Question 5 of 7: Single-, Double- and Triple-Differencing in GPS Surveys

Question 5: Single-, Double- and Triple-Differencing in GPS Surveys (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Carrier-phase observations collected simultaneously by two receivers ($A$, $B$) tracking common satellites in a relative (differential) GPS survey.

Find. A sketch and a principle statement for each of single-, double- and triple-differencing.

Carrier-phase positioning is far more precise than code positioning but is corrupted by three dominant nuisance terms: the satellite-clock error, the receiver-clock error, and the integer ambiguity $N$ (the unknown whole number of cycles at lock-on). Differencing forms linear combinations of the raw phase observations $\Phi$ that algebraically remove these terms one layer at a time. The essential requirement is simultaneity: the same satellites are observed by both receivers at the same epochs, so the shared errors are common and cancel.

Single differencing (between receivers)

Subtract the simultaneous phase of the same satellite $k$ observed at the two receivers: $$\Phi^{k}_{AB} = \Phi^{k}_{A} - \Phi^{k}_{B}$$ Because both receivers see the identical satellite clock at that instant, the satellite-clock error cancels. Most of the correlated atmospheric (ionospheric/tropospheric) delay also cancels over short baselines. The receiver-clock difference and the ambiguity difference remain.

Double differencing (between receivers and satellites)

Difference two single differences formed to two satellites $k$ and $l$: $$\nabla\!\Delta\Phi^{kl}_{AB} = \left(\Phi^{k}_{A}-\Phi^{k}_{B}\right) - \left(\Phi^{l}_{A}-\Phi^{l}_{B}\right)$$ The second subtraction removes the receiver-clock errors as well (they are common to both single differences). What is left is the geometry plus an integer double-differenced ambiguity $\nabla\!\Delta N^{kl}_{AB}$, which is a true integer and can be "fixed." The double difference is the fundamental observable of precise relative GPS.

Triple differencing (between epochs)

Difference two double differences formed at two epochs $t_1$ and $t_2$: $$\delta\nabla\!\Delta\Phi^{kl}_{AB} = \nabla\!\Delta\Phi^{kl}_{AB}(t_2) - \nabla\!\Delta\Phi^{kl}_{AB}(t_1)$$ Provided no loss of lock occurred, the ambiguity $N$ is constant between epochs, so it cancels entirely. The triple difference is therefore ambiguity-free and ideal for an initial approximate solution and for detecting cycle slips (a slip appears as a spike), at the cost of higher noise and correlated errors — hence it seeds, but does not replace, the double-difference fixed solution.