Performance of Receiver Autonomous Integrity Monitoring (RAIM) for vertically guided approaches

Transcription

1 erormance o Receiver Autonomous Integrity Monitoring (RAIM or vertically guided approaches Anaïs Martineau, Christophe Macabiau, Igor Nikiorov, Benoit Roturier o cite this version Anaïs Martineau, Christophe Macabiau, Igor Nikiorov, Benoit Roturier. erormance o Receiver Autonomous Integrity Monitoring (RAIM or vertically guided approaches. ENC-GNSS 008, Conérence Européenne de la Navigation, Apr 008, oulouse, France <hal- 0006> AL Id hal-0006 https//hal-enac.archives-ouvertes.r/hal-0006 Submitted on 30 Sep 04 AL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. he documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire AL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 erormance o Receiver Autonomous Integrity Monitoring (RAIM or Vertically Guided Approaches A. Martineau, C. Macabiau, I. Nikiorov, B. Roturier 3 Ecole Nationale de l'aviation Civile, oulouse, France Université echnologique de royes, royes, France 3 Direction Générale de l'aviation Civile, oulouse, France BIOGRAY Anaïs Martineau graduated in July 005 as an electronics engineer rom the Ecole Nationale de l Aviation Civile (ENAC in oulouse, France. She is now working as a h.d. student at the signal processing lab o the ENAC where she carries out research on integrity monitoring techniques. Christophe Macabiau graduated as an electronics engineer in 99 rom the ENAC in oulouse, France. Since 994, he has been working on the application o satellite navigation techniques to civil aviation. e received his h.d. in 997 and has been in charge o the signal processing lab o the ENAC since 000. is research now also applies to vehicular, pedestrian and space applications, and includes advanced GNSS signal processing techniques or acquisition, tracking, intererence and multipath mitigation, GNSS integrity, as well as integrated GNSS inertial systems and indoor GNSS techniques. Igor Nikiorov received his M.S. degree in automatic control rom the Moscow hysical echnical Institute in 974, and the h.d. in automatic control rom the Institute o Control Sciences (Russian Academy o Science, Moscow, in 98. e joined the University o echnology o royes (U in 995, where he is roessor and ead o the Institute o Computer Sciences and Engineering o royes. is scientiic interests include statistical decision theory, ault detection/ isolation/ reconiguration, signal processing and navigation. Benoît Roturier graduated in Engineering rom the ENAC in 985 and obtained a hd diploma in Electronics rom Institut National olytechnique o oulouse in 995. is activities are within the France aviation administration (DGAC since 987, where he has been successively managing installation o Instrument landing Systems (ILS at SNA, head o the research laboratory on CNS systems o ENAC, head o satellite navigation subdivision (GNSS within DSNA/DI (Direction des Services de la Navigation Aérienne/Direction de la echnique et de l Innovation. Since 007, he is the project manager o satellite navigation (GNSS and area navigation (RNAV implementation or DSNA/DI. INRODUCION Receiver Autonomous Integrity Monitoring (RAIM is a simple and eicient solution to check the integrity o GNSS in civil aviation applications such as Non recision Approaches (NA. In the next ten years, in a multi constellation context implying a large number o satellites and new signals, more demanding phases o light such as Approach with Vertical guidance (AV operations could be targeted using RAIM to check GNSS integrity. Considering those expectations, it is needed to precisely determine what are the vertically guided approaches that can be achieved. Globally, the improvement in the number and quality o measurements (dual requency measurements, better clock and ephemeris inormation, better ranging signals enhances position estimation and autonomous integrity monitoring perormance. owever, the beneit or position integrity needs to be quantiied, as a larger number o available measurements also implies a larger number o potential aulty measurements or the receiver. Moreover, the targeted phases o light are characterized by smaller horizontal and vertical tolerable position errors compared to NA, and by lower acceptable probabilities or the corresponding alert limits to be exceeded. hereore, the threatening range errors that need to be detected by the ault detection algorithm have to be reconsidered, since they could have smaller amplitude, and a probability o occurrence that is not clearly deined currently. he aim o this study is to evaluate the potential