Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 328

The DOP values can be expressed in terms of the components of ( H T H ) −1 as follows:

PDOP = DD D

11

+ + (7.50)

22 33

HDOP = DD

11

+ (7.51)

22

VDOP = D 33 (7.52)

TDOP = Dc

44

(7.53)

(In some treatments of DOP, TDOP is defined by the formula σ ct b = TDOP × σ UERE .In

this case, (7.53) takes the simpler form TDOP = D 44 . The variable ct b represents a

range equivalent of the time bias error, and σ ct b is its standard deviation. In the cur-

rent formulation, TDOP is defined so that when multiplied by σ UERE , the standard

deviation of the time bias error is obtained directly. This is the more relevant formu-
lation if actual time accuracy is of interest. The linear relationship between t b and ct b
yields the formula c t

between their standard deviations, and one can easily
convert between the formulations.)

σσ =

ct

b b

7.3.2 Accuracy Metrics
The formulas derived in Section 7.3.1 allow one to compute 1-sigma horizontal, ver-
tical, or three-dimensional position errors as a function of satellite geometry and the
1-sigma range error. They also allow one to compute 1-sigma user clock errors. It is
important to recall that these formulas were derived under the assumptions that
range errors are zero mean with a Gaussian distribution and that range errors are
independent from satellite to satellite. Oftentimes, other metrics besides 1-sigma
position errors are used to characterize GPS accuracy performance. Some common
metrics are derived and discussed in this subsection.
If pseudorange errors are Gaussian-distributed, (7.36) tells us that vertical posi-
tion errors also have a Gaussian distribution:

N

∑

dz = K d ρ (7.54)

3 , m m

m 1

where dz is the error in the vertical component of the computed position. This result
is obtained by noting that a linear function of Gaussian random variables is itself a
Gaussian random variable. One common measure of vertical positioning accuracy is
the error magnitude that 95% of the measurements fall within, which is approxi-
mately equal to the 2-sigma value for a Gaussian random variable. Thus:

95 % vertical position accuracy ≈ 2 σ = 2 ⋅ VDOP ⋅ σ (7.55)

dz UERE

assuming that pseudorange errors are additionally zero mean and independent
among satellites. Using a global average value of 1.6 for VDOP for the nominal
24-satellite GPS constellation and the UERE values from Tables 7.3 and 7.4 yields
95% vertical position accuracy of 4.5m for the PPS and 22.7m for the SPS, respec-