## 3 Simulation Results
### Simulating the True Total DC  
The true total DC vector \(\boldsymbol{\xi}\) is simulated considering a first order Gauss-Markov process with a decay rate parameter of \(1/D\) and process variance equal to the sum of the individual variances \(\sigma_{CE}^{2}=0.85^{2}\)\(m^{2}\), \(\sigma_{I}^{2}=9^{2}\)\(m^{2}\) and \(\sigma_{T}^{2}=1\)\(m^{2}\) taken to be representative of worst-case values [7]. This is equivalent to generating different values of the error sources at each RS. Moreover, the simulated vector as such yields the exponential decay rate function that characterizes the true total DC covariance matrix as in Equation (5). The mean of the simulated vector as such is zero, however the pseudorange DC for a standard user is around 3-10 m, reflecting the ionosphere, troposphere, unmodeled effects and modeling/measurement errors. Therefore, a constant error of \(\sigma_{\xi}\) m is manually added to the simulated DC vector to take into account biases such as satellite ephemeris and clock errors, as well as propagation errors (ionosphere and troposphere), while keeping the structure of the correlation matrix \(\boldsymbol{R_{\xi}}\). It is worthnoting that adding a bias to the simulated process vector \(\mathbf{\xi}\) as such maintains the diversity of the errors at each RS.  
Alternatively, the true total DC vector \(\mathbf{\xi}\) is simulated as a sum of three first order Gauss-Markov processes having the same decay rate parameter value of \(1/D\) but different process variances \(\sigma_{CE}^{2}\), \(\sigma_{I}^{2}\) and \(\sigma_{T}^{2}\). In this case, \(\mathbf{\xi}\) can be seen as the output of the sum of three white noise signals \(\mathbf{w_{i}}\sim\mathcal{N}(\mathbf{0},\mathbf{\sigma_{w_{i}}})\) passed through a low pass filter H(z):  
\[H(z)=\frac{d\sqrt{2\alpha}}{1+\alpha d-z^{-1}}=\frac{a}{1-bz^{-1}} \tag{6}\]  
with \(a=\frac{d\sqrt{2\alpha}}{1+\alpha d}\), \(b=\frac{1}{1+\alpha d}\), \(\alpha=1/D\), \(d=80\) km which represents the RS distance separation and \(\mathbf{\sigma_{w_{i}}}\) represent the aforementioned variances relative to the satellite clock and ephemeris error, the ionosphere and troposphere errors. The sum of the output of these three filters can then be written as:  
\[\xi[n]=\sum_{l=-\infty}^{\infty}h[l]\sum_{i=1}^{3}w_{i}[n-l]=\sum_{l=-\infty} ^{\infty}h[l]w[n-l] \tag{7}\]  
with \(h[n]=ab^{n}\) is the Z inverse transform of \(H(z)\) and \(w[n]\) a white noise signal with a normal distribution \(\mathcal{N}(\mathbf{0},\sqrt{\sum\sigma_{w_{i}}^{2}})\). This proves that the true total DC vector generated as such is still a Gauss-Markov process, with the same decay rate parameter and variance \(\sigma_{\xi}^{2}=\sqrt{\sum\sigma_{w_{i}}^{2}}\).  
On the other hand, as previously mentioned, the vector of noise terms \(\mathbf{\epsilon}\) is modeled as a zero mean jointly Gaussian random process with a diagonal covariance matrix defined by \(\mathbf{R_{w}}=\sigma_{w}^{2}\mathbf{I}\) with \(\sigma_{w}^{2}=\sigma_{m}^{2}+\sigma_{r}^{2}\) and \(\sigma_{m}=1.5\) m and \(\sigma_{r}=0.35\) m as approximate maximum 1-Sigma error values on code phase measurements. The aforementioned error values \(\sigma_{m}\) and \(\sigma_{r}\) are due to multipath and receiver noise respectively considering a stand-alone user (not using Differential GNSS (DGNSS) corrections) [6]. It is worth noting that such values are independent between user and reference receivers and correspond to clean or moderate reflecting source conditions around user and reference receivers. Moreover, it is assumed that a default level of carrier smoothing is present. Finally, a vector of correlation distance \(\mathbf{D}\) values and variance values \(\mathbf{\sigma^{2}}\) is considered in generating the true total DC.