AC 20-191 Appendix B Figure B-7 Cumulative probability of reported Mean Wind, and Head Wind, Tail Wind Cross Wind Components when landing.
403020100 Total Wind Head Wind ComponentCross Wind Component Tail Wind Component
WIND SPEED KNOTS Cumulative probability of reported Mean Wind, and Head Wind, Tail Wind Cross Wind Components, when landing.
P R O B A B I L I T Y O F E X C E E D I N G G I V E N W I N D S P E E D
0.01
0.05
0.1
0.2
0.5 1 2 8 10 20 30 40 50 60 70 80 90 95 98 99
99.8
99.9
99.99 NOTE: This data is based on world wide in-service operations of UK airlines (sample size about 2000) B-12
AC 20-191 Appendix B B.5 Acronyms and Definitions for the Wind Models.
h Height AGL in ft.
Lu Turbulence scale factor RMS Root Mean Square VT Aircraft speed in ft/sec.
Vwref Mean wind speed measured at h ft V20 Mean Wind speed measured at 20 ft. AGL w Temporal frequency in radians/sec, σw RMS turbulence scale in knots W Spatial frequency in radians/ft.
Φw(Ω) Spectral density of the vertical component of turbulence in (ft/sec)2 B-13
AC 20-191 Appendix C APPENDIX C. GBAS PERFORMANCE MODEL FOR APPROACH AND LANDING
SIMULATION C.1 GBAS Performance Analysis.
In carrying out the performance analysis the following model of GBAS Performance may be used. The GBAS Performance Model simulates the outputs of a fault-free GBAS airborne receiver when used in conjunction with a GBAS ground station categorized as either Facility Approach Service Type (FAST) C or FAST D.
C.1.1 GBAS Performance Model.
C.1.1.1. The architecture of the GLS Model is illustrated in Figure C-1. The GLS Model includes a Navigation System Error (NSE) Generator which generates NSE representative of a GBAS providing approach service type C or D as defined by the requirements [1, 2, 3]. The Position Calculator adds NSE to the true position of the GLS Reference Point (GRP). The deviation calculator computes the deviations of the GRP given the Final Approach Segment (FAS) data. A latency model is applied to each output of the GLS Model.
C.1.1.2. Development of all components of the GLS Model is documented in [4,5] and [6]. The NSE Generator and NSE Step Generator are discussed below.
C.1.2 GBAS NSE Generator.
The GBAS NSE Generator produces NSEs in the along-track, cross-track and vertical directions in meters. The block diagram of the GBAS NSE Generator is shown in Figure C-2. The Gaussian White Noise (GWN) Generator produces three independent noise sequences with zero mean and unity variance. Each sequence is filtered by a second-order Butterworth filter. The compensation gain which brings the RMS of the filtered noise back to unity is obtained as:
2 2Gx = ∆Tωn [1] The filter output is scaled by NSE scale factors Katrk, Kxtrk, and Kvert. At the beginning of each run, the NSE generator filter must be initialized at a value sampled from a Gaussian distribution consistent with these scale factors.
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AC 20-191 Appendix C Figure C-1 GBAS Signal Model NSE Generator GRP Position Calc Deviation Calc Deviations GRP Pos (LLH) GRP
P (ECEF) True Ant Pos FAS Data, GERP Flag, Dev Scale Flags FAS Attitudes, Body Rates, GRP Offset, Offset Flag GAST Seed & ∆ T NSE Horiz Dist to LTP Latency Latency Latency Figure C-2 GLS NSE Generator Σ Σ Σ GAST Limit Case Fault Mode Generator Step Event Time Malfunction Event Time Model Type (nominal vs.
Limit Geometry) Compensation Gain Compensation Gain Compensation Gain 2nd Order Filter 2nd Order Filter 2nd Order Filter GWN Generator
Seq1 Seq2 Seq3 Seed , ∆ T K Value Selector X X X σ =1 σ =1 σ =1 NSE atrk NSE xtrk NSE vert
K atrk K xtrk K vert Step Function Generator Σ Σ Σ C-2
AC 20-191 Appendix C C.1.3 Second-Order Filter of GBAS NSE Generator.