o GS/Galileo RAIM or AV operations. his paper investigates the extent to which the augmentation o the number o satellites and the improvement o pseudorange measurements quality could enable the use o RAIM or both horizontal and vertical guidance. he paper is organized as ollows. In a irst part, every assumption that has been made or this study is reviewed. hus target operational requirements are ormulated and particularly the way these requirements are interpreted to obtain the probability o missed detection is detailed. he way RAIM perormance is evaluated is also recalled, and a complete set o models and values are proposed. In particular, measurements quality parameters such as the resented at ENC'GNSS 008

3 UERE are discussed. hen a second part recalls some RAIM techniques such as classical least square residual algorithm and solution separation method. he last part o the study is dedicated to GS/Galileo RAIM simulations that have been conducted using a proposed pseudorange model o smoothed GS L/L5 and Galileo E/E5b measurements. he dierent RAIM algorithms previously described are evaluated comparing their perormance to achieve operations with vertical guidance. I- SE OF ASSUMIONS A AVE BEEN ADOED I- Expected perormance bounds For those RAIM simulations, operations with vertical guidance are targeted and more particularly AV I operations which requirements are described in the ollowing table. AV I Alert limits Integrity risk Maximum allowable alse alert rate AL40 m VAL50 m 0 / per sample.6 0 per sample But other inputs are necessary to monitor GNSS integrity with RAIM algorithms such as the targeted probability o missed detection that depends on the probability o satellite ailure. his aspect reers to the threat model and particularly needs to be detailed. I--- robability o satellite ailure wo main types o probabilities are available to characterize GS satellite ailure probability - the probability o occurrence o satellite ailure larger than 30 m (Major Service Failure which corresponds to 3events per year [] where > > > is the probability o ailure o one satellite corresponds to the ault ree case corresponds to the aulty case Critical biases calculation is done or a given user position at a given moment by - Computing the probability to exceed the alert limit in the ault ree case > and > - For each available pseudorange measurement, computing the smallest additional bias that lead to a probability > or > such as > > > > he computations o the probabilities and are detailed in appendix and do not depend on any detection algorithm. But it can be seen that they depend on the ailure probability o occurrence. Considering a double constellation GS/Galileo and AVI requirements, critical biases have been computed or a probability o satellite ailure occurrence o 0 /h (corresponding to the category o small ailures, the smallest obtained values are represented on the ollowing igure - the probability o occurrence o satellite ailure larger than 3.6 m which is per approach per satellite [], corresponding or an average o 7 visible satellites to,.75 0 /h First o all, we need to know the minimal amplitude o single pseudorange ailure that leads to an unacceptable positioning error or AV I operations and thus the minimal bias amplitude that needs to be detected by RAIM algorithms. A ault γ is considered as a horizontal positioning ailure i its impact violates the integrity risk, that is to say i > > > A ault γ is considered as a vertical positioning ailure i its impact violates the integrity risk such as Figure - Smallest Critical Bias or AV operations, 0 /h hereore only considering the single ailure case, it can be seen that the smallest single pseudorange ailures that lead to an inacceptable positioning error or a probability o occurrence o 0 /h are between 35 and 70 meters. hese critical biases systematically have an amplitude larger than 30 m and belong to the «Major Service Failure» category, that is to say a signal in space ranging error exceeding 30 meters. resented at ENC'GNSS 008

