# Cloud-Based Single-Frequency Snapshot RTK Positioning
## 2 Methodology
### SRTK Performance
#### 2.4.1 RTK Filter  
There are usually different algorithm configurations for RTK depending on the baseline distance. For long baseline scenarios, typically the following DD measurements models are used [27].  
\[\begin{split}\
abla\Delta\text{P}_{i}=\
abla\Delta\rho+\
abla \Delta I+\
abla\Delta T+\
abla\Delta\epsilon_{i,p},\\ \
abla\Delta\lambda_{i}\phi_{i}=\
abla\Delta\rho+\lambda_{i} \
abla\Delta N_{i}-\
abla\Delta I+\
abla\Delta T+\
abla\Delta\epsilon_{i, \phi},\end{split} \tag{16}\]  
where:  
* \(\
abla\Delta\) is the DD operator;
* \(P\) and \(\phi\) are the pseudorange in the m and carrier phase measurement in cycles, respectively;
* \(\lambda_{i}\) is the signal wavelength for carrier frequency \(i\), in m;
* \(T\) and \(I\) are the Troposphere and Ionosphere delay, respectively, in m;
* \(\epsilon\) represents the measurement noise, in m;
* \(N\) stands for the carrier phase integer ambiguity.  
In these cases, the vector of unknowns includes the parameters for Ionosphere and Troposphere delays. For the Ionosphere delay, this parameter can be the Total Electron Content (TEC). Moreover, a variant strategy is to build the Ionosphere Free linear combination of measurements from different carrier frequencies [28], however, this is not applicable to the present research as our focus in on single frequency snapshots. As for Troposphere delay, it can be estimated as a Zenith Total Delay (ZTD) and the North and East components of the Troposphere gradient [29].  
However, for short baseline cases, the Ionosphere and Troposphere delays and other minor correction terms such as the phase center variations are assumed to be eliminated through the DD operation [30]. As a result, the models presented in Equation (16) can be simplified as follows:  
\[\begin{split}\
abla\Delta\text{P}_{i}=\
abla\Delta\rho+\
abla \Delta\epsilon_{i,p},\\ \
abla\Delta\lambda_{i}\phi_{i}=\
abla\Delta\rho+\lambda_{i}\
abla \Delta N_{i}+\
abla\Delta\epsilon_{i,\phi}.\end{split} \tag{17}\]  
In this research, only the short baseline model is considered for simplicity and also due to the fact that only single frequency snapshots are considered. This means that the residual DD Ionosphere and Troposphere delays that are actually not fully cancelled out are lumped into the DD measurement noise term \(\
abla\Delta\epsilon\); thus, it can be expected that the RTK performance will degrade when the baseline grows larger.  
After the float solutions are generated using these measurements, the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) method is applied for the IAR procedure [31]. Due to the nature of snapshot positioning, only a single epoch of data is available. Thus, these measurements are processed in instantaneous mode. To determine if the current ambiguities are fixed, the distance between the best integer estimates to the float solution is calculated. This process is repeated for the second best integer set and then the ratio between these two distances are computed, resulting in the so called LAMBDA Ratio Factor (LRF). By comparing it to a defined threshold value, we can decide whether the fix is successful. Note that this may not be the optimal acceptance test in terms of failure rate or probability of producing a wrong fix [32], but is the simplest to implement. In this research, the RTK performance is mainly defined by the mean LRF value and the fix rate, i.e., the percentage of epochs that have a LRF value higher than a certain threshold.