## 3 Double Difference Observation Modelling
### DD Computation Algorithm  
To analyse the quality of carrier phase observations using DDs, additional processing steps are required. All the steps for the computation of DD from OMC, to the presentation of the results are depicted in the flow chart in Figure 7.  
After computing the DDs (cf. Equation (2)) the kinematic DD bias correction term is subtracted from the DD to obtain \(\bar{\Phi}_{AB}^{k}\) (cf. Equation (4)). Carrier phase observables are affected by cycle slips, which mainly occur due to low signal strength, signal losses and re-acquisition of the signal by the  
Figure 6: Kinematic double difference (DD) bias. (**a**) Terrestrial experiment: Kinematic DD bias of GPS L1 signal exemplarily shown for receiver combination UBK 0867–UBK 1771 and satellite combination PRN 29–PRN 26 (**b**). Aerial experiment: Kinematic DD bias of GPS L1 signal exemplarily shown for receiver combination JVD 0081–JVD 0082 and satellite combination PRN 3–PRN 1.  
Figure 7: Double difference processing procedure.  
receiver [35; 36]. In this algorithm, cycle slips are detected and subsequently repaired using triple differences. In order to obtain a true indication on the statistical behaviour of the signals, an outlier detection is applied based on a threshold detection method. Epochs wherein the triple difference is exceeding the threshold of 1.2 cm (i.e., the 3-\(\sigma\) value of the optimal carrier phase noise [36]) are deleted and hence, outliers are eliminated. Further, the DD integer ambiguity term needs to be computed and subtracted for each satellite arc. Due to the zero moving baseline, the integer ambiguity can be determined by just rounding to the nearest integer and is expressed as:  
\[N_{AB}^{ik}=\text{round}\left(\frac{\text{med}\left(\Phi_{AB}^{ik}\right)}{ \lambda}\right) \tag{5}\]  
where \(\text{med}\left(\Phi_{AB}^{ik}\right)\) is the median of the DD of one continuous arc and \(\lambda\) the wavelength of the signal.  
After subtraction of DD integer ambiguities, the GPS and Galileo DD can be used to assess the noise behaviour. Since GLONASS distinguishes satellites based on frequency-division multiple access (FDMA) technology, each satellite has a unique frequency. This leads to an additional bias referred to as single difference bias term when combining observations from two different satellites [37; 38]. This bias is considered in the computation algorithm.  
By applying all the steps as explained in the flowchart, DDs are obtained which have approximately a zero-mean for the complete time series. Statistics related to the computation of DD for two receiver combinations from each experiment (terrestrial and aerial) are summarised in Table 4. The larger number of ambiguities set up in the terrestrial experiment corresponds to more data gaps underlining that the signal reception conditions have a big impact on the quality of the observations. This can be seen clearly in the statistics as the number of cycle slips, outliers and ambiguities are higher whereas the number of observations is lower in the case of urban environments. The Galileo L1 signal seems more stable in the urban setup. In the case of the flight experiment, the count of cycle slips, outliers and ambiguities are considerably larger for receiver combination JVD 0993-0346 compared to JVD 0081-0082. On the contrary, the number of observations is almost similar for both the combinations.