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Hazardously Misleading Information Analysis for Loran LNAV
Dr. Ben Peterson, Peterson Integrated GeopositioningDr. Per Enge, Dr. Todd Walter, Dr. Sherman Lo and Lee Boyce
Stanford UniversityRobert Wenzel, Booz Allen Hamilton
Mitchell Narins, U. S. Federal Aviation Administration
Loran Integrity Performance Panel (LORIPP)Stanford University, July 24, 2002
Key Assumptions/Requirements• Integrity requirement is 99.99999% for all conditions/locations;
not an average, prove with analysis, not statistics• All-in-view receiver w/H field, software steered, antenna• TOE vice SAM control• Signal in space integrity > 99.99999%
– RAIM does not have to detect transmitter timing error
• Cross rate cancelled– Or blanked, but getting enough pulses to average a problem
• Modulation, if present, does not affect navigation performance• Integrity requirement met once at start of approach, then if
signal lost, receiver checks accuracy requirement • One time calibration of ASF, periodic validation by periodic
flight inspection, no real time airport monitors• Not an attempt to certify existing receivers
40 45 50 55 60 65 70 75 80 85
2
6
10
14
18
22
Noise - dB re 1 uv/m in 30 kHz NEBW
Loca
l tim
e of
day
Atmospheric Noise at Dana, Mean = 61.7 dB, Max = 76.2 dB,
WinterSpringSummerFall
95% Levels by Time of Day and Season of Year
34 dB
50 55 60 65 70 75 80 85 90
10-4
10-3
10-2
10-1
100
dB re 1uv/m in 30 kHz NEBW
1-C
umul
ativ
e di
strib
utio
n
Dana all seasons & times: 95% - 66.5dB, 99% - 73.7dB, 99.9% - 82.3dB
All time periodsSummer @1400
-15 -10 -5 0 5 10 1510
-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
SNR = 6dB
SNR = -6dB
Probability density of TOA for average over 500 pulses
usec relative to selected zero crossing
Typical Distributions of TOA MeasurementBlue - Low SNR, Red - High SNR
Prob
abil
ity
Den
sity
of
TO
A
Accuracy = fn(Phase
uncertainty)
Pcycle error = fn(Envelope uncertainty)
Phase Error Terms
• Noise terms– Transmitter jitter (6 meters, one )– Noise at the receiver
• Bias terms not correlated from signal to signal– Transmitter offset– Errors in predicting ASF (modeled as % of predicted ASF)
• ASF seasonal variation correlated from signal to signal
Number of Pulses Averaged
Stan
dard
Dev
iati
on o
f Ph
ase
Mea
sure
men
ts
Phase Measurement Error in usec Due to Noise and Interference
101
102
103
104
10-2
10-1
100
-15dB
-10dB
-5dB
0dB
5dB
10dB
Number of pulses averaged
Sta
ndar
d de
viat
ion
of p
hase
mea
sure
men
t-us
ec
SNR-dB in 30kHz NEBW
Accuracy of LORAN phase measurement vs averaging time and SNR
Gain realized by clipping 15% of the samples(Discrete points from: Enge & Sarwate, “Spread-Spectrum Multiple-Access
Performance of Orthogonal Codes: Impulsive Noise,” IEEE Tr. Comm., Jan. 1988.)
0 2 4 6 8 10 12 14 16 18-5
0
5
10
15
20
25
30
35
Gai
n du
e to
clip
ping
- d
B
Vd - dB
Min Avg Max
Gain due to clipping2 * Vd - 2.5
We are temporarily using 15dB
EXAMPLE OF LORAN SEASONAL ASF VARIATION CORRELATION
Correlation Coefficient Calculated Over 2.8 Years: 0.978
8970 M: Dana, IN, X: Seneca, NY, Y: Baudette, MN
8970-Y A-2 Data From PlumbrookAvg. = 50395.82 s.d. = 0.463
4.7
4.9
5.1
5.3
5.5
5.7
5.9
6.1
6.3
1/1
/1999
2/1
/1999
3/1
/1999
4/1
/1999
5/1
/1999
6/1
/1999
7/1
/1999
8/1
/1999
9/1
/1999
10/1
/1999
11/1
/1999
12/1
/1999
1/1
/2000
2/1
/2000
3/1
/2000
4/1
/2000
5/1
/2000
6/1
/2000
7/1
/2000
8/1
/2000
9/1
/2000
10/1
/2000
11/1
/2000
12/1
/2000
1/1
/2001
2/1
/2001
3/1
/2001
4/1
/2001
5/1
/2001
6/1
/2001
7/1
/2001
8/1
/2001
8970-X at PlumbrookAverage = 31373.44 S.D. = 0.311
2.3
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
1/1
/1999
2/1
/1999
3/1
/1999
4/1
/1999
5/1
/1999
6/1
/1999
7/1
/1999
8/1
/1999
9/1
/1999
10/1
/1999
11/1
/1999
12/1
/1999
1/1
/2000
2/1
/2000
3/1
/2000
4/1
/2000
5/1
/2000
6/1
/2000
7/1
/2000
8/1
/2000
9/1
/2000
10/1
/2000
11/1
/2000
12/1
/2000
