## 3 Simulation Results
### Estimating the True Total DC  
Assuming that one has a simultaneous access to the corrections generated at each RSs, the true total DC vector \(\mathbf{\xi}\) is estimated using the LMMSE as in Equation (4). It is assumed that \(\mathbf{\mu_{\xi}}=\mathbf{\mu_{\eta}}\) as previously mentioned, and that a relatively good estimate of the true total DC covariance matrix \(\hat{\mathbf{R_{\xi}}}\) modeled as a single exponential function is obtained. However, it is essential to assess the impact of the spatial correlation matrix modeling errors on the true total DC estimation quality. For that end, a range of modeling errors \(\Delta\mathbf{D}=0\) to 2000 km are added to the vector \(\mathbf{D}\) which holds the true correlation distance values considered in simulating the true total DC vector. Moreover, a range of true total DC variance modeling errors \(\mathbf{\Delta\sigma^{2}}=0\) to \(25\,\mathrm{m}^{2}\) is added to the true value \(\mathbf{\sigma^{2}}\) which in these simulations coincides with \(\sigma_{\xi}^{2}\). The results presented herein are relative to a simulated true total DC. In order to assess the LMMSE estimation quality, a Monte-Carlo simulation of 500 runs has been carried out: the true total DC vector is simulated 500 times, the corresponding LMMSE estimates are generated, and the estimation error is computed. Figure 5 shows on the same plot, the histogram of both the estimation error distribution and the true noise distribution for two individual cases: (a) a true distance correlation of 100 km and a modeling error \(\Delta D=200\) km in Figure 5a and a corresponding (b) 900 km and a modeling error of 1800 km in Figure 5b. Both the estimation error and true noise refer to all RSs which are separated between themselves by a constant distance \(d=80\) km. It can be concluded from these two histograms that the LMMSE estimation yields no bias errors and generally reduces the noise affecting the true total DC, as the probability of estimation error lying around the 0 bin is higher than that corresponding to the true error. Furthermore, this probability is slightly higher for a larger correlation distance value of 900 km as seen in Figure 5b compared to 100 km Figure 5a. This is further demonstrated in Figure 6 where the  
Figure 5: Superimposed histogram of estimation error and true error distributions for a single combination of modeling errors (**a**) \(\Delta D=200\) km on top of \(D=100\) km and (**b**) \(\Delta D=1800\) km on top of \(D=900\) km. Figure 6: Probability of estimation error lying below half a meter for different correlation distance values and corresponding modeling errors. probability of the estimation error lying below half a meter is plotted with respect to various combinations of correlation distance and corresponding modeling error. The DC variance modeling error is taken to be 25 \(m^{2}\). It can be seen that this probability is higher as the correlation distance increases irrespective of the modeling error magnitude. After this initial test on the estimation quality, the variance of the estimation error is numerically computed from the Monte-Carlo simulations, and is plotted as a function of the ratio between correlation distance values and a constant distance separation of 80 km. Multiple curves are overlayed on the same figure to indicate the behavior of the various modeling errors. Figure 7 shows that although the estimation error variance decreases with increasing ratio of correlation to separation distance values, it does not change much after a certain ratio value. This is true for all cases of correlation distance modeling errors. The true error or noise is also plotted, and it can be seen that for correlation distance modeling errors  
Figure 8: Estimation error variance corresponding to a single Gauss-Markov process and different \(\Delta\sigma^{2}\) modeling errors. Figure 7: Estimation error variance corresponding to a single Gauss-Markov process and different \(\Delta D\) modeling errors.