# Noises in Double-Differenced GNSS Observations
## 2 Materials and Methods  
### Zero-Baseline Double-Differenced Observable  
The noise parameters of the GNSS observations were derived using double-differenced (DD) measurements from a zero-length baseline (ZB) carried out by two receivers of the same type. The ZB DD observation combination effectively separates the measurement noise from the deterministic portion (e.g., atmospheric delays, clock and orbital errors, multipath effects) and does not de-correlate the time series, which is essential for autocorrelation analysis.  
The simplified code (P) and carrier phase (L) observation equations are represented by the following formulas:  
\[\begin{split} P&=R+c\delta t+\delta T+\delta I+ \delta M_{P}+\delta K_{P}+\varepsilon_{P}\\ L&=R+c\delta t+\delta T-\delta I+\delta M_{L}+ \delta K_{L}+\lambda B+\varepsilon_{L}\end{split} \tag{1}\]  
where \(R\) is the geometric distance of the satellite receiver (including the satellite orbital errors, antenna phase centre offset for the satellite, and receiver as well as carrier phase wind-up errors), \(\delta t\) denotes the difference in the receivers satellite clock error, \(\delta T\) and \(\delta I\) denote the troposphere and ionosphere errors, \(\delta M_{P}\) and \(\delta M_{L}\) are the multipath effects for code and carrier phase, respectively, \(\delta K_{P}\) and \(\delta K_{L}\) denote differences in the receiver satellite hardware biases for the code and carrier phase observations, \(B\) is the integer for carrier phase ambiguity, and \(\varepsilon_{P}\) and \(\varepsilon_{L}\) are random errors for the code and carrier phase measurements. All of the parameters are expressed in units of length, except for the clock errors, which are expressed in units of time, and the integer carrier phase ambiguity, which is denoted in wave cycles. Additionally, \(c\) and \(\lambda\) denote the speed of light and carrier wavelength, respectively.  
The ZB DD combinations are computed by differencing each type of observation (for the same frequency/signal type) from the two receivers connected to the same antenna to the two satellites during the same measurement epoch. For such a combination, all of the geometrical parameters are eliminated, except for the double-differenced (denoted by the symbol \(\
abla\Delta\)) carrier phase ambiguity parameter, which can be easily resolved if the precise coordinates of the antenna are known. The ZB DD observables for the code (\(\
abla\Delta P\)and carrier phase (\(\
abla\Delta L\)) (after subtracting the DD carrier phase ambiguity) only contain the double-differenced measurement noise, which expresses the magnitude of the random errors in the observed signal. Assuming that the combinations are only computed for intra-system differences and that the inter-frequency bias (IFB) is eliminated for the GLONASS system because it uses Frequency Division Multiple Access (FDMA), the expectation (\(E\{\Delta\}\)) and the dispersion (\(D\{\Delta\}\)) of the ZB DD combination for a pair of homogeneous receivers can be expressed as the equations [47]:  
\[E\{\
abla\Delta L\}=0 \tag{2}\] \[D\{\
abla\Delta L\}=4(1-\rho_{\Delta})\sigma_{L}^{2}\]  
ZB DD dispersion describes the measurement noise with a variance \(\sigma_{L}^{2}\), which is affected by the cross-correlation \(\rho_{\Delta}\) due to the simultaneous use of the same antenna (in a zero-length baseline setup) with a common Low-Noise Amplifier (LNA) [48; 49]. Observation differencing eliminates common LNA noise in the ZB DD time series, causing the dispersion value to be significantly reduced. However, when assuming a constant LNA noise value over time, \(\rho_{\Delta}\) does not affect the determined noise parameters. The equations relate to the carrier phase observations, but they have an analogous form with the parameter \(\sigma_{P-}\), which is used for code observations.