AC 20-191 Appendix C addition to the fault-free NSE and FTE distributions. This must be done for the full range of relevant error sizes to form the total conditional probability of an unsuccessful landing as a function of the error. The conditional unsuccessful landing probability is expressed as follows for the land short and land long cases:
Land Short ∫ LSC 𝑃𝑃(𝑈𝑈𝑈𝑈|𝐸𝐸)(𝐸𝐸) = p TSE_LON|E(x, E)dx−∞ [12] Land Long 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) =∫ ∞ p (x, E)dxLLC TSE_LON|E [13] Land with wheels less than 5 ft from the edge of the runway:
( ) ∫ ∞ ( ) +∫ −RWE+GW/2 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸 𝐸𝐸 = p x, E dx p (x, E)dxRWE−GW/2 TSE_LAT|E −∞ TSE_LAT|E [14] Where:
LSC – is the land short criteria of section 5.2.9.6 (i.e. 200 ft) LLC – is the land long criteria of section 5.2.9.6 (i.e. 3000 ft) RWE – is the lateral landing criteria as defined in section 5.2.9.6. (i.e. 70 ft) GW – is the lateral distance between the main landing gear pTSE_LON|E(x, E) - is the probability density function for the longitudinal touchdown given a bias of magnitude E.
pTSE_LAT|E(x, E) - is the probability density function for the lateral touchdown given a bias of magnitude E.
Note: Care must be taken to insure consistency of units when making these calculations.
C.1.10 Computing Pmd for Ranging Source Errors.
A bound on the probability of missed detection for the ranging source error, 𝑷𝑷𝒎𝒎𝒎𝒎(𝑬𝑬𝑹𝑹) is defined by the performance constraint region given in Annex 10, appendix B, section
3.6.7.3.3.2. The Pmd performance must lie below the curve defined by Table B-76A in the SARPs repeated here in Table C-7 for convenience.
Table C-7 Pmd_limit Parameters Probability of Missed Detection Pseudorange Error (meters) Pmd_limit ≤ 1 0 ≤ |Er| < 0.75 Pmd_limit ≤ 10(-2.56×|Er| + 1.92) 0.75 ≤ |Er| < 2.7 Pmd_limit ≤ 10-5 2.7 ≤ |Er| < ∞ C-11
AC 20-191 Appendix C For example, in the case of the longitudinal touchdown requirement the vertical position error has the largest effect on the touchdown location. The worst-case projection of a range error into vertical error max (|SAprvert.i|), may be used to determine the resulting limit on Pmd(EV) by substituting ER = EV /max(|S Aprvert.i|).
Figure C-7 illustrates the relationship between Pmd_limit and the Pmd (EV) for max(|SAprvert.i| )= 5.
Figure C-7 Example of the Satellite Ranging Source Pmd in the Range domain and Position Domain.
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AC 20-191 Appendix C Computing Pmd for Reference Receiver Fault Monitoring (RRFM).
The Pmd for the RRFM is given by:
T ( V ) BAC PmdApprox E = ∫ pBmd(x, EV )dx 0 [15] Where:
TBAC is the maximum threshold for the RRFM monitor given by:
0.0842×VAL 2 TBAC = 5.5× + 0.42 Meters [16] m−1 Where:
VAL – is the vertical alert limit that is used by airborne equipment to screen geometry expressed in meters.
And pBmd(x,EV) is the probability density function (pdf) of |Bj,vert(EV)| in the faulted circumstance given by:
0.0842×VAL dnorm x, EV , M −1 M 0.0842×VAL ( ) x ≥ 0; + dnorm− x, EBmd x E = V , p , V M −1 M [17] x< 0; 0 Where dnorm(x,µ,σ) is the Gaussian pdf ( ) 2x−µ− 2 ( ) e 2σdnorm x,µ,σ = [18] 2πσ Bj,vert(EV) is the broadcast difference between the broadcast pseudo-range corrections and the corrections obtained by excluding the jth reference receiver measurement projected in the vertical direction across all of the ranging sources used in the position solution. In this case, it is written as a function of the size of the bias in the vertical error, Ev, due to the single reference receiver fault/ For a derivation of these expressions see Reference [9].
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AC 20-191 Appendix C C.1.11 Example Assessments.
