Compliance of Single Frequency Ionospheric Delay Estimation and Cycle Slip Detection with Civil Aviation Requirements

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1 Compliace of Sigle Frequecy Ioospheric Delay Estimatio ad Cycle Slip Detectio with Civil Aviatio Requiremets Christophe Ouzeau, ENAC/TéSA/DTI Christophe Macabiau, ENAC Ae-Christie Escher, ENAC Beoît Roturier, DSNA-DTI BIOGRAPHY ABSTRACT Christophe OUZEAU graduated i 5 with a Ioosphere is a dispersive medium that ca master i astroomy at the Observatory of Paris. He strogly affect GPS ad GALILEO sigals. started the same year his Ph.D. thesis o degraded Ioospheric delay affectig the GPS ad GALILEO modes resultig from the multicostellatio use of sigle frequecy pseudorage measuremets is the GNSS, supported by DTI ad supervised by ENAC. largest source of ragig error. I additio, this perturbatio is difficult to model ad thus difficult Christophe MACABIAU graduated as a to predict. Nomial dual frequecy measuremets electroics egieer i 99 from the ENAC i provide a good estimatio of ioospheric delay. I Toulouse, Frace. Sice 994, he has bee workig additio, the combiatio of GPS ad GALILEO o the applicatio of satellite avigatio techiques avigatio sigals at the receiver level is expected to civil aviatio. He received his PhD i 997 ad to provide importat improvemets for civil has bee i charge of the sigal processig lab of aviatio. It could, potetially with augmetatios, the ENAC sice. provide better accuracy ad availability of ioospheric correctio measuremets. Ideed, GPS Ae-Christie ESCHER graduated as a users will be able to combie GPS L ad L5 electroics egieer i 999 from the ENAC i frequecies, ad future GALILEO sigals will Toulouse, Frace. Sice, she has bee brig their cotributio as some of them will be workig as a associated researcher i the sigal trasmitted at the same frequecies as the GPS processig lab of the ENAC. She received her sigals. However, if affected by radio frequecy Ph.D. i 3. iterferece, a receiver ca lose oe or more frequecies leadig to the use of oly oe Beoît ROTURIER graduated as a CNS frequecy to estimate ioospheric code delay. systems egieer from Ecole Natioale de Therefore, it is felt by the authors as a importat l Aviatio Civile (ENAC), Toulouse i 985 ad task to ivestigate the performace of techiques obtaied a PhD i Electroics from Istitut tryig to sustai multi-frequecy performace whe Natioal Polytechique de Toulouse i 995. He a multi-costellatio receiver istalled i a aircraft was successively i charge of Istrumet Ladig loses dual frequecy capability, durig critical Systems at DGAC/STNA (Directio Géérale de phases of flight. l Aviatio Civile/Service Techique de la Navigatio Aériee), the of research activities o After a loss of several frequecies leadig to a CNS systems at ENAC. He is sice head of sigle frequecy degraded mode, a receiver ca use GNSS Navigatio subdivisio at DGAC/DTI code ad carrier phase pseudorages made o oly (Directio de la Techique et de l Iovatio, oe carrier frequecy to estimate the ioospheric formerly kow as STNA) ad is ivolved i the delay. To achieve this estimatio, the receiver ca developmet of civil aviatio applicatios based o use the differece betwee code ad carrier phase GPS/ABAS, EGNOS ad GALILEO. He is also measuremets. Ideed, this quatity ca be curretly ivolved i stadardizatio activities o modelled as twice the ioospheric delay plus oise, future multicostellatio GNSS receivers withi multipath, ad the carrier phase ambiguity. The Eurocae WG6 ad is the chairma of the techical ioospheric delay ca the be extracted from this, group of ICAO Navigatio Systems Pael. provided the ambiguity is properly removed. This ca be achieved after covergece of a Kalma 96

2 Filter for example, but the cycle slips eed to be moitored. The probability of a cycle slip to occur is low but ot egligible for civil aviatio purposes. Several causes of cycle slips may be idetified. For istace multipath, dyamics, sigal blockage ad ioospheric scitillatio may be sources of this type of rupture i carrier phase measuremets. Cycle slips may have radom magitudes. Those oes have to be detected ad corrected with a performace compliat with civil aviatio requiremets for itegrity, cotiuity, accuracy ad availability. This problem of cycle slip detectio is a priority before aalyzig the accuracy of the sigle frequecy ioo corrected pseudorage. We propose to follow the methodology exposed below to assess the performace of potetial algorithms of detectio (ad estimatio) of cycle