4 his is why only Major Service Failure events are considered or this study and this assumption leads to the ollowing process. Let s us denote the individual major satellite ailure probability and N the number o satellite in view, then the probability o having k simultaneous ailures among N satellites is,, According to the GS signal speciication 3 major ailures are allowed per year and per constellation which correspond to major ailure per hour or a constellation o 4 satellites such as,, /h.43 0 /h It is assumed that a Galileo satellite will have the same probability o ailure than a GS satellite. For a dual constellation, i 0 satellites are in view, the probability o one satellite ailure is,,.85 0 /h For a dual constellation, i 7 satellites are in view, the probability o one satellite ailure is,,.43 0 /h Considering this probability o satellite ailure occurrence o.43 0 /h, critical biases have been computed again and the smallest obtained values are represented on the ollowing igure Figure - Smallest Critical Bias or AV operations,,.43 0 /h hus it can be veriied that the smallest single pseudorange ailure that lead to an inacceptable positioning error or a probability o occurrence o.43 0 /h are between 40 and 75 meters and that they eectively belong to the «Major Service Failure» category. I--- robability o missed detection Only considering the single ailure case, the probability o missed detection shall be lower than the integrity risk requirement divided by the probability o ailure o one satellite among the all satellites in view. For example i 7 satellites are in view and inally,, For this study, the same probability is allocated or vertical and horizontal ailure, the integrity risk that it is taken into account in this ormula is 0 per approach or the vertical risk and 0 per approach or the horizontal one. I- erormance evaluation wo types o RAIM algorithm have been tested the classical LSR RAIM and the Solution Separation RAIM. he way they are implemented is detailed in section II. As it will be detailed in section II, RAIM tests are built to detect ailures that are abnormally large above the assumed noise level. he smallest bias that the test can detect is then projected in the position domain to inally obtain the protection level. It has been decided or this study to also observe the test ability to detect dangerous biases and thus to measure the eective. his is why RAIM availability has been observed through two methods - orizontal and Vertical rotection Level have been computed and compare to the corresponding Alert Limit - Critical biases o size presented in the lat section have been added to pseudo range measurements through Monte Carlo simulations and the capacity o RAIM algorithm to detect them has been measured Concerning the Monte Carlo simulations, or every user position at every epoch simulation period, critical biases have been successively added on each available measurement. Only one critical bias was added at the same time on the measurements. For each pseudorange, the number o simulation iterations has been designed to be signiicant with respect to required probability o missed detection such as 0 In this way, the is estimated with a number o digits equal to the number o digits o th required. RAIM unction that has been tested was ault detection unction. Simulations have been made or a user grid with a latitude step o 0 and a longitude step o 0. For each resented at ENC'GNSS 008

5 position o the user grid, a test has been made every 30 minutes. Each Galileo satellite has an approximate period o 4 hours and 5 minutes which corresponds to 5 revolutions in three days. hree days also correspond to 6 GS satellites periods. hereore the simulation time o three days has been chosen. his represents a total o 4948 dierent satellite-user geometries to compute protection levels and to test critical bias detection capability. I-3-- I-3 Internal RAIM parameters Geometrical considerations, osition solution estimation Satellite Constellations that have been considered are an optimized 7 satellites Galileo constellation and an optimized 4 satellites GS constellation. A 5 degree mask angle has been used or GS satellites and a 0 degree mask angle has been used or the Galileo satellites. hese assumptions lead to an average o 7 visible satellites on Earth (see igure 3. intererence can be assimilated to white noise and or Early Minus Late ower discriminator (or example [4] σ EML B L B / ( 0.5BL G( sin ( πcs B / B / C π G N0 B / B / G B / B / C G N0 B / ( sin( πc ( cos ( πc S S ( cos( πc d d, where d S d B L the one sided bandwidth o the equivalent loop ilter the data period the power spectrum density o the signal the signal to noise ratio the chip spacing the two sided bandwidth o the ront end ilter Without considering the temporal repetition period o the N sequence, the power spectrum density expression o the BSK signal is sin with the code period. his expression is used or GS L, GS L5 and GALILEO E5b code tracking loop error variance. For Galileo E, the normalized power spectrum density o the BOC(, is equal to cos Figure 3-Average number o visible satellites Only 4 unknowns have been taken or the position solution computation, that