Differential GPS Part 1

The error sources previously described limit the accuracy attainable using GPS. Several of the sources of error (ionospheric delay, tropospheric delay, satellite ephemeris and clock error), referred to as the common mode errors, are spatially and temporally correlated. Therefore, if the errors could be estimated by one base receiver and promptly broadcast to other roving receivers, then the GPS positioning accuracy at each of the roving receivers could be substantially improved. This is the basic principle of how Differential GPS (DGPS) works. A typical DGPS setup is illustrated in Figure 8.13.

The following subsections will discuss a few DGPS methods. In each case, DGPS will involve a GPS receiver/antenna at a location p0, a receiver/antenna at an unknown possibly changing position pr, a satellite at the calculated position pand a communication medium from the first receiver to the second receiver. In the discussion, the former receiver will be referred to as the base. The latter receiver will be referred to as the rover. Although the discussion is phrased in terms of one base and one rover, the number of roving receivers is not limited.

Relative DGPS

This section has three parts. Section 8.8.1.1 considers relative DGPS based on the pseudorange observables. Section 8.8.1.2 considers relative DGPS based on the phase observables. Section 8.8.1.3 analyzes the effect of base-rover separation on the computation of the range difference via projection of the base-rover offset vector onto the satellite line-of-sight vector.


Pseudorange

The correction equations of this subsection are derived for a single satellite, but the process is identical for all satellites of interest. The GPS pseu-dorange observables, measured simultaneously at the base and rover are, respectively,

tmp18514_thumb

wheretmp18515_thumbrepresents the sum of the ionospheric and tropospheric errors, the other error terms are all defined as in eqn. (8.6), and the subscripts o and r denote the error terms corresponding to the base and rover, respectively.

If the base pseudorange observables were communicated to the rover, then the single-differential measurements

tmp18517_thumb

could be formed. From eqns. (8.115-8.116), this differential measurement is modeled as

tmp18518_thumb

where

tmp18519_thumb

The transition from eqn. (8.118) to eqn. (8.119) is often invoked, but it is not exact. The analysis of the incurred error is presented in Section 8.8.1.3. The satellite clock errors are identical for all receivers making simultaneous measurements; therefore, they cancel in eqns. (8.118-8.119). The correlation of the atmospheric error term between the base and receiver locations depends on the various factors, the most important of which is the base-to-rover distancetmp18520_thumbThe satellite position error terms decorrelate with distance due to the change of the satellite position error that projects onto the line-of-sight vector. The effect of satellite position error on differential measurements is bounded (see p. 3-50 in [108]) bytmp18521_thumb where r is the range to the satellite and 5r is the magnitude of the satellite position error. This bound is about 2cm for d = 40km, andtmp18522_thumb

Although, for a vector of measurements, eqn. (8.119) would appear to have the form of a least squares estimation problem, strictly speaking it is not a LS problem due to the dependence of h* on pr. However, eqn. (8.118) is easily solved by a few iterations of the algorithm described in Section 8.2.2. After convergence, the resulttmp18523_thumbis the relative position and relative receiver clock bias between the rover and the base receivers. In navigation applications, the relative clock error is usually inconsequential.

For some applications, for example maneuvering relative to a landing strip or formation flight, relative position may be sufficient. In such applications, the base station position po is not required. In other applications where the Earth relative position is required and the base position is known, it is straightforward to compute the Earth relative rover position as the sum of the base position and the estimated relative position vector. Note that errors in the estimate of the base position vector directly affect the estimated rover position vector.

Since the non-common mode noise n has range errors between 0.1 and 3.0m, depending on receiver design and multipath mitigation techniques, DGPS position accuracy is much better than SPS GPS accuracy. Cancellation of the common mode noise sources in eqn. (8.119) assumes that the rover is sufficiently near (within 10-50 mi.) the base station and that the corrections are available at the rover in a timely fashion.

Carrier Phase

In non-differential operations, the common mode errors corrupting the phase observables limit the utility of the phase observables. In differential mode, the common mode errors are essentially removed. Assuming that the integer ambiguities can be identified, the carrier observables then enable position estimation with accuracy approximately 100 times better than is possible with pseudorange alone.

For the carrier phase observables, the relative DGPS process is nearly identical to that described for the pseudorange in Section 8.8.1.1. Let the phase observables be modeled as

tmp18528_thumb

where the symbolstmp18529_thumbare used to distinguish the phase atmospheric delays from the similarly denoted pseudorange atmospheric delays in Section 8.8.1.1. The code and phase atmospheric delays are not the same as discussed in Section 8.4.4.1.

If the base observables were communicated to the rover, then by methods similar to those in Section 8.8.1.1, the differential measurements

tmp18531_thumb

could be formed and modeled as

tmp18532_thumb

where

tmp18533_thumb

is still an integer and

tmp18534_thumb

The common mode errors cancel in the same sense as was discussed in Section 8.8.1.1. Often the residual ionospheric error is modeled as an additional term in eqns. (8.123-8.124) when its effect is expected to be significant, as can be the case in integer ambiguity resolution over between receivers separated by large distances.

The n term in eqns. (8.118-8.119) is expected to have magnitude on the meter scale, while the 3 term of eqns. (8.123-8.124) is expected to have magnitude on the centimeter scale. However, the challenge to using eqns. (8.123-8.124) as a range estimate for precise point positioning is that the integer Nro is unknown. Integer ambiguity resolution is discussed in Section 8.9. Carrier-smoothing between eqns. (8.118) and (8.123) is also an option during the time interval that the integer search is in progress.

Rover-Base Separation

The transition from eqn. (8.118) to eqn. (8.119) involves the approximation

tmp18535_thumb

This section first motivates the approximation from a physical perspective and then shows the result mathematically.

Relative GPS range.

Figure 8.14: Relative GPS range.

For the physical description, consider the geometry depicted in Figure 8.14. The figure shows signals arriving at the base and rover locations. The geometry of the figure assumes that (1) the satellite is sufficiently far away for the wave fronts to appear as planes that are indicated by dashed lines; and, (2) the line-of-sight vectors for the rover and base are identical. From the figure geometry it is clear that subject to the two assumptions, eqn. (8.125) is an accurate approximation. The analysis below shows that the accuracy of the approximations is approximatelytmp18537_thumbwhich has magnitude 2.5cm for d = 1km and 2.5m at d = 10km.

The following mathematical derivation will allow quantitative analysis. The derivation will start from the following facts:

tmp18539_thumb

wheretmp18540_thumbare represented as row vectors so that the superscript T for transpose can be dropped. Todecrease the notation slightly, we will use the following symbols:tmp18541_thumbandtmp18542_thumbThe analysis is as follows:

tmp18546_thumb

Figure 8.15 shows the notation and geometry that will be useful for the analysis of the last two terms in eqn. (8.126). From the figure it is clear that

tmp18547_thumb

 

 

Notation and geometry for analysis of rover to base separation.

Figure 8.15: Notation and geometry for analysis of rover to base separation.

Therefore,

tmp18549_thumb

Using the approximationtmp18550_thumb(accurate to third order intmp18551_thumb) we have

tmp18554_thumb

Using this information, eqn. (8.128) reduces to

tmp18555_thumb

In the transition from eqn. (8.118) to eqn. (8.119) this error term is incorporated into the measurement error. The error can be substantial for d > 1km.

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