slips. First, the cycle slip detectio ad correctio ability will be defied by the smallest cycle slip detectable with a required probability of missed detectio. This smallest detectable cycle slip implies a bias o positio error depedig o geometry. Therefore, availability of protectio agaist cycle slips compatible with APV ad APV for istace, depeds o geometry ad must be computed at every secod. The mai goal of this paper is to kow exactly the impact of the capability of cycle slip detectio algorithms o the availability of reliable sigle frequecy ioo corrected pseudorages. I. INTRODUCTION I case of radiofrequecy iterferece (RFI), the loss of oe frequecy may be a problem if oe wats to keep the same ioospheric correctio quality as i the omial dual frequecy case. It would be ecessary to use alterate techiques to estimate the ioospheric error i a degraded sigle frequecy case. Total Electroic Cotet estimatios provided by GPS Klobuchar ad Galileo NeQuick models do ot provide a sufficiet accurate ioospheric delay estimatio at ay receivers positio (latitude, logitude, altitude, see [NATS, 3]), for ay geometry of satellites i view. Cosequetly, it is ecessary to employ a other techique i a sigle frequecy case. Code ad carrier phase measuremets vary differetly with ioosphere; they ca be joitly used to extract the ioospheric delay. This is the traditioal code mius carrier techique [Shau- Shiu Ja, 3]. Ideed, this differece betwee code ad phase measuremets ca be modelled as twice the ioospheric delay plus oise, multipath, ad the carrier phase ambiguity. The ioospheric delay ca the be extracted from this, provided the ambiguity is properly removed. This ca be achieved after covergece of a Kalma Filter for example, but the cycle slips eed to be moitored i order ot to bias the ioo corrected measuremet. As the capability of cycle slip detectio is ot exactly kow, our priority is to determie that performace, before aalyzig the accuracy of the obtaied sigle frequecy ioo corrected pseudorage. Carrier phase measuremets are provided by carrier trackig loops. Measuremets quality depeds o the loop ability to give a approximate value close to the actual value of the carrier phase. I a omial case, this loop is able to quickly follow the time evolutio of the icomig phase, ad measuremet errors are small. However, the trackig loop may lose the sigal durig a short period ad the re-acquire it just after the break. Such a pheomeo causes a phase jump (cycle slip) ad ca be modelled by a rupture, a sudde chage from oe measuremet to the ext oe. The phase ambiguity will therefore vary i this case. This problem, called cycle slip, occurs whe the C/N is low (blockage, multipath) or whe the receiver has too importat ad upredictable movemets. The magitude of those cycle slips will be a multiple of half the carrier wavelegth ad multiple of wavelegth after half cycle ambiguity resolutio through parity check. Makig the differece betwee code ad carrier phase measuremets i a sigle frequecy case is therefore a cadidate techique for a good estimatio of ioospheric error for civil aviatio, as metioed i [NATS, 3]. I the case where a cycle slip occurs i sigal carrier phase, this differece is biased by the magitude of the cycle slip. We will focus o this code mius carrier divergece techique i the followig. II. KALMAN FILTERING FOR SINGLE FREQUENCY IONOSPHERIC ESTIMATION The iput state of the Kalma filter we metioed i itroductio ca take ito accout the two costellatios (GPS ad GALILEO) space vehicles. It will estimate both ioospheric delay ad ambiguities of all satellites i view [Ouzeau, 6]. As the covergece time of the filter reaches several tes of secods (larger tha the TTA for our simulatios), it is preferable to iitialize it i omial dual frequecy mode. Oe should take ito accout the estimatio of dyamics parameters as dyamics will have importace i cycle slippig as we will see further i this paper. But, eve if those parameters are limited (see GALILEO MOPS, bouded values of acceleratio ad jerk), additioal iput state parameters will decrease its robustess, the umber of i-view satellites icreasig the umber of iput 97

