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High-accuracy distributed temperature measurement using difference sensitive-temperature compensation for Raman-based optical fiber sensing

Jian Li, Qian Zhang, Yang Xu, Mingjiang Zhang, Jianzhong Zhang, Lijun Qiao, Mehjabin Mohiuddin Promi, and Tao Wang

Open Access Open Access

Abstract

In the temperature monitoring field, the Raman Distributed Temperature Sensor (R-DTS) is required with a temperature accuracy better than 1 °C over a long distance. This paper proposes and experimentally demonstrates an R-DTS system based on the difference sensitive-temperature compensation to optimize the temperature accuracy with the enhanced temperature sensitivity of backscattered spontaneous Raman scattering. While operating in the experiment, the distributed temperature measurement and theory analysis use the dual-demodulation, self-demodulation and double-end configuration principles for R-DTS are demonstrated. The experimental results show that the temperature accuracy is 12.54 °C, 8.53 °C and 15.00 °C along the 10.8 km under the standard R-DTS systems, respectively. Further, we analyze and recalibrate the intensity of the Raman scattering signal in theory, and substitute the sensitive-temperature factor (M(L)) into the Raman scattering signal. Finally, a novel temperature demodulation method with difference sensitive-temperature compensation is applied to the dual-demodulation, self-demodulation, and double-end configuration R-DTS systems. After compensation, the temperature accuracy can be optimized to 0.38 °C, 0.36 °C and 0.56 °C at the same position. It proves that the proposed method can make the temperature accuracy better than 1 °C for these three demodulation systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Raman Distributed Temperature Sensor (R-DTS) exploits specific optical effects along the sensing fiber to obtain a spatially distributed temperature profile. It offers unique attributes and capabilities compared to conventional discrete sensing methods [1]. In R-DTS system, the principle of spontaneous Raman scattering is based on the energy exchange in sensing fiber. When the pulsed light quantum and fiber material molecule cause an inelastic collision in optics-fiber, this will produce an anti-Stokes light with a central wavelength of 1450 nm and a Stokes light with 1650 nm. The anti-Stokes light is sensitive to the surrounding temperature, and it is used to modulate the environmental temperature based on the principle of Raman scattering [2]. In recent years, the R-DTS system has been widely used in the temperature safety monitoring applications due to the advantages of distributed measurement [3], long distance [4], and high spatial resolution [5], such as transport infrastructure, smart grid [6] and gas pipeline [7], etc.

The R-DTS system with high-performance that measures surrounding temperature must be selected according to many different criteria, such as temperature accuracy, temperature resolution and spatial resolution [8–10]. In the industrial temperature measurement system, the accurately temperature data can reflect the structure's internal information. For example, the carrier density in the power cable is estimated by using a specific temperature [6]. Moreover, the specific temperature along the gas pipeline can also be used to locate the position of pipeline leakage [7]. For the above applications, the temperature profiles along the fiber need to be calculated with high-accuracy better than 1 °C. The groundwater influx even is required to be monitored with a temperature accuracy of 0.01 °C or better [11]. However, the intensity of Raman backscattered light is about 30 dB weaker than the Rayleigh light, which brings the R-DTS to have a weaker signal-to-noise ratio (SNR) [12]. This phenomenon causes its temperature measurement accuracy to be less than 1 °C along a long distance. It leads to the most of R-DTS systems are used for fire temperature security detection [4,13–15], which requires the fiber-based sensors with a higher warning-time and spatial resolution [4,9], but has a minor requirement for temperature accuracy, which limits the application with a high temperature accuracy for R-DTS systems.

To solve this problem, the straightforward method is to increase the average times of Raman signal. But it will deteriorate the measurement time of the R-DTS system [16]. In recent years, several signal processing technologies, such as pulsed modulation with coding [17–18], denoising of the wavelet transform modulus maxima [19–20], 2D and 3D image restoration [1], Rayleigh noise suppression [12] and dispersion compensation [21] have been demonstrated to improve the temperature measurement accuracy. These above methods are mainly based on the techniques of signal denoising, that is, using a denoising method to improves the SNR of the Raman signal in the extracted signal. However, the temperature accuracy of the long-distance R-DTS (10 km or more) cannot be better than 1 °C as far as we know.

In this letter, we propose and experimentally demonstrate a R-DTS system based on difference sensitive-temperature compensation to optimize the temperature accuracy with the enhanced temperature sensitivity of backscattered spontaneous Raman scattering. In the experiment, the distributed temperature measurement and theory analysis are carried out by using the graded-index multimode sensing fiber with 10.8 km. The experimental results show that the temperature accuracy is 12.54 °C, 8.53 °C and 15.00 °C over the 10.8 km under the standard dual-demodulation, self-demodulation and double-end configuration systems, respectively. For solving this problem, we analyze and recalibrate the intensity of Raman scattering signal in theory, and design the difference sensitive-temperature compensation methods with three types of demodulation principles for R-DTS systems. After compensation, the temperature accuracy can be optimized to 0.38 °C, 0.36 °C and 0.56 °C at the same position. The research content can be applied in the temperature safety monitoring with a high temperature accuracy.

2. Novel temperature demodulation principles and results

At present, there are three types of temperature demodulation principles in the R-DTS system, which include dual-demodulation [22–24], self-demodulation [25] and double-ended configuration principles [26–28]. The method based on dual-demodulation uses the intensity ratio of anti-Stokes over Rayleigh or Stokes light for detecting the surrounding temperature. The self-demodulation method only uses the Raman anti-Stokes backscattered light to extract the temperature profiles. The R-DTS based on the dual-demodulation and self-demodulation principles are all used in the single-ended configuration (where the fiber is interrogated from one fiber end only) sensing fiber to get a directional Raman backscattered signal. The method based on the double-ended configuration principle uses the forward and backward Raman backscattered signal to extract the temperature information along the sensing fiber. The above R-DTS system with different principles can be applied to different applications according to its special advantages (It is described in detail in the discussion.). These three methods are described in detail below.