The second order filter to be implemented in the GBAS NSE Generator is characterized by:
ω 2 H (s)= n [2] s2 + 2ωn s+ω 2n where ωn is the natural frequency given by For GAST C: ωn= 0.01 rad/sec For GAST D: ωn = 0.033 rad/sec C.1.4 Noise Scale Factor.
The model accounts for the variation in accuracy due to satellite geometry by setting the noise scale factor to a constant which is sampled from a distribution. For each run, the value of Kvert is first determined by selecting a sample, x, from a uniform distribution between 0 and 1. The value of Kvert is then given by the following function of x:
Kvert f(x) a= = a1 + a2 x− 3 [3] x−1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-1.
Table C-1: GAST Dependent Parameters for Kvert Service Type a1 a2 a3 Kvert_max GAST C 0.4 0.2 0.006 10/5.762=1.736 GAST D 0.52 0.47 0.005 10/5.762=1.736 C.1.4.1 If the random pick from a distribution between 0 and 1 results in K vert >Kvert_max, then the value should be discarded and another sample should be selected from the uniform distribution set to the maximum value from the table 2. An alternative acceptable means for computing the NSE scale factors is given in section C.2 below.
C.1.4.2 For each run, the value of K xtrk is determined by selecting a sample, x1, from a uniform distribution between 0 and 1. For each run, the value of Katrk is determined by selecting a sample, x2, from a uniform distribution between 0 and 1. The cross-track and along track scale factors are then computed by:
3This corresponds to the case where VPL>10m and therefore the system is not available.
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AC 20-191 Appendix C aK f ( ) = 3xtrk = x1 a1 + a2 x1 − [4] x1 −1 = ( ) aK f x 3atrk 2 = a1 + a2 x2 − [5] x2 −1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-2:
Table C-2: GAST Dependent Parameters for Kxtrk and Katrk Service Type a 1 a2 a3 Kxtrk_max, Katrk max GAST C 0.2 0.1 0.003 40/5.762=6.942 GAST D 0.21 0.12 0.003 17/5.762=2.951 C.1.4.3. If the random pick from a distribution between 0 and 1 results in Kxtrk>Kxtrk_max or Kaxtrk>Kaxtrk_max, then the value should be discarded and another sample should be selected from the uniform distribution.
C.1.5 NSE Step Generator.
The NSE step generator is illustrated in Figure C-3. Step errors occur when the satellite geometry changes, either due to the removal of an individual satellite from the position solution (e.g. a satellite fails and stops transmitting or the user receiver stops tracking a satellite for any reason) or due to an individual satellite rising or setting. The step generator produces representative NSE step errors in the vertical, along-track and crosstrack directions in meters. This is accomplished by scaling a unit step function by factors that are derived from representative statistical distributions. First, three random samples, one for each axis, are selected from a zero mean unit variance normal distribution. Then these samples are multiplied by scale factors that are chosen to simulate the statistical variation in the size of an error that would result from normal variations in the relative geometry between the user and the satellites. Finally, the resultant constant factors are multiplied with a unit step function time sequence.
Injecting one step in each simulated approach at a random time should reasonably model the effects of geometry changes.
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AC 20-191 Appendix C Figure C-3 NSE Step Generator
NSE Step_atrk NSE Step_xtrk NSE Step_vert GWN Generator
Seq Scale Factor Computation X X X K Step_atrk K Step_xtrk K Step_vert Unit Step Function Generator Uniform Random Number Generator Step Event Time X X X Seed Seed x Kstep = f(x) One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 t = event time 0 t = event time 0 One sample from Uniform distribution, 0≤x≤1 C.1.6 NSE Step Generator Scale Factor Computation.
For each run, the value of σ step_vert is determined by selecting a sample, x, from a uniform distribution between 0 and 1. The value of σstep_vert is then given by the following function of x:
( ) bσ step _ vert = f x = b1 + b2 x− 3 [6] x−1.01 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-3.
