BASIC PRINCIPLE OF POSITION FIXING
Position fixing in three dimensions may involve the measurement of distance (or range) to at least three satellites whose X, Y and Z position is known, in order to define the user’s Xp, Yp and Zp position. In its simplest form, the satellite transmits a signal on which the time of its departure (tD) from the satellite is modulated. The receiver in turn notes the time of arrival (tA) of this time mark. Then the time which it took the signal to go from satellite to receiver is (tA − tD) = t called the delay time. The measured range R is obtained from
R1 = (tA − tD)c = tc
where c = the velocity of light.
Whilst the above describes the basic principle of range measurement, to achieve it one would require the receiver to have a clock as accurate as the satellite’s and perfectly synchronized with it. As this would render the receiver impossibly expensive, a correlation procedure, using the pseudo-random binary codes (P or C/A), usually ‘C/A’, is adopted. The signal from the satellite arrives at the receiver and triggers the receiver to commence generating its own internal copy of the C/A code. The receiver-generated code is cross-correlated with the satellite code (Figure 9.10). The ground receiver is then able to determine the time delay (t) since it generated the same portion of the code received from the satellite. However, whilst this eliminates the problem of the need for an expensive receiver clock, it does not eliminate the problem of exact synchronization of the two clocks. Thus, the time difference between the two clocks, termed clock bias, results in an incorrect assessment oft. The distances computed are therefore called ‘pseudo-ranges’. The use of four satellites rather than three, however, can eliminate the effect of clock bias. A line in space is defined by its difference in coordinates in an X, Y and Z system:
If the error in R, due to clock bias, is δR and is constant throughout, then:
where Xn, Yn, Zn = the coordinates of satellites 1, 2, 3 and 4 (n = 1 to 4)
Xp, Yp, Zp = the coordinates required for point P
Rn = the measured ranges to the satellites
Solving the four equations for the four unknowns Xp, Yp, Zp and δR also solves for the error due to clock bias.
Correlation of the pseudo-binary codes
Whilst the use of pseudo-range is sufficient for navigational purposes and constitutes the fundamental approach for which the system was designed, a much more accurate measurement of range is required for positioning in engineering surveying. Measuring phase difference by means of the carrier wave in a manner analogous to EDM measurement does this. As observational resolution is about 1% of the signal wavelength λ, the following table shows the reason for using the carrier waves; this is referred to as the carrier phase observable.
Carrier phase is the difference between the incoming satellite carrier signal and the phase of the constantfrequency signal generated by the receiver. It should be noted that the satellite carrier signal when it arrives at the receiver is different from that initially transmitted, because of the relative velocity between transmitter and receiver; this is the well-known Doppler effect. The carrier phase therefore changes according to the continuously integrated Doppler shift of the incoming signal. This observable is biased by the unknown offset between the satellite and receiver clocks and represents the difference in range to the satellite at different times or epochs. The carrier phase movement, although analogous to EDM measurement, is a one-way measuring system, and thus the number of whole wavelengths (N) at lock-on is missing; this is referred to as the integer or phase ambiguity. The value of N can be obtained from GPS network adjustment or from double differencing or eliminated by triple differencing.