3 states (ambiguities) depedig o ambiguity resolutio. The Kalma filter may be hybridized with models, Klobuchar for GPS costellatio ad NeQuick for Galileo oe. Two differet approaches may be cosidered, o the oe had, the use of a uique filter with a state vector composed of ioospheric estimatio ad ambiguities of all satellites i view from the two costellatios. O the other had, two idepedet filters may be employed idepedetly cosiderig separately GPS ad Galileo costellatios. The first approach allows cosiderig a virtual sigle costellatio. The less states i the filter, the more robust the filter will be. III. CIVIL AVIATION REQUIREMENTS AND CYCLE SLIP DETECTION I all cases, eve i a omial dual frequecy case, cycle slips have a ifluece o code-carrier smoothig as carrier phase measuremets are used to compute smoothed code measuremets as described further i this paper. I a sigle frequecy case, with sigle frequecy ioospheric correctio, it is particularly importat to be able to detect ad correct cycle slips. Ideed, code mius carrier divergece seems to be the most iterestig techique i case of loss of frequecy for civil aviatio [NATS, 3]. The impact of cycle slippig is cosequetly ot egligible as it will be a large source of error i code mius carrier ioospheric estimatio, ad detectio algorithms have to be implemeted ad tested agaist civil aviatio requiremets. For civil aviatio, these detectio algorithms are importat, mostly for critical approaches. ICAO requiremets are defied for those phases of flight ad are partially recalled further i this paper. These requiremets are accuracy, itegrity, cotiuity ad availability. The itegrity risk iduced by cycle slip detectio ability is the product of the probability of occurrece of cycle slips by the probability of missed detectio of those jumps i carrier phase measuremets. From itegrity ad cotiuity ICAO requiremets, cycle slip miimum detectable magitude is determied computig the probability of cycle slip missed detectio ad false alarm rate as described i [Ouzeau, 6]. This methodology ad the values used i simulatios are recalled i sectio 5 of this paper. We focus o the APV phases of flight; we wat to determie if the levels of performace required by ICAO ca be met with our proposed algorithms described latter. Ideed, the large sigma values ivolved whe usig Klobuchar or NeQuick estimatios of ioospheric code delay do ot allow supportig flight operatios that require vertical protectio level computatio. We wat to kow precisely if our proposed code mius carrier plus cycle slip detectio techique will allow supportig those operatios. The first phase of flight to cosider is cosequetly APV I. As a cosequece, if the results obtaied usig code mius carrier divergece ad cycle slip detectio ad correctio allow satisfyig ICAO requiremets, the mai default of code mius carrier divergece techique will be removed. The value of the probability of missed detectio is deduced from the itegrity risk ad the probability of occurrece of cycle slips. For APV ad APV approaches, the Sigal I Space (SIS) itegrity risk equals * -7 /approach. Allowed false alarm probability is determied from cotiuity ICAO requiremets. I [RTCA, 6], the SIS false alert probability is set to.6-5 /sample for APV phases of flight. IV. CYCLE SLIP DETECTION ALGORITHMS PROPOSED We propose here cycle slip detectio algorithms. There are may possible cycle slip detectio techiques. This ca be doe durig data parity check, through a compariso betwee Dopplerpredicted phase ad real-time obtaied phase, compariso betwee smoothed ad raw pseudorages ad fially, usig a geeralized likelihood ratio test at the Kalma filter output. A Kalma filter may estimate ioospheric delay ad the ambiguities of all satellites i view, makig the differece betwee code ad carrier phase measuremets as preseted i sectio II. A Geeralized Likelihood Ratio test at the output of this filter ca be implemeted to detect ad estimate cycle slips. It is based o a multiple hypothesis test examiig all the possible mea jumps at each istat. A decisio is take comparig the likelihood ratio for the chose magitude ad time idex with a defied threshold. A time-slidig widow is used to make the hypothesis test. If a cycle slip rupture is detected, Kalma filter estimatios are corrected to take ito accout the iduced error. A computatio of cycle slip amplitude is iheret to detectio. We wo t describe here the Kalma filter ad Geeralized Likelihood Ratio i details, the iterested reader may fid those iformatios i [Ouzeau, 6]. We have here metioed may detectio algorithms, other oes may be implemeted ad a combied use of those algorithms has to be tested. Detectio usig predicted phase measuremets 98