Table C-3: GAST Dependent Parameters for σ step _ vert Service Type b 1 b2 b3 max_ _vertstepσ GAST C 0.4 0.8 0.07 FASVAL/2 GAST D 0.5 0.8 0.05 3.5
C.1.6.1. If the random pick from a distribution between 0 and 1 results in σ step_vert >σstep_vert_max, then the value should be discarded and another sample should be selected from the uniform distribution.
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AC 20-191 Appendix C C.1.6.2. For each run, the value of σ step_xtrk is determined by selecting a sample, x1, from a uniform distribution between 0 and 1. The value of σstep_atrk is determined by selecting a sample, x2, from a uniform distribution between 0 and 1. The cross-track and along track NSE step scale factors are then computed by:
σ step _ xtrk = f( bx1)= b + b x− 31 2 1 [7] x1 −1 σ step trk ( ) b_ a = f x2 = b1 + b 32 x2 − [8] x2 −1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-4.
Table C-4: GAST Dependent Parameters for σ step _ xtrk or σ step _ atrk Service Type b 1 b2 b3 max_ _axtrk stepσ GAST C 0.32 0.32 0.05 20m GAST D 0.35 0.35 0.35 5.5m C.1.6.3. If the random pick from a distribution between 0 and 1 results in σstep_xtrk >σstep_xtrk_max, or σstep_atrk>σstep_axtrk_max, then the value should be discarded and another sample should be selected from the uniform distribution.
C.1.6.4. The scale factor, Kstep_vert, of the magnitude of the vertical step error is then selected from a normal distribution with 0 mean and a standard deviation of σstep_vert. The scale factor, Kstep_xtrk, of the cross-track deviation is selected from a normal distribution with 0 mean and a standard deviation of σstep_xtrk.
C.1.7 Latency Model.
The latency of the GLS output should be delayed for 400 msec.
C.1.8 Fault Mode Generator.
The fault mode generator as depicted in Figure C-4 can be used for modeling undetected, non-aircraft system error conditions (monitor “limit” cases) or any worstcase undetected error (monitor “malfunction” cases). The fault mode generator produces a ramp error with characteristics as illustrated in Figure C-5. The effect of a malfunction is modeled as a ramp, with a start time, a ramp rate and a total exposure time, Tmax. The maximum value of the ramp depends on the ramp rate and time to alert.
The ramp is assumed to increase to the level of the maximum value and then to exceed that value for a period equal to the time-to-detect and mitigate the failure. The erroneous satellite is isolated and the error returns to the nominal value (i.e. the fault error is set to zero). The model may alternatively produce step errors where the C-6
AC 20-191 Appendix C maximum change in error due to the step is specified rather than the ramp rate. (See reference [7] for more details regarding GLS fault modeling).
Figure C-4 Limit or Malfunction Generator or both NSE LMF_atrk NSE LMF_xtrk NSE LMF_vert GWN Generator
Seq Set Limits X X X K Limit_atrk K Limit_xtrk K Limit_vert Unit Ramp Function Generator Uniform Random Number Generator Ramp Event Time X X X Seed Seed x Kother = f(x) One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 t = event time 0 e 0 One sample from Uniform distribution, 0≤x≤1 Limit or Malfunction Axis to Apply Ramp Rate Ramp Duration From Figure C-5, it can be seen that for the ramp:
EffectiveVALTMAX = TTD+ [9] RR EMAX and the Effective VAL depend on the type of malfunction. For Satellite Ranging sources, the Effective VAL is a function of the maximum error allowable with probability greater than 1x10-9 by the Pmd performance constraint with conditional probability (ICAO Annex 10 appendix B Section 3.6.7.3.3.3 ), (i.e., 1.6 meters) multiplied by the geometry screening limit.
EMAX =1.6⋅ Svert _ max [10] Where:
Svert _ max is the maximum vertical projection for any satellite allowed by geometry screening. This airframer limits the size of EMAX by specifying a maximum Svert for satellites used in the position solution as described in reference [8].