is to say that the GS/Galileo time dierence is not considered as an unknown. I-3-- seudo range measurement error he pseudo range measurement error variances rom dierent sources are gathered in the User Equivalent Range Error UERE. he contributions that have to be considered are orbit determination and synchronisation error, troposphere residual error, ionosphere residual error, multipath residual error and receiver noise residual error. Receiver noise residual error Computation o error variance o a code-tracking loop he error variance o the code-tracking loop will depend on the choice o the discriminator. Assuming that he error variance o the code tracking loop, error due to noise, can be thus computed or dierent kind o signals. For those simulations, the ollowing values have been used GS L GS L5 GalileoE GalileoE5b C B B 6 0 z 0 0 z 0 0 z 4 0 z C N 35 dbz 9 dbz 36.5 dbz 9.7 dbz 0.0 s 0.0 s 0. s 0. s Note that worst case C N are considered and not typical values. Iono ree measurements In nominal mode, the pseudorange measurements that are available to the aircrat receiver are the GS L, GS L5, GALILEO E, GALILEO E5a, GALILEO E5b code and resented at ENC'GNSS 008

6 phase measurements. But or uture civil aviation GNSS receivers complying with EUROCAE requirements, dual requency measurements will be combined into a single composite measurement called the iono-ree measurement, corrected or ionospheric error. hereore, rom GS L L5, and rom GALILEO E E5b, two distinct iono-ree measurements are built. Denoting the measurement at the instant k (representing the code measurement or the phase measurement and E L L L 5 E E5b.6, L5. 6,.4 L 5 E5b E 5 b L E.4 No signiicant correlation actor can be expected or the noise and multipath error aecting the dierent measurements made on the our carrier requencies. his is why the standard deviation o the error aecting the iono-ree measurement is modelled as Smoothing Once elaborated, these two GS and GALILEO iono-ree measurements are then smoothed to reduce the inluence o noise and multipath [6] where is the time smoothing constant in seconds is the raw code pseudorange measurement error variance is the smoothed code pseudorange measurement error variance Finally, the receiver noise residual error variance is obtained. It corresponds to the receiver noise, thermal noise, inter channel bias and processing error. where is the elevation angle in degree o the considered satellite. his was validated and adopted or GS L C/A. It is also assumed here or GS L5, Galileo E and E5b although smaller error can be anticipated [7]. Ionospheric residual error In the case o a dual requency receiver with ionospheric correction the ionospheric residual error is not considered as signiicant 0 ropospheric residual error he model or the residual error or the tropospheric delay estimate is where is the elevation angle sin 0. his model was adopted or GS L C/A and is assumed or GS L5 and Galileo E and E5b. User equivalent range error he User Equivalent Range Error is the value relecting the error budget and it is based on the computation o the ollowing contributions orbit determination and synchronisation error, troposphere residual error, ionosphere residual error, multipath residual error and receiver noise residual error. / It is supposed that 0.75 and 0. he igure represents the obtained Galileo smoothed iono ree UERE or dierent elevation angles Multipath error he smoothed multipath error or the airborne equipment is described by Fig 4 GS L/L5 and Galileo E/E5b smoothed ionoree UERE resented at ENC'GNSS 008

7 hose values are gathered in the ollowing table UERE (m Elevation angle ( GS III L/L Galileo E/E5b II- RAIM ECNIQUES wo types o RAIM algorithm have been tested in this study the classical LSR RAIM and the Solution Separation RAIM. he aim o this part is to briely recall the way they have been implemented or this study. II- LSR RAIM he classical LSR RAIM method is based on the comparison between a test statistic depending on the prediction error vector and a given threshold. II-- Implemented Detection unction Let s consider the measurement residual ΔY (also called the prediction error vector which can be expressed thanks to a linear relationship the measurement error vector E, its covariance matrix Σ and the observation matrix Δ Σ Σ he LSR RAIM test is then deined by 4 where. he detection threshold is obtained by considering the test statistic in the ault ree case I the measurement error E is noise only such as with ~0, hereore, is chi-squared distributed with N-4 degrees o reedom, ~, that is to say,, ~0, he probability o alse alarm is used to