# Development of a Submillimetric GNSS-Based Distance Meter for Length Metrology +
## 2 GNSS Data Processing Approach
### Functional Model for the Double Differences  
The functional model used in our approach is specifically tailored to length determination and differs from the corresponding geodetic approach in four ways: only L1/E1 double differences are processed, ambiguity determination is not required, and both ionospheric and tropospheric corrections are not included.  
Since L1/E1 carrier phase observables are less noisy than L2, L5, E5a or E5b observables, or any combination of them [33], double differences of L1/E1 carrier phase are the preferred equations to obtain the final solution.  
A free ambiguity approach reduces the number of unknowns to determine thus strengthening the model [34]. The only requirement for using the free-ambiguity method is a prior determination of the absolute coordinates of both ends with an accuracy of a few centimetres, which can be achieved by means of a PPP static processing, thus obtaining \((X_{i}^{(0)},Y_{i}^{(0)},Z_{i}^{(0)})\) and \((X_{j}^{(0)},Y_{j}^{(0)},Z_{j}^{(0)})\)  
Given the double difference equation  
\[\lambda\varphi_{ij}^{kl}=\rho_{ij}^{kl}+\lambda N_{ij}^{kl}+\epsilon_{ij}^{kl} \tag{6}\]  
for substraction with the order  
\[\varphi_{ij}^{kl}=(\varphi_{j}^{l}-\varphi_{i}^{l})-(\varphi_{j}^{k}-\varphi_ {i}^{k})=\varphi_{j}^{l}-\varphi_{i}^{l}-\varphi_{j}^{k}+\varphi_{i}^{k} \tag{7}\]  
and expanding around \((X_{j}^{(0)},Y_{j}^{(0)},Z_{j}^{(0)})\) with \((X_{i}^{(0)},Y_{i}^{(0)},Z_{i}^{(0)})\) held fixed, Equation (6) can be rewritten as  
\[\frac{\lambda\varphi_{ij}^{kl}-\rho_{ij}^{kl}}{\lambda} = \frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial X_{j}} dX_{j}+\frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial Y_{j}}dY_{j} \tag{8}\] \[+ \frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial Z_{j}} dZ_{j}+N_{ij}^{kl}+\frac{1}{\lambda}\epsilon_{ij}^{kl}\]  
which can be subsequently arranged as  
\[\frac{\lambda\varphi_{ij}^{kl}-\rho_{ij}^{kl}}{\lambda}-int(\frac {\lambda\varphi_{ij}^{kl}-\rho_{ij}^{kl}}{\lambda}) = \frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial X_{j}} dX_{j}+ \tag{9}\] \[+\frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial Y_{j}} dY_{j}+\frac{1}{\lambda}\frac{\partial\rho_{ij}^{kl}}{\partial Z_{j}} dZ_{j} + \frac{1}{\lambda}\epsilon_{ij}^{kl}\]  
where the only unknowns to be solved are \(dX_{j},dY_{j},dZ_{j}\).  
Regarding tropospheric delays, our current model assumes that they are negligible for distances below 1 km. Note that existing metrological facilities are small and points of interest are usually at about the same altitude. Consequently, the path taken by a GNSS signal through the atmosphere to each end of a baseline is almost the same, especially for those satellites with higher elevation. In addition, we collected GNSS data in the nighttime. Thus, only residual atmospheric delay errors with minor influence on the distance are expected.  
For longer distances, ionospheric delay can be determined using a combination of carrier waves [28] and subsequently use them to correct L1/E1 carrier phase observables. On the contrary, the influence of residual tropospheric delays should be taken into account when both ends of the measured baseline have a substantial height difference which can happen in some local ties or geodetic control networks. Since the tropospheric errors expected in the double difference equations are very small and also dependant on local meteorological parameters, global models may not be accurate enough for submillimetric length determination.