1/1
/2001
2/1
/2001
3/1
/2001
4/1
/2001
5/1
/2001
6/1
/2001
7/1
/2001
8/1
/2001
-160 -140 -120 -100 -80 -6020
25
30
35
40
45
50
55
60
65
70
1.5
1.5
1.51.5
1.5
1.5
1.5
2.5
2.5
2.5
2.52.5
2.5
2.52.5
2.52.5
2.5
3.5
3.53.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
3.5
4.54.5
4.54.5
4.5
4.5
4.5
4.54.5
4.5
4.5
4.5
5.5
5.55.5
5.5 5.55.5
5.5
5.5
5.55.5
5.5
5.5
5.5
340ns/Mm
140ns/Mm
90ns/Mm
40ns/Mm
140ns/Mm
0ns/Mm
Regions of rate of seasonal variation in ASF(ns/Mm = nanoseconds/Megameter)
Bias due to seasonal asf variations in meters W = R-1, R = Rnoise + 0 x correlation of bias terms
0 10 20 30 40 50 60 70 80 90 100
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
50
Bias due to seasonal asf variations in meters W = R-1, R = Rnoise + 1 x correlation of bias terms
0 10 20 30 40 50 60 70 80 90 100
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
50
Loran Cycle Error Analysis compared to GPS RAIM
• Signal in space integrity better than 99.99999%– Allow for finite but small probabilities for
• Signal out of tolerance w/o blink
• Signal out of tolerance w/ blink & blink not detected
– Future effort to validate/quantify
• Algorithm detects receiver cycle selection failure (3,000 m) not Loran transmitter timing errors
• Large variation in reliability of cycle selection• Need to be able to detect multiple errors
101
102
103
104
105
10-1
100
101
-15dB
-10dB
-5dB
0dB
5dB
10dB
Number of pulses averaged
Sta
ndar
d de
viat
ion
of E
CD
mea
sure
men
t-us
ec
SNR-dB in 30kHz NEBW
Cycle Slip #1:
Envelope TOA Versus SNR and Averaging(Austron 5000 method, new technology may be 30% or more better)
Number of Pulses Averaged
Stan
dard
Dev
iati
on o
f E
CD
Mea
sure
men
ts-u
sec
Cycle slip #2: Calculation of Probability of cycle error (Pcycle)
Pcycle = red areas under curve = normcdf(-5, ECDbias, ) + normcdf(-5,- ECDbias, ) Where: = K/sqrt(N * SNR), K = 42 usec for Austron 5000 method,
present technology may be 3dB or more better N = number of pulses averaged, 1000 is used ECDbias = bound on constant errors such as propagation
uncertainty, receiver calibration, & transmitter offset
-10 -5 0 5 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Cycle slip #3: Loran Cycle Integrity Equations
G is the usual 3 x 3 matrix of direction cosines. The weighted least squares solution is:
xwls = (GT W G)-1 GT W y = K y
W is the weighting matrix given by W = R-1
R is the covariance matrix of the pseudorange errors
y is the pseudorange measurements, and
K (GT W G)-1 GT W
Predicted y = G xwls
Prediction error: w = y - predicted y = [I – G K] y
w = [I – G K] ( is the vector of pseudorange errors)
Cycle slip #4: Loran Cycle Integrity Equations
Positive definite test statistic:
WSSE = wT W w = T[I – G K]T W [I – G K] = T
Where
M [I – G K]T W [I – G K]
The expected distributions of WSSE are chi square with N-3 degrees of freedom for the non fault case and chi square with a non zero non-centrality parameter with N-3 degrees of freedom for the faulted case.
Cycle slip #5: LORAN Cycle Integrity Equations
For Pfalse_alarm = 10-3, Threshold = chi2inv(0.999, N-3)
Where N = # of signals
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
WSSE
Threshold
No Fault
Fault
Pmissed_detection Pfalse_alarm
Need to investigate tradeoffs among:
• Pmissed_detection
• Continuity
• Pfalse_alarm
• Frequency of cycle integrity calculation
• Correlation time of underlying errors
• Power of slip detector after a trusted fix
Cycle slip #5: Loran Cycle Integrity Equations
Pmissed_detection = ncx2cdf(threshold,N-3, )
(ncx2cdf = Noncentral chi-square cdf)
Where
= Mii * [300 * (10 – PhaseBiasi)]2
for a single cycle error on the ith signal
= Mii * [300 * (10 – PhaseBiasi)]2
+ Mjj * [300 * (10 – PhaseBiasj)]2
+/- 2 * Mij * [300 * (10 – PhaseBiasi)] * [300 * (10 – PhaseBiasj)
For a double cycle error on the ith & jth signals, +/- depends on relative signs of the cycle errors.
