# Estimation of Temperature and Associated Uncertainty from Fiber-Optic Raman-Spectrum Distributed Temperature Sensing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Estimation of Temperature from Stokes and Anti-Stokes Scatter

## 3. Integrated Differential Attenuation

#### 3.1. Single-Ended Measurements

#### 3.2. Double-Ended Measurements

## 4. Estimation of the Variance of the Noise in the Intensity Measurements

## 5. Single-Ended Calibration Procedure

## 6. Double-Ended Calibration Procedure

## 7. Confidence Intervals of the Temperature

#### 7.1. Single-Ended Measurements

#### 7.2. Double-Ended Measurements

## 8. Python Implementation

## 9. Example

#### 9.1. Setup and Data Collection

#### 9.2. Estimation of the Temperature and the Associated Uncertainty

#### 9.3. Effect of Parameter Uncertainty

#### 9.4. Effect of Difference in Reference Temperatures

## 10. Discussion

#### 10.1. Improved Temperature Estimation for Double-Ended Setups

#### 10.2. Calibration to Reference Sections

## 11. Conclusions

- Dataset license: GPL-3.0-or-later

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Intensity-Dependent Variance of the Noise in the Intensity Measurements

## Appendix B. Correlation Stokes and Anti-Stokes Residuals

**Figure A1.**Stokes residuals plotted against the anti-Stokes residuals. The Stokes and anti-Stokes intensity recorded by the DTS instrument have arbitrary units that are linearly related to the power of the scattered signals.

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**Figure 3.**(

**a**) Temperature with its 95% confidence intervals at the first time step. (

**b**) Differences between the estimated temperature and the reference temperature at the first time step.

**Figure 4.**Spatial variation of the standard uncertainty of the estimated temperature, and the mean and standard deviation of the differences between the estimated and reference temperature.

**Figure 5.**Temporal variation of the standard uncertainty of the estimated temperature, and the mean and standard deviation of the differences between the estimated and reference temperature.

**Figure 6.**Synthetic example of the standard uncertainty of the estimated temperature using arithmetic mean and the inverse-variance weighted mean.

Name | Fiber Section (m) | Average Temperature (°C) | Number of Measurement Locations | Notes |
---|---|---|---|---|

Cold 1 | 7.5–17.0 | 4.35 | 37 | Used for calibration |

Warm 1 | 24.0–34.0 | 18.52 | 39 | Used for calibration |

Ambient | 40.0–50.0 | 12.62 | 39 | |

Cold 2 | 70.0–80.0 | 4.35 | 39 | |

Warm 2 | 85.0–95.0 | 18.52 | 39 |

Cold 1 | Warm 1 | Ambient | Cold 2 | Warm 2 | Total |
---|---|---|---|---|---|

95.6% | 95.0% | 92.3% | 94.7% | 94.3% | 94.4% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

des Tombe, B.; Schilperoort, B.; Bakker, M.
Estimation of Temperature and Associated Uncertainty from Fiber-Optic Raman-Spectrum Distributed Temperature Sensing. *Sensors* **2020**, *20*, 2235.
https://doi.org/10.3390/s20082235

**AMA Style**

des Tombe B, Schilperoort B, Bakker M.
Estimation of Temperature and Associated Uncertainty from Fiber-Optic Raman-Spectrum Distributed Temperature Sensing. *Sensors*. 2020; 20(8):2235.
https://doi.org/10.3390/s20082235

**Chicago/Turabian Style**

des Tombe, Bas, Bart Schilperoort, and Mark Bakker.
2020. "Estimation of Temperature and Associated Uncertainty from Fiber-Optic Raman-Spectrum Distributed Temperature Sensing" *Sensors* 20, no. 8: 2235.
https://doi.org/10.3390/s20082235