4 This algorithm is based o a predictio of future phase measuremets with Doppler measuremets: ˆ ( t) = ( t t) + f d t t Where d ( ) t f is the Doppler frequecy ad t is the time delay betwee the previous ad the curret measuremet. The the differece betwee phase measuremets ad predicted phase measuremets is compared to a threshold which has to be fixed: The threshold is a fuctio of false alarm probability with regards to APV phase of flight requiremet ad o the oise affectig this criterio. Detectio comparig smoothed to raw pseudorages Cycle slips o carrier phase measuremets will have a impact o the smoothig process. More precisely, code-carrier smoothig is expected to reduce multipath ad receiver oise o the pseudorages ad will help us detectig potetial cycle slips. Ideed, i this purpose, we compare output ad iputs of the smoothig filter described below: P proj ( t ) = P ( t ) + ( ( t ) ( t )) P ( t ) = αρ ( t ) + ( α ) P proj ( t ) P is the projected pseudorage i meters proj ˆ ( t ) ( t) > Threshold P (t) is the carrier smoothed pseudo rage i meters at time t (t) is the accumulated carrier phase measuremet i meters at time t ρ (t) is the raw pseudo rage i meters α is the weightig fuctio of the filter, uit less, it is defied by the ratio of the sample time to the smoothig time costat. The secod term of P is the differece proj betwee two cosecutive phase measuremets. If a cycle slip occurs at t i the phase measuremets, this phase variatio will be biased, so the differece betwee smoothed ad raw pseudorages will abormally vary. V. MINIMUM DETECTABLE BIAS To determie the smallest detectable bias with the required missed detectio probability P MD, we lauched simulatios to determie the performace of some cycle slip detectio algorithms. Differet magitudes of cycle slips must be simulated, ad we have to compute o-detectio probability ad to determie whether the obtaied values are acceptable as a fuctio of magitude of cycle slips. From ICAO cotiuity requiremets, we tested differet cycle slip magitudes over a sufficiet umber of samples to reach a required probability of false alarm (.6-5 /sample for APV ). For each cycle slip magitude, over / (false alarm probability) samples were itetioally biased by that cycle slip ad tested with our detectio algorithm was tested. A too low umber of samples wo t provide sigificat results, as the expected false alarm rate wo t be reached with a isufficiet umber of samples. I priciple, the more biased samples are tested, the more sigificat will be the results. However, we do ot dispose of a ifiite simulatio time. Cosequetly, simulatios were coducted for thirty times the miimum required umber of samples (3/false alarm probability). This provided us a threshold of cycle slip magitudes. The, from itegrity requiremets (* -7 per approach for APV) ad the cycle slip occurrece probability, we determied the required missed detectio probability. We tested varyig cycle slip magitudes with regards to the obtaied threshold over a sufficiet umber of samples as for false alarm probability (3/missed detectio probability), util we reached the required P MD. This provided us the miimum detectable bias with the employed detectio algorithm. This study was made for ormal ad abormal aircraft dyamics defied i [MOPS, 6]. Ifluece of dyamics Dyamics of the oboard receiver differs from case to case. Those values are provided by [MOPS, 6] white paper ad are recalled below: 99

5 Groud speed Horizotal acceleratio Vertical acceleratio NORMAL DYNAMICS ABNORMAL DYNAMICS 8 Kt 8 Kt.58 g. g.5 g.5 g Total jerk.5 g/s.74 g/s Table : Normal ad abormal aircraft dyamics, [MOPS, 6]. Where g = 9.8m/s² ad Kt are Kots. Ifluece of sigal type o detectio The probability of occurrece of cycle slips due to dyamics differs from oe sigal to aother oe. We recall here how we computed this probability for each sigal. We defie a flow of evets which are cycle slips occurrig successively ad separated by radom time itervals. This process has Poisso characteristics. So the probability of occurrece durig will be: t P OCC = exp( T t) where T is the cycle slip rate ad t is the exposure time. T is the mea time betwee two cycle slips. The cycle slip rate (or cycle slip mea time) is computed usig the followig formula; see [Holmes, 99] for a complete demostratio: I + π πγ T = tah I ( ) BLγ = + γ where I are Bessel fuctios of order. is the phase loop oise, its value depeds o the type of loop employed: For a Costas loop, BL = + C C T N N For a classical PLL, D = B L C N C N is the carrier to oise desity ratio T D is the coheret itegratio time B L is the loop badwidth We ca ote that this last value of does ot deped o itegratio time for a classical PLL, but i reality, the itegratio will play a role for the ormalizatio of