# Evaluating Optical Clock Performance for GNSS Positioning
## 4 Clock Stability Analysis  
A clock stability analysis is carried out to compare the three successfully developed optical clocks; CROC, Iodine MTS, and Sr Lattice clock, against current GNSS satellite atomic clocks. It is important to note that the ADEV values of the optical clocks are sourced from the relevant literature [11; 12; 13], which presented the clock direct measurement outcomes. In contrast, we are limited to obtaining such information for the GNSS atomic clocks. Therefore, we can only deduce the clock errors indirectly from 24 h of IGS clock products with 30 s intervals to estimate the ADEV values. So, IGS clock products would necessarily incorporate other noise sources and estimation biases. On the other hand, IGS clock products are derived from a weighted ensemble of satellite clocks and the global infrastructure of monitoring stations in reference to the IGS time scale, which is determined with a two-state polynomial model driven by a mainly white noise process [53]. So, for simplicity, it is assumed that no modelling errors or reference time scale biases are found in the clock products. In total, 12 IGS clock products are used in this study spanning from 1 January 2022 to 1 December 2022 (the first day of each month in 2022). The overlapping ADEV method is used as it has the advantage of maximizing the use of all possible combinations in the data set compared to the other non-overlapping methods [51]. The overlapping ADEV, \(\sigma_{y}(\tau)\), as formulated by [54], with the averaging time \((\tau)\) reads,  
\[\sigma_{y}(\tau)=\sqrt{\frac{1}{2(N-2m)\tau^{2}}\sum_{i=1}^{N-2m}(x_{i+2m}-2x _{i+m}+x_{i})^{2}} \tag{1}\]  
where \(N\) is the number of original time residual measurements, \(m\) is averaging factor defined as \(m=\tau/\tau_{0}\), \(\tau_{0}\) is the original sampling intervals (seconds), and \(x_{i}\) is the \(i\)-th of \(N\) values. The averaging times \((\tau)\) are expressed in seconds at 30, 60, 120, 240, 480, 900, 1800, 3600, 7200, 14,400, and 28,800. The 30 s and 900 s are the main interest of this study, given 30 s is the shortest averaging time estimated from the clock products, and 900 s, under which white noises drive most satellite clocks. Also, 900 s is the sampling interval of IGS Ultra-Rapid precise orbit and clock products for GNSS real-time users [55]. In light of the recent formation of four fully operational GNSS constellations, we investigated quantifying and differentiating the satellite clocks of each constellation to improve the data preparation and processing. As a result, according to the IGS clock products, there were 122 satellites in operation from the four main GNSS constellations in 2022. We classified the satellites into different groups as per the satellite vehicle types (or generations) and estimated the total number of atomic clocks, assuming that the satellite clock types are the same in each satellite generation. As per the details in Table 2, there were 439 atomic clocks onboard GNSS satellites maintaining the continuous operation of the four constellations in 2022, 260 of which are Rubidium Atomic Clocks (Rb), 110 are Passive Hydrogen Maser Clocks (PHM), and the remaining 69 clocks are Caesium Atomic Clocks (Cs). Rb clocks are predominantly employed for GNSS satellite missions. However, it is found that different atomic clocks are assembled and employed on the same satellite together. Thus, satellite types or Pseudo-Random Noise (PRN) are referred to as the ensemble of atomic clocks in Table 2.