Cycle slip # 6: Loran Cycle Integrity Equations
Pwc = Pcycle (i) Pmissed_detection (i)
+ Pcycle (i) Pcycle (j) Pmissed_detection (i,j)
+ terms for 3 or more cycle errors
If N = 3, then Pwc = Pcycle (i)
Pwc must be < 10-7 - Probability that a signal was out of tolerance w/o blink - Probability that a signal was out of tolerance w/ blink and blink was not detected
j = i
i = 1:N
i = 1:N
Probability of undetected cycle error Pwc is probability error occurred x probability it was not detected summed over all possible combinations of errors
HPL #1:
Horizontal Protection Limit (HPL) Calculations
• If Pwc satisfies integrity criterion (i.e. we have > 99.99999% confidence in cycle selection and signal in space)– 1. Calculate one sigma noise contribution using weighted
least squares, multiply by 5.33– 2. Add vectors associated with phase bias terms for all
combinations of signs– 3. Calculate bias annual variation assuming correlation from
signal to signal– 4. Add terms in #2 & #3 assuming worst combination of
signs (analogous to absolute value in VPL)– 5. Add this to #1 linearly
-200 -100 0 100 200 300
-400
-350
-300
-250
-200
-150
-100
-50
0
50
5.33 x 0ne sigma noise error
HPL
HPL #3: Combining Bias and Noise in Calculation of HPL
Choose sign of bias term for
each pseudorange that maximizes HPL (red lines)
Bias term for seasonal variation
ECD Noise = 29 usec/sqrt(Nenv*SNR), 3dB better than Austron 5000Nenv = 4000, Nph = 500, Clipping Credit = 15dB
0 0.1 0.2 0.3 0.4 0.5
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
50HPL in nm w/max ASF errors = 0.3 x predicted, SNR threshold = -25 dB, Noise 99%, clipping cred 15 dB
0.1
0.10.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.20.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.20.2
0.2
0.2
0.2 0.20.2
0.2
0.30.3
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.60.6
0.6
0.6
0.6
0.6
0.6
0 0.1 0.2 0.3 0.4 0.5
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
50HPL in nm w/max ASF errors = 0.15 x predicted, SNR threshold = -25 dB, Noise 99%, clipping cred 15 dB
0.1
0.1
0.1
0.1
0.10.1
0.1
0.1
0.1
0.1
0.10.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.50.5
0.50.5
0.5
0.5
0.5
0.50.6
0.6 0.6
0.6 0.6
0.6
0.6
0.6
0.6
5 10 15 20 25 30 35 40 45 50
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
502drms accuracy not including ASF terms, SNR threshold = -20 dB, Noise 95%, clipping cred 15 dB
1515 15
15
151515
15
15
15
15
15
15
20
2020
20
20
20
20
20
2020
20
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50
5 10 15 20 25 30 35 40 45 50
-120 -110 -100 -90 -80 -70 -6025
30
35
40
45
502drms accuracy not including ASF terms, SNR threshold = -25 dB, Noise 95%, clipping cred 15 dB
10
10
15
15
15
20
20
20
20
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25 25
25
25
30
30
30
3035
35 35
3540
40 40
45
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50
50
5050
50
ECD Noise Sigma = 42 usec/sqrt(Nenv*SNR)Nenv = 4000, Nph = 500, Clipping Credit = 15dB
0
0.1
0.2
0.3
0.4
0.5
-165 -160 -155 -150 -145 -140 -135 -130 -12550
55
60
65
70
75
80HPL in nm w/max ASF errors = 0.3 x predicted, SNR threshold = -20 dB, Noise 99%, clipping cred 15 dB
0.1
0.10.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.20.2
0.2
0.3
0.3
0.3
0.3 0.3
0.3
0.30.3
0.3
0.3
0.3 0.3
0.4
0.4
0.4
0.4
0.4
0.4
0.40.4
0.4
0.4
0.4
0.5
0.5
0.5
0.50.50.5
0.5
0.5
0.6
0.6
0.6
0.6
0.6 0.60.6
0.6
0.6
Single Point Analysis
• The software then permits the user to click on a particular point of interest– Shows plots of stations used, noise an ASF’s
– Analyzes Pwc and HPL with signals removed one at a time
– 2nd version analyzes weighted vs unweighted test statistics
Example where removing single station helps integrity calculation
-170 -160 -150 -140 -13045
50
55
60
65
70
1 Williams L
2 Shoal Cove
3 Port Hardy
4 St. Paul
5 Port Clarn
6 Kodiak
7 Tok
61N, 148.75W
0 2 4 6 810
0
101
102
103
Signal #
Met
ers
Black - noise sigma, Green - ASF bound
0 2 4 6 810
-80
10-60
10-40
10-20
100
Pwc with one signal removed
Signal # removed0 2 4 6 8
340
360
380
400
420
440
460HPL (assuming correct cycles) with one signal removed
Signal # removed
Met
ers
Alaska w/one station removed at a time and using best combination
0
0.1
0.2
0.3
0.4
0.5
-165 -160 -155 -150 -145 -140 -135 -130 -12550
55
60
65
70
75
80min HPL in nm w/max ASF errors = 0.3 x predicted, SNR threshold = -20 dB, Noise 99%, clipping cred 15 dB
0.1
0.10.1
0.1
0.10.1
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.6
Weighted vs Unweighted Test Statistics• In GPS, we want to detect a ranging error large enough to cause
a significant position error. If a particular SV is weighted out of the solution, using a weighted RAIM test statistic makes sense because even if that particular error is large, we don't care.