the PLL discrimiator. γ represets the costat phase trackig error due to the receiver dyamics (rad). Accordig to [Hegarty, 997], for a third order PLL, the imum value of γ is provided by: 5.67 j γ λb L j is the imum expected jerk i g/s B L is the loop badwidth λ is the carrier wavelegth For GNSS sigals, we first compared the probabilities of occurrece of cycle slips usig a Costas loop. Table summarizes the values obtaied: Sigal type Itegratio time Probability of occurrece withi 5s GPS L ms 5.3 e-4 C/A GPS L5 ms 8. e-4 GALILEO ms 5.3 e-4 L GALILEO E5b ms 7.7 e-4 Table : Probability of occurrece of cycle slips for each sigal withi critical phases of fight (5 secods), durig ladig. These probabilities are computed cosiderig a Costas PLL, with Hz badwidth, jerk of.5 g/s (ormal maoeuvres), as a fuctio of coheret itegratio time. The depedece of these probabilities upo the itegratio time is egligible ( -6 imum differece o the probability for itegratio times betwee ms ad ms). From the required itegrity risk, with these probabilities of occurrece, the missed detectio probability should be betwee 4* -4 ad 4* -4 for ormal maoeuvres. Fially, to overboud the missed detectio probability, the chose values are 3

6 -5 for ormal maoeuvres ad -6 for the abormal maoeuvres case. Geeratio of pseudorages ad Doppler measuremets for simulatios For the ext parts, we geerate pseudorages ad Doppler measuremets allowig us to cotrol the perturbatios affectig sigals ad dyamics parameters of the receiver. We geerate code, phase ad Doppler measuremets. Those measuremets were geerated takig ito accout dyamics: Φ ( t ) = ρ + v t + a 9.8 t + j 9. 8 t + b + multipath + atmosphere + oise P ( t ) = ρ + v t + a 9.8 t + j 9. 8 t + b + multipath + atmosphere + oise is the phase measuremet i meters P is the code pseudo rage measuremet i meters ρ is a typical costat rage (ex: km) v is the rage rate, take here to be m/s (worst case rage rate due to satellite ad aircraft movemet durig a approach). a is the acceleratio i g. It is take accordig to table for ormal ad abormal maeuvers (further i this paper). j is the jerk i g/s. It is take accordig to table. b is the receiver clock bias geerated as described i [Wikel, ]. The Doppler measuremets were geerated as a first order derivative of the previously defied phase but the additive oise is provided by Gaussia radom values multiplied by a FLL sigma value defied i [Kapla, 996] istead of a PLL sigma value: 8W (Hz) L FLL = + π T C C D T D N N The multipath for code measuremets is geerated by drawig Gaussia radom values with a sigma correspodig to the worst case sigma at 5 elevatio (for Galileo, this value of mask agle will be degrees) usig [SARPs, 6] formula: E e = * (m) multipath The imum carrier phase multipath error cosidered here does ot exceed oe quarter of a carrier cycle. Ideed, this is the imum phase trackig error, assumig there is oly oe sigal replica with a magitude of. So the correspodig 3 3 chose sigma value is set to this correspodig imum. The atmospheric effects are geerated by multiplyig Gaussia radom values by a sigma correspodig to troposphere ad ioosphere, with a elevatio agle E of 5 degrees [Shau-Shiu Ja, 3]. The sigma for the troposphere is:. *. tropospher e =. + si( E) The sigma for the ioosphere [Shau-Shiu Ja, 3]: =.m ioosphere_ phase ioosphere_ code =.83m The error due to oise o the phase ad code is geerated as radom Gaussia values with a sigma correspodig to the stadard deviatio of the trackig error due to oise. For phase lock loop, we chose the followig value (Costas trackig loop): BL = + C C TD N N C N is the carrier to oise desity ratio, we chose C N = 3 db Hz. T D = ms is the coheret itegratio time. B = Hz is the loop badwidth. L For the delay lock loop, we chose DLL =.4 m. VI. SIMULATION RESULTS Smallest detectable cycle slip I our simulatios, usig Doppler predictio of phase measuremets, the smallest detectable cycle slip has a amplitude of 3 meters for ormal dyamics ad 6 meters for abormal maeuvers [Ouzeau, 6]. The test betwee smoothed ad raw pseudorages does ot provide a better performace of cycle slip detectio tha Doppler estimatios, as the results obtaied are a little more tha 5.8 meters for ormal dyamics ad for a widow of secods before the time of estimatio. Those values seem to be high, but this may be explaied by the fact we geerated pseudo rages for our tests with worst case sigma values as described i sectio V of this paper (ioosphere, troposphere, drawig Gaussia radom values with sigmas). 3

