On Difference Variances as Residual Error Measures in Geolocation

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On Difference Variances as Residual Error Measures in Geolocation Victor S. Reinhardt Raytheon Space and Airborne Systems El Segundo, CA, USA ION National Technical Meeting January 28-30, 2008 San Diego, California x(t n ) Trajectory Res Error t x(t n ) x(t n +) ()x(t n ) 

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### On Difference Variances as Residual Error Measures in Geolocation

Victor S. ReinhardtRaytheon Space and Airborne SystemsEl Segundo, CA, USA

ION National Technical MeetingJanuary 28-30, 2008San Diego, California

x(tn)

Trajectory

ResError

t

x(tn)

x(tn+)

()x(tn)

Two Types of Random Error Variances Used in Navigation
• Residual error (R) variancesare used in measuringgeolocation error

= Mean Sq (MS) ofdifference between position or time data x(tn) and a trajectory est from data

• Mth order difference() variances used inmeasuring T&F error

 MS of Mth order difference of data x(tn) over 

• 1st order difference ()x(tn) = x(tn+) - x(tn)

x(tn)

Trajectory

ResError

t

()2x(tn)

()x(tn)

()x(tn+)

Two Types of Random Error Variances Used in Navigation
• Residual error (R) variancesare used in measuringgeolocation error

= Mean Sq (MS) ofdifference between position or time data x(tn) and a trajectory est from data

• Mth order difference() variances used inmeasuring T&F error

 MS of Mth order difference of data x(tn) over 

• 1st order difference ()x(tn) = x(tn+) - x(tn)
• MS of ()2x(tn) Allan variance (of x)
• MS of ()3x(tn)  Hadamard variance (of x)
R-variances not known for good convergence properties
• When negative power law (neg-p) noise is present
• Neg-p noise  PSD Lx(f)  f p with p < 0
• p generally -1, -2, -3, -4 for T&F sources
• R-variances are the proper residual error measures in geolocation
• Despite any such convergence problems
• Address statistical questions being posed
• -variances known for good convergence properties when neg-p noise present
• But -variances don’t seem to relate to residual geolocation error as R-variances do
Paper Will Show
• -variances do measure residual geolocation error under certain conditions
• Mainly when an (M-1)th order polynomial is used to estimate the trajectory
•  & R variances equivalent for these conditions
• R-variances do have good convergence properties for neg-p noise
• Because trajectory estimation process highpass (HP) filters the noise in the data
• True under more general conditions than for equivalence between  & R variances

x(tn)

TrueTrajectoryxc(t)

Model FnEst xw,M(t,A)

True Noisexp(t)

t

N Data Samples over T = N∙o

Residual Errors in Geolocation Problems
• Have N data samples x(tn) over interval T
• Data contains a (true) causal trajectory xc(t)that we want to estimate from the data
• And data also contains neg-p noise xp(t)
• Assume we estimate xc(t) by fitting
• A model function xw,M(t,A) to the data
• Through adjustment of M parameters A = (ao,a1,…aM-1)

x(tn)

ObservableRes (R) Error xj(tn)

t

N Data Samples over T = N∙o

Observable Residual (R) Error (of Data from Fit)

xj(tn) = x(tn) - xw,M(tn,A)

• Define point R variance at x(tn)  E{xj(tn)2}
• E{…} = Ensemble average over random noise
• Average R variance x-j2 Average of E{xj(tn)2} over N samples
• Average can be uniformly or non-uniformly weighted (depending on weighting used in fit)

xc(t)

xw,M(t,A)

x(tn)

True Function(W) Error xw(tn)

ObservableRes (R) Error xj(tn)

t

N Data Samples over T = N∙o

xw(tn) = xw,M(tn,A)- xc(tn)

• True measure of fit accuracy but not observable from the data
• Point W variance  E{xw(tn)2}
• Average W variance  x-w2

xc(t)

xw,M(t,A)