2.1 The experimental setup and results based on dual-demodulation principle

The experimental setup based on dual-demodulation (Stokes demodulate anti-Stokes) principle is shown in Fig. 1. The dual-demodulation R-DTS system consists of a pulsed laser (Connect Laser, wavelength: 1550 nm), a Raman filter (Xufeng Photoelectric, 1550 nm/1450 nm/1650 nm), APD (FBY Photoelectric, bandwidth:100 MHz), amplifiers (REBES, bandwidth:100 MHz), DAC (Jemetech, bandwidth:100 MHz, 4 channels), computer, reference fiber and Fiber Under Tests (FUTs). In the experiment, the FUTs consist of three sections with a length of 10 m (FUT 1), 10 m (FUT 2) and 20 m (FUT 3). Among them, the pulsed laser directly operates at about 1550 nm wavelength with a pulse width of 10 ns (Its spatial resolution is 1 m.) and a repetition rate of 4 kHz. A pulsed laser coupled in one end of the sensing fiber (common multimode fiber, 62.5/125 um), propagates along the sensing fiber with its intensity partially backscattered and guided back to the launching end. Then the backscattered signals from the Raman filter are separated into anti-Stokes and Stokes signals. The anti-Stokes and Stokes signals are then detected simultaneously by two low-noise APDs and amplifiers. And the electrical signals are collected by a high-speed Data Acquisition Card (DAC). Finally, they are transmitted to the computer for temperature demodulation.

 figure: Fig. 1.

Fig. 1. The experimental setup based on dual-demodulation principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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In the R-DTS system based on dual-demodulation principle, the temperature information is extracted from the intensity ratio of anti-Stokes over Stokes. The intensity ratio of anti-Stokes over Stokes backscattering is referred as follows:

$$\frac{{{\phi _a}}}{{{\phi _s}}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - \frac{{h\Delta v}}{{kT}}} \right)\exp \left[ {\int_0^L {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
Where ${\phi _a}$ and ${\phi _s}$ are the intensity of anti-Stokes and Stokes respectively, ${K_a}$ and ${K_s}$ are the coefficient of anti-Stokes and Stokes scattering cross section, ${\upsilon _a}$ and ${\upsilon _s}$ are the frequency of anti-Stokes and Stokes, h is the Planck’s constant, $\Delta v$ is the Raman frequency shift, k is the Boltzmann constant, T is the absolute temperature, ${\alpha _a}$ and ${\alpha _s}$ are the attenuation coefficient of anti-Stokes and Stokes respectively, and L is the position along the sensing fiber. Compared to the tradition dual-demodulation R-DTS system, we set up a reference fiber unit at the front end of the sensing fiber for eliminating the influence of the APD fluctuation and inherent parameters on the temperature demodulation results [10]. In the experiment, the reference fiber is placed in a high-precision thermostat (Nanjing Kenfan Electronics, KSC-15 T) to control the temperature of the reference fiber. And the temperature fluctuation range of high-precision thermostat is 0.05 °C. The intensity ratio of anti-Stokes over Stokes backscattering at the reference fiber is referred as follows:
$$\frac{{{\phi _{ac}}}}{{{\phi _{sc}}}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - \frac{{h\Delta v}}{{k{T_c}}}} \right)\exp \left[ {\int_0^{{L_c}} {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
Where ${\phi _{ac}}$ and ${\phi _{sc}}$ are the intensity of anti-Stokes and Stokes at the reference fiber respectively, Lc and Tc are the position and temperature of the reference fiber. After the Eq. (1) divided by the Eq. (2), it can be seen from the Eq. (3) that the inherent parameters (${K_s},{K_a},{\upsilon _s},{\upsilon _a}$) can be eliminated.
$$\frac{{{\phi _a}}}{{{\phi _s}}}\frac{{{\phi _{sc}}}}{{{\phi _{ac}}}} = \exp \left[ { - \frac{{h\Delta v}}{k}\left( {\frac{1}{T} - \frac{1}{{{T_0}}}} \right)} \right]\exp \left[ {\int_{{L_c}}^L {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
In order to compensate the fiber attenuation $({{\alpha_s}(L) - {\alpha_a}(L)} )$ for extracting the temperature information along the sensing fiber, the sensing fiber must be calibrated before the temperature measurement (pre-calibration stage [10]). In the pre-calibration stage, the Raman intensity ratio of sensing fiber and reference fiber can be expressed as:
$$\frac{{{\phi _{ao}}}}{{{\phi _{so}}}}\frac{{{\phi _{sco}}}}{{{\phi _{aco}}}} = \exp \left[ { - \frac{{h\Delta v}}{k}\left( {\frac{1}{{{T_o}}} - \frac{1}{{{T_{co}}}}} \right)} \right]\exp \left[ {\int_{{L_c}}^L {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
Where the ${\phi _{aco}}$ and ${\phi _{sco}}$ are the anti-Stokes and Stokes intensity at the reference fiber respectively, To and Tco are the temperature of the FUT and reference fiber. Using the Eq. (3) and (4), we can get the temperature demodulation algorithm (Eq. (5)). From the Eq. (5), the temperature profiles are only modulated by the Raman signal, and it can be directly extracted the temperature profiles along the sensing fiber.
$$T = \frac{1}{{\left( {\frac{1}{{{T_c}}} + \frac{1}{{{T_o}}} - \frac{1}{{{T_{co}}}}} \right) - \frac{k}{{h\Delta v}}\ln \left( {\frac{{{\phi_{sc}}_o{\phi_a}_o{\phi_{ac}}{\phi_s}}}{{{\phi_{aco}}{\phi_{so}}{\phi_{sc}}{\phi_a}}}} \right)}}.$$
The distributed temperature measurement based on dual-demodulation is carried out by using the graded-index multimode fiber (62.5/125 um). The multimode fiber distribution is shown in Fig. 1. The multimode fiber with 11.0 km range has been used. The fiber under tests (FUTs) consist of three sections with a length of 10.0 m, 10.0 m, and 20.0 m. The temperature of FUTs are set at 42.64 °C, 52.64 °C, 61.97 °C, 71.95 °C, 81.33 °C and 90.60 °C by the temperature-control chamber (TCC, BILON, temperature accuracy: 0.10 °C) and the rest of the sensing fiber is maintained at room temperature (the room temperature is about 27.00 °C). The temperature measurement result of overall distribution based on the Eq. (5) are obtained in Fig. 2. Figure 2(a)–(f) stand for the overall distribution of the temperature measurement results (The results are averaged 10 k times). Figure 2(g)–(i) stand for the temperature results of FUTs under the control of the TCC. It can be observed that the proposed temperature demodulation method can distinguish between the temperature of TCC and room temperature, and the R-DTS can accurately measure the temperature distribution along the FUT.

 figure: Fig. 2.