7 Our strategy was first to determie a detectio threshold compliat with cotiuity ICAO requiremets ad the to determie the smallest detectable bias with regard to itegrity requiremets. Sice itegrity defiitios are all i the positio domai, we have to switch from the pseudo rage variace domai to the positio variace domai. Cycle slip detectio availability calculatio Before discussig the accuracy of code mius carrier plus detectio algorithm, we choose to represet the availability maps of the Doppler predicted phase detectio ability for GPS ad Galileo stadaloe costellatios ad for the combiatio of the two. Ideed, if the availability obtaied usig our proposed algorithms is isufficiet for civil aviatio, it wo t be ecessary to discuss the accuracy of estimatios. I additio, the Doppler algorithm provided us the best results i detectio, that is to say, the smallest miimum detectable bias. That is why we must kow if this techique is available eough to meet ICAO requiremets. The smallest detectable cycle slip we will be able to detect will give us a idea of the positio error iduced by the udetectable error i the sigle frequecy ioospheric corrected rage. But this error will deped upo the geometry defied by the positios of the satellites ad the receiver. So, we will multiply the bias iduced by the iability of our algorithms to detect cycle slips by imum geometrical factors described latter, i order to compare it with alert limits preseted i Table 3, for APV phases of flight. HAL VAL TTA APV 4 m 5 m s APV 4 m m 6 s Table 3: Itegrity requiremets, [MOPS, 6]. HAL stads for Horizotal Alert Limit, VAL for Vertical Alert Limit ad TTA for Time To Alert. The miimum detectable bias iduces a error i code mius carrier (CMC) estimatio, ad cosequetly, a error i positioig. Before discussig the accuracy of our proposed CMC + detectio algorithm, we focus o the availability of such a method over Europe. The bias iduced by the capability of the algorithm to detect small cycle slips is itroduced i the followig equatios i order to kow precisely the error iduced i positioig for a give costellatio, cosiderig the worst-positioed satellites. 3 As described i detail i [Macabiau, 5], we assume the receiver makes pseudo rage measuremets collected i a vector oted Y. The measuremet vector Y ad the state vector X composed of positios ad clock bias are liked by: Y = g ( X ) + E E beig the measuremet error, due to multipath, oise, possible cycle slips, atmospheric effects ad satellite clock residuals. X = [ x y z b] T is composed of positios (x, y ad z) ad clock bias b. Let us describe the least squares avigatio solutio of the equatio. If we deote X ˆ a iitial estimate of X, we ca the ote X = X ˆ + X. The measuremet model ca be rewritte as: Y = g ( Xˆ + X ) + E This expressio may be liearized aroud ˆX, the estimate of X : g Y g ( Xˆ + Xˆ ) ( ). X + E X g ( Xˆ ) = G X g ( X ˆ ) ( X ) ( X ) ( X ) x y z b = g ( X ) ( ) ( ) ( ˆ ) X X X x y z b We ca rewrite the liearized model as: Y g Xˆ ) = G. X + E ( Y = G. X + E Y = Y g ( Xˆ ) is the deviatio betwee measuremets ad oiseless predicted measuremets if positio ad clock delay were ˆX. From this liear relatio betwee Y ad X, we deduce the least squares estimate of X : X T T = [ G G ] G. Y () ad Xˆ = Xˆ + X ˆ. The residual Y cosiderig Xˆ may be expressed as: Y = Y g ( Xˆ ) = g ( X ) g ( Xˆ ) + E Y = g ( Xˆ + X g Xˆ + Xˆ ) ( ) + E As described i [Macabiau, 5], we liearize the previous expressio: Y g( Xˆ ) G X G Xˆ + E = G( X Xˆ ) + E () however, ˆ T T X = [ G G] G [ Y g( Xˆ )], therefore: X ˆ T [ ] T = G G G [ G X + E ], which is equivalet to: Xˆ = X T T + [ G G ] G E

8 X Xˆ T T = [ G G ] G E (3) Cosequetly, from (), we deduce: Y Y g Xˆ T T = ( ) = ( I G [ G G ] G ) E ad so from (): X ˆ T T T T = ( G G ) G [ I G ( G G ) G ] E E cotais all the measuremet errors ad we ca ote there is a relatioship betwee the measuremet error vector E ad the predictio error vector Y or the least squares estimate of X. From (3), we deduce the projected errors i horizotal ad vertical plaes defied by the positio of each satellite i view, ad i particular, the bias iduced by the cycle slip detectio algorithm. The projectio for each visible satellite is the liear relatioship existig betwee the estimated horizotal or vertical positio error ad the pseudo rage error to a give satellite. I other words, it is the liear relatioship betwee biases i each of the visible satellites ad the iduced positio error. These projectios are characterized by the geometry matrix G ad vary with time; it differs