## 3 Double Difference Observation Modelling
### Time Synchronisation  
To compute DD, observations from two different receivers are combined. Each receiver is driven by a different oscillator that realises a different time scale [31]. Depending on the technology and processing philosophy of the manufacturer, these time scales can be steered internally to a GPS time scale at different accuracy levels (e.g., \(\mu\)s or even better). Sometimes, the oscillators are completely free running or with an internal adjustment to GPS time within \(\pm\) 0.5 ms. When the clock error reaches this threshold, a 1 ms jump is introduced. A special case occurs for the high sensitivity UBX receivers: the time tag of the raw observations is not an integer second but can be an arbitrary time offset in the range of milliseconds. The impact of an asynchronism between the receiver clocks needs to be considered in relative positioning techniques as stated in the different texts, for example, [32] or [33]. The bias due to poor time synchronisation can be resolved by using extrapolation [22] or by aligning the receiver time scales [21; 34].  
For kinematic cases, the asynchronism between the receiver clocks induces additional baseline components depending on the dynamics of the platform. For our applications and encountered receiver clock offsets, the following linear correction is sufficient:  
\[b_{AB}^{ik}=\left(\mathbf{s}_{B}^{i}(t)-\mathbf{s_{B}}^{k}(t)\right)^{\intercal }\cdot\mathbf{v}(t)\cdot(t_{A}-t_{B}) \tag{3}\]  
where \(\mathbf{s}_{B}^{i}(t)\) and \(\mathbf{s_{B}}^{k}(t)\) denote the line of sight unit vectors between the platform and the satellites i and k at the nominal reception time t, respectively, \(\mathbf{v}(t)\) is the velocity of the platform at that time and T represents the transpose operator. The correction term depends further on the relation of the actual reception times of the receivers \(t_{A}\) and \(t_{B}\) that includes the receiver clock biases. When applying the correction to the computed DD, the kinematic DD bias needs to be subtracted:  
\[\tilde{\Phi}_{AB}^{ik}=\Phi_{AB}^{ik}-b_{AB}^{ik} \tag{4}\]  
In Figure 6, two examples of the computed kinematic DD bias are shown, one for each experiment. For the urban scenario, the kinematic DD bias is computed for the UBX receiver combination 0867-1771 and GPS satellite pair PRN 29-26. For the flight experiment, the kinematic DD bias is computed for the geodetic-grade receiver combination 0081-0082 and GPS satellite pair PRN 3-1. The magnitude of the correction term highly depends on the velocity of the moving platform and the relative receiver clock biases. Hence, the correction values are almost zero in stand still phases of the urban area test drive. In epochs where the velocity is about 15 m/s, the correction values increase to more than about 10 cm. The kinematic DD bias values for the high sensitivity equipment are quite high due to the large time offsets as stated earlier.  
The magnitude of correction during the flight experiment is lower even though the aircraft is moving at speeds of more than 100 m/s during some phases. This is specifically due to the smaller relative clock biases of the two JVD receivers, which were connected to external atomic clocks. However, changes in the values still follow a behaviour, which can be explained by the dynamic manoeuvres and consequential changes of the heading and the change in the speed of the aircraft.  
These results clarify that the computation of the kinematic DD bias correction term is essential for precise relative positioning.