Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

contains 16 components whose meaningneeds to be clarified. The matrix A in
its component form can be presented as follows:
⎡
⎤
A =
⎢
⎣
A11 A12 A13 A14
A21 A22 A23 A24
A31 A32 A33 A34
A41 A42 A43 A44
⎥
⎦
(5.19)
The covariance of the positioningestimate can in turn be presented as follows:
⎡
⎤
cov(ɛx) =
⎢
⎣
σ 2 xu σ 2 xuyu σ2 xuzu σ2 xud
σ 2 xuyu σ 2 yu σ 2 yuzu σ2 yud
σ 2 xuzu σ2 yuzu σ 2 zu σ 2 zud
σ 2 xud σ 2 yud σ 2 zud σ 2 d
⎥
⎦
(5.20)
What really interests us are the variances σ2 xu , σ2 yu , σ2 zu and σ2 d . Using5.18, 5.19
and 5.20 one obtains expressions for the precedingvariances:
σ 2 xu = A11 × σ 2 ˙ρ (5.21)
σ 2 yu = A22 × σ 2 ˙ρ (5.22)
σ 2 zu = A33 × σ 2 ˙ρ (5.23)
σ 2 d = A44 × σ 2 ˙ρ (5.24)
When evaluatingpositioningperformance it is more natural to use standard deviations
σ rather than variances σ 2 because standard deviations are physically
much more intuitive. Mathematically transformingbetween the two does not
cause any problems since the standard deviation is naturally the square root of
the variance. Usingthe ratios of different positioningaccuracy standard deviations
and the measurement accuracy standard deviation we can define Doppler
dilution of precision (Doppler DOP, DDOP) measures. These measures indicate
the quality of the positioninggeometry. They are defined as follows:
• Doppler position dilution of precision (DPDOP)
σ2 xu + σ2 yu + σ2 zu
= DPDOP =
σ ˙ρ
A11 + A22 + A33
• Doppler horizontal dilution of precision (DHDOP)
σ2 xu + σ2 yu
= DHDOP =
σ ˙ρ
A11 + A22
• Doppler vertical dilution of precision (DVDOP)
σ2 zu
= DVDOP =
σ ˙ρ
A33
38
(5.25)
(5.26)
(5.27)