Figure C-8 illustrates a landing short assessment for a hypothetical aircraft with a nominal longitudinal touchdown point of 1500 ft from the threshold and a dispersion that can be bounded by a Gaussian distribution with σ=220 ft. Also, a max( SAprvert. i ) of 5, VAL of 10 meters and GPA of 3 degrees is used. Rearranging equation [11]:
10−5 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) < [19] 𝑃𝑃𝑚𝑚𝑚𝑚(𝐸𝐸) Hence by dividing 10-5 by the Pmd curves for satellite ranging sources and RRFM, the grey “keep out regions” shown in Figure C-8 can be obtained. The assessment is then simple. If the curve for 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) does not enter the keep out regions, then the requirement that 𝑃𝑃 (𝐸𝐸) < 10−5𝑈𝑈𝑈𝑈 is met for all values of E.
An alternative approach to the analysis is illustrated in Figure C-9 where the probability of an unsuccessful landing is explicitly calculated for both monitor types (ranging sources and RRFM).
Extension of these examples to the land long and lateral cases is straight forward.
Figure C-8 Example Assessment of Landing Short Performance C-14
AC 20-191 Appendix C Figure C-9 Explicit Calculation of PUL for the Land Short Example Above.
C.2 Alternative Method for Calculating and Using NSE Model Scale Factors.
C.2.1 Estimation of 10-pt piecewise linear interpolation of GBAS NSE autoland simulator histograms – GAST D.
An alternative method to use the NSE model is to compute, before launching any run of the Monte-Carlo autoland simulations, the distributions of the scale factors, Kvert and Kxtrk = Katrk. For these two last quantities, we conservatively allocate the worst horizontal sigma. These distributions have been computed using assumptions described in [11] and [12]. Then for each run of the Monte-Carlo simulations, we draw, from these two distributions, a Sigma_vert= Kvert and a worst horizontal sigma for Kxtrk and Katrk, at the beginning of the approach which are kept constant during the approach.
In order to facilitate the use of these distributions, the 10 point piecewise linear interpolation of Kvert and Kworst_horizontal_sigma are provided on the histograms and using a dual entry table X-axis corresponding to sigma_vert or worst_sigma_horizontal in meters.
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AC 20-191 Appendix C C.2.2 Sigma Vert.
Figure C-10 Sigma Vertical Samples Table C-8 Sigma_vert Values and Number of Occurrences Sigma_vert Number of occurrences
0.485000000000000 0.000429191837945
0.543326533206011 0.002257116788039
0.655055242735278 0.045229563598357
0.699018742036146 0.043376936959747
0.731326218889690 0.035448466874264
0.860858976217526 0.008926572687034
1.070145122860493 0.001211926592757
1.409975954322260 0.000073333137778
1.573104102697827 0.000010035060959
1.735000000000000 0.000008491205427 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05 nb of occurrences sigma vert (meters)
bins width = 0.01 m C-16
AC 20-191 A ppendix C C.2.3 Worst Horizontal Sigma.
As with the NSE noise scale factors, an alternative way of using the step function is to derive scale factors sigma_step_vert and worst_sigma_step horizontal to take into account steps induced by loss of tracking of satellites by the receiver due to geometry (rise/set of satellites) or due to satellite failures. The distributions and the 10 point piecewise linear interpolations are provided below for these two cases for the two dimensions.
Figure C-11 Worst Horizontal Sigma Samples 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.02
0.04
0.06
0.08
0.1
0.12nb of occurrences worst horizontal sigma (meters)
bins width = 0.01 m C-17
AC 20-191 Appendix C Table C-9 Sigma_Worst_Horizontal Values and Number of Occurrences Sigma_worst_horizontal Number of occurrences
0.005000000000000 0
0.257665870406134 0.001235856353506
0.293780182441118 0.021311381766140
0.341769349550846 0.107172907188037
0.398519584681962 0.038574002399152
0.489966724012817 0.005669037514146
0.501954763905412 0.003652762189125
0.733255508608094 0.000223859052165
1.294724573608414 0.000000771927766
1.705000000000000 0.000000771927766 C.2.4 Rise/set Sigma Step Vertical.
