## 4 Virtual Station Localization Algorithm Based on Scattering Signal Identification  
On the basis of the identification of a one-bounce scattering signal from the same scatterer, we can easily locate scatterers. Once the scatterers are located, the distance between the scatterer and BS can be computed, and then the distance between the scatterer and MS can be computed based on the TOA distance from MS to BS. Regarding scatterers as virtual stations, MS can be located with the existing LOS localization algorithm.  
Misidentification of scattering signal will lead to significant scatterer localization error, which may result in a much larger MS localization error. To eliminate MS localization error resulting from the misidentification of scattering signal, in the localization procedure of MS, we adopt four scatterers as four virtual stations, in GNSS-denied urban or indoor environments, four or more scatterers can be achieved. The four virtual stations are divided into four virtual station combinations, each virtual station combination has three virtual stations for LOS localization of MS. Consequently, there are four estimated positions of MS. As long as the probability of scattering signal misidentification of one scatterer is small enough, the probability of scattering signal misidentification of two scatterers will be even smaller. For instance, if the probability of scattering signal misidentification of a scatterer is less than 10%, then the corresponding probability of two scatterers will be less than 10% * 10% = 1%, such that we can consider that there is at most one scatterer whose scattering signal is misidentified in the four scatterers.  
Furthermore, due to there being at most one scatterer whose scattering signal is misidentified in the four scatterers, there is at most one scatterer that has a significant localization error in the four scatterers. Therefore, among the four virtual station combinations with every three virtual stations, at most one virtual station combination has a scatterer that has a significant localization error. So, we can consider that there is at most one virtual station combination that has a significant MS localization error, which results from one of its three scatterer's scattering signals being misidentified. Hence, in the four estimated positions of MS, corresponding to the four virtual station combinations, at most one estimated position is far away from the MS. The estimated position, which is relatively far away from the other three positions, should be discarded as an outlier, and the mean of the rest three positions is adopted as the estimated position of MS ultimately. The discarding of outliers eliminates the error resulting from scattering signal misidentification, this kind of error may lead to huge errors in the MS localization procedure. Averaging the other three estimated positions of MS helps further reduce MS localization errors. The virtual station localization algorithm can be summarized as algorithm 4.  
Figure 2: Flow chart of two-step one-bounce scattering signal identification algorithm used for localization of scatterer.  
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Input: four groups scattering signal corresponding to four scatterers. Output: localization of MS. for each group scattering signal do  Estimate position of scatterer. endfor  Create four virtual station combinations with every three scatterers. for each virtual station combination do  Estimate position of MS with LOS algorithm. endfor  Discard the outlier in four estimated positions of MS.  Select the mean of the rest three positions as MS position ultimately.
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**Algorithm 4** Virtual station localization algorithm