dilution of precision (GDOP) (approximately), we obtain
Known are δZρ and H[1] from the pseudorange, satellite position, and nominal value of the user's position. The correction δx is the unknown vector.
If we premultiply both sides of Eq. (2.28) by H[1]T, the result will be
Then, we premultiply Eq. (2.29) by (H[1]T H[1])−1:
(2.30)
If δx and δZρ are assumed random with zero mean, the error covariance (E = expected value)
The pseudorange measurement covariance is assumed uncorrelated satellite to satellite with variance σ2:
Substituting Eq. (2.32) into Eq. (2.31) gives
(2.33)
for
and
and the covariance matrix becomes
(2.34)
We are principally interested in the diagonal elements of
(2.35)
that represent the DOP of range measurement error to the user solution error (see Figure 2.4):
Hence, all DOPs represent the sensitivities of user solution error to pseudorange errors. Figure 2.4 illustrates the relationship between the various DOP terms.
Figure 2.4 DOP hierarchy.
2.3.4 Example Calculation of DOPs
2.3.4.1 Four Satellites
For simplicity, consider four satellite measurements. The best accuracy is found with three satellites equally spaced on the horizon, at minimum elevation angle, with the fourth satellite directly overhead, as listed in Table 2.1.
The diagonal of the unscaled covariance matrix (H[1]T H[1])−1 then has the terms
where
Table 2.1 Example with four satellites.
| Satellite location | ||||
| 1 | 2 | 3 | 4 | |
| Elevation (°) | 5 | 5 | 5 | 90 |
| Azimuth (°) | 0 | 120 | 240 | 0 |
Typical example values of H[1] for this geometry are
The GDOP calculations for this example are
Gdop.m calculates the GDOP for the chosen constellation for GPS_perf.m by calculating H[1] matrix calcH. See Appendix A on www.wiley.com/go/grewal/gnss.
2.4 Time and GPS
2.4.1 Coordinated Universal Time (UTC) Generation
Coordinated universal time (UTC) is the timescale based on the atomic second but is occasionally corrected by the insertion of leap seconds so as to keep it approximately synchronized with the Earth's rotation. The leap second adjustments keep UTC within 0.9 seconds of UT1, which is a timescale based on the Earth's axial spin. UT1 is a measure of the true angular orientation of the Earth in space. Because the Earth does not spin at exactly a constant rate, UT1 is not a uniform timescale [5].
2.4.2 GPS System Time
The timescale to which GPS signals are referenced is referred to as GPS time. GPS time is derived from a composite or “paper” clock that consists of all operational monitor station and satellite atomic clocks. Over the long run, it is steered to keep it within about 90 nanoseconds (1σ) of UTC, as maintained by the master clock at the