Fig. 2. The temperature measurement results of overall distribution based on dual-demodulation principle under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature demodulation results at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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The temperature measurement accuracy represents the difference between the values of actual measured temperature and the standard temperature. Then we analyze the temperature measurement accuracy of the traditional method (Eq. (5)) and the proposed method (Eq. (7)). The temperature measurement accuracy based on Eq. (5) are shown in Fig. 3(a1)–(a3). It can be observed that the temperature measurement accuracy is 8.30 °C, 11.73 °C and 12.54 °C at the sensing distance of 1.5 km, 9.6 km and 10.8 km by measuring the average of all data. (We performed a cumulative average of 10,000 times on the Raman signal, and also used the wavelet modulus maximum denoising method [19].) It cannot meet the requirements of temperature monitoring with high-accuracy. We think that the sensitive-temperature effect of backscattered spontaneous Raman to fiber is decreasing along with the increase of distance. Further, the temperature accuracy of R-DTS system is affected by the sensitive-temperature effect and fiber cable insulation. That means the temperature modulation function of the Raman intensity ($\exp ({{{h\Delta v} \mathord{\left/ {\vphantom {{h\Delta v} {kT}}} \right.} {kT}}} )$) is not only modulated by the temperature. The traditional temperature demodulation methods don’t consider the influence of the sensitive-temperature effect on the temperature measurement. Therefore, the measurement of the Raman intensity ratio is not the theoretical value presented in Eq. (1). For solving this problem, we set the M(L) as the sensitive-temperature factor, and it is related to the sensing distance. The intensity ratio of Raman Stokes light over anti-Stokes light along the sensing fiber is:

$$\frac{{{\phi _a}}}{{{\phi _s}}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - M(L )\frac{{h\Delta v}}{{kT}}} \right)\exp \left[ {\int_0^L {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
After calculated in the pre-calculation stage, the actual temperature demodulation equation based on dual-demodulation is:
$$T = \frac{{M(L )}}{{\left( {\frac{{M({{L_c}} )}}{{{T_c}}} + \frac{{M(L)}}{{{T_o}}} - \frac{{M({{L_{co}}} )}}{{{T_{co}}}}} \right) - \frac{k}{{\textrm{h}\Delta \nu }}\ln \left( {\frac{{{\phi_{sco}}{\phi_{ao}}{\phi_{ac}}{\phi_s}}}{{{\phi_{aco}}{\phi_{so}}{\phi_{sc}}{\phi_a}}}} \right)}}.$$

 figure: Fig. 3.

Fig. 3. The experimental results of temperature measurement accuracy at the (a1) 1.5 km, (a2) 5.6 km and (a3) 10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3) 10.8 km after calibration.

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The $M(L )$ can be obtained by using the Eq. (8). In order to get the mathematical function of $M(L )$, we have designed a calibration experiment. In experiment, the correction fibers (FUT, 10 m) at the different distance were placed in TCC with a constant temperature. Then the intensity of Raman backscattered light (${\phi _a}_o,{\phi _{so}},{\phi _s},{\phi _a}$) at the FUT are acquired by the DAC. Finally, we can calculate the data of M. The data of $M$ at the different sensing distances are shown in Table 1.

$$M(L )\textrm{ = }{{\left( {\frac{{{\phi_a}{\phi_{so}}}}{{{\phi_s}{\phi_{ao}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{{\phi_a}{\phi_{so}}}}{{{\phi_s}{\phi_{ao}}}}} \right)} {\left( {\frac{{\textrm{h}\Delta \nu }}{{k{T_o}}} - \frac{{\textrm{h}\Delta \nu }}{{kT}}} \right)}}} \right. } {\left( {\frac{{\textrm{h}\Delta \nu }}{{k{T_o}}} - \frac{{\textrm{h}\Delta \nu }}{{kT}}} \right)}}.$$
After fitting the data of M, the mathematical algorithm of M is: $M\textrm{ = } - 0.007L + 0.7864$, then substitute it into Eq. (7) to obtain the actual temperature demodulation equation. The temperature accuracy measurement results based on Eq. (7) are shown in Fig. 3(b1)–(b3). The experimental results show that the temperature accuracy based on novel dual-demodulation is 0.16 °C,0.31 °C and 0.38 °C at a sensing distance of 1.5 km, 9.6 km and 10.8 km. Table 2 stands for the experimental results of the temperature accuracy. (Among them, the gray area is the temperature measurement result after calibration.) It can be observed that the temperature accuracy is improved compared to the traditional demodulation system.

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Table 1. The data of M at the difference sensing distances

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Table 2. The temperature accuracy based on dual-demodulation

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In addition, the positioning method of the R-DTS system is based on the OTDR principle. So, the spatial resolution depends on the pulse width of the system. In R-DTS system by we proposed, the pulse width of the laser is 10 ns, so the spatial resolution of these three methods is 1 m. In addition, the few-mode-fiber can optimize the spatial resolution by allowing more coupling light and reducing the pulse broadening effect [9]. In some cases where the higher space-resolution is required, the few-mode-fiber can be used as the sensing fiber of the R-DTS system.

2.2 The experimental setup and results based on self-demodulation principle

The R-DTS system based on the self-demodulation principle only uses the Raman anti-Stokes backscattered light (1450 nm) to extract the temperature profile. Figure 4 shows the experimental setup based on the self-demodulation principle. Compared to the dual-demodulation R-DTS, the self-demodulation system contains only one APD and an amplifier. This means that the WDM only needs to filter out the anti-Stokes signal. The traditional temperature demodulation algorithm based on the self-demodulation principle is:

