## 2 Principle of the Commonly Used Temperature Demodulation  
When a light pulse propagates along the FUT, it interacts with the optical fiber material, and Raman scattering light is generated. For Raman OTDR, the power of the Raman scattering light received at the fiber front end can be expressed as follows (Formula (1)):  
\[P_{s}(L,T)=P_{0}K_{s}v_{s}^{4}R_{s}(T)\exp[-(\alpha_{R}+\alpha_{s})L)] \tag{1}\]  
where \(P_{s}(L,T)\) is the power of the Raman Stokes light from position \(L\); the temperature value is \(T\); \(P_{0}\) is a coefficient that linearly relates to the power of the light pulse and the fiber capturing factor of the back-scattering light; \(v_{s}\) is the frequency of the Raman Stokes light; \(R_{s}(T)\) is the factor that indicates the relationship between temperature and light power; and \(\alpha_{R}\) and \(\alpha_{s}\) are the fiber attenuation coefficients relating to the Rayleigh light and Raman Stokes light, respectively. The \(R_{s}(T)\) is described in Formula (2), where \(h\) is [PERSON]'s constant; \(\Delta\
u\) is the bandwidth of the Raman scattering spectrum (approximately 13.2 THz); and \(k_{B}\) is the [PERSON]'s constant. \[R_{s}(T)=\frac{1}{1-\exp(-h\Delta\
u/k_{B}T)} \tag{2}\]  
Similarly, the Raman anti-Stokes light can be expressed as the Formula (3),  
\[P_{a}(L,T)=P_{0}K_{a}v_{a}^{4}R_{a}(T)\exp[-(\alpha_{R}+\alpha_{a})L)] \tag{3}\]  
where \(P_{a}(L,T)\) is the power of the Raman anti-Stokes light from position \(L\); the temperature value is \(T\); \(v_{a}\) is the frequency of the Raman anti-Stokes light; and \(R_{a}(T)\) is the factor that indicates the relationship between temperature and Raman anti-Stokes light power. The \(R_{a}(T)\) is described in Formula (4). \[R_{a}(T)=\frac{1}{\exp(h\Delta\
u/k_{B}T)-1} \tag{4}\]  
To demodulate the temperature \(T\) at each position \(L\) along the FUT, the power and temperature information should be calibrated first, meaning the data corresponding to \(P_{s}(L,T_{0})\) and \(P_{a}(L,T_{0})\) should be obtained with temperature fixed at \(T_{0}\). Moreover, \(P_{s}(L,T_{0})\) and \(P_{a}(L,T_{0})\) are described in Formulas (5) and (6). \[P_{s}(L,T_{0})=P_{0}K_{s}v_{s}^{4}R_{s}(T_{0})\exp[-(\alpha_{R}+\alpha_{s})L)] \tag{5}\]  
\[P_{a}(L,T_{0})=P_{0}K_{a}v_{a}^{4}R_{a}(T_{0})\exp[-(\alpha_{R}+\alpha_{a})L)] \tag{6}\]  
By using Formulas (1), (3), (5) and (6), the Formulas (7) and (8) can be obtained as the following. \[r(T,L)=\frac{P_{a}(L,T)}{P_{s}(L,T)}=\frac{K_{a}}{K_{s}}(\frac{v_{a}^{4}}{v_{ s}^{4}})\exp(-\frac{h\Delta\
u}{K_{B}T})\exp[(\alpha_{s}-\alpha_{a})L] \tag{7}\]  
\[r(T_{0},L)=\frac{P_{a}(L,T_{0})}{P_{s}(L,T_{0})}=\frac{K_{a}}{K_{s}}(\frac{v_{ a}^{4}}{v_{s}^{4}})\exp(-\frac{h\Delta\
u}{K_{B}T_{0}})\exp[(\alpha_{s}- \alpha_{a})L] \tag{8}\]  
The ratio between Formulas (7) and (8) is described in expression (9). \[I=\frac{r(L,T)}{r(L,T_{0})}=\exp[-\frac{h\Delta\
u}{K_{B}}(\frac{1}{T}-\frac{ 1}{T_{0}})] \tag{9}\]Finally, by Formula (9), the temperature \(T\) at each position \(L\) along the FUT can be demodulated as shown in expression (10). \[\frac{1}{T}=\frac{1}{T_{0}}-\frac{K_{B}}{h\Delta v}\ln[\frac{\tau(L,T)}{r(L,T_{0 })}] \tag{10}\]  
Hence, from the above description, it is known that the traditional temperature demodulation method needs two data channels: the Raman Stokes data and the Raman anti-Stokes data. However, as the wavelengths of the Raman Stoke light and the Raman anti-Stokes light are approximately 200 nm apart, the walk-off effect is unavoidable. Therefore, the positional information \(L\) in Formulas (7)-(10) is not the same because the Raman Stokes signal and the Raman anti-Stokes signal collected directly by the DAQ are marked by the same time intervals, and the speed of light multiplied by the time represents the position. If the walk-off effect is not corrected before the temperature demodulation by the above formula, the measured results may be wrong, especially at positions far from the fiber front end.