Precise Definition of Mth Order-Variance for Paper
• Overlapping arithmetic average of square of ()Mx(tn) over data plus E{…}
• Not discussing total or modified averages
• M  All orders equal for white (p=0) noise
• Can show Mth order -Variance HP filters Lx(f) with 2Mth order zero (at f = 0)

x,1()2  MS Time Interval Error  2nd Order zero

x,2()2  Allan variance of x  4th Order zero

x,3()3  Hadamard var of x  6th Order zero

-Variancesas Measures of Residual Error in Geolocation
• Can prove for N = M + 1 data points that
• R-variance = Mth order -Variance with  = T/Mx-j2 = x,M(T/M)2when
• xa,M(t,A) is (M-1)th order polynomial
• Uniform weighted Least SQ Fit (LSQF) is used
• x-j2 “unbiased” MS ( sum sq by N – M)
• Well-known for Allan variance of x
• Equivalent to 3-sample x-j2when time & freq offset (1st order polynomial in x) removed
• Hadamard variance of x equivalent to
• 4-sample x-j2 when time & freq offset & freq drift (2nd order polynomial in x) removed

Errors vs N(M=2)

f0 Noise

2

RMS{xj}

1

x-w

0

1

10

100

1K

Samples N 

N

2

x-w

f -2 Noise

N-M

x-j2(N)  x,M(T/M)2

1

RMS{xj}

0

2

f -4 Noise

x-w

1

RMS{xj}

0

 N = M+1

Can Extend Equivalence to Any N as Follows
• “Biased” x-j RMS{xj}
• “Biased”   sum sq by N
• Can show RMS{xj}  Constant as N varies (while T remains fixed)
• Thus for “unbiased” x-j2
• Exact relationship exists for each p & N
• Similar Allan-Barnes “bias” functions for Allan variance
Consequences of Equivalence Between  & R Variances
• -variances measure res geolocation error
• When xw,M(t,A) is poly & uniform LSQF used
• For non-uniform weighting (Kalman?) x,M(Teff/M) should also be estimate of x-j
• Teff Correlation time for non-uniform fit
• Don’t have to remove xw,M(t,A) from data if use x,M(Teff/M) to estimate x-j
• Because ()Mxw,M(t,A) = 0when xw,M(t,A) = (M-1)th (or lower) order poly
• Explains sensitivity of Allan variance to causal frequency drift & insensitivity of Hadamard to such drift
HP Filtering of Noise in R-Variances
• Paper proves fitting process
• HP filters Lx(f) in R-variances
• HP filtering order depends on complexity of model function xw,M(tn,A) used
• R-variances guaranteed to converge if free to choose model function
• True under very general conditions
• Fit solution is linear in data x(tn)
• Fit is exact solution when no noise is present & xw,M(tn,A) is correct model for xc(tn)
• True even when x,M() not measure of x-j
• Applies to any weighting, LSQF, Kalman, …

long term error-1.xls

Fit Solutions for Various p

f0 Noise

f0 Noise

x(tn)

x(tn)

f -2 Noise

f -2 Noise

xj

xj

xw

xw

f -4 Noise

f -4 Noise

xc

xc

xa,M

xa,M

T

• For white (p=0) noise the fit behaves in classical manner
• As N   xw,M(tn,A)  xc(tn)& x-w 0
• Again T is fixed as N is varied
• But for neg-pnoise
• As N  xw,M(tn,A) not  xc(tn)
• Because fit solution necessarily tracks highly correlated low freq (LF) noise components in data

long term error-1.xls

Fit Solutions for Various p

f0 Noise

x(tn)

f -2 Noise

f -2 Noise

xj

vj

xw

vw

f -4 Noise

f -4 Noise

xc

vc

xa,M

va,M

T

This tracking causes HP filtering of Lx(f) in R-Variances
• With HP knee fT
• fT 1/T (uniform weighted fit)
• fT 1/Teff (non-uniform)
• True for all noise
• Implicit in fitting theory for correlated noise
• Can’t distinguish correlated noise from causal behavior
• Only apparent for neg-p noise because most power in f< fT
• While for white noise power uniformly distributed over f

