## 3 The Simulation and Its Results  
Six sets of Brillouin spectra were simulated using a computer experiment according to Equation (1) to validate the proposed method. Each set imitated a spectrum matrix measured using a BOTDA with a certain scanning step: 0.5, 1, 2, 4, 8, and 16 MHz. For each scanning step, a generated matrix had a fixed signal-to-noise ratio (SNR) of spectra, ranging from 2 to 12 dB, with a step of 1 dB. Given that these spectra have been artificially simulated, the true BFS for each spectrum is known in advance.  
Figure 4 illustrates the dependence of the error (accuracy) in determining the position of the BGS maximum on the number of spectra involved in the averaging procedure for modeled spectra with an SNR of 2 dB.  
The upper curve corresponds to the results of spectra with a frequency step of 16 MHz processing. The subsequent curves, from top to bottom, depict the results of processing spectra with steps of 8, 4, 2, 1, and 0.5 MHz, respectively. The procedure for determining the gain in accuracy is illustrated for a step of 16 MHz. The dashed line depicts the error in determining BFS by the classical LCF method without averaging neighboring spectra. The green arrow represents the difference between the BFS extraction errors by the proposed method and LCF for 24 spectra involved in the correlation process. The number of correlation spectra was chosen to reduce the processing time while maintaining high accuracy. The final dependence of the gain on the frequency step is shown in the inset, which is calculated with the formula:  
\[\Delta=E_{LCF}-E_{IP}, \tag{5}\]  
where \(\Delta\) is the gain in accuracy of BFS extraction; \(E_{LCF}\) is the BFS extraction error of the LCF method; \(E_{IP}\) is the BFS extraction error of the IP method.  
As illustrated in Figure 4, the proposed algorithm yields superior outcomes compared to the LCF method, starting with eight averaged spectra. At approximately 20 spectra, the gain reaches its maximum value, and with an increase in the number of spectra, the gain remains relatively constant. Figure 5 presents similar results, but when processing spectra with an SNR of 11 dB. It can be observed that the algorithm approaches the accuracy of the classical LCF with a large number of averaged spectra yet does not outpace it.  
Figure 4: Dependences of the BFS extraction accuracy on the number of averaged spectra for spectra with SNR = 2 dB. Individual curves correspond to different scanning steps (16, 8, 4, 2, 1, and 0.5 MHz from top to bottom). Green arrow shows gain in BFS extraction accuracy. Inset: BFS extraction accuracy gain as a function of frequency step.  
Figure 6 illustrates the discrepancy in the efficacy of the proposed algorithm and the LCF method. For the sake of clarity, the outcomes are presented as a surface, and the higher the point on the surface, the greater the gain of the proposed algorithm.  
As illustrated in Figure 6, the algorithm demonstrates a clear advantage over LCF in the context of low SNR and coarse scanning steps over the spectrum.  
Precision outcomes for the second selected metric are similar (Figure 7). The algorithm outperforms LCF at low SNR and large scanning steps. The optimal value of the averaged spectra is approximately 20.  
Figure 5: Dependence of the BFS extraction accuracy on the number of averaged spectra for spectra with SNR=\(11\) dBm. Red arrow shows loss in BFS extraction accuracy. Inset: BFS extraction accuracy gain as a function of the scan step.  
Figure 6: Dependence of the gain/loss in the accuracy of determining the BFS of the presented algorithm over LCF when processing the simulated spectra.