## 3 Instantaneous Multi-Constellation Relative Positioning: Methodology and Performance
### Loosely Combined Relative Positioning Model with Parameterised Ionospheric Delays
The equations are presented for dual frequency phase and code observables; however, the model may be extended for multi-frequency signals accordingly [32]:  
\[\left\{ \begin{array}{c}\lambda_{f_{1}}\varphi_{kl,f_{1}}^{ij}=\rho_{kl}^{ij}+ T_{kl}^{ij}-I_{kl}^{ij}+\lambda_{f_{1}}N_{kl,f_{1}}^{ij}+\epsilon_{kl,\varphi,f_{1}} ^{ij}\\ P_{kl,f_{1}}^{ij}=\rho_{kl}^{ij}+T_{kl}^{ij}+I_{kl}^{ij}+\epsilon_{kl,P,f_{1}} ^{ij}\\ \lambda_{f_{2}}\varphi_{kl,f_{2}}^{ij}=\rho_{kl}^{ij}+T_{kl}^{ij}-\mu I_{kl}^{ ij}+\lambda_{f_{2}}N_{kl,f_{2}}^{ij}+\epsilon_{kl,\varphi,f_{2}}^{ij}\\ P_{kl,f_{2}}^{ij}=\rho_{kl}^{ij}+T_{kl}^{ij}+\mu I_{kl}^{ij}+\epsilon_{kl,P,f_{ 2}}^{ij}\end{array} \right. \tag{5}\]  
where \(\lambda\) is the signal wavelength, \(\varphi\) is the carrier phase observable in cycles, \(P\) is the code pseudorange, \(\rho\) denotes the geometric satellite to receiver range, \(N\) is the integer ambiguity, superscripts \(i\), \(j\) and \(k\), \(l\) states for satellites and stations, respectively, subscripts \(f_{1}\) and \(f_{2}\) correspond to the first and second applied frequencies, \(I\) is the ionospheric delay at the first frequency, \(\epsilon\) denotes multipath and observation noise. \(T\) is the tropospheric delay, the double differenced value of which can be denoted as:  
\[T_{kl}^{ij}=\left(\alpha_{k}^{i}ZTD_{k}-\alpha_{k}^{j}ZTD_{k}-\alpha_{l}^{i} ZTD_{l}+\alpha_{l}^{j}ZTD_{l}\right), \tag{6}\]  
where \(ZTD\) is the tropospheric total zenith delay and \(\alpha\) is the troposphere mapping function coefficient. \(ZTD\)s may be introduced as known values derived from global tropospheric models (troposphere fixed model) or treated as constrained parameters whose a priori values are corrected via estimation of the corrections (troposphere-weighted model). Since the linearized equations are formed independently for each constellation, there are no inter-system DD observables. The vector of unknowns is expressed as follows:  
\[x=\left[\delta X_{k},\delta Y_{k},\delta Z_{k},\delta X_{l},\delta Y_{l}, \delta Z_{l},\delta ZTD_{k},\delta ZTD_{l},N_{kl}^{ij}\ldots N_{kl^{\prime}} ^{ij},\iota_{kl}^{ij}\ldots I_{kl}^{ij}\right]^{T}, \tag{7}\]  
where \(\left\{\delta X_{k},\delta Y_{k},\delta Z_{k},\delta X_{l},\delta Y_{l}, \delta Z_{l}\right\}\) are the corrections to a priori geocentric coordinates of the stations, \(\left\{\delta ZTD_{k},\delta ZTD_{l}\right\}\) are the residual zenith tropospheric delays, \(\left\{N_{kl}^{ij}\ldots N_{kl}^{ij}\right\}\) denotes the set of phase ambiguities for selected DD intra observables and, finally, \(\left\{I_{kl}^{ij}\ldots I_{kl}^{ij}\right\}\) denotes the set of epoch-specific DD ionospheric delays. The final solution with resolved double-differenced ambiguities is performed via a four-step procedure: (1) the float solution; (2) the ambiguity resolution; (3) the ambiguity validation; (4) the fixed solution obtained via parameter update, taking into account fixed ambiguities from step (2). The details of the processing strategy are listed in Table 3.