determine the normalised detection threshold a such as where max, > hus, a ault is detected i the chi-squared variable is abnormally large above the assumed noise level. Finally, the threshold that it is compared to our criteria is h 4 II--- rotection levels computation he protection levels derive rom the smallest bias the algorithm is able to detect satisying the alse alarm and the missed detection requirement. Let s consider that he measurement error E is noise and a bias b on one satellite j such as 0 0 In this case, SSE is chi-squared distributed with N-4 degrees o reedom and non-centrality parameter λ such as SSE~,,, ~, he non centrality parameter is computed in order to satisy the md requirement such as, he obtained non centrality parameter is the smallest that can be detected by the test. It does not depend o any pseudorange. As Σ Σ Δ, the relation between the smallest detectable bias on the pseudorange j and the test statistic is simpliied as where B Σ Σ is the smallest detectable non-centrality parameter previously obtained he smallest detectable measurement bias b on satellite j can be then expressed as resented at ENC'GNSS 008

8 he relationship between the position error and the measurement error is with Σ Σ hereore the impact o the bias in position domain is obtained by, Δ, 0,, 0 hen, A, A, b A, Denoting, we obtain Denoting, A, A, B p A, p B,,,, he protection levels are computed reerring to the worst satellite And II- max max Solution Separation RAIM he solution separation method is based on the observed separation between the position estimate generated by the ull-set ilter (using all the satellite measurements and that generated by each one o the subset ilters (each using all but one o the satellite measurements. he separation between each pair o the estimates (the ull ilter estimate and each sub- ilter estimate orms a test statistic and each test statistic is compared to its respective detection threshold which is determined to meet the maximum allowable rate requirement II-- Implemented detection unction Let be the true user position at the instant k and the LSR user position estimation at the instant k hen the relationship between the position error and the measurement error is with Σ Σ For,, let be the LSR user position estimation at the instant k do not considering the pseudo range obtained rom the satellite i. he solution separation discriminators are 4 vectors linearly depending on the error measurement such as heir covariance matrix is given by Σ For the horizontal part, computations that are not described here show that or the criteria, where, a threshold satisying the probabity o alse alarm can be deined such as where is the largest eingenvalue o the covariance matrix,, For the vertical part o the detection, we obtain or, the threshold such as Or such as where 3,3 II-- rotection level computation For,, let s assume that there is a bias on the pseudorange i and that it is not detected by the corresponding criteria. For the horizontal aspect that means that,, resented at ENC'GNSS 008

9 Since hereore, Since the aulty measurement has been removed rom computation, the vector corresponds to a ault ree case situation. So let s consider the distribution o this vector which is the position error resulting rom the sub solution that doesn t take into account the pseudo range Figure 5- orizontal rotection Level Computations that are not described here show that in this case is bounded by with the probability hereore, non detected bias on the i pseudorange And a class o horizontal protection levels can be deined as max, For the vertical aspect,,, and, can be easily bounded with the probability. A bound is obtained such as A class o vertical protection levels can be deined as max, III- SIMULAIONS RESULS III- LSR RAIM Figure 6-Vertical rotection Level - Monte-Carlo simulations Monte Carlo simulations have been perormed by adding on each available pseudorange the smallest bias that will lead to a positioning ailure. he algorithm ability to detect it has been measured. For every user position at every epoch o 3-days simulation period, biases have been successively added on each available smoothed GS L/L5 or Galileo E/E5b.pseudorange measurement. Only one critical bias was added at the same time on the measurements. For each pseudorange, the number o simulation iteration has been designed to be signiicant with respect to required probability o missed detection such as 0 he way this critical bias is computed or every pseudorange is detailed in appendix. he average value o this critical bias is represented on the ollowing igure - rotection level Vertical and horizontal protection levels have been computed or each point o our user grid. As it can be seen on the ollowing igures the protection are much lower than the corresponding alert limit. It results that the LSR RAIM is 00% o the time available or AVI operation or each point o our user grid. Figure 7- Average critical bias resented at ENC'GNSS 008

