## 2 LMMSE Estimator within GNSS Augmentation Systems
### Estimating True Total DC  
The true total DC vector \(\mathbf{\xi}\), characterized by a covariance matrix \(\mathbf{R}_{\mathbf{\xi}}\), can be estimated using a classical LMMSE algorithm, as in [13]. It is worth noting that a similar approach is applied in [11], where the LMMSE algorithm is applied to estimate vertical ionospheric delays at the considered user position starting from vertical ionospheric delays measured at various WAAS RSs. In this paper, the user is not taken into consideration, but only the network of RSs. This study intends to explore the quality of the true DC estimates that can be reached by harnessing the pseudorange DC generated by this network of RSs with fixed surveyed positions and a-priori knowledge of their mutual second order statistics. In fact, the approach in [11] makes a key assumption, mainly that the correlation of the vertical ionospheric delay between two points of the ionospheric grid depends only on the distance between these two points. The same is true for satellite ephemeris errors and propagation errors such as troposphere delay [6]. By noting that the different error sources acting on the true total DC are independent, the corresponding total covariance matrix can be written as the sum of the individual covariance matrices:  
\[\mathbf{R}_{\mathbf{\xi}}=\sigma_{CE}^{2}\exp\left(-\mathbf{x}/D_{E}\right)+\sigma_{I}^{ 2}\exp\left(-\mathbf{x}/D_{I}\right)+\sigma_{T}^{2}\exp\left(-\mathbf{x}/D_{T}\right) \tag{3}\]  
where \(D_{E}\), \(D_{I}\) and \(D_{T}\) are the correlation distances relative to the correlation functions characterizing the satellite ephemeris, ionosphere and troposphere corrections at different RSs respectively. The satellite clock error is the same for all users and thus is discarded in the covariance matrix. The spatial decorrelation value or the ionosphere gradient, is reported to be in the range of 1-4 mm/km in active but not stormy ionosphere conditions where it can go up to 400 mm/km [6]. The ionosphere correlation distance, which is the reciprocal of the spatial decorrelation, can thus be deduced to be in the range of 250-1000 km in normal conditions and go down as low as 2.5 km during large geomagnetic storms. Likewise, the ephemeris gradient is lower than 0.5 mm/km while the worst-case tropospheric decorrelation value is reported to be around 5 mm/km in [6]. Going back to Equation (2), the LMMSE estimate of a vector of random variables buried in noise is [13]:  
\[\boldsymbol{\hat{\xi}}=\boldsymbol{\mu_{\xi}}+\boldsymbol{R_{\xi}}( \boldsymbol{R_{\xi}}+\boldsymbol{R_{w}})^{-1}(\boldsymbol{\eta}-\boldsymbol{ \mu_{\eta}}) \tag{4}\]  
where it is assumed that the modeling errors and the true DC are uncorrelated and independent Gaussian Random Variable (RV). Given that the outcome of the estimation algorithm is most adversely affected by the narrowest correlation function, the sum of the three correlation functions can be approximated to be equal to one correlation function considering the lowest correlation distance, and with a variance coefficient equal to the sum of the three variances. This motivates the fact that it is commonplace to model the true total DC covariance matrix by a single exponential function characterized by a constant correlation distance \(D\) and a pseudorange DC standard deviation \(\sigma\)[10, 11] as:  
\[\boldsymbol{R_{\xi}}=\sigma^{2}\exp{(-\boldsymbol{x}/D)} \tag{5}\]  
where \(\boldsymbol{x}\) is a 2-Dimensional matrix considering all distance combinations between RSs spread along the rectilinear railroad track. The behavior of this spatial correlation for a fixed distance separation and a variable correlation distance is shown in Figure 4. It can be concluded that, for a fixed distance separation, the spatial correlation value increases with the correlation distance following a horizontal asymptote. However, the LMMSE estimation method can also be applied considering the sum of individual covariance matrices as shown in Equation (3).