3.3 Coordinates, Observables and Error Sources
3.3.4 Combinations of observables
3.3.4.1 Single dierencing
A key method used to reduce the impact of these errors is by combining observables. The simplest form is single dierencing, where a pseudorange from one satellite (which we will call satellite 1 ) is subtracted from the pseudoranges at all other satellites, in order to eliminate receiver clock bias; the eect on equation 3.6 on page 46 is:
Rsr−R1r = %sr−%1r−Irs+Ir1+Trs−Tr1+c∆δr−c∆δr (3.8)
+dr−dr−ds+d1+νrs−ν 1 r
= %sr−%1r−Irs+Ir1+Trs−Tr1−ds+d1+νrs−νr1 (3.9)
Using fs−1 to signify the dierence between f for satellite s and satellite 1, we can
dene Rs−1 r =Rsr−R1r; %s −1 r =%sr−%1r; Is −1 r =Irs−Ir1; Ts −1 r =Trs−Tr1; ds −1=ds+d1; and νs−1
r =νrs−νr1, allowing us to rewrite equation 3.9 as:
Rrs−1 =%rs−1−Irs−1+Trs−1−ds−1+νrs−1 (3.10) Likewise, by dening Ns−1 r = Nrs −Nr1; Ds −1 = Ds − D1; ϕs−1 0 = ϕ s 0 − ϕ 1 0; and µs−1
r =µsr−µ1r, equation 3.7 on page 46 can be rewritten as:
λΦsr−1 =%rs−1+Irs−1+Trs−1+λNrs−1 −Ds−1+ϕs−1
0 +µ
s−1
r (3.11)
In other words, single dierencing eliminates errors common among satellites - namely receiver clock error and receiver hardware delays.
3.3.4.2 Double dierencing
To eliminate errors from sources such as the ionosphere and troposphere, to allow high-precision positioning, a xed base station at a known location takes measure-
CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 49
Base station at
known location
Roving receiver
Ionosphere
Troposphere
Ephemeris errors
Satellite hardware delay
Satellite initial phase
Signal errors are +2.13m and -1.31m I'll subtract 2.13m and add 1.31m
Figure 3.12: Use of a xed base station to eliminate common errors on GPS measure- ments.
ments and, as its location is known, can work out the delay on each signal. This data can then be broadcast to a roving GPS receiver, where the errors can be subtracted. This is shown in gure 3.12. When the roving receiver is within a few kilometres of the base station, the errors are common, allowing them to be completely elimina- ted. This technique is often called Dierential GPS (DGPS) when applied to the code measurement, and Carrier Dierential GPS (CDGPS) when applied to the carrier mea- surement. When single-dierenced measurements from a base station are subtracted from single-dierenced measurements from a roving receiver, the results are known as double-dierenced measurements. For example, equation 3.10, after the subtraction of the measurements at base station b, would be:
Rsr−1−Rsb−1 = %sr−1−%sb−1−Irs−1+Ibs−1+Trs−1−Tbs−1 (3.12)
−ds−1+ds−1+νrs−1−νbs−1
= %sr−1−%sb−1+νrs−1−νbs−1 (3.13)
CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 50 dene Rsr−−1b = Rsr−1 −Rsb−1; %sr−−1b = %rs−1 −%sb−1; and νrs−−b1 = νrs−1 −νbs−1 giving the
double-dierenced code measurement:
Rsr−−1b =%rs−−1b+νrs−−b1 (3.14)
Likewise, for equation 3.11 on page 48 we can dene Φsr−−1b = Φsr−1 −Φsb−1; Nrs−−b1 =
Nrs−1−Nbs−1; andµrs−−b1 =µrs−1−µsb−1 producing:
λΦsr−−1b =%sr−−1b+λNrs−−b1+µsr−−1b (3.15)
In summary, when a roving receiver is within a few kilometres of a base station, double dierencing can counteract several major sources of inaccuracy.
In applications using xed base stations there are two common congurations: Either the user owns and operates the xed base station, or the user purchases access to a network RTK system. Network RTK uses a network of base stations around the country, interpolating between the nearest base stations to determine errors at the user's location.
For applications using dual-frequency receivers costing several thousand pounds, net- work RTK subscriptions costing one to two thousand pounds a year can be cost- eective, and for applications working beyond the range of a single base station, net- work methods are vital. One example of this arrangement is Cai et al. (2011). For golf course applications, only a single base station at the clubhouse would be required, and if a low cost single frequency receiver was used for it the savings on subscription costs would pay for the hardware within months. This was the route we chose to explore. It is possible in the future subscription costs will come down, or a special price could be negotiated; should this happen, using a reference network may become cost-eective.
CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 51 3.3.4.3 Wide lane combination
When dual frequency data is available, through the use of a high precision receiver, a wide lane combination can be formed. The L1 (1575.42 MHz, 19.0 cm wavelength) signal and the L2 (1227.60 MHz, 24.4 cm wavelength) signal can be multiplied together to produce a beat frequency:
sin (1575.42×t)×sin (1227.60×t) = 12 cos (1575.42×t−1227.60×t) + 12cos (1575.42×t+ 1227.60×t) = 1 2cos (347.82×t) + 1 2cos (2803.02×t) (3.16) Band pass ltering leaves only the 347.82 MHz component, with a wavelength of 86.2 cm. This simplies integer ambiguity resolution because with an ambiguity distance of 86.2 cm a range error of 43cm will still be nearest to the correct ambiguity, while for a single-frequency measurement with a 19.0 cm ambiguity distance, a range error of just 9.5 cm would be required to produce the same result.
3.3.4.4 Ionosphere free combination
Ionospheric interference depends on Total Electron Content (TEC), and varies with frequency. When dual frequency data is available, rst-order eects can be removed using a so-called ionosphere free combination. The model of ionospheric delay used for this is
∆Iono = 1 cosz
40.3
f2 T V EC (3.17)
wherez is the zenith angle to the satellite,f is the signal frequency and T V EC is the
total vertical electron count, in units of 1016 electrons/m2. The measurements at two
frequencies may be combined like
RIono =RL1− fL1 fL2
CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 52 where RIono, RL1 and RL2 are ionosphere-free, L1 and L2 pseudoranges respectively,
while fL1 and fL2 are the L1 and L2 frequencies respectively. The name of the ionos-
phere free combination is not strictly correct as it doesn't entirely eliminate ionospheric errors, but it is reported to provide a substantial reduction.