Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 330

(The 1 ellipse is defined here as a specific cut through the pdf and is not to be con-
fused with 1 containment. The latter curve is the locus of points, one point on each
ray from the origin, where the points are at a distance of 1 for the ray's direction. In
general, the 1 containment curve is a figure-eight-shaped curve that encloses the 1
ellipse.) If the major and minor axis of the ellipse are aligned with the x and y axes,
the equation for the ellipse reduces to x . In general, however, the

off-diagonal terms in cov( d R ) are nonzero, and the elliptical contours for the density
function are rotated relative to the x and y axes. Denote the major and minor axes of
the 1

2

/ σ

2

+ y

2

/ σ

2

= 1

x y

σ S (“long” and “short”), respectively. In general, the 1
ellipse is contained in a box of width

ellipse by σ L and

σ y centered on the ellipse. Figure
7.6 illustrates graphically the relationship between the ellipse and the parameters

σ x and height

σ x ,

σ y , σ S .

The probability that the error lies within the elliptical contour defined for a spe-
cific value of m is1

σ L , and

2

− m / 2

− e . In particular, the probability of being in the 1 ellipse ( m

= 2) is 0.86. (These values

are in contrast to the one-dimensional Gaussian result that the probability of being
within

1) is 0.39; the probability of being in the 2 ellipse ( m =

1 of the mean is 0.68.)
Several parameters are in common use that characterize the magnitude of the
horizontal error. The distance root mean square (drms) is defined by the formula

±

y

T

1

σ

ellipse: xy , pairs u = ( , ) xy

T − 1

satisfying u [cov(d )] R u = 1

σ y

σ L

σ S

σ x

x

2

+=+

2 2 2

σσ σ σ
x y L S

= DRMS 2

Figure 7.6 Relationship between 1 ellipse and distribution parameters.