Robust Vector BOTDA Signal Processing with Probabilistic Machine Learning
2. Mathematical Background
2.1. Curve Fitting Approach
2.2. Machine Learning Approach
2.3. Comparison of Curve Fitting vs. Machine Learning for BFS Extraction
- Data Processing Time: The computational advantage of the ML approach over the CF approach is evident when comparing the schematics of both approaches, as shown in Figure 2. The CF approach requires repeating the optimization iterations for each BGS, while the ML approach can predict the BFS and FWHM directly from the BGS measurements using Equation (7), once the ML model is trained offline.
- Interpretability: However, the CF approach is more interpretable since the function can be chosen based on the underlying optical physics knowledge. On the other hand, no such reasoning exists to construct in the ML approach and several different ML models have been proposed in the literature.
- Robustness: A key feature of a robust signal processing algorithm is its ability to accurately quantify the confidence/uncertainty in predictions. As mentioned earlier, curve fitting approaches use regression to estimate parameters and can yield CIs of the parameter estimates. However, the ML approach (Equation (5)) can not be used directly to estimate the CIs since the term does not represent the sensor measurement error. Instead, in Equation (5) should be interpreted as prediction error with an unknown probability distribution due to the nonlinear nature of . Subsequently, the ML approach provides BFS estimates without providing a measure of uncertainty/confidence (i.e., confidence intervals or error bars) of the BFS predictions. With the increasing adoption of deep neural networks for BOTDA processing, it is even more crucial that the BFS be estimated along with its confidence level, in order to avoid over-fitting.
3. Proposed Probabilistic Machine-Learning-Based BFS Extraction
- Mean vector directly predicts the means of BFS () and FWHM (w) from BGS/BPS measurements using a suitable ML model
- Standard deviation matrix quantifies the uncertainty in estimates of BFS () and FWHM (w) due to the noise in underlying measurements
- Robustness: The PML approach prevents overfitting that arises when using ML and DNN models to represent .
- Speed: It inherits the computational advantages of the ML approach and enables fast processing of BOTDA data with simultaneous assessment of prediction uncertainties.
4. PML Model Development and Training
4.1. PML Model Training
- Uniformly sample s, , and w from the bounds in Equation (19) to obtain
- Simulate gain and phase values for each of the n frequencies and for using a suitable spectrum model.This work has chosen Lorentzian BGS and BPS  (Equations (22) and (23)) given by the following:
- Sample and add Gaussian noise corresponding to the noise amplitude to obtain training dataset sample
4.2. PML Model Architecture
5. Experimental Setup
6. Results and Discussions
6.1. Custom BOTDA System Using 10 km Long Sensing Fiber
6.2. Custom VBOTDA System Using 25 km Long Sensing Fiber
Data Availability Statement
Conflicts of Interest
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Venketeswaran, A.; Lalam, N.; Lu, P.; Bukka, S.R.; Buric, M.P.; Wright, R. Robust Vector BOTDA Signal Processing with Probabilistic Machine Learning. Sensors 2023, 23, 6064. https://doi.org/10.3390/s23136064
Venketeswaran A, Lalam N, Lu P, Bukka SR, Buric MP, Wright R. Robust Vector BOTDA Signal Processing with Probabilistic Machine Learning. Sensors. 2023; 23(13):6064. https://doi.org/10.3390/s23136064Chicago/Turabian Style
Venketeswaran, Abhishek, Nageswara Lalam, Ping Lu, Sandeep R. Bukka, Michael P. Buric, and Ruishu Wright. 2023. "Robust Vector BOTDA Signal Processing with Probabilistic Machine Learning" Sensors 23, no. 13: 6064. https://doi.org/10.3390/s23136064