long term error-1.xls

Spectrally RepresentingR-Variances
• Gj(t,f) & Kx-j(f)  HP filtering due to fit
• To understand what Gj(t,f) & Kx-j(f) areconsider the following
• Can write fit solution in terms of Green’s function gw(t,t’) because assumed fit is linear in x(tn)
Spectrally RepresentingR-Variances
• Gw(t,f) = Fourier transform of gw(t,t’) over t’
• Hs(f)Xp(f) = Fourier transform of xp(t)
• Green’s fn for xj(t)  gj(t,t’) = (t-tn) - gw(t,t’)
• Fourier transform  Gj(t,f) = ejt - Gw(t,f)
Spectrally RepresentingR-Variances
• Kx-j(f)  Average of |Gj(t,f)|2 over t (data)
• c2 & x-c2 Modeling error terms
• Generated when xw,M(tn,A) can’t follow xc(tn) over T
Spectrally RepresentingR-Variances
• The paper proves the following
• |Gj(t,f)|2 & Kx-j(f)  f 2M(f<<1)
• When xa,M(t,A) is (M-1)th order polynomial
• |Gj(t,f)|2 & Kx-j(f) at least  f 2(f<<1)
• For any xa,M(t,A) with DC component
• So R-variances guaranteed to converge
• If free to choose model function for estimating the trajectory

Kx-j(f) in dB forUniform Weighted Fit

Kx-j(f) in dB forNon-Uniform Weighted Fit

M = 1  f 2

M = 1

f = 1/Teff

M = 2  f 4

f = 1/T

M = 3  f 6

M = 2

Weighting

M = 5  f 10

M = 4  f 8

Teff

M = 3

M = 4

M = 5

T

1

Log10(fT)

Log10(fT)

(N =1000)

(N =1000)

Kx-j(f) Calculated for Polynomial xw,M(t,A) & LSQF

xc(t)

xw,M(tn,A)

x(tn)

xw(t)

xj(tn)

xp(t)

Spectral Equations for True Function & Model Errors
• We note that xj(tn) + xw(tn) = xp(tn)
• So noise must be LP filtered in xw(tn) because noise in xj(tn) is HP filtered
• In paper derive spectral equations for
• W-variances E{xw(tn)2} & x-w2 in terms of Lx(f)
• Model error variances c2 & x-c2 in terms of dual freq Loève Spectrum Lc(fg,f) of xc(t)
• Note Hs(f) appears in all spectral equations
• So what is this Hs(f)?

Topological Hs(f) for 2-Way Ranging

d

~

D

Xponder

x(t)

x(t-d)

|Hs(f)|2 = 4sin2(fd)

x

Hs(f) = System Response Function(See Reinhardt, FCS, 2006)
• Models filtering action of system on x(t)
• Generated by actual filters in system & topological structures (such as PLLs)
• Acts on all variables the same way xp(t), xc(t), xj(t), xw(t)
• Hs(f) can HP filter as well as LP filter Lx(f)
• 2-way ranging Hs(f) generates 2nd order 0 at f=0
• So Hs(f) helps both R & W variances converge

 f 2 (fd <<1)

− Obs Residual

− True Fn Error

f

fT

f

fT

fl

fh

fl

fh

• At the knee freq fT the fitting process
• HP filters the obs residual error (R-variances)
• LP filters the true fn error (W-variances)
• Hs(f) filters both the same  fl = HP fh = LP
• As Teff (fT << fl)true fn error  0
• If Hs(f) alone can overcome pole in Lx(f)
• Then W-variances also converge for neg-p noise
• Transition to stationary but correlated statistics
• Confirms that R-variances measure consistency not accuracy for small Teff
• Can view control segment operations as PLL-like Hs(f) with HP cutoff fl = 1/TGPS
• TGPS determined by time constant of satellite parameter correction loops
• Can assume true function errors (W-variances) converge for this Hs(f)
• System tied to known ground sites
• Drift of timescale doesn’t effect nav accuracy
• R-variances measure true accuracy whenTeff >> TGPS
Final Summary & Conclusions
• -variances can be used for R-variancesin some geolocation problems
• Mainly when model function is polynomial
• R-variances HP filter noise in data due to trajectory estimation process
• True under very general conditions
• R-variances guaranteed to converge if free to choose model function
• R-variances represent true errors for large T when Hs(f) makes W-variances converge
• Preprints: www.ttcla.org/vsreinhardt/