the avalablty to the temporal range error, a safety factor of s appled to the maxmum temporal range error. Even wth the.0 m/s temporal range error, the avalablty for a short reacquston tme (less than 3 s) does not change much. We can stll expect more than 99.% avalablty, f the L/L correlaton s about 0% and a recever s reacquston tme s or s even under the severe scntllaton scenaro that we generated. Fgure. Avalablty of LPV-00 wth the new VPL boundng twce of the maxmum temporal range error (.0 m/s). New Requrement for Future Dual Frequency Avaton Recever Performance Standards The current WAAS MOPS ncludes the followng note about scntllaton, There s nsuffcent nformaton to characterze scntllaton and defne approprate requrements and tests for ncluson n ths MOPS. New requrements may be defned when onospherc effects can be adequately characterzed []. Although the current WAAS MOPS does not have any specfc performance requrement to mtgate scntllaton mpact, we observe that the current Satellte Reacquston Tme requrement s related to ths mtgaton. The current requrement says, For satellte sgnal outages of 30 seconds or less the equpment shall reacqure the satellte wthn 0 seconds from the tme the sgnal s rentroduced []. Ths requrement consders Rado Frequency Interference (RFI), but sgnal outages under - September 009, Savannah, GA

8 strong scntllaton are much shorter than 30 s and are actually shorter than s. In order to provde hgh navgaton avalablty under strong scntllaton, we suggest an addtonal requrement for a future dual frequency MOPS. The new requrement that we propose s, For satellte sgnal outages of second or less the equpment shall reacqure the satellte wthn second from the tme the sgnal s rentroduced [7]. If a recever s propagatng ts trackng loop for bref outages nstead of gong through ts full reacquston procedures, t can mmedately retrack the lost channel when the sgnal comes back. Hence, fast reacquston after a bref outage s techncally possble. However, f a recever performs extensve safety checks before rentroducng the lost channel nto poston solutons, the reacquston tme can be much longer. Therefore, ths new proposal needs to be evaluated aganst RFI threats as well. Ths new requrement was proposed at the RTCA Specal Commttee-9 (GPS) Workng Group- (GPS/WAAS) Meetng on 4 June 009. CONCLUSION The expected avalablty beneft from the frequency dversty of the future GPS s parametrcally studed n ths paper. For the parametrc study, we propose a new sample correlaton coeffcent and a method to generate L/L fadng processes wth an arbtrary correlaton coeffcent. The avalablty result shows that hgh navgaton avalablty (more than 99.% avalablty) of LPV-00 s attanable even under the severe scntllaton scenaro that we generated, f a recever s reacquston tme s short. Wthout ths requrement, a certfed avaton recever may provde less than 0% avalablty durng severe scntllaton. Therefore, f a future dual frequency MOPS mandates fast reacquston after bref loss of lock, LPV- 00 can acheve hgh avalablty durng strong scntllaton of solar maxma n the equatoral area. APPENDIX Ths appendx proves Cov[L (t), L (t)] = Var[C(t)], where L (t), L (t), C(t) are gven as n Fgure. Snce X (t), X (t), C(t) are all ndependent, E[L(t) L (t)] = E[{X (t) + C(t)}{X (t) + C(t)}] = E[X(t)X (t) + X (t) C(t) + X (t) C(t) + C(t) ] = E[X(t)] E[X (t)] + E[X(t)] E[C(t)] + E[X (t)] E[C(t)] + E[C(t) ] E[L (t)] E[L (t)] = E[X (t) + C(t)] E[X (t) + C(t)] = {E[X (t)] + E[C(t)]}{E[X (t)] + E[C(t)]} = E[X (t)] E[X (t)] + E[X (t)] E[C(t)] Hence, + E[X (t)] E[C(t)] + E[C(t)]. = E[C(t) ] E[C(t)] = Var[C(t)]. Cov[L (t), L (t)] = E[L (t) L (t)] E[L (t)] E[L (t)] ACKNOWLEDGMENTS The authors gratefully acknowledge the Federal Avaton Admnstraton (FAA) CRDA 08-G-007 for supportng ths research and Theodore Beach, AFRL (Ar Force Research Lab), for provdng the data sets. The authors also apprecate the valuable comments from Sam Pullen, Stanford Unversty, and Seebany Datta-Barua, ASTRA (Atmospherc and Space Technology Research Assocates). The opnons dscussed here are those of the authors and do not necessarly represent those of the FAA or other afflated agences. REFERENCES [] Enge, P., T. Walter, S. Pullen, C. Kee, Y.-C. Chao, and Y.-J. Tsa, Wde area augmentaton of the global postonng system, Proceedngs of the IEEE, 84, , 996. [] Cabler, H., and B. DeCleene, LPV: New, mproved WAAS nstrument approach, Proceedngs of the th Internatonal Techncal Meetng of the Satellte Dvson of the Insttute of Navgaton, Portland, OR, 4-7 September, 00. [3] Walter, T., P. Enge, J. Blanch, and B. Pervan, Worldwde vertcal gudance of arcraft based on modernzed GPS and new ntegrty augmentatons, Proceedngs of the IEEE, 96, 98-93, 008. [4] Hegarty, C. J., and E. Chatre, Evoluton of the Global Navgaton Satellte System (GNSS), Proceedngs of the IEEE, 96, 90-97, 008. [] Datta-Barua, S., T. Walter, J. Blanch, and P. Enge, Boundng hgher-order onosphere errors for the - September 009, Savannah, GA

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