Global Positioning System Reference
In-Depth Information
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[14
Lin
-2.
——
Nor
PgE
Figure 4.8
Standard deviation curve.
ar e the projections of the ellipse in the directions of the x 1 and x 2 axes. This is shown
in Figure 4.9. Equations (4.317) and (4.318) follow from the fact that the diagonal
el ements of the covariance matrix are the variances of the respective parameters.
Eq uation (4.316) confirms for
90° that the axes a and b equal
th e maximum and minimum standard deviations, respectively. The rectangle formed
by the semisides
ψ =
0 and
ψ =
[14
σ x 2 encloses the ellipse. This rectangle can be used as an
ap proximation for the ellipses. The diagonal itself is sometimes referred to as the
m ean position error
σ x 1 and
σ
,
x 2 0 q x 1 +
2
x 1
2
σ =
σ
q x 2
(4.319)
The points of contact between the ellipse and the rectangle in Figure 4.9 are
functions of the correlation coefficients. For these points, the tangent on the ellipse is
either horizontal or vertical in the (y i ) coordinate system. The equation of the ellipse
in the (y) system is, according to (4.294),
[ y 1 y 2 ] q x 1
1 y 1
y 2
q x 1 ,x 2
2
0
(4.320)
q x 1 ,x 2
q x 2
By replacing the matrix by its inverse, the expression becomes
[ y 1 y 2 ] q x 2
y 1
y 2
q x 1 ,x 2
= q x 1 q x 2
q x 1 ,x 2 σ
2
0
(4.321)
q x 1 ,x 2
q x 1
 
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