from oe satellite to aother oe. We ote: The: Xˆ Where A = ( G T G ) G X Xˆ = AE + X Xˆ Xˆ = AE X X ˆ = AE is a classical matrix orm. The differece betwee X ad its estimate is thus described as a fuctio of the measuremet error E. The horizotal ad vertical projectios of the previous relatio are computed usig the followig equatios: fori =. H ( i) = Ai + A i H = ( H ( i)) i V V ( i) = A = i 3 i T ( V ( i)) Figure : Computed positio error for all satellites i view. As we ca see figure, the more importat the geometrical factor is, the higher the positio error is. So we cosider the imum geometrical factor so as to study the worst satellite positio case. The horizotal impact of the udetected bias is thus modeled by: H bias H impact = A similar relatioship ca be used to defie the vertical oe: V bias V impact = H ad V are the horizotal ad vertical projectios that iduce the imum positio errors i horizotal ad vertical plaes respectively, cosiderig all satellites i view. H ( V ) is the horizotal (vertical) value of the satellite whose bias is the most difficult to detect. Note that i the followig simulatios, we chose to itroduce the bias obtaied with our Doppler-estimatio algorithm. Whe the computed impacts ( H impact, V ) impact are over alert limits (horizotally ad vertically), the detectio algorithm is declared uavailable. Whe those impacts are uder or equal to alert limits, it is declared available. Availability maps for GPS ad Galileo costellatios ad aircraft dyamics ifluece We focus o APV phase of flight; we draw the availability maps of detectio algorithms over Europe idepedetly for ormal ad abormal imum dyamics. Simulatios over 4 hours for GPS ad over days for Galileo, show that usig GPS or Galileo costellatios i stadaloe mode, the availability of cycle slip Fault Detectio fuctio is far from beig sufficiet for ormal aircraft dyamics whe usig oly detectio usig Doppler measuremets. It is also cosequetly the case for abormal dyamics; we oly represet here a ormal maeuver case for each costellatio sice as we ca see, the availability requiremet is ot fulfilled for stadaloe modes. Positio error Alert limit H or V H or V Bias Measuremet error 33

9 plus cycle slip detectio is maily due to the largest impact of the bias o the vertical error. This is due T T to the fact that the geometrical factor ( G G ) G is larger o the vertical axis tha o the horizotal plae. Recall that the bias icluded i the availability computatio is calculated for the higher value of dyamics. That is to say the availability is calculated for imum jerks ad acceleratios of a aircraft i a ormal dyamics case. We thus cosidered the least favorable case for each calculatio of availability. Figure : Availability of proposed cycle slip detectio algorithm over Europe cosiderig GPS costellatio oly ad ormal aircraft dyamics, for APV alert limits. Figure 3: Availability of proposed cycle slip detectio algorithm over Europe cosiderig Galileo costellatio oly ad ormal aircraft dyamics for APV alert limits. As we ca see i figure, for GPS, assumig ormal dyamics, a imum value of % availability is reached but oly i Norther Europe. The miimum value obtaied is 97.%. The mea value is 99.5%. This is felt as ot sufficiet for aviatio purposes. This meas that sigle frequecy GPS measuremets corrected from ioospheric delay through CMC techique plus cycle slip detectio ca ot be used i stadaloe mode, as the miimum required cotiuity value durig APV I phases of flight is 99.99%. As see i figure 3, for Galileo costellatio, assumig ormal dyamics, the results obtaied show that availability is sufficiet over a large part of Europe but ot eough over Norway for istace. The miimum availability is 97.8%, the mea value is 99.63% ad the imum is %. Cycle slip risk i sigle frequecy ioo corrected measuremets is cosequetly also ot covered by Galileo i a stadaloe mode for APV I. Note that, this lack of availability for GPS ad GALILEO sigle frequecy ioospheric correctio However, whe combiig GALILEO ad GPS costellatios, as we ca expect, the availability map shows a cosiderable improvemet of the availability (% over Europe for both APV ad APV, for ormal ad abormal maoeuvres). As we are i the particular case of a sigle frequecy mode, this dual costellatio case will occur whe oly oe frequecy is available for both GPS ad Galileo costellatios. However, the combied use of GPS ad Galileo iduces failure modes which are still badly aalyzed. This combied mode must be cosidered with simplified hypothesis. Therefore, if we wat to use oe of the