Data Processing for Positioning - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 127

If we want to improve the likelihood that a new sample lies within the ellipsoid,
we may increase the probability 1 − α , the axes of the confidence ellipsoid, the

c -value, or all of them.

We start from the covariance matrix of the least-squares problem (8.28):

⎡
⎣

⎤
⎦

2
X

σ σ XY σ XZ

σ X , cdt
σ Y , cdt
σ Z , cdt

2
Y

σ YX σ σ YZ

ECEF =

2
Z

. (8.41)

σ ZX σ ZY σ

2
cdt

σ cdt , X σ cdt , Y σ cdt , Z σ

The law of covariance propagation transforms ECEF into the covariance ma-

trix expressed in a local system with coordinates ( e , n , u ) . The interesting 3 by 3

submatrix S of ECEF is shown in (8.41). After the transformation with F ,the

submatrix becomes

⎡
⎣

⎤
⎦ =

e

σ σ en σ eu

2
n

F T SF

enu = .

σ ne σ σ nu (8.42)

u

σ ue σ un σ

In practice we meet several forms of the dilution of precision (abbreviated
DOP):

tr

2
cdt

e n u

σ + σ + σ + σ ( ECEF )

σ

Geometric: GDOP = = ,

2
0

2
0

σ

σ

e

+ σ

n

Horizontal: HDOP = ,

2
0

σ

tr

2 2 2

Z

e n u

σ + σ + σ σ

X + σ Y + σ

( enu )
σ

Position: PDOP = = = ,

0 0 0

σ σ

Time: TDOP = σ cdt /σ 0 ,

Vertical: VDOP = σ U /σ 0 .

Note that all DOP values are dimensionless. They multiply the range errors to
give the position errors (approximately). Furthermore, we have

GDOP 2 PDOP 2 TDOP 2 HDOP 2 VDOP 2 TDOP 2

= + = + + .

Some satellite constellations are better than others and the knowledge of the
time of best satellite coverage is a useful tool for anybody using GPS. Experience
shows that good observations are achieved when PDOP < 5 and measurements

come from at least five satellites .