## 4 Data Processing
### Time Domain Averaging after Normalization  
Since these results show weak dependence on the number of averaging points, the analysis procedure is modified. After dividing the signals into sections corresponding to separate mode pulses, we also separate each section into several intervals. Figure 7 shows the separation of the \(i\)-th mode pulse into 10 equal intervals with green vertical lines. Thus, the mode pulse consisting of \(J\) points is divided into 10 intervals, within which preliminary averaging was performed. As a result, each of the mode-pulses was connected with two sets of ten average values. If we make \(J\) a multiple of 10 through eliminating excessive points, the averaging procedure is performed as follows:  
\[\begin{split} I_{i}^{(1)}=\left<I_{ij}\right>_{j=0,...,\frac{l_{0 }}{10}}\\ \cdots\\ I_{i}^{(n)}=\left<I_{ij}\right>_{j=\frac{(l-1)}{10},...,l_{0}} \\ \cdots\\ I_{i}^{(10)}=\left<I_{i,j}\right>_{j=(J-\frac{l}{10}),...,J}.\end{split} \tag{6}\]  
The blue circles in Figure 7a,b show the average values for the second interval of the \(i\)-th mode pulse (\(I_{i}^{(2)}\) and \(R_{i}^{(2)}\), respectively). Next, 10 data arrays corresponding to different intervals of the mode pulses are constructed for the reference and MZI output signals, and the corresponding normalization of the signals are made:  
\[S_{i}^{(n)}=I_{i}^{(n)}/R_{i}^{(n)}. \tag{7}\]  
Figure 8: (**a**) Time domain averaging before normalization: reflectograms for different numbers of averaging points; (**b**) dependence of NFL on number of averaging points.  
Therefore, the normalization is performed in this case for individual parts of the mode pulses, and we produce 10 interferograms on the basis of these \(S_{i}^{(n)}\) values, taking every \(n\)-th section from each of the pulse. Individual reflectograms corresponding to the interferograms are calculated through taking the FFT: \(F^{n}=\hat{F}\left(S_{i}^{(n)}\right)\). The calculation result is presented as a heat map in Figure 9a. The horizontal and vertical axes correspond to the coordinate along the LUT (\(z\)) and the number of the interval used for the reflectogram construction, respectively, while color corresponds to the amplitude. It can be noted that while the part of the reflectogram corresponding to the noise floor changes quite chaotically with variation in the mode pulse section, the other part at \(z<2.2\) m (corresponding to the RLS) only slightly changes. There are vertical bands in the RLS parts of the reflectograms, indicating that, for a fixed LUT coordinate, the signal remains constant, regardless of the number of a particular interval used for the calculations.  
Since the signal corresponding to the LUT does not change, it is possible to average the reflectograms corresponding to different sections of the mode pulses (i.e., using the number n). The result is represented via the red curve in Figure 9b. For comparison, the signal obtained after the averaging over the entire mode pulse is also shown. It can be seen that the signal for z \(<2.2\) m is the same for signals shown in Figures 8a and 9b. In turn, the NFL increases, while fluctuations of the noise signal decrease.