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12.540 Principles of the Global
Positioning System
Lecture 12
Prof. Thomas Herring
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Estimation
? Summary
– Examine correlations
– Process noise
?White noise
? Random walk
? First-order Gauss Markov Processes
– Kalman filters – Estimation in which the parameters
to be estimated are changing with time
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Correlations
? Statistical behavior in which random variables tend to
behave in related fashions
? Correlations calculated from covariance matrix.
Specifically, the parameter estimates from an
estimation are typically correlated
? Any correlated group of random variables can be
expressed as a linear combination of uncorrelated
random variables by finding the eigenvectors (linear
combinations) and eigenvalues (variances of
uncorrelated random variables).
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Eigenvectors and Eigenvalues
? The eigenvectors and values of a square matrix
satisfy the equation Ax=λx
? If A is symmetric and positive definite (covariance
matrix) then all the eigenvectors are orthogonal and
all the eigenvalues are positive.
? Any covariance matrix can be broken down into
independent components made up of the
eigenvectors and variances given by eigenvalues.
One method of generating samples of any random
process (ie., generate white noise samples with
variances given by eigenvalues, and transform using
a matrix made up of columns of eigenvectors.
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Error ellipses
? One special case is error ellipses. Normally
coordinates (say North and East) are correlated and
we find a linear combinations of North and East that
are uncorrelated. Given their covariance matrix we
have:
σ
n
2
σ
ne
σ
ne
σ
e
2
?
?
?
?
?
?
Covariance matrix;
Eigenvalues satisfy λ
2
?(σ
n
2
+σ
e
2
)λ+(σ
n
2
σ
e
2
?σ
ne
2
)=0
Eigenvectors:
σ
ne
λ
1
?σ
n
2
?
?
?
?
?
?
and
λ
2
?σ
e
2
σ
ne
?
?
?
?
?
?
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Error ellipses
? These equations are often written explicitly as:
? The size of the ellipse such that there is P (0-1)
probability of being inside is
λ
1
λ
2
?
?
?
=
1
2
σ
n
2
+σ
e
2
± σ
n
2
+σ
e
2
()
2
? 4 σ
n
2
σ
e
2
?σ
ne
2
()
?
?
?
?
?
tan2φ=
2σ
ne
σ
n
2
?σ
e
2
angle ellipse make to N axis
ρ= ?2ln(1? P)
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Error ellipses
? There is only 40% chance of being in 1-sigma
error (compared to 68% of 1-sigma in one
dimension)
? Commonly see 95% confidence ellipse which
is 2.45-sigma (only 2-sigma in 1-D).
? Commonly used for GPS position and velocity
results
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Example of error ellipse
-8
-6
-4
-2
0
2
4
6
8
-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
Var2
Var1
Error Ellipses shown
1-sigma 40%
2.45-sigma 95%
3.03-sigma 99%
3.72-sigma 99.9%
Covariance
2 2
2 4
Eigenvalues
0.87 and 3.66,
Angle -63
o
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Process noise models
? In many estimation problems there are
parameters that need to be estimated but
whose values are not fixed (ie., they
themselves are random processes in some
way)
? Examples include for GPS
– Clock behavior in the receivers and satellites
– Atmospheric delay parameters
– Earth orientation parameters
– Station position behavior after earthquakes
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Process noise models
? There are several ways to handle these types of
variations:
– Often, new observables can be formed that eliminate the
random parameter (eg., clocks in GPS can be eliminated by
differencing data)
– A parametric model can be developed and the parameters of
the model estimated (eg., piece-wise linear functions can be
used to represent the variations in the atmospheric delays)
– In some cases, the variations of the parameters are slow
enough that over certain intervals of time, they can be
considered constant or linear functions of time (eg., EOP are
estimated daily)
– In some case, variations are fast enough that the process can
be treated as additional noise
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Process noise models
? Characterization of noise processes
– Firstly need samples of the process (often not easy
to obtain)
– Auto-correlation functions
– Power spectral density functions
– Allan variances (frequency standards)
– Structure functions (atmospheric delays)
–(see Herring, T. A., J. L. Davis, and I. I. Shapiro,
Geodesy by radio interferometry: The application of
Kalman filtering to the analysis of VLBI data, J.
Geophys. Res., 95, 12561–12581, 1990.
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Characteristics of random processes
? Stationary: Property that statistical properties
do no depend on time
Autocorrelation ?(t
1
,t
2
) = x
1
x
2
x
1
x
2
∫∫
f (x
1
,t
1
;x
2
,t
2
)dx
1
dx
2
For stationary process only depends of τ = t
1
? t
2
?
xx
(τ) = limT →∞
1
2T
x(t)x(t +τ)dt
∫
PSD Φ
xx
(ω) = ?
xx
(τ)
?∞
∞
∫
e
?iωτ
dτ
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Specific common processes
? White-noise: Autocorrelation is Dirac-delta
function; PSD is flat; integral of power under
PSD is variance of process (true in general)
? First-order Gauss-Markov process (one of
most common in Kalman filtering)
?
xx
(τ) =σ
2
e
?β τ
Φ
xx
(ω) =
2βσ
2
ω
2
+β
2
1
β
is correlation time