• In Loran integrity analysis we are trying to detect cycle errors of 3,000m. These don’t show up in weak stations when using a weighted test statistic.
• The expected unweighted test statistic is not chi square with N-3 degrees of freedom, but that of the sum of normal rv's with different variance or a convolution of chi square distributions each with one degree of freedom & different scale parameters.
Example where cycle error detectability enhanced with unweighted test statistic
-180 -170 -160 -150 -140 -13040
50
60
70
1 Williams L
2 Shoal Cove
3 Port Hardy
4 St. Paul
5 Attu
6 Port Clarn
7 Kodiak
8 Tok
61.25N, 152.75W
0 2 4 6 810
0
101
102
103
Signal #
Met
ers
Black - noise sigma, Green - ASF bound
100
101
102
10-5
100
Praim(1 1) = 1.3156e-006, Pcycle = 0.038436, WSSE: No Fault-Black, Fault-Blue, Pmd = 0, SSE: No Fault-Red, Fault-Green, Pmd = 0
WSSE/SSE
Threshold
Example of Bad Detection Geometry
-100 -90 -80 -7030
35
40
45
1 Malone 2 Grangevlle
3 Baudette
4 Dana
5 Seneca
42.25N, 87W
1 2 3 4 510
0
101
102
103
Signal #
Met
ers
Black - noise sigma, Green - ASF bound
100
101
10-4
10-3
10-2
10-1
100
Praim(3 3) = 5.0741e-005, Pcycle = 5.6117e-005, WSSE: No Fault-Black, Fault-Blue, Pmd = 0.86396, SSE: No Fault-Red, Fault-Green, Pmd = 0.84492
WSSE/SSE
Threshold
Conclusions to this Point I
• We are quite confident that Loran will be able to provide RNP 0.3 integrity over virtually all of CONUS and much of Alaska– To get availability north of Brooks Ranges requires
additional transmitter, probably @ Prudhoe Bay
– Because main limit is cycle integrity, RNP 0.5 and RNP 0.3 availability/coverage not significantly different
Conclusion to this Point II
• Key assumptions– Analysis assumes ASF error is 30% of whole value.
– Most likely way to implement is one time calibration of each airport.
– Periodic validation by periodic flight inspection.
– Temporal variation not needed
– Early airports will need more intense calibration.
– With experience, later airports will need no more than a one time calibration (and perhaps less).
Where do we go from here?• Validate/revise each part of the analysis, assumption,
parameter, etc.– Credit for impulse nature of noise
• Revised noise model for RF simulator
– Sensitivity to size of ASF error– Bounds on ASF estimates, transmitter timing offsets, ECD
predictions, transmitted ECD errors– Bounds on probability of signal out of tolerance w/o blink,
probability of missed blink detection– Averaging time constants in receivers – Investigate areas that are counter-intuitive– Implement algorithms in receiver/validate actual performance
Where do we go from here? -2• Can we do better in either integrity of accuracy by
elimination of some signals from the solution?– If so, what is criteria for eliminating signals?
• Would an unweighted test statistic give better detectability of cycle errors and thus better availability?
• Use the analysis software to see where we need to allocate effort to get the availability we need– How far down do we need to beat the bounds on ASF errors?
– Are more stations required?
– User receiver performance
– Transmitter performance
• Establish work plan for LORIPP
• Maintain list of new monitoring requirements