10 hese simulations have demonstrated that the implemented classical LSR RAIM was always able to detect the smallest dangerous biases showing an availability o 00% or AVI operation or each point o our user grid. III- Solution Separation RAIM Vertical and horizontal protection levels have been computed or each point o our user grid. Figure 8- orizontal rotection Level It also has been seen that the improvement in the quality o measurements (dual requency measurements, better clock and ephemeris inormation, better ranging signals has signiicantly decreased the user equivalent range error variance. Considering that UERE is the major parameter o position estimation and autonomous integrity monitoring perormance, great RAIM availability could be expected rom an UERE standard deviation o approximately one meter. hen classical LSR and Solution Separation RAIM availabilities have been computed or AVI approaches using both GS L/L5 and Galileo E/E5b pseudorange measurements. An availability o 00% has been obtained or the both algorithms. For the LSR RAIM and the Solution Separation RAIM, all computed xl were below the corresponding xal or every point o the user grid and or each epoch. Moreover, the LSR RAIM has been able to detect every single critical bias that has been added on each available pseudorange. Nevertheless the threat model that has been used in this study still needs to be consolidated since it does not consider the multiple ailure case. Even or the single ailure case, the threat model should be completed in order to take into account potential nominal biases due to signal deormation and antenna bias. hese nominal biases are not correctly bounded with zero-mean Gaussian distributions which are currently used or modeling the error measurement in the ault ree case. his parameter should be included in uture protection level calculation. Concerning the detection o multiple ailures, Solution Separation RAIM algorithm seems to be a promising method but complete studies need to be conducted using a consolidated threat model. Figure 9-Vertical rotection Level As it can be seen the protection levels are much lower than the corresponding alert limit. It results that the Solution Separation RAIM is 00% o the time available or AVI operation or each point o our user grid. CONCLUSION A complete review o the assumptions that are made in RAIM simulations has been irst proposed in this paper. It has been demonstrated, or the single ailure case using GS Galileo constellations, that the amplitude o pseudo range additional biases that lead to a positioning ailure are systematically larger than 30 meters or AV I operations. hereore even i the targeted phases o light are characterized by smaller horizontal and vertical tolerable position errors compared to NA, this eect is mitigated by the great number o available measurements that reduce the impact a o single satellite bias on the global positioning error. hus only Major Service Failures are taken into account or the single ailure case in this study. It is also important to keep in mind that only integrity aspects have been addressed through this paper. Continuity issue also needs to be studied beore considering RAIM as a uture mean or perorming integrity monitoring in AV operations. hus, urther studies are needed to deinitively conclude on the potential use o RAIM or approaches with vertical guidance even i these results seem promising. AENDIX CRIICAL BIAS his part is dedicated to the computation or each pseudo range i o the bias b that will lead to a positioning i ailure with a probability corresponding to the integrity risk. Let us consider the case where there is a bias on the pseudo range i, he error in the position domain is ε t t ( Σ Σ ( ξ B pos, WGS 84 resented at ENC'GNSS 008

