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Appendix I / Utilities / Covariance
_ _
The estimated angular error is bounded by,
B 2
2 2
( δ ξ ) : = 2 ⋅ max ( σ , σ )
max E A
Θ Ψ
22.13-10
Equation 22.13-47
To account for the cross-correlation term [A] is diagonalised, eigenvalues
(λ1 , λ2) being the major and minor axes of the resulting uncertainty ellipse.
A − λ ⋅ I 3 : =
2
2
2
( − λ ) ⋅ ( σ − λ ) − σ : = 0
σΘ Ψ
ΘΨ
2 2
2 2 2
( σ + σ ) + ( σ ⋅ σ − σ ) : = 0
2
λ − λ ⋅ Θ Ψ
Θ Θ ΘΨ
2 ⋅
( λ , λ )
1
2 2
2 2 2
2 2 2
( σ + σ ) ± ( σ + σ ) − 4 ⋅ ( σ ⋅ σ − σ )
Θ
Ψ
Θ
Ψ
2
0
: =
Θ
Ψ
Equation 22.13-48
Equation 22.13-49
Equation 22.13-50
ΘΨ
Equation 22.13-51
The estimated pointing error is the spectral radius of this quadratic form,
B 2 ( δ ξ ) : = ρ ( A ) : = max ( λ , )
E λ
A
Since by definition the square root has only positive arguments,
⋅
B 2
2 2
2 2 2 2
( δξ
) : = σ + σ + ( σ − σ ) − 4 ⋅ σ
1
2
Equation 22.13-52
2 E A
Θ Ψ
Θ Ψ
ΘΨ
Equation 22.13-53
When the eigenvalues are different the spectral radius is a pessimistic
estimate of the pointing error and the CEP defined in §22.13.17 should be
used,
B 2
MD
MD
( δ ξ ) : = − 2 ⋅ λ ⋅ λ ⋅ log ( 1 − pr ( δξ
∈ [ 0 , E ( δ ) ] ) )
E ξ
A
1
2
e
C
C
Equation 22.13-54
From Figure 22-16, if the true error is less than the expected pointing error
997 times in 1000 samples then pr(…) := 0.997 and,