stadaloe costellatios, the detectio algorithm would have to be improved so as to keep the cotiuity of the sigle frequecy code mius carrier divergece techique. VII. CONCLUSION AND PERSPECTIVES The objectives of this paper were to kow if our proposed sigle frequecy CMC ad detectio algorithms allow maitaiig the dual frequecy performaces durig APV. Because the aalyzed CMC techique is sesitive to cycle slips, we first focused o the cycle slip detectio capability. We therefore aalyzed the availability of the detectio method over Europe for GPS ad Galileo costellatios for all kids of aircraft dyamics. From our simulatios, it is show that the miimum detectable bias iduced by cycle slip detectio algorithms does ot allow to support flight operatios that require vertical protectio level computatio (APV), usig oe of the two costellatios (GPS or Galileo) i stadaloe mode. This is represeted by the plotted availability maps over Europe (97.8% miimum availability for Galileo ad 97.% for GPS costellatio i stadaloe mode). Whe combied, the availability obtaied is % for ormal ad abormal aircraft dyamics. However, further ivestigatios have to take ito accout the potetial itegrity problems due to 34

10 the combied use of GPS ad Galileo costellatios. We have aalysed the availability of the proposed detectio techiques, followig the itegrity ad cotiuity ICAO requiremets. Now the accuracy of the proposed algorithms could be discussed regards to civil aviatio requiremets. Also, after detectio, a reparatio algorithm may be employed. The magitude estimatio of cycle slips may be provided by a Geeralized Likelihood Ratio algorithm as metioed i [Ouzeau, 6]. Aother poit to uderlie is the hybridizatio of a Kalma filter with Klobuchar or NeQuick models to perform the estimatio of ioospheric delay. This hybridizatio will deped upo the cosidered costellatio. Oe of the questios after discussig the sigle frequecy estimatio is how to bridge gaps betwee omial dual frequecy mode ad degraded sigle frequecy mode. Oe of the resposes to brig to this questio is the use of a Kalma filter ru with dual frequecy measuremets i the omial mode, ad used i the degraded mode with sigle frequecy measuremets oly. REFERENCES [Bastide, ] Note o Acquisitio, Trackig ad Demodulatio Thresholds Computatio Spectrum Subgroup meetig, Washigto DC, USA, April 8 th th,, Frederic Bastide, Beoît Roturier. [Hegarty, 993] Hegarty Christopher J., "A Assessmet of Alterative Methods of Mitigatig Ioospheric Errors for WADGPS Users", MITRE Techical Report MTR93W, The MITRE Corporatio, McLea, Virgiia, 993. [Hegarty, 997] Christopher Hegarty: Aalytical Derivatio of Maximum Tolerable I- Bad Iterferece Levels for Aviatio Applicatios of GNSS, Navigatio, The Joural of the ION, Vol 44, N, Sprig 997. [Holmes, 99] Coheret Spread Spectrum Systems, Robert E. Krieger Publishig Compay, Malabar, Florida, Holmes, 99. Receiver, Lauret Lestarquit, Norbert Suard, Jea- Luc Issler, CNES, 995. [Macabiau, 5] RAIM Performace i Presece of Multiple Rage Failures, C. Macabiau, B. Gerfault, I. Nikiforov, L. Fillatre, B. Roturier, E. Chatre, M. Raimodi, A-C. Escher, ION Natioal Techical Meetig 5. [MOPS, 6] Iterim Miimum Operatioal Performace Specificatio For Airbore Galileo Satellite Receivig Equipmet, White Paper, EUROCAE, May 6. [NATS, 3] Sustaiable Mobility ad Itermodality Promotig Competitive ad Sustaiable Growth, Maitaiig Dual Frequecy Navigatio Performace i the Presece of Iterferece, 3. [Niser, 995] GPS Ioosphere Determiatio Usig L Oly, Paul Niser, UK Natioal Air Traffic Services, Mike Trethewey, Sigal Computig Ltd, ION 995. [Ouzeau, 6] Ioospheric Code Delay Estimatio i a Sigle Frequecy Case for Civil Aviatio, proceedigs of the ION GNSS, Fort Worth, 6. [RTCA, 6] Radio Techical Commissio for Aeroautics, do9c, 6. [SARPs, 6] Stadards ad Recommeded Practices, ICAO, 6. [Shau-Shiu Ja, 3] Aircraft Ladig Usig a Moderized Global Positioig System Ad The Wide Area Augmetatio System, Shau-Shiu Ja, May 3. [Willsky, 976] A.S. Willsky ad H. L. Joes (976). A geeralized likelihood ratio approach to the detectio ad estimatio of jumps i liear systems. IEEE Tras. Aut. Cotrol,, pp. 8-. [Wikel, ] PhD Thesis dissertatio,. [Kalma, 993] Kalma Filterig, Theory ad Practice, Mohider S.Grewal, Agus P. Adrews, Pretice Hall Iformatio ad System Scieces Series, Thomas Kailath, Series Editor. [Kapla, 996] Uderstadig GPS, Priciples Ad Applicatios, Elliott D. Kapla, Artech House, Bosto, Lodo, 996. [Lestarquit, 995] Determiatio of the Ioosphere Error Usig Oly L Frequecy GPS 35