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

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/1999

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/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

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/1999

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/1999

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/1999

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/1999

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/2000

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11/1

/2000

12/1

/2000

1/1

/2001

2/1

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3/1

/2001

4/1

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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

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0.20.2

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0.2 0.20.2

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0.30.3

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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

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0.1

0.10.1

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0.6

5 10 15 20 25 30 35 40 45 50

-120 -110 -100 -90 -80 -70 -6025

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502drms accuracy not including ASF terms, SNR threshold = -20 dB, Noise 95%, clipping cred 15 dB

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502drms accuracy not including ASF terms, SNR threshold = -25 dB, Noise 95%, clipping cred 15 dB

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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

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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

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0.3

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0.3 0.3

0.3

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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

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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

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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

pdf

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

pdf

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