## 3 Results
### LRNTS Observables  
LRNTS observables are code pseudoranges and carrier phases, the same as with any GNSS. Regarding different Lunar conditions (geometric and environmental) compared to Earth GNSS, standard GNSS observable equations change in some terms.  
A simplified form of a code pseudorange between two transceivers can be expressed as:  
\[P=c(T_{r}-T_{s})=c\Delta T \tag{1}\]  
where \(T_{r}\) and \(T_{s}\) are signal reception and signal emission times, respectively, while c stands for the speed of light in a vacuum. Readings of the times \(T_{r}\) and \(T_{s}\) correspond to the receiving (\(r\)) and transmitting (\(s\)) transceiver, which yields:  
\[P=c[(t_{r}+\delta_{r})-(t_{s}+\delta_{s})]=c\Delta t+c\Delta\delta=\rho+c\Delta\delta \tag{2}\]  
With:  
* \(t_{r}\), \(t_{s}\)--readings of the received and transmitted signals, respectively;
* \(\delta r\), \(\delta_{s}\)--biases of the receiver and the transmitter clocks, respectively;
* \(\rho\)--the geometric distance between transceivers \(s\) and \(r\).  
The biases \(\delta_{r}\), \(\delta_{s}\) are calculated per transceiver and are available via a navigation message. The complete pseudorange equation introduces other error sources:  
\[P=\rho+c\Delta\delta+D_{r}+D_{s}+M_{rs}+\epsilon_{rs} \tag{3}\]  
where:  
* \(D_{r}\), \(D_{s}\)--hardware delays for the receiver r and the transmitter s, respectively;
* \(M_{rs}\)--multipath term between \(r\) and \(s\);
* \(\epsilon_{rs}\)--remaining random error of the measured pseudorange.  
If a signal is transmitted at the frequency \(f\), then the basic carrier phase equation is written as:  
\[\Delta\varphi_{rs}=\varphi_{r}-\varphi_{s}=-f\frac{\rho}{c}-f\Delta\delta \tag{4}\]  
where \(\varphi_{s}\) is the phase angle of the sent signal and \(\varphi_{r}\) is the received phase angle of the same signal. The carrier phase \(\Delta\Phi_{rs}\) is expressed in cycles, taking values within one cycle, i.e., [0, 2\(\pi\)]. In order to measure the cumulative phase angle \(\Delta\Phi_{rs}\), the number of whole cycles, known as the integer ambiguity \(N_{0}\), is involved. If the frequency \(f\) is expressed as a function of \(c\) and signal wavelength \(\lambda\) using the relation \(f=\frac{c}{\lambda}\), the standard form of the carrier phase equation becomes:  
\[\Delta\Phi_{rs}=-\Delta\varphi_{rs}=\frac{\rho}{\lambda}+\frac{c}{\lambda} \Delta\delta+N_{0} \tag{5}\]  
Elements of (5) are expressed in meters by multiplying it by \(\lambda\):  
\[\lambda\Delta\Phi_{rs}=-\Delta\varphi_{rs}=\rho+c\Delta\delta+\lambda N_{0} \tag{6}\]  
The final form of the carrier phase measurements also contains hardware delays (\(D_{r}\)), multipath (\(D_{s}\)), and a random measurement error (\(\epsilon_{rs}\)), yielding:  
\[\lambda\Delta\Phi_{rs}=\rho+c\Delta\delta+\lambda N_{0}+D_{r}+D_{s}+M_{rs}+ \epsilon_{rs} \tag{7}\]  
\(N_{0}\) corresponds to the number of full cycles in the first epoch. It is unknown, as in the case of GNSS. However, because LRNTS transceivers are static, the number of full cycles for a pair (transmitter, static receiver) does not change in time and can be determined after the second measuring epoch, because \(N_{0}=N_{1}=N_{2}=\ldots\).