# Evaluation of BDS-2 and BDS-3 Satellite Atomic Clock Products and Their Effects on Positioning
## 2 Methods
### Satellite Clock Stability Assessment and BDS-2/BDS-3 Clock Datum Discrepancy  
Due to different strategies for handling satellite clock offset, the time datum of satellite clock products of different institutions may be different. Thus, considering the different time datum, the satellite clock offset can be given by the following:  
\[dt^{s}_{AC}=\Delta t^{s}+\widehat{clk}^{sys}_{AC} \tag{1}\]  
where the superscript \(s\) and \(sys\) refer to satellites and GNSS systems, respectively; \(\Delta t^{s}\) is the \"true\" clock offset relative to the ideal time reference; the subscript \(AC\) refers to the analysis center; \(dt^{s}_{AC}\) is the estimated clock offset in products; \(\widehat{clk}^{sys}_{AC}\) is the clock datum with respect to the ideal time reference in products. The Allan deviation (ADEV)--or its square, the Allan variance (AVAR)--is a common time domain measure of frequency stability. Compared with the standard variance, the AVAR has the advantage of being convergent for most types of clock noise, such as white and flicker phase noise [34]. It is defined as the infinite average of the difference in adjacent fractional frequency values averaged over the time interval \(\tau\) and can be expressed as follows:  
\[\sigma^{2}_{y}(\tau)=\langle\frac{1}{2}(\overline{y}^{\tau}_{i+1}-\overline{y }^{\tau}_{i})^{2}\rangle \tag{2}\]  
where \(\sigma^{2}_{y}(\tau)\) is the Allan variance for the interval of interest \(\tau\), \(\langle\cdot\rangle\) represents an infinite time average operation and \(\overline{y}^{\tau}_{i}\) denotes the average fractional frequency over the averaging time \(\tau\). In reality, the AVAR is usually calculated with phase data as follows:  
\[\sigma^{2}_{y}(\tau)=\frac{1}{2(N^{\prime}-2)\tau^{2}}\sum_{i=1}^{N^{\prime}-2 }\left(x_{1+(i+1)\cdot m}-2x_{1+(i)\cdot m}+x_{1+(i-1)\cdot m}\right)^{2} \tag{3}\]  
where \(m\) is the averaging factor and equals averaging time, \(\tau\), divided by basic sampling interval \(\tau_{0}\), \(x_{i}\) is the i-th of \(N\) phase values and \(N^{\prime}=\text{int}(N/m)\) with the round down operation \(\text{int}(\cdot)\). Several improved versions of the AVAR are put forward, and one of the mostly used version now is the overlapping Allan variance, which can make maximum use of a data set and provide better statistical confidence [35]. In terms of phase data, the overlapping Allan variance used in this paper is defined as follows:  
\[\sigma^{2}_{y}(\tau)=\frac{1}{2(N-2m)\tau^{2}}\sum_{i=1}^{N-2m}(x_{i+2m}-2x_{i +m}+x_{i})^{2} \tag{4}\]  
The ADEV \(\sigma_{y}\) is a more common expression of the result. In addition, gross errors are detected and eliminated by the median absolute deviation (MAD) method. Though satellite clock stability can be assessed by substituting \(dt^{s}_{AC}\) into Equation (4) for different product, the clock datum discrepancy between BDS-2 and BDS-3 cannot be confirmed directly with a single satellite clock offset product. Hence, the clock datum difference between different products for BDS-2 and BDS-3 is estimated firstly as follows:  
\[\Delta\widehat{clk}^{sys}=\frac{\sum_{s=1}^{n^{sys}}\left(dt^{s}_{AC_{1}}-dt^ {s}_{AC_{2}}\right)}{n^{sys}}=\widehat{clk}^{sys}_{AC_{1}}-\widehat{clk}^{ sys}_{AC_{2}} \tag{5}\]  
where \(\widehat{\Deltaclk}^{sys}\) is the datum difference of the two products generated by \(AC_{1}\) and \(AC_{2}\), and \(n^{sys}\) is the number of satellites. The difference between \(\Delta\widehat{clk}^{sys}\) of BDS-2 and BDS-3 can thus be expressed as the difference between the BDS-2/BDS-3 clock datum discrepancy in different products, as follows:  
\[ \begin{array}{rl}\Delta\widehat{clk}^{BDS-2}-\Delta\widehat{clk}^{BDS-3}& =(\widehat{clk}^{BDS-2}_{AC_{1}}-\widehat{clk}^{BDS-2}_{AC_{2}})-( \widehat{clk}^{BDS-3}_{AC_{2}}-\widehat{clk}^{BDS-3}_{AC_{2}})\\ &=(\widehat{clk}^{BDS-2}_{AC_{1}}-\widehat{clk}^{BDS-3}_{AC_{1}})-(\widehat{clk }^{BDS-2}_{AC_{2}}-\widehat{clk}^{BDS-3}_{AC_{2}})\end{array}