Figure C-12 Rise/set Sigma Step Vert. Error Samples 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.005
0.01
0.015
0.02
0.025
0.03
0.035nb of occurrences Rise/set - sigma step vert (meters)
bins width = 0.01 m C-18
AC 20-191 Appendix C Table C-10 Rise/Set Sigma_step_vert and Number of Occurrences Rise/Set Sigma_step_vert Number of occurrences
0.005000000000000 0.000102160167567
0.072435248412769 0.000251314012215
0.173069321361968 0.007640899466239
0.183661728498818 0.014862132304962
0.266851710720950 0.032471159430220
0.360116883106780 0.022183880145120
0.560386526870370 0.010022935263488
0.707103250848933 0.002542678888367
1.168108921989317 0.000590218235325
1.605000000000000 0 C.2.5 Rise/set Worst Sigma Step Horizontal.
Figure C-13 Rise/set Worst Sigma Step-horizontal Error Samples 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.01
0.02
0.03
0.04
0.05
0.06nb of occurrences Rise/set - worst sigma step hor (meters)
bins width = 0.01 m C-19
AC 20-191 Appendix C Table C-11 Rise/Set Worst Sigma_Step_Hor and Number of Occurrences Rise/Set Worst Sigma_Step_Hor Number of occurrences
0.005000000000000 0.000002897188944
0.041875996448845 0.004326488398194
0.054920944044943 0.005392271389607
0.109250543156205 0.058667264538146
0.163832898257929 0.050878995869548
0.205763067528231 0.030017730600579
0.339826367851692 0.005884118967452
0.481495931510671 0.001622467406624
0.537593983881708 0.001038030498786
1.425000000000000 0.000000090537154 C.2.6 Signal Loss Sigma step vertical.
Figure C-14 Signal Loss- Sigma Step Vert Error Samples 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.005
0.01
0.015
0.02
0.025nb of occurrences Signal Loss - sigma step vert (meters)
bins width = 0.01 m C-20
AC 20-191 Appendix C Table C-12 Signal loss Sigma_Step_Vert and Number of Occurrences Signal loss Sigma_Step_Vert Number of occurrences
0.005000000000000 0
0.175788354389016 0.000006172962943
0.341225896607212 0.001849289916083
0.501345966346395 0.016428884464905
0.509272880272156 0.014817130740563
0.585415172749149 0.024277512141267
0.634728223925139 0.019985074857159
0.971964045257760 0.004855442265040
0.993665191757176 0.004618156897366
1.565000000000000 0 C.2.7 Signal Loss Worst Sigma Step Horizontal.
Figure C-15 Signal loss worst sigma step hor. Samples 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04 nb of occurrences Signal Loss - worst sigma step hor (meters)
bins width = 0.01 m C-21
AC 20-191 Appendix C Table C-13 Signal Loss Worst Sigma_Step_Hor and Number of Occurrences Signal Loss Worst Sigma_Step_Hor Number of occurrences
0.005000000000000 0.000476007789371
0.115610839554299 0.025790440689983
0.180148721789614 0.034439075056633
0.320833114781513 0.015472307517217
0.586810140697307 0.003675003851236
0.880744610706960 0.000358424380494
3.908283843494504 0.000000000123437
3.975118428756177 0.000000000123437
4.601736622281720 0
4.995000000000000 0.000000016910875 C.3 References for GBAS Performance Model.
• ICAO NSP Mar 2009 WGW/WP 30 "SARPS Support for Airworthiness Assessments GLS Signal Modeling", presented by Tim Murphy, Bretigny, 17-27 March 2009.
• ICAO NSP May10 WGW/WP16, "Computation of Maximum Undetected Error for RRFM in Support of Airworthiness Assessments".
• ICAO NSP May 2010 WGW WP 19, " SARPS Support for Airworthiness Assessments - More on GLS Signal Modeling" Prepared by Tim Murphy.
• ICAO Paper GNSSP-WP-8, Validation of GBAS CAT I Accuracy: A GLS Model and Autoland Simulations for Boeing Airplanes, presented at the ICAO Global Navigation Satellite Systems Panel, Working Group B Meeting, Seattle, WA, May 29 - June 9, 2000.
• ICAO Standards and Recommended Practices (SARPs) for the Global Navigation Satellite System (GNSS). Annex 10 to the Chicago Convention, Vol 1.
• M. Harris, T. Murphy, "Geometry Screening for GBAS to Meet CAT III Integrity and Continuity Requirements", Proceedings of the Institute of Navigation International Technical Meeting 2007.
• Neri, P., Macabiau, C., Azoulai, L., "Study of a GBAS Model for CAT II/III Simulations", Proceedings of the 22nd International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009.
• RTCA / DO-245, Minimum Aviation System Performance Standards (MASPS) for the Local Area Augmentation System (LAAS).
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