APPENDIX 1 TO AMC AWO.B.CATIII.115 Performance demonstrations

1 Limit case analysis

Demonstration of compliance with paragraph 4.1 of AMC AWO.B.CATIII.115 may be done by analysis to show that for all possible sizes of navigation error, the joint probability that the error is not detected and that the error results in the aeroplane landing outside the safe landing box as defined in CS AWO.A.ALS.106 is less than 1 × 10–5. The analysis uses the nominal touchdown distributions (lateral and longitudinal) along with the geometry factors (Svert and Slat), and the maximum allowable Pmd performance of the monitors for satellite ranging source failures and for the reference receiver fault monitor (RRFM). The nominal touchdown distribution is used to compute the probability of an unsuccessful landing given a particular size of error P_(UL|E)(E) . This probability is then multiplied by the probability of an error not being detected as a function of E, Pmd(E). The probability of an unsuccessful landing given in error is the joint probability that the fault that causes an error, E, is not detected and the landing will be unsuccessful given an error, E:

P_(UL)(E)=P_(UL|E)(E)∙P_(md)(E)<10⁻⁵ [1]

To form the conditional unsuccessful landing probability , P_(UL|E)(E) , a conditional touchdown distribution should be used that would result from a constant bias error in addition to the fault-free NSE and flight technical error distributions. This should 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 P_((UL|E)(E))=∫_(-∞)^(LSC) p_(TSE_LON|E)(x,E)dx [2]

Land long P_(UL|E)(E)=∫_(LLC)^(∞) p_(TSE_LON|E)(x,E)dx [3]

Land with wheels less than 5 ft from the edge of the runway:

P_(UL|E)(E)=∫_(RWE-GW/2)^(∞) p_(TSE_LAT|E)(x,E)dx+∫_(-∞)^(-RWE+GW/2) p_(TSE_LAT|E)(x,E)dx [4] where:

LSC is the land-short criteria (i.e. 200 ft);

LLC is the land-long criteria (i.e. 3 000 ft);

RWE is the lateral landing criteria (i.e. 70 ft);

GW is the lateral distance between the main landing gear;

p_(TSE_LON|E)(x,E) is the probability density function for the longitudinal touchdown given a bias of magnitude E; and p_(TSE_LAT|E)(x,E) is the probability density function for the lateral touchdown given a bias of magnitude E.

Note: Care should be taken to ensure consistency of units when making these calculations.

1.1 Computing Pmd for ranging source errors

A bound on the probability of missed detection for the ranging source error, P_(md)(E_(R)) , is defined by the performance constraint region given in ICAO Annex 10 Appendix B Section 3.6.7.3.3.2. The Pmd performance should lie below the curve defined by Table B-76A in the SARPs, repeated here for convenience.

[Figure or form omitted from this preview — available in the Avioverse workspace library.]

Source transcription: Probability of missed detection / Pseudo-range error (metres):

P_md_limit ≤ 1 for 0 ≤ |E_r| < 0.75.

P_md_limit ≤ 10^(−2.56 × |E_r| + 1.92) for 0.75 ≤ |E_r| < 2.7.

P_md_limit ≤ 10^(−5) for 2.7 ≤ |E_r| < ∞.

Table B-76 A: Pmd_limit parameters

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(|S_(Apr_(vert).i)|) , may be used to determine the resulting limit on P_(md)(E_(V)) by substituting E_(R)= E_(V)/(max(|S_(Apr_(vert).i)|)) .

Figure 1 illustrates the relationship between Pmd_limit and the P_(md)(E_(V)) for max(|S_(Apr_(vert).i)|) = 5.

[Figure or form omitted from this preview — available in the Avioverse workspace library.]

Figure 1: Example of the satellite ranging source Pmd in the range domain and position domain

1.2 Computing Pmd for reference receiver fault monitoring

The Pmd for the RRFM is given by:

[Figure or form omitted from this preview — available in the Avioverse workspace library.] (P_mdApprox(E_V) = ∫ from 0 to T_BAC p_Bmd(x, E_V) dx) [5] where:

[Figure or form omitted from this preview — available in the Avioverse workspace library.] (T_BAC) is the maximum threshold for the RRFM monitor given by:

[Figure or form omitted from this preview — available in the Avioverse workspace library.] (T_BAC = 5.5 × √((0.0842 × VAL / √(m − 1))² + 0.4²)) Metres [6] where:

VAL is the vertical alert limit that is used by airborne equipment to screen geometry expressed in metres.

And pBmd(x,EV) is the probability density function (pdf) of |Bj,vert(EV)| in the faulted circumstance given by:

[Figure or form omitted from this preview — available in the Avioverse workspace library.] (p_Bmd(x, E_V) = dnorm(x, E_V, 0.0842 × VAL / (√(M − 1) × √M)) + dnorm(−x, E_V, 0.0842 × VAL / (√(M − 1) × √M)) for x ≥ 0; p_Bmd(x, E_V) = 0 for x < 0) [7] where dnorm(x,μ,σ) is the Gaussian pdf

[Figure or form omitted from this preview — available in the Avioverse workspace library.] (dnorm(x, μ, σ) = exp(−(x − μ)² / (2σ²)) / (√(2π) × σ)) [8] For a derivation of these expressions, see reference [¹] of Appendix 1 to AMC to Subpart A.

1.3 Example assessments

Figure 2 illustrates a landing-short assessment for a hypothetical aeroplane with a nominal longitudinal touchdown point of 1 500 ft from the threshold and a dispersion that can be bounded by a Gaussian distribution with σ = 220 ft. Also, a max(|S_(Apr_(vert).i)|) of 5, VAL of 10 metres and GPA of 3 degrees is used. Rearranging equation [1]:

P_(UL|E)(E)<10⁻⁵/(P_(md)(E)) [9]

Hence, by dividing 10–5 by the Pmd curves for satellite ranging sources and RRFM, the grey ‘keep-out regions’ shown in Figure 2 can be obtained. The assessment is then simple. If the curve for P_(UL|E)(E) does not enter the keep-out regions, then the requirement that P_(UL)(E)<10⁻⁵ is met for all values of E.

An alternative approach to the analysis is illustrated in Figure 3 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 straightforward.

[Figure or form omitted from this preview — available in the Avioverse workspace library.]Figure 2: Example assessment of landing-short performance

[Figure or form omitted from this preview — available in the Avioverse workspace library.]Figure 3: Explicit calculation of PUL for the land-short example above

[Issue: CS-AWO/2]

¹ ICAO Standards and Recommended Practices (SARPs) for the Global Navigation Satellite System (GNSS). Annex 10 to the Chicago Convention, Vol 1.