## 4 Integrity Monitoring
### Overbounding CDF  
Due to the existence of multipath and the atmospheric residuals, the empirical distributions of the between-receiver phase and code residuals are neither symmetric nor have a zero-mean. In the IM, as it is difficult to model the exact empirical distribution, an overbounding normal distribution is thus needed, so that the CDF overbounds that of the empirical distribution. In this contribution, the overbounding CDF is searched for the weighted between-receiver phase (\(\delta\phi_{v,j}\)) and code (\(\delta p_{v,j}\)) residuals, which are formulated as:  
\[\delta\phi_{v,j} = \delta\phi_{DDj}^{1s}\times\sqrt{w_{\phi,j}^{1s}} \tag{31}\] \[\delta p_{v,j} = \delta p_{DDj}^{1s}\times\sqrt{w_{p,j}^{1s}} \tag{32}\]  
where the DD weighting functions \(w_{\phi,j}^{1s}\) and \(w_{p,j}^{1s}\) are computed applying the variance rule:  
\[\frac{1}{w_{\phi,j}^{1s}} = \frac{1}{w_{\phi,j}^{1}}+\frac{1}{w_{\phi,j}^{s}} \tag{33}\] \[\frac{1}{w_{p,j}^{1s}} = \frac{1}{w_{p,j}^{1}}+\frac{1}{w_{p,j}^{s}} \tag{34}\]  
In such a case, the mean (\(\bar{m}_{\phi,j}\), \(\bar{m}_{p,j}\)) and standard deviations (\(\bar{\sigma}_{\phi,j}\), \(\bar{\sigma}_{p,j}\)) of the searched overbounding CDF refer to the between-receiver level at the zenith direction and with a C/N\({}_{0}\) of \(\infty\). Note that the residuals screened out during the \(\chi^{2}\)-test (Equations (20) and (21)) and the time epochs with PDOP higher than 10 are not used for computing the overbounding CDF, as these observations are excluded in the FDE procedure of the RTK processing. According to [44], the EMC overbounding strategy does not only have no shape restriction on the empirical distribution, but is also evidenced for the conservatism during the transformation from the observation-domain to the position-domain. Compared with the general paired overbounding strategy [45], the EMC allows an excess mass \(\epsilon\), i.e., a total mass \(1+\epsilon\) over 1. This relieves the strict search process at the tails. The empirical distribution, denoted by \(G_{a}\), is then bounded between the left overbounding CDF \(G_{L}\) and the right overbounding CDF \(G_{R}\), such that it is always below the \(G_{L}\) and above the \(G_{R}\), which are given as:  
\[G_{L}(x) = \int_{-\infty}^{x}(1+\epsilon)\mathcal{N}(-\bar{m},\bar{\sigma}) dx\geq G_{a}(x) \tag{35}\] \[G_{R}(x) = \int_{-\infty}^{x}(1+\epsilon)\mathcal{N}(\bar{m},\bar{\sigma}) dx-\epsilon\leq G_{a}(x) \tag{36}\]  
for which the normal distribution with mean \(\bar{m}\) and standard deviation \(\bar{\sigma}\) is denoted by \(\mathcal{N}(\bar{m},\bar{\sigma})\). As examples, Figure 6 shows the empirical CDF \(G_{a}\) and its overbounding CDFs \(G_{L}\) and \(G_{R}\) for the weighted code residuals on L1 \(\delta p_{v,1}\) in the open-sky (a) and the suburban (b) scenarios. To satisfy Equations (35) and (36), for code residuals, the overbounding mean and standard deviations are searched from 0.05 m to 1 m with a step of 0.01 m, and for phase residuals from 0.001 m to 0.03 m with a step of 0.001 m. The excess mass \(\epsilon\) is set to 0.01 in this contribution and will be considered by computing the protection levels in Section 4.3. The search begins from the smallest overbounding mean value, and for each mean value, the algorithm searches again from the smallest overbounding standard deviation. The search stops when Equations (35) and (36) are fulfilled. The overbounding mean and standard deviations for the weighted between-receiver phase and code residuals are given in Table 4 for all observation types, for the open and suburban scenarios. The overbounding mean and standard deviations are in general higher in the suburban scenario than in the open-sky scenario. The multipath under the open-sky and the suburban scenarios, as well as the atmospheric residuals under quiet conditions are included in the weighted between-receiver residuals (\(\delta\phi_{v,j}\) and \(\delta p_{v,j}\)) and are covered by the overbounding standard deviations and mean values.