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Abstract
We report, to our knowledge for the first time on simultaneous distributed temperature and strain sensing in a standard telecom optical fiber using a machine learning assisted Brillouin frequency domain analysis (BOFDA) system. The well-known temperature and strain cross-sensitivity problem is addressed by developing a BOFDA system with a high signal-to-noise ratio and applying machine learning. The spectrum consists of four highly resolved peaks, whose Brillouin frequency shifts are extracted and serve as features for the machine learning algorithms. The spectra result from a 450-m standard SMF-28 optical fiber, and particularly from a segment of 30 m. This fiber segment is coiled around a stretcher and placed in a climate chamber. The applied temperature and strain values range from 20 °C to 40 °C and from 0 µɛ to 1380 µɛ, respectively. The total measurement time to achieve a high SNR and resolve four peaks with a spatial resolution of 6 m is 16 min. To discriminate temperature and strain effects, simple frequentist and more sophisticated Bayesian-based algorithms are employed with the powerful Gaussian process regression (GPR) delivering the best performance in terms of temperature and strain errors, which are found to be 2 °C and 45 µɛ, respectively. These errors are calculated using leave-one-out cross-validation, so that an unbiased estimation of the sensor’s performance is provided.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Brillouin distributed fiber optic sensors for temperature and strain sensing have attracted a lot of interest in the last decades and have been employed in a wide range of applications, mostly in the field of civil and geotechnical engineering [1]. Brillouin optical frequency domain analysis (BOFDA) is among the well-known and established techniques providing ultra-long range up to 100 km and high spatial resolution even on cm and mm-scale [2–4]. However, simultaneous strain and temperature monitoring using standard optical fibers remains a challenge not only for BOFDA but also for other widely used Brillouin-based fiber optic sensors, such as Brillouin optical time domain analysis (BOTDA) [5] and Brillouin optical correlation domain analysis (BOCDA) [6].
Cross-sensitivity of temperature and strain on the Brillouin signal does not allow the discrimination