# Galileo: The Added Value for Integrity in
## 3 Integrity Algorithm
### Forward-Backward  
Only configurations characterized by strong geometry conditions, _i.e._, characterized by a WARP lower than a fixed threshold, are then checked using the FB algorithm. After the preliminary check, the FB method is implemented [9; 12; 17]. In the first section, the forward phase, a GT, a LT and a separability test, are used. In the second phase, the backward step, only the GT is used.  
After the preliminary check, the first step of the FB algorithm is the GT. In this case, the test decision statistic, \(D\), is a quadratic form of the residuals, \(\mathbf{r}\), defined in Equation (7):  
\[D=\mathbf{r}^{T}W\mathbf{r} \tag{19}\]  
Under the assumption that the observation errors are normally distributed, \(D\) follows a central chi-square distribution with \(m-n\) degrees of freedom. \(m\) is the number of measurements, and \(n\) is the number of unknowns, that in the multi-constellation case considered here is five, _i.e._, the number of elements of Equation (3). \(D\) is compared to the threshold, \(T_{G}\), defined as:  
\[T_{G}=\chi_{1-\alpha,(m-n)}^{2} \tag{20}\]  
where \(\chi_{1-\alpha,(m-n)}^{2}\) is the abscissa corresponding to a probability value \(1-\alpha\) of a \(\chi^{2}\) distribution with \((m-n)\) degrees of freedom. \(\alpha\) is the target false alarm probability, _i.e._, the probability of incorrectly accepting a measurement. Note that \(T_{G}\) was already used in Equation (15) for the definition of the WARP parameter. The GT is applied to the whole set of measurements: if the measurement set is declared not to be consistent, then an LT has to be carried out to identify outliers. In this case, the standardized residuals are used as decision variables and are defined as:  
\[w_{i}=\left|\frac{r_{i}}{\sqrt{(C_{r})_{ii}}}\right|\qquad i=1,\ldots,m \tag{21}\]  
where \((C_{r})_{ii}\) is the \(i\)-th diagonal element of the residual covariance matrix \(C_{r}\), which is determined, starting from Equation (7), using the general variance-covariance propagation law [12]. The standardized residuals are assumed to be normally distributed and, thus, are compared to a threshold, \(T_{L}\), defined as:  
\[T_{L}=\mathcal{N}_{(1-\alpha/2)} \tag{22}\]  
where \(\mathcal{N}_{(1-\alpha/2)}\) is the inverse cumulative density function (CDF) of a Gaussian random variable computed at \(1-\alpha/2\). The measurement corresponding to the standardized residual exceeding \(T_{L}\) is indicated as an outlier and, thus, should be excluded from the navigation solution computation. Before measurement rejection, a further test is carried out to analyze the correlation among observables. In particular, a large blunder can lead to abnormal residuals also in other measurements, causing erroneous measurement rejection. Hence, a test to verify the separability of the measurements is performed. In this case, the decision variable adopted is the correlation coefficient, \(\gamma_{ij}\), of the standardized residuals [15]:  
\[\gamma_{ij}=\frac{(C_{r})_{ij}}{\sqrt{(C_{r})_{ii}(C_{r})_{jj}}} \tag{23}\]  
The coefficients are compared with respect to a threshold, if there is at least one coefficient exceeding the threshold, the measurement identified as an outlier by the LT is not rejected, and the solution is flagged as unreliable. Only if there is no correlation coefficient exceeding the threshold, the measurement is rejected. The former test, which is not present in the classical FB algorithm, concludes the forward phase. If more than one measurement is rejected, the backward phase is implemented to re-introduce measurements erroneously excluded from the navigation solution.