Question 4 of 9: Construction of the Double-Differenced Observable

Question 4: Construction of the Double-Differenced Observable (15 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.

Approach. Start from the undifferenced pseudorange model, difference between the two receivers to remove the satellite-dependent errors, then difference between the two satellites to remove the receiver-dependent errors; finally track what happens to each error term.

Undifferenced model. For receiver \(A\) and satellite \(j\) at epoch \(t\), the L1 C/A pseudorange is

$$\rho_A^j(t)=r_A^j(t)+c\big[dt_A(t)-dT^j(t)\big]+I_A^j(t)+T_A^j(t)+\varepsilon_A^j(t),$$

where \(r_A^j\) is the true geometric range, \(dt_A\) the receiver-\(A\) clock error, \(dT^j\) the satellite-\(j\) clock error, \(c\) the speed of light, \(I_A^j\) the ionospheric delay, \(T_A^j\) the tropospheric delay, and \(\varepsilon_A^j\) the combined multipath-plus-receiver-noise term. Analogous equations hold for \(\rho_B^j,\ \rho_A^k,\ \rho_B^k\).

Step 1 — Single difference (between receivers). Subtracting the two receivers’ observations of the same satellite \(j\) removes the satellite clock error \(dT^j\), which is common to both:

$$\Delta\rho_{AB}^{j}(t)=\rho_B^j-\rho_A^j=\big(r_B^j-r_A^j\big)+c\big(dt_B-dt_A\big)+\big(I_B^j-I_A^j\big)+\big(T_B^j-T_A^j\big)+\big(\varepsilon_B^j-\varepsilon_A^j\big).$$

The same single difference is formed for satellite \(k\): \(\Delta\rho_{AB}^{k}=\rho_B^k-\rho_A^k\).

Step 2 — Double difference (between satellites). Subtracting the two single differences (satellite \(k\) minus satellite \(j\)) removes the receiver clock difference \(c(dt_B-dt_A)\), which is common to both single differences:

$$\nabla\!\Delta\rho_{AB}^{jk}(t)=\Delta\rho_{AB}^{k}-\Delta\rho_{AB}^{j}=\big(\rho_B^k-\rho_A^k\big)-\big(\rho_B^j-\rho_A^j\big),$$

which expands to

$$\boxed{\;\nabla\!\Delta\rho_{AB}^{jk}=\nabla\!\Delta r_{AB}^{jk}+\nabla\!\Delta I_{AB}^{jk}+\nabla\!\Delta T_{AB}^{jk}+\nabla\!\Delta\varepsilon_{AB}^{jk}\;}$$

The operator \(\nabla\!\Delta(\cdot)\) denotes the between-satellite (\(\nabla\)) difference of the between-receiver (\(\Delta\)) difference; \(\nabla\!\Delta r_{AB}^{jk}=(r_B^k-r_A^k)-(r_B^j-r_A^j)\) is the double-differenced geometric range that carries the baseline information.

Error behaviour. The differencing acts on each error term as follows:

• Satellite clock \(dT^j\): cancelled exactly by the single (receiver) difference — identical for both receivers viewing the same satellite.
• Receiver clocks \(dt_A,dt_B\): cancelled exactly by the double (satellite) difference — identical in both single differences.
• Orbit (ephemeris) error: strongly reduced — the range-error component from an orbit error is nearly common to two nearby receivers, so most of it differences away on short baselines.
• Ionospheric \(I\) and tropospheric \(T\) delays: strongly reduced on short baselines because they are spatially correlated (the two receivers look through nearly the same atmosphere); the residual grows with baseline length and must be modelled on long baselines.
• Multipath and receiver noise \(\varepsilon\): increased — these are uncorrelated between stations and satellites, so they do not cancel. Combining four independent observations makes the double-differenced noise variance \(4\sigma^2\) (standard deviation \(2\sigma\)).

The net effect is that double differencing eliminates both clock unknowns and cancels or greatly reduces the spatially-correlated systematic errors, leaving a clean geometric observable at the cost of a modest (factor-2) increase in random noise — the basis of precise relative (and carrier-phase) GNSS positioning.