$$T\textrm{ = ln}{\left\{ {\frac{{[{\exp ({{\raise0.7ex\hbox{${h\Delta \nu }$} \!\mathord{\left/ {\vphantom {{h\Delta \nu } {k{T_o}}}} \right.}\!\lower0.7ex\hbox{${k{T_o}}$}}} )- 1} ][{\exp ({{\raise0.7ex\hbox{${h\Delta \nu }$} \!\mathord{\left/ {\vphantom {{h\Delta \nu } {k{T_c}}}} \right.}\!\lower0.7ex\hbox{${k{T_c}}$}}} )- 1} ]}}{{[{\exp ({{\raise0.7ex\hbox{${h\Delta \nu }$} \!\mathord{\left/ {\vphantom {{h\Delta \nu } {k{T_{co}}}}} \right.}\!\lower0.7ex\hbox{${k{T_{co}}}$}}} )- 1} ]\left( {\frac{{{\phi_a}{\phi_{aco}}}}{{{\phi_a}_o{\phi_{ac}}}}} \right)}} + 1} \right\}^{ - 1}}({{\raise0.7ex\hbox{${h\Delta \nu }$} \!\mathord{\left/ {\vphantom {{h\Delta \nu } k}} \right.}\!\lower0.7ex\hbox{$k$}}} )$$
The experimental results based on the self-demodulation principle are shown in Fig. 5. The temperature of FUTs are also set at 42.64 °C, 52.64 °C, 61.97°C, 71.95 °C, 81.33 °C and 90.60 °C by the TCC and the rest of the sensing fiber is maintained at room temperature. Figure 5(a)–(f) stand for the distributed temperature measurement results of overall distribution. Figure 5(g)–(i) stand for the temperature measurement results of FUTs at 1.51 km (FUT 1), 9.60 km (FUT 2) and 10.80 km (FUT 3). It can be observed that the temperature measurement curves (FUT 1-FUT 3) based on the self-demodulation principle are relatively smooth. There are no temperature mutations in the temperature of the FUTs compared to the dual-demodulation principle (Fig. 2). Because the dual-demodulation system is affected by the fiber dispersion (the wavelength difference of Stokes and anti-Stokes). The position of Stokes and anti-Stokes light are misplaced when the temperature profiles are extracted, which ultimately leads to an accuracy deterioration in the temperature measurement. In addition, the system noise carried by the Stokes channel is also substituted into the sensing fiber. Therefore, the temperature measurement accuracy of the self-demodulation is higher than the dual-demodulation system. However, due to the calibration of the Stokes channel, the temperature stability of R-DTS based on the dual-demodulation principle is better than the R-DTS based on self-demodulation.

 figure: Fig. 4.

Fig. 4. The experimental setup based on self-demodulation principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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 figure: Fig. 5.

Fig. 5. The temperature measurement result of overall distribution based on self-demodulation under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature measurement result at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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Then we also analyze the temperature accuracy of the R-DTS based on self-demodulation principle. The temperature measurement accuracy is shown in Fig. 6(a1)–(a3). It can be observed that the temperature measurement accuracy is 4.39 °C, 7.26 °C and 8.73 °C at the sensing distance of 1.5 km, 9.6 km and 10.8 km by measuring the average of all data. For compensating the sensitive-temperature effect on the Raman signal. The intensity of anti-Stokes light after modulated can be expressed as:

$${\phi _a} = {K_a}{\upsilon _a}^4\exp {\left( {M(L )\frac{{h\Delta v}}{{kT}} - 1} \right)^{ - 1}}\exp \left[ { - \int_0^L {({{\alpha_a}(L) + {\alpha_o}(L)} )\textbf{d}L} } \right].$$
Where the ${\alpha _o}(L)$ stands for the transmission loss of incident light in the sensing fiber. In self-demodulation system, the temperature information is extracted by the intensity ratio (R(T)), between anti-Stokes signal (${\phi _a}_o$) at the pre-calibration stage and the measured anti-Stokes (${\phi _\textrm{a}}$), which is expressed as:
$$R(T) = \frac{{{\phi _a}}}{{{\phi _a}_o}} = \frac{{\exp ({M(L ){\raise0.7ex\hbox{${h\Delta \upsilon }$} \!\mathord{\left/ {\vphantom {{h\Delta \upsilon } {k{T_0}}}} \right.}\!\lower0.7ex\hbox{${k{T_0}}$}} - 1} )}}{{\exp ({M(L ){\raise0.7ex\hbox{${h\Delta \upsilon }$} \!\mathord{\left/ {\vphantom {{h\Delta \upsilon } {kT}}} \right.}\!\lower0.7ex\hbox{${kT}$}} - 1} )}}.$$

 figure: Fig. 6.

Fig. 6. The experimental results of temperature accuracy based on self-demodulation at (a1) 1.5 km, (a2) 5.6 km and (a3)10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3)10.8 km after calibration.

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In order to eliminate the inherent parameters in fiber, a reference fiber is introduced as correction fiber. The reference fiber is in front of sensing fiber. Then, the anti-Stokes intensity of reference fiber (${\phi _{aco}},{\phi _{ac}}$) are substituted into the Eq. (11), After calculation, the actual temperature demodulation equation based on self-demodulation principle is:

$$T\textrm{ = ln}{\left\{ {\frac{{\left[ {\exp \left( {M(L )\frac{{h\Delta \nu }}{{k{T_o}}}} \right) - 1} \right]\left[ {\exp \left( {M({{L_c}} )\frac{{h\Delta \nu }}{{k{T_c}}}} \right) - 1} \right]}}{{\left[ {\exp \left( {M({{L_{co}}} )\frac{{h\Delta \nu }}{{k{T_{co}}}}} \right) - 1} \right]\left( {\frac{{{\phi_a}{\phi_{aco}}}}{{{\phi_a}_o{\phi_{ac}}}}} \right)}} + 1} \right\}^{ - 1}}\left( {M(L )\frac{{h\Delta \nu }}{k}} \right).$$
In order to get the mathematical function of M, we have also designed a calibration experiment. In the calibration experiment, the correction fiber (FUT, 10 m) at 1.2 km, 2.4 km, 4.8 km, 7.2 km, and 10.8 km were placed in TCC. Then the Raman scattered light intensities (${\phi _a}_{o,}{\phi _a}$) at the FUT are acquired. Finally, we can calculate the value of M based on the Eq. (11). The data of M at the different sensing distances are shown in Table 3. After fitting the data of M, the mathematical function of M is: M(L)= −0.011L + 0.8619, then substitute it into the Eq. (12) for obtaining the actual temperature demodulation equation. The experimental results of the temperature accuracy after calibration are shown in Fig. 6(b1)–(b3). The experimental results show that the temperature measurement accuracy based on self-demodulation are 0.27 °C, 0.48 °C and 0.36  °C at a sensing distance of 1.5 km, 9.6 km and 10.8 km. Table 4 stands for the experimental results of the temperature accuracy, and the gray area is the temperature measurement result after calibration. The results show that the temperature accuracy based on self-demodulation after calibration is improved, and it can be better than 1 °C along the 10.8 km.

