## 3 Simulations of System Measurement Method in Noise Conditions
### Analysis of Anti-noise Performance of the Temperature/Strain Demodulation Algorithm  
As the CRN changes with time, it is necessary to analyze the anti-noise performance of the temperature/strain demodulation algorithm in Formula (1). First, a Gaussian noise array is built for reference. Considering the temperature/strain measurement range of the system, for example, temperature measurement range \(-45\)\({}^{\circ}\)C\(\sim\)+155 \({}^{\circ}\)C, 301 frequency sweeping points are selected as the spectrum reference. Second, the array for Rayleigh frequency shift identification should be determined. This array is the \(p_{v}(f_{i}+\Delta f,L)\) in Formula (1) and its length, M, should also be considered. The 2D Rayleigh spectrum at each position along the sensing fiber obeys statistical characteristics, so it can be equivalent to standard data in the reference array with noise added to it. The array lengths chosen are 11, 21, 31, 51, and 101, respectively, as shown in Figure 5. Third, the array \(p_{v}(f_{i}+\Delta f,L)\) slides left and right relative to the reference array to determine the array \(p_{u}(f_{i},L)\) for contrast, and then the relevant cross correlation coefficients are computed by Formula (1). Finally, the Rayleigh frequency shift \(\Delta f\) at each position is the difference between the frequency corresponding to the maximum cross-correlation coefficient and the initial frequency. Figure 6 shows the maximal cross-correlation coefficients with a fiber length of 10 m and the frequency position shifts from the initial frequency position 101 to position 151, which means that the Rayleigh frequency shift \(\Delta f\) is 50 frequency sweeping intervals. At last, by Formulas (2) and (3), the temperature/strain data at each position along the sensing fiber can be obtained. Figure 4: The probability density distribution of the CRN along OTDR trace. Figure 5: The arrays with different lengths for temperature/strain demodulation. As is known, the temperature/strain demodulation algorithm in Formula (1) depends on the data of two arrays from two independent measurements and the data are the values of OTDR traces obtained by the frequency sweeping method. The CRN fluctuates in different independent measurements, so the data of the later measurement can be regarded as the data of the former measurement with Gaussian noise added. Therefore, the Gaussian noise with amplitudes of 0.2, 0.4, 0.6, 0.8, 0.9, and 1.0 times are added, respectively, in the later arrays \(p_{v}(f_{i}+\Delta f,L)\) with lengths (number of frequency sweeping points) of 11, 21, 31, 51, and 101, respectively. The Rayleigh frequency shift \(\Delta f\) is fixed at 50 frequency sweeping intervals, where the maximal cross-correlation coefficients appear. By Formula (1), the results are shown in Figures 7-11. At different positions along the sensing fiber, the maximal cross-correlation coefficients may not be the same, but they should be at the same frequency point. Figure 7 indicates that with 0.2 times of noise amplitude added, the array with a length of 11 brings many errors because the maximal cross-correlation coefficients appear at other frequency points, which is unacceptable. In Figure 8, with 0.4 times of noise amplitude added, the array with a length of 21 is also improper. Moreover, the array with a length of 31 cannot resist 0.6 times of noise amplitude, as shown in Figure 9. Figure 10 shows that with 0.8 times of noise amplitude added, the array with a length of 51 should not be adopted because of measurement errors. Figure 6: The cross-correlation coefficients corresponding to different fiber positions. Figure 7: The maximal cross-correlation coefficients after adding the Gaussian noise with an amplitude of 0.2 times to the initial signal: **(a)**\(11\) frequency points; **(b)**\(21\) frequency points; **(c)**\(31\) frequency points; **(d)**\(51\) frequency points; **(e)**\(101\) frequency points.