11 ξ and where N( 0 ~ 4,Σ 0 B b i 0 I the matrix is expressed in the local geographic rame such as cos Ei cos Ai cos En cos An cos E cos E i n sin A i sin A n sin E sin E i n hen the positioning error is directly expressed in the local geographic rame ε t t ( Σ Σ ( ξ B pos, local he covariance matrix C o the error is such as t [ ε pos, local. pos, local ] C E ε t t t ( Σ Σ Σ ( Σ t C ( Σ t ( Σ t he horizontal positioning error is a two dimensions vector which ollows a Gaussian bi-dimensional law o mean b i, local, the projection o b i in the horizontal plane and o covariance matrix C, such as (, i, b ( t t C C, b ( Σ Σ and b i, local, i, local Its density unction is ε π pos, local det C ( X exp local ( X b C ( X b i, local, i, local, where X is expressed in the Nord East local rame such as xn X xe Since C is a covariance matrix, C is a positive deinite matrix, it is diagonalizable and its eigenvalues are all positive. In particular we can ind an orthonormal basis ( e, e e e, i, i B that is composed o eigenvectors, corresponding to the eigenvalues λ and λ and such as where, C λ,λ diag( is the diagonal matrix whose elements are the eigenvalues o C is the projection matrix whose columns are the eigenvectors e,e. In particular is orthogonal det λ λ C, hen we have ( C ( X bi, local, C ( X bi, local, ( X bi, local, ( X bi, local, ( X b X b [ ] [ ( ] i, local, i, local, And we have X X and Ω bi, local, X where is the vector X expressed in the new local rame and Ω is the vector b i, local, in the new local rame. 0 ( X exp π λ λ. ( x Ω ( y Ω he probability that a couple ( x, y be such that x y AL is the probability that y x AL and considering the distribution o the horizontal positioning error, this probability is ( X D D exp π λ λ denoting D the domain such as λ λ ( x Ω ( y Ω λ x λ y AL. dxdy Let s make a change o coordinates such as we could have ( x Ω ( y Ω r. λ λ We re-write x, this way ( y x Ω r λ cosθ y Ω r λ sinθ he equation x y AL that deines the boundaries o the integration domain becomes y ( Ω r λ cosθ ( Ω r λ sinθ x resented at ENC'GNSS 008

12 Ω r λ sin r r λ cos θ Ω r θ Ω r λ cos θ Ω λ sin θ AL ( λ cos θ λ sin θ r( Ω λ cosθ Ω λ sinθ ( Ω Ω AL 0 Solving this equation, two roots r ( θ and ( θ θ [ 0,π ] are obtained such as x Ω r ( θ λ cosθ, θ [ 0, π ] y Ω r ( θ λ sin θ x Ω r ( θ λ cosθ, θ [ 0, π ] deine y Ω r ( θ λ sin θ boundaries o the integration domain. r or and the he jacobian o this transormation is computed to make our change o coordinates J r λ λ, and r ( X D exp( r drdθ π D' where the new domain D is deined by ( r r ( θ ( r r ( θ 0. θ [ 0, π ] Considering properties o second order polynomials θ π r r ( θ ( X D r ( r drdθ π exp θ 0 r r ( θ Assuming or example that r 0 r, ( X D π θ π r r r 0 r r dz r r dr d θ r r r exp exp 0 0 ( ( θ θ π r r π θ 0 and this last integral is computed numerically. ( X D exp exp dθ _ i A, and pseudo pos_ East ( i An equivalent analysis o the vertical risk (which is easier in one dimension must also be done. hen by comparing successively the obtained probabilities with the integrity risk or dierent bias amplitudes, the minimum bias which leads to a positioning ailure with a probability equal to the integrity risk is inally obtained. REFERENCES [] Global ositioning System Standard ositioning Service Signal Speciication (995, Second Edition, June 995 [] RCA/ DO 45A (004, Minimum Aviation System erormance Standards or Local Area Augmentation System (LAAS, RCA, Inc., Washington D.C., USA. [3] RCA/ DO 9D (006, Minimum Operational erormance Standards or Global ositioning Systems / Wide Area Augmentation System Airborne Equipment, RCA, Inc., Washington D.C., USA. [4] egarty C. (996, Analytical Derivation o Maximum olerable In-Band Intererence Level or Aviation Applications o GNSS [5] Julien O. (005, Design o Galileo LF tracking loops, h.d. thesis, University o Calgary, Department o Geomatics Engineering [6] Martineau, A., C. Macabiau, (008, Computation o the smallest bias that lead to a positioning ailure, ENAC internal report [7] Macabiau, C. (007, GNSS Integrity Course, GNSS Solutions utorials ION GNSS 007 [8] Macabiau C., L. Moriella, M. Raimondi, C. Dupouy, A. Steingass, A. Lehner (006, GNSS Airborne Multipath Errors Distribution Using the igh Resolution Aeronautical Channel Model and Comparison to SARs Error Curve, ION NM 006, January. hus the probability that the point ( y circle o radius AL is θ π π x, is out o the ( X D exp exp dθ θ 0 r r In order to pass rom a bias b on a given pseudo range to an error vector in the local horizontal plane, projections are made using linear relations. Denoting A ( we deine or i [, N ] pseudo_ pos _ North( i A, i [9] Brown R. G., G. Y. Chin (997, GS RAIM calculation o threshold and protection radius using chisquare methods a geometric approach, Global ositioning System Institute o Navigation, vol. V, pp , 998 [0] Brenner M. (990, Implementation o a RAIM Monitor in a GS Receiver and an Integrated GS/IRS, ION 990 [] Brenner M. (996, Integrated GS/inertial detection availability, Journal o he Institute o Navigation, Vol. 43, No., Summer 996 resented at ENC'GNSS 008