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Table 3. The data of M at the difference sensing distances

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Table 4. The temperature accuracy based on self-demodulation

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2.3 The experimental setup and results based on double-ended principle

In the temperature monitoring based on optics-fiber sensors, the fiber loss is not constant during the sensor lifetime. So, the Raman backscattered intensity in the fiber is also modulated by the local external attenuation (fiber fusion, bending, fiber aging, etc.) [27]. It will deteriorate the temperature accuracy of the system. In addition, the above R-DTS based on the dual-demodulation and self-demodulation principles does not solve the problem of local external attenuation on measurement results. For avoiding this measurement error, the double-ended configuration for R-DTS is proposed [26–28], which can obtain the forward and backward Raman backscattered signal by using an optical switch. The experimental setup based on the double-ended configuration principle is shown in Fig. 7, and the traditional temperature demodulation algorithm based on the double-ended configuration principle is:

$$\frac{1}{T}\textrm{ = }\left[ {\ln \left( {\frac{{{R_{Loop}}(T,L)}}{{{R_{Loop}}({T_o},{L_o})}}} \right)\left( { - \frac{k}{{h\Delta v}}} \right)} \right] + \frac{1}{{{T_o}}}.$$
The experimental results based on double-ended demodulation configuration are shown in Fig. 8. The temperature of FUTs are also set at 42.64 °C, 52.64 °C, 61.97°C, 71.95 °C, 81.33 °C and 90.60 °C by the TCC and the rest of the sensing fiber are maintained at room temperature. Figure 8(a)–(f) stands for the distributed temperature measurement result of overall distribution (with 10 k time-averaged traces). The Fig. 8(g)–(i) stand for the temperature measurement results of FUTs at 1.51 km (FUT 1), 9.60 km (FUT 2) and 10.80 km (FUT 3). In the double-ended demodulation R-DTS, the temperature demodulation method requires the forward and backward Raman backscattered signal to extract the temperature information. In the experimental results, the SNR of Raman signal deteriorates accompanied by the distance of sensing fiber. The measurements exhibit a worse temperature resolution in the proximity of both fiber ends compared to the midway fiber resign. The double-ended configuration can eliminate the influence of the fiber loss on measurement results, which can improve the engineering applicability. However, the optical switch is introduced into the double-ended R-DTS, compared to the single-ended R-DTS, it requires twice the length of the fiber at the same sensing distance.

 figure: Fig. 7.

Fig. 7. The experimental setup based on the double-ended principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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 figure: Fig. 8.

Fig. 8. The temperature measurement result of overall distribution based on double-ended demodulation under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature measurement results at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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In addition, we analyze the temperature accuracy of the R-DTS based on the double-ended principle. The temperature measurement accuracy (without correction) are shown in Fig. 9(a1)–(a3). It can be observed that the temperature measurement accuracy is 5.78 °C, 14.10 °C and 15.00 °C at the sensing distance of 1.5 km, 9.6 km and 10.8 km by measuring the average of all data. For compensating the effect of sensitive-temperature, the M(L) is substituted into the temperature function of Raman light. In fact, the intensity ratio of Stokes over anti-Stoke light at the forward and backward can be expressed as:

$$\frac{{{\phi _a}^B}}{{{\phi _s}^B}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - M(L )\frac{{h\Delta v}}{{kT}}} \right)\exp \left[ {\int_0^L {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
$$\frac{{{\phi _a}^{_F}}}{{{\phi _s}^{_F}}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - M({l - L} )\frac{{h\Delta v}}{{kT}}} \right)\exp \left[ {\int_L^l {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$

 figure: Fig. 9.

Fig. 9. The experimental results of temperature measurement accuracy based on double-ended demodulation at (a1) 1.5 km, (a2) 5.6 km and (a3)10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3) 10.8 km after calibration.

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Among them, ${\phi _a}^B$ and ${\phi _s}^B$ are the anti-Stokes and Stokes intensity of the backward direction, ${\phi _a}^{_F}$ and ${\phi _s}^{_F}$ are the anti-Stokes and Stokes intensity of the forward direction. l stands for the whole length of the sensing fiber. The Raman intensity in the double-ended configuration ${R_{Loop}}(T,L)$ can be obtained from the geometric mean of the normalized single-ended configuration in both forward and backward directions according to

$${R_{Loop}}(T,L) = \sqrt {\frac{{{\phi _a}^B}}{{{\phi _s}^B}} \cdot \frac{{{\phi _a}^{_F}}}{{{\phi _s}^{_F}}}} = \frac{{{K_a}}}{{{K_s}}}{\left( {\frac{{{\upsilon_a}}}{{{\upsilon_s}}}} \right)^4}\exp \left( { - \frac{{h\Delta v}}{{kT}}({M(L) + M(l - L)} )} \right)\exp \left[ {\int_0^l {({{\alpha_s}(L) - {\alpha_a}(L)} )\textbf{d}L} } \right].$$
After calibration at the pre-calibration stage, the actual temperature demodulation equation based on double-ended configuration demodulation is:
$$\frac{1}{T} = \frac{{M(L )+ M({l - L} )}}{{\{{{{[{M(L )+ M({l - L} )} ]} \mathord{\left/ {\vphantom {{[{M(L )+ M({l - L} )} ]} {{T_o}}}} \right.} {{T_o}}}} \}- \{{\textrm{l}n[{{{{R_{Loop}}({T,L} )} \mathord{\left/ {\vphantom {{{R_{Loop}}({T,L} )} {{R_{Loop}}({{T_o},L} )}}} \right.} {{R_{Loop}}({{T_o},L} )}}} ]({{k \mathord{\left/ {\vphantom {k {h\Delta \nu }}} \right.} {h\Delta \nu }}} )} \}}}.$$
In order to obtain the mathematical function of M, the correction fiber (FUT, 10 m) at 1.2 km, 2.4 km, 4.8 km, 7.2 km, and 10.8 km were placed in TCC. Then the Raman scattered light intensity $({{{{R_{Loop}}(T,L)} \mathord{\left/ {\vphantom {{{R_{Loop}}(T,L)} {{R_{Loop}}({T_o},{L_o})}}} \right.} {{R_{Loop}}({T_o},{L_o})}}} )$ at the FUT is acquired, then we can calculate the value of sensitive-temperature factor based on the double-ended demodulation. The data of the sensitive-temperature factor ($M$) at the different sensing distances is shown in Table 5. After fitting the data of M in Table 3, the mathematical function equation of M is: M(L) = −0.0167L + 0.84. Then substitute it into the Eq. (17) to obtain the actual temperature demodulation equation. The experimental results of the temperature accuracy after correction are shown in Fig. 9(b1)–(b3). It can be observed that the temperature accuracy is 0.26 °C, 0.46 °C and 0.56 °C at a sensing distance of 1.5 km, 9.6 km and 10.8 km. Table 6 stands for the experimental results of the temperature accuracy, and among them, the gray area is the temperature measurement result after calibration.

