Robust Vector BOTDA Signal Processing with Probabilistic Machine Learning
Abstract
:1. Introduction
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 ${f}_{c}$ can be chosen based on the underlying optical physics knowledge. On the other hand, no such reasoning exists to construct ${f}_{d}$ 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 $\mathbf{e}$ does not represent the sensor measurement error. Instead, $\mathbf{e}$ in Equation (5) should be interpreted as prediction error with an unknown probability distribution due to the nonlinear nature of ${f}_{d}$. 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 overfitting.
3. Proposed Probabilistic MachineLearningBased BFS Extraction
 Mean vector ${\mathit{\mu}}_{\mathit{\lambda}}$ directly predicts the means of BFS (${\nu}_{B}$) and FWHM (w) from BGS/BPS measurements $\mathbf{X}$ using a suitable ML model
 Standard deviation matrix ${\mathbf{\Sigma}}_{\mathit{\lambda}}$ quantifies the uncertainty in estimates of BFS (${\nu}_{B}$) and FWHM (w) due to the noise in underlying measurements $\mathbf{X}$
 Robustness: The PML approach prevents overfitting that arises when using ML and DNN models to represent ${f}_{d}$.
 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, ${\nu}_{B}$, and w from the bounds in Equation (19) to obtain $\left({s}_{j},{{\nu}_{B}}_{j},{w}_{j}\right)$
 Simulate gain and phase values for each of the n frequencies and for $\left({{\nu}_{B}}_{j},{w}_{j}\right)$ using a suitable spectrum model.$$\begin{array}{cc}\hfill {g}_{j}\left({\nu}_{i}\right)& =g\left({\nu}_{i};{{\nu}_{B}}_{j},{w}_{j}\right),\phantom{\rule{0.166667em}{0ex}}\forall i=1\dots K\hfill \end{array}$$$$\begin{array}{cc}\hfill {\varphi}_{j}\left({\nu}_{i}\right)& =\varphi \left({\nu}_{i};{{\nu}_{B}}_{j},{w}_{j}\right),\phantom{\rule{0.166667em}{0ex}}\forall i=1\dots K\hfill \end{array}$$This work has chosen Lorentzian BGS and BPS [7] (Equations (22) and (23)) given by the following:$$\begin{array}{cc}\hfill g\left(\nu ;{\nu}_{B},w\right)& ={g}_{0}\frac{{w}^{2}}{4{\left(\nu {\nu}_{B}\right)}^{2}+{w}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \varphi \left(\nu ;{\nu}_{B},w\right)& ={g}_{0}\frac{2w\left(\nu {\nu}_{B}\right)}{4{\left(\nu {\nu}_{B}\right)}^{2}+{w}^{2}}\hfill \end{array}$$
 Sample $e\sim \mathcal{N}\left(0,1\right)$ and add Gaussian noise corresponding to the noise amplitude ${s}_{j}$ to obtain training dataset sample ${\mathbf{X}}_{j}$$$\begin{array}{c}\hfill {\mathbf{X}}_{j}={\left[\begin{array}{ccc}{\nu}_{1}& \cdots & {\nu}_{K}\\ {g}_{j}\left({\nu}_{1}\right)+{s}_{j}e& \cdots & {g}_{j}\left({\nu}_{K}\right)+{s}_{j}e\\ {\varphi}_{j}\left({\nu}_{1}\right)+{s}_{j}e& \cdots & {\varphi}_{j}\left({\nu}_{K}\right)+{s}_{j}e\end{array}\right]}^{\u22ba}\end{array}$$
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
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Gain Spectrum  Function  Parameters 

${\mathit{f}}_{\mathit{c}}\left(\mathit{\nu};\mathit{\lambda}\right)$  $\mathit{\lambda}$  
Lorentzian 
$${f}_{c}\left(\nu ;\mathit{\lambda}\right)=\frac{{g}_{0}}{1+4{\xi}^{2}},\phantom{\rule{1.em}{0ex}}\xi =\frac{\left(\nu {\nu}_{B}\right)}{w}$$

$$\mathit{\lambda}\equiv \left[{g}_{0},{\nu}_{B},w\right]$$

Gaussian 
$${f}_{c}\left(\nu ;\mathit{\lambda}\right)={g}_{0}exp\left[4ln2{\xi}^{2}\right],\phantom{\rule{1.em}{0ex}}\xi =\frac{\left(\nu {\nu}_{B}\right)}{w}$$

$$\mathit{\lambda}\equiv \left[{g}_{0},{\nu}_{B},w\right]$$

PseudoVoigt 
$${f}_{c}\left(\nu ;\mathit{\lambda}\right)={g}_{0}\left[\frac{p}{1+4{\xi}^{2}}+\left(1p\right)exp\left(4ln2{\xi}^{2}\right)\right]$$

$$\mathit{\lambda}\equiv \left[p,{g}_{0},{\nu}_{B},w\right]$$

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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/s23136064
Chicago/Turabian StyleVenketeswaran, 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