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AC 20-191 Appendix C C.1.8.1. For ground segment reference receiver failures, the effective VAL will depend on the geometry screening applied in the airborne equipment. If no additional geometry screening is applied other than VPL<VAL, the maximum effective VAL is 9.35 [m] (see Table C-5). If additional geometry screening is applied, a lower effective VAL may result. Reference [8] explains how to compute the effective VAL given additional geometry screening. Figure C-6 shows a plot of maximum vertical and lateral errors as a function of vertical and lateral Alert Limit screening. The calculations to produce the plot in Figure C-6 are described in detail in Reference [9].
Figure C-5 Malfunction Transient time Position Error on Axis Of Choice Effective VAL TTD Slope Characterized by Ramp Rate [ft/s] Total Exposure Time - Tmax EMAX C.1.8.2. For ionospheric anomalies, the maximum vertical error Emax is limited to a specified maximum allowable position error for the airborne installation for each axis, vertical (maxEV) and lateral (maxEL), as a part of the satellite geometry screening in the avionics [10]. These values along with broadcast information provided by the ground station determine the geometry screening.
C.1.8.3. Table C-5 and Table C-6 give the characteristics for transient errors in the vertical and horizontal directions respectively for each of the three major identified fault types.
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AC 20-191 A ppendix C Table C-5 Malfunction Transient Characteristics in the Vertical Direction Fault Type Service Type Ramp Rates [m/s] Effective VAL [m] Emax [m] Time to Detect (TTD) [s] Ranging Source Failures GAST C 0 - ∞ 10 Dependent 6 GAST D 0 - ∞ 1.6 × Svert Dependent 2.5 Iono Anomaly GAST C 0 – 4 N/A N/A N/A GAST D 0 – 4 N/A maxEV
N/A Single Ref Receiver Failure GAST C 0 - ∞ 10 Dependent 6 GAST D 0 - ∞ 9.35 [note] Dependent 2.5 Note: This value is an absolute worst case assuming no additional geometry screening is afforded based on reference receiver fault monitoring using TBAC. Only geometry screening of VPL<VAL=10m is assumed here. Smaller maximum values can be obtained by using additional geometry screening per reference [9].
Table C-6 Malfunction Transient Characteristics in the Lateral Direction Fault Type Service Type Ramp Rates [m/s] Effective LAL [m] Emax [m] Time to Detect (TTD) [s] Ranging Source Failures GAST C 0 - ∞ 40 Dependent 6 GAST D 0 - ∞ 1.6 × Slat Dependent 2.5 Iono Anomaly GAST C 0 – 4 N/A N/A N/A GAST D 0 – 4 N/A maxEL
N/A Single Ref Receiver Failure GAST C 0 - ∞ 40 Dependent 6 GAST D 0 - ∞ 35.9 [note] Dependent 2.5 Note: This value is an absolute worst case assuming no additional geometry screening is afforded based on reference receiver fault monitoring using TBAC. Only geometry screening of LPL<LAL=40m is assumed here. Smaller maximum values can be obtained by using additional geometry screening per reference [9].
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AC 20-191 Appendix C Figure C-6 Maximum Error (E) and Monitor Threshold (TBAC) as a function of Alert Limits for Various Numbers of Ground Reference Receivers (M).
C.1.9 Compatibility with Rare Undetected Non-Aircraft System Error Conditions.
Demonstration of compliance with section 5.2.9.6, of this AC 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 aircraft landing outside of the safe landing box as defined in section 5.2.10.1.6 is less than 1x10-5. 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 a probability of an unsuccessful landing given a particular size of error PUL|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 an 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:
𝑃𝑃𝑈𝑈𝑈𝑈(𝐸𝐸) =𝑃𝑃 −𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸)∙𝑃𝑃𝑚𝑚𝑚𝑚(𝐸𝐸) < 10 5 [11] To form the conditional unsuccessful landing probability,𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸), a conditional touchdown distribution must be used that would result from a constant bias error in C-10