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Preliminary experiments were performed with the algorithms of Buist et al. (2009) and Pinchin et al. (2008). Both Buist et al. and Pinchin et al. report success rates for single-epoch, single-frequency ambiguity resolution (one entirely simulated and one with high quality antennas) but in our experiments with patch antennas the success rate for this was low; multi-epoch ambiguity resolution was investigated instead, with ambiguity resolution times and success rates as measures of algorithm quality.

5.2.1 Float solution dierence test

It was found that multipath was a common cause of incorrect solutions, but that when present it is rarely identical at both receivers, leading to the two roving receivers' oat solutions not matching up. The following test was proposed (using the terminology of equation 5.3 on page 114) to detect this situation:

max abs(ˆa23−(ˆa13+a12))< k (5.6)

where max abs(a) means to take the largest absolute value from the vector a; and k

is a threshold value (1.5 L1 cycles in our experiments). When the largest dierence between oat solutions at the two roving receivers exceeds k it is taken as a sign of

multipath, and the solution is not accepted until the dierence has reduced to below

k.

5.2.2 Fixed length baseline residual in ratio test

In line with previous chapters, an F-ratio test (See section 3.4.4.3 on page 65) is used during multi-epoch ambiguity resolution, in order to choose whether to accept a solu- tion or gather more data before deciding. However, instead of a ratio of the squared

CHAPTER 5. MULTIPLE RECEIVER CONFIGURATIONS 120 residuals of the two best solutions for the xed length baseline, it was proposed to add to them the squared residual of the best solution for the known-length baseline. When multipath at the roving receivers is low, the residual between the two roving receivers should be low, making the new ratio test less conservative, while higher multipath will lead to a greater residual, making the new ratio test more conservative.

In other words, while the standard F-ratio test is dened by equation 3.73 on page 65 as: keˆk2_Q y +kaˆ−a2k 2 Qˆa keˆk2_Q y +kaˆ−a1k 2 Qˆa > k

the proposed test

keˆk2_Q y +kaˆ−a2k 2 Qˆa +keˆf ixedk 2 Qy +kaˆf ixed−aˇf ixedk 2 Qˆa kˆek2_Q y+kˆa−a1k 2 Qˆa+kˆef ixedk 2 Qy+kˆaf ixed−aˇf ixedk 2 Qˆa > k (5.7) wherekˆef ixedk 2

Qy is the oat solution residual for the xed-length baseline and similarly

kaˆf ixed−aˇf ixedk 2

Qˆa is the squared norm between the oat solution and the chosen xed solution.

5.2.3 Performance comparison criteria

As both the modications outlined in previous sections will lead to variations in both success rate and ambiguity resolution time, it was decided to gather data for a range of F-ratio thresholds, and to make comparisons between ambiguity resolution times with a success rate of 99.5% or higher.

5.3 Experimental Conguration

Section 4.3 on page 99 describes the collection of 24 hours of 10Hz GPS data from a stationary vehicle in a moderate multipath environment. Two single-frequency GPS receivers were present on the vehicle, and a third receiver, 280 metres away, served as a xed base station. The distance between the two antennas on the vehicle was 1.9 metres, as shown in gure 5.3 . The baseline between the roving receivers ran from

CHAPTER 5. MULTIPLE RECEIVER CONFIGURATIONS 121

Figure 5.3: Test vehicle tted with two single frequency GPS receivers 1.9 metres apart. north to south.

The recorded data was processed by performing ambiguity resolution starting every 30 seconds throughout the data, taking as long as needed to meet the acceptance threshold, and recording the ambiguity resolution times and success rates. This processing was performed using several dierent algorithms:

ˆ A single receiver, and an F-ratio threshold between 1.0 and 3.0

ˆ Dual receivers using Buist et al. (2009)'s suboptimal algorithm, an F-ratio thre- shold between 1.0 and 3.0, with and without the oat dierence test of equa- tion 5.6 on page 119.

ˆ Dual receivers using Pinchin et al. (2008)'s algorithm, an F-ratio threshold bet- ween 1.0 and 3.0, with and without the oat dierence test.

CHAPTER 5. MULTIPLE RECEIVER CONFIGURATIONS 122 0 100 200 300 400 500 600 700 800 900 1000 95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100

Mean ambiguity resolution time (s)

% Correctly resolved

Resolution time vs success rate for different dual−receiver configurations

Single receiver

Buist et. al. (2009) without float difference test Buist et. al. (2009) with float difference test

Buist et. al. (2009) with fixed length baseline residual added Pinchin et. al. (2008) without float difference test Pinchin et. al. (2008) with float difference test

Pinchin et. al. (2008) with fixed length baseline residual added Zheng and Gebre−Egziabher (2009) without float difference test Zheng and Gebre−Egziabher (2009) with float difference test

Figure 5.4: Graph of resolution time vs. success rate as F-ratio threshold is varied, for several algorithms.

norm ratio threshold between 1.0 and 6.0, with and without the oat dierence test.

ˆ Dual receivers using Buist et al. (2009)'s suboptimal algorithm with the ratio test with xed length baseline residual added, thresholds between 1.0 and 3.0. ˆ Dual receivers using Pinchin et al. (2008)'s algorithm with the ratio test with

xed length baseline residual added, thresholds between 1.0 and 3.0. Results graphs and tables are presented in the following section.