Question 5 of 9: Counting Measurements and Unknowns — Static vs Kinematic

Question 5: Counting Measurements and Unknowns — Static vs Kinematic (10 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.

Find. The number of measurements and the number of unknown parameters (baseline coordinates + integer ambiguities), separately for static (receivers stationary) and kinematic (one receiver moving) positioning.

Approach. Count the independent double differences per epoch, multiply by the number of epochs for the measurements; then count the unknowns — ambiguities are one per double-differenced pair and constant over an uninterrupted session, while the coordinate unknowns are constant in static positioning but repeat every epoch in kinematic positioning.

1. Count the independent double differences per epoch. With \(n_s\) satellites and one receiver pair, one satellite is chosen as reference, so the number of independent double differences per epoch is $$n_{DD}=n_s-1=7-1=6.$$ Each of the 7 undifferenced-per-receiver observations is not independent once clocks are removed; 6 is the count that carries independent baseline information.
2. Total number of measurements. The same 6 double differences are formed at every one of the 300 epochs: $$N_{meas}=n_{DD}\times n_e=6\times 300=\boxed{1800\ \text{DD phase measurements}}.$$ This total is the same for static and kinematic positioning — the observation schedule is identical; only the unknowns differ.
3. Count the integer ambiguities. Each double-differenced satellite pair carries one integer ambiguity \(\nabla\!\Delta N^{jk}\). Because tracking is continuous (no cycle slips), each ambiguity is constant for the whole session: $$N_{amb}=n_s-1=6\quad(\text{shared by both modes}).$$
4. Static unknowns. The baseline vector \((\Delta X,\Delta Y,\Delta Z)\) is constant because both receivers are stationary, so there are just 3 coordinate unknowns plus the 6 ambiguities: $$U_{static}=3+N_{amb}=3+6=\boxed{9\ \text{unknowns}}.$$ Redundancy \(=N_{meas}-U_{static}=1800-9=1791\) — a very strongly over-determined solution.
5. Kinematic unknowns. The moving receiver has a new position at every epoch, so the 3 coordinates repeat \(n_e\) times; the 6 ambiguities remain constant: $$U_{kin}=3\,n_e+N_{amb}=3(300)+6=900+6=\boxed{906\ \text{unknowns}}.$$ Redundancy \(=1800-906=894\). A single epoch alone (6 measurements, 3+6 unknowns) is under-determined; the solution is possible only because the ambiguities are common to all 300 epochs, tying the epochs together.