# Monitoring Vehicles on Highway by Dual-Channel \(\varphi\)-OTDR
## 3 Methods of Vibration Positioning
Then, the multiplication in Equation (2) can be performed in the space domain with good spatial matching, when the position abscissas for both the accumulated signals of \(\mu_{1}\) and \(\mu_{2}\) are characterized by \(z_{\mathrm{m,i}}=c\times t_{\mathrm{m,i}}/2n_{\mathrm{m}}\) (m = 1, 2); here, \(t_{\mathrm{m,i}}\) is the round-trip time from input to the \(i\)-th scattering element in fiber m.  
The CDM shows good reliability in terms of positioning the vibration signal, but it is very sensitive to the selection of the differential step (i.e., k in Equation (1)). An inappropriate k value would fail to position the vibration or lead to a wrong location information. In practice, the value of k is highly dependent on the frequency of vibration signals and the repetition rate the probe pulses (and hence the sensing distance). Generally, a smaller k is preferable for a higher-frequency vibration signal, and larger for lower-frequency signal. On the highway, the k-value is determined by considering that the vehicle-caused vibration frequency is also closely related to the driving speed, the type of the vehicle, and the operation condition of the highway. Therefore, it is necessary to propose an improved signal demodulation method to reduce the impact of k-value sensitivity on the vibration location of running vehicles on the highway. It is known that external vibrations would generate localized modulation in the effective refractive index of the sensing fiber, causing a change in the intensity of the backward-scattered Rayleigh light. Compared to the locations where no vibration is present, the Rayleigh traces continuously measured within a certain time would show a large fluctuation in intensity at the vibration points. Although each trace seems to change randomly, the mean and variance provide a deep insight into the overall variations of tens or hundreds of Rayleigh traces. Figure 2 shows the measured Rayleigh scattering traces at two different situations. Figure 2a is obtained in the lab with a piezo-electric transducer (PZT) at \(\sim\)1 km being used to simulate a stable vibration of 40 Hz, in which 80 consecutive traces are superimposed. This frequency is comparable to the vibration frequency of a driving vehicle on the highway. We also measured 100 consecutive Rayleigh traces on the highway, where the vibration was caused by a running vehicle at \(\sim\)8.2 km, and the result is given in Figure 2c. Figure 2b,d show their mean values and standard variances, respectively. From these figures, it is evident that the traces are more divergent and show larger variances at the vibration positions relative to other places. What we should notice is that it is not enough to obtain trustworthy vibration positions solely by relying on the means and variances in complex environments. It is thus necessary to present a more robust signal processing technique by combing both the advantages of the CDM and the variances of Rayleigh scattering traces. For convenience and distinction, the proposed algorithm will be referred to as CDM&V in the following description. In the CDM&V, we similarly suppose \(D[j]\) be the \(j\)-th of the consecutive N raw scattering traces and \(\overline{D}\) be the mean value for the N traces. For each sensing channel, the summed differential signal, \(\sigma\), is changed to be:  
\[\sigma_{m}=\sum_{j=k+1}^{N}\left|\left\{D[j]-D[j-k]\right\}\times\left\{D[j-k ]-\overline{D}\right\}\right|_{m}, \tag{3}\]  
Figure 2: (**a**) Superimposition of 80 raw backscattered traces with PZT modulation at 1 km; (**b**) mean and standard variance of (**a**); (**c**) superimposition of 100 raw traces with a running vehicle at 8.2 km; (**d**) mean and standard variance of (**c**).