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Table 5. The data of sensitive-temperature factor at the difference sensing distances

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Table 6. The temperature accuracy based on double-ended demodulation

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3. Discussion and conclusion

In the temperature demodulation system based on R-DTS, the temperature accuracy and resolution of the self-demodulation is higher than the dual-demodulation system due to the signal-to-noise ratio. However, due to the calibration of the Stokes channel, the temperature stability of R-DTS based on the dual-demodulation principle is better than the self-demodulation system [10]. Therefore, the most dual-demodulation R-DTS system is applied to the actual industrial temperature monitoring. In addition, the double-ended configuration for R-DTS can avoid the measurement error based on the change of local external attenuation. And it needs the double-length sensing fibers to build the loop configuration.

Moreover, in the R-DTS system, the limitation of temperature accuracy is the signal-to-noise ratio. If the system's signal-to-noise ratio is increased, the temperature accuracy will also be optimized. In the experiment, the denoising method of cumulative average and wavelet transform modulus maximum are applied to the Raman signal. In order to further improve the R-DTS system performance, we will optimize the system's signal-to-noise ratio (such as eliminate the environmental noise) in the next step. The difference sensitive-temperature compensation by we proposed can apply to the system with single mode fibers (or other fiber type). However, the R-DTS system needs to be recalibrated before replacing the single mode fibers for temperature measurement.

In this paper, the distributed temperature measurement and theory analysis using the dual-demodulation, self-demodulation and double-end configuration methods for R-DTS are demonstrated with relevant experiments. The experimental results indicate that the temperature accuracy is 12.54 °C, 8.53 °C and 15.00 °C along the 10.8 km under the standard R-DTS system, respectively. It cannot meet the requirements of the temperature safety monitoring with high-accuracy. In the experiment, the sensitive-temperature property of multimode fiber shows a different state in different sensing distance, it deteriorates the temperature accuracy of the R-DTS system. For optimizing the temperature accuracy, we analyzed and recalibrated the intensity equation of Raman scattering signal in theory, and designed a difference sensitive-temperature compensation method to the R-DTS systems with self-demodulation, dual-demodulation and double-ended configuration principles. After compensation, the experimental results show that the temperature accuracy can be optimize to 0.38 °C, 0.36 °C and 0.56 °C. It can achieve the temperature measurement accuracy better than the 1 °C over the 10.8 km. It proves that the proposed method has a better measurement effect than the conventional one compared to the original data, which proved to be a powerful tool to enhance the performance of R-DTS system. In addition, the system can be further improved by optimizing the proposed method combined with the advanced coding and denoising methods.

Funding

National Natural Science Foundation of China (61527819, 61875146); Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (Program for Sanjin Scholar); Key Research and Development Program of Jiangxi Province (201803D121064); Shanxi Scholarship Council of China (2016-036, 2017-052); Shanxi Provincial Key Research and Development Project; Shanxi Provincial Key Research and Development Project (2019BY033).

Disclosures

The authors declare no conflicts of interest.

References

1. M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016). [CrossRef]  

2. “Technology Focus: Optical-fiber sensors,” Nat. Photonics2(3), 143–148 (2008).

3. M. Wang, H. Wu, M. Tangg, Z. Zhao, Y. Dang, C. Zhao, R. Liao, W. Chen, S. Fu, C. Yang, W. Tong, P. P. Shum, and D. Liu, “Few-mode fiber based Raman distributed temperature sensing,” Opt. Express 25(5), 4907–4916 (2017). [CrossRef]  

4. J. Li, B. Q. Yan, M. J. Zhang, J. Z. Zhang, B. Q. Jin, Y. Wang, and D. Wang, “Long-Range Raman Distributed Fiber Temperature Sensor with Early Warning Model for Fire Detection and Prevention,” IEEE Sensors J. 19(10), 3711–3717 (2019). [CrossRef]  

5. J. D. Zhang, T. Zhu, H. Zhou, S. H. Huang, M. Liu, and W. Huang, “High spatial resolution distributed fiber system for multi-parameter sensing based on modulated pulses,” Opt. Express 24(24), 27482–27493 (2016). [CrossRef]  

6. A. P. Engelhardt, J. S. Kolb, and F. Roemer, “Temperature-dependent investigation of carrier transport, injection, and densities in AlGaAs-based multi-quantum-well active layers for vertical-cavity surface-emitting lasers,” Opt. Eng. 54(1), 016107 (2015). [CrossRef]  

7. L. Ren, T. Jiang, and Z. G. Jia, “Pipeline corrosion and leakage monitoring based on the distributed optical fiber sensing technology,” Measurement 122, 57–65 (2018). [CrossRef]  

8. D. Hwang, D. J. Yoon, I. B. Kwon, D. C. Seo, and Y. Chung, “Novel auto-correctin method in a fiber-optic distributed-temperature sensor using reflected anti-Stokes Raman scattering,” Opt. Express 18(10), 9747–9754 (2010). [CrossRef]  

9. Y. Chen, M. Lin, W. Tong, W. J. Tong, and Z. Y. He, “Long-range Raman distributed temperature sensor with high spatial and temperature resolution using graded-index few-mode fiber,” Opt. Express 26(16), 20562–20571 (2018). [CrossRef]  

10. J. Li, B. Q. Yan, M. J. Zhang, J. Z. Zhang, L. J. Qiao, and T. Wang, “Auto-correction method for improving temperature stability in a long-range Raman fiber temperature sensor,” Appl. Opt. 58(1), 37–42 (2019). [CrossRef]  

11. K. P. Hilgersom, N. C. van de Giesen, P. G. B. de Louw, and M. Zijlema, “Three-dimensional dense distributed temperature sensing for measuring layered thermohaline systems,” Water Resour. Res. 52(8), 6656–6670 (2016). [CrossRef]  

12. B. N. Sun, J. Chang, J. Lian Z, L. Wang, G. P. Lv, X. Z. Liu, W. J. Wang, S. Zhou, W. Wei, S. Jiang, Y. N. Liu, S. Luo, X. H. Liu, Z. Liu, and S. S. Zhang, “Accuracy improvement of Raman distributed temperature sensors based on eliminating Rayleigh noise impact,” Opt. Commun. 306, 117–120 (2013). [CrossRef]  

13. M. Sun, Y. Q. Tang, Y. Shuang, J. Li, M. W. Sigist, and F. Z. Dong, “Fire Source Localization Based on Distributed Temperature Sensing by a Dual-Line Optical Fiber System,” Sensors 16(6), 829 (2016). [CrossRef]  

14. B. Torres, “Optical fiber sensors embedded in concrete for measurement of temperature in a real fire test,” Opt. Eng. 50(12), 124404 (2011). [CrossRef]  

15. T. Sun, Z. Y. Zhang, and K. T. V. Grattan, “Frequency-domain fluorescence based fiber optic fire alarm system,” Rev. Sci. Instrum. 72(4), 2191–2196 (2001). [CrossRef]  

16. M. K. Saxena, S. D. V. S. J. Raju, R. Arya, S. V. G. Ravindranath, S. Kher, and S. M. Oaka, “Optical fiber distributed temperature sensor using short term Fourier transform based simplified signal processing of Raman signals,” Measurement 47(1), 345–355 (2014). [CrossRef]  

17. G. Bolognini and M. A. Soto, “Optical pulse coding in hybrid distributed sensing based on Raman and Brillouin scattering employing Fabry-Perot lasers,” Opt. Express 18(8), 8459–8465 (2010). [CrossRef]  

18. M. A. Soto, T. Nannipieri, A. Signorini, A. Lazzeri, F. Baronti, R. Roncella, G. Bolognini, and F. D. Pasquale, “Raman-based distributed temperature sensor with 1 m spatial resolution over 26 km SMF using low-repetition-rate cyclic pulse coding,” Opt. Lett. 36(13), 2557–2559 (2011). [CrossRef]  

19. J. Li, Y. T. Li, M. J. Zhang, Y. Li, J. Z. Zhang B, Q. Yan, D. Wang, and B. Q. Jin, “Performance improvement of Raman distributed temperature system by using noise suppression,” Photonic Sens. 8(2), 103–113 (2018). [CrossRef]  

20. Z. L. Wang, J. Chang, S. Zhang, S. Luo, C. W. Jia, S. Jiang, B. Sun, Y. N. Liu, and G. P. Lv, “An Improved Denoising Method in RDTS Based on Wavelet Transform Modulus Maxima,” IEEE Sens. J. 15(2), 1061–1067 (2015). [CrossRef]  

21. J. Li, Y. Xu, M. J. Zhang, J. Z. Zhang, L. J. Qiao, and T. Wang, “Performance improvement in double-ended RDTS by suppressing the local external physics perturbation and intermodal dispersion,” Chin. Opt. Lett. 17(7), 070602 (2019). [CrossRef]  

22. M. K. Saxena, S. J. Raju, R. Arya, R. B. Pachor, S. Pavindranath, S. Kher, and S. M. Oak, “Empirical Mode Decomposition-Based Detection of Bend-Induced Error and Its Correction in a Raman Optical Fiber Distributed Temperature Sensor,” IEEE Sens. J. 16(5), 1243–1252 (2016). [CrossRef]  

23. P. C. Wait, K. D. Souza, and T. P. Newson, “A theoretical comparison of spontaneous Raman and Brillouin based fiber optic distributed temperature sensors,” Opt. Commun. 144(1-3), 17–23 (1997). [CrossRef]  

24. A. L. Chakraborty, R. K. Sharma, M. K. Saxena, and S. Kher, “Compensation for temperature dependence of Stokes signal and dynamic self-calibration of a Raman distributed temperature sensor,” Opt. Commun. 274(2), 396–402 (2007). [CrossRef]  

25. J. Park, G. Bolognini, D. Lee, P. Kim, P. Cho, F. D. Pasquale, and N. Park, “Raman-based distributed temperature sensor with simplex coding and link optimization,” IEEE Photon. Technol. Lett. 18(17), 1879–1881 (2006). [CrossRef]  

26. Z. L. Wang, S. S. Zhang, J. Chang, G. P. Lv, W. J. Wang, S. Jiang, X. Z. Liu, X. H. Liu, S. Luo, B. N. Sun, and Y. N. Liu, “Attenuation auto-correction method in Raman distributed temperature measurement system,” Opt. Quantum Electron. 45(10), 1087–1094 (2013). [CrossRef]  

27. M. A. Soto, A. Signorini, T. Nannipieri, S. Faralli, and G. Bolognini, “High-Performance Raman-Based Distributed Fiber-Optic Sensing Under a Loop Scheme Using Anti-Stokes Light Only,” IEEE Photon. Technol. Lett. 23(9), 534–536 (2011). [CrossRef]  

28. M. A. Soto, A. Signori, T. Nannipieri, S. Faralli, G. Bolognini, and F. D. Pasquale, “Impact of Loss Variations on Double-Ended Distributed Temperature Sensors Based on Raman Anti-Stokes Signal Only,” J. Lightwave Technol. 30(8), 1215–1222 (2012). [CrossRef]  

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Figures (9)
Fig. 1.
Fig. 1. The experimental setup based on dual-demodulation principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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Fig. 2.
Fig. 2. The temperature measurement results of overall distribution based on dual-demodulation principle under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature demodulation results at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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Fig. 3.
Fig. 3. The experimental results of temperature measurement accuracy at the (a1) 1.5 km, (a2) 5.6 km and (a3) 10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3) 10.8 km after calibration.

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Fig. 4.
Fig. 4. The experimental setup based on self-demodulation principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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Fig. 5.
Fig. 5. The temperature measurement result of overall distribution based on self-demodulation under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature measurement result at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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Fig. 6.
Fig. 6. The experimental results of temperature accuracy based on self-demodulation at (a1) 1.5 km, (a2) 5.6 km and (a3)10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3)10.8 km after calibration.

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Fig. 7.
Fig. 7. The experimental setup based on the double-ended principle. APD: avalanche photodiode; Amp: amplifier; DAC: high-speed data acquisition card.

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Fig. 8.
Fig. 8. The temperature measurement result of overall distribution based on double-ended demodulation under the (a) 42.64 °C, (b) 52.64 °C, (c) 61.97°C, (d) 71.95 °C, (e) 81.33 °C and (f) 90.60 °C. The temperature measurement results at the position of (g) 1.5 km, (h) 9.6 km, and (i) 10.8 km.

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Fig. 9.
Fig. 9. The experimental results of temperature measurement accuracy based on double-ended demodulation at (a1) 1.5 km, (a2) 5.6 km and (a3)10.8 km before calibration. The experimental results of temperature measurement accuracy at (b1) 1.5 km, (b2) 5.6 km and (b3) 10.8 km after calibration.

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Tables (6)

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Table 1. The data of M at the difference sensing distances

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Table 2. The temperature accuracy based on dual-demodulation

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Table 3. The data of M at the difference sensing distances

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Table 4. The temperature accuracy based on self-demodulation

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Table 5. The data of sensitive-temperature factor at the difference sensing distances

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Table 6. The temperature accuracy based on double-ended demodulation

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Equations (17)

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ϕ a ϕ s = K a K s ( υ a υ s ) 4 exp ⁡ ( − h Δ v k T ) exp ⁡ [ ∫ 0 L ( α s ( L ) − α a ( L ) ) d L ] .
ϕ a c ϕ s c = K a K s ( υ a υ s ) 4 exp ⁡ ( − h Δ v k T c ) exp ⁡ [ ∫ 0 L c ( α s ( L ) − α a ( L ) ) d L ] .
ϕ a ϕ s ϕ s c ϕ a c = exp ⁡ [ − h Δ v k ( 1 T − 1 T 0 ) ] exp ⁡ [ ∫ L c L ( α s ( L ) − α a ( L ) ) d L ] .
ϕ a o ϕ s o ϕ s c o ϕ a c o = exp ⁡ [ − h Δ v k ( 1 T o − 1 T c o ) ] exp ⁡ [ ∫ L c L ( α s ( L ) − α a ( L ) ) d L ] .
T = 1 ( 1 T c + 1 T o − 1 T c o ) − k h Δ v ln ⁡ ( ϕ s c o ϕ a o ϕ a c ϕ s ϕ a c o ϕ s o ϕ s c ϕ a ) .
ϕ a ϕ s = K a K s ( υ a υ s ) 4 exp ⁡ ( − M ( L ) h Δ v k T ) exp ⁡ [ ∫ 0 L ( α s ( L ) − α a ( L ) ) d L ] .
T = M ( L ) ( M ( L c ) T c + M ( L ) T o − M ( L c o ) T c o ) − k h Δ ν ln ⁡ ( ϕ s c o ϕ a o ϕ a c ϕ s ϕ a c o ϕ s o ϕ s c ϕ a ) .
M ( L )  =  ( ϕ a ϕ s o ϕ s ϕ a o ) / ( ϕ a ϕ s o ϕ s ϕ a o ) ( h Δ ν k T o − h Δ ν k T ) ( h Δ ν k T o − h Δ ν k T ) .
T  = ln { [ exp ⁡ ( h Δ ν / h Δ ν k T o k T o ) − 1 ] [ exp ⁡ ( h Δ ν / h Δ ν k T c k T c ) − 1 ] [ exp ⁡ ( h Δ ν / h Δ ν k T c o k T c o ) − 1 ] ( ϕ a ϕ a c o ϕ a o ϕ a c ) + 1 } − 1 ( h Δ ν / h Δ ν k k )
ϕ a = K a υ a 4 exp ⁡ ( M ( L ) h Δ v k T − 1 ) − 1 exp ⁡ [ − ∫ 0 L ( α a ( L ) + α o ( L ) ) d L ] .
R ( T ) = ϕ a ϕ a o = exp ⁡ ( M ( L ) h Δ υ / h Δ υ k T 0 k T 0 − 1 ) exp ⁡ ( M ( L ) h Δ υ / h Δ υ k T k T − 1 ) .
T  = ln { [ exp ⁡ ( M ( L ) h Δ ν k T o ) − 1 ] [ exp ⁡ ( M ( L c ) h Δ ν k T c ) − 1 ] [ exp ⁡ ( M ( L c o ) h Δ ν k T c o ) − 1 ] ( ϕ a ϕ a c o ϕ a o ϕ a c ) + 1 } − 1 ( M ( L ) h Δ ν k ) .
1 T  =  [ ln ⁡ ( R L o o p ( T , L ) R L o o p ( T o , L o ) ) ( − k h Δ v ) ] + 1 T o .
ϕ a B ϕ s B = K a K s ( υ a υ s ) 4 exp ⁡ ( − M ( L ) h Δ v k T ) exp ⁡ [ ∫ 0 L ( α s ( L ) − α a ( L ) ) d L ] .
ϕ a F ϕ s F = K a K s ( υ a υ s ) 4 exp ⁡ ( − M ( l − L ) h Δ v k T ) exp ⁡ [ ∫ L l ( α s ( L ) − α a ( L ) ) d L ] .
R L o o p ( T , L ) = ϕ a B ϕ s B ⋅ ϕ a F ϕ s F = K a K s ( υ a υ s ) 4 exp ⁡ ( − h Δ v k T ( M ( L ) + M ( l − L ) ) ) exp ⁡ [ ∫ 0 l ( α s ( L ) − α a ( L ) ) d L ] .
1 T = M ( L ) + M ( l − L ) { [ M ( L ) + M ( l − L ) ] / [ M ( L ) + M ( l − L ) ] T o T o } − { l n [ R L o o p ( T , L ) / R L o o p ( T , L ) R L o o p ( T o , L ) R L o o p ( T o , L ) ] ( k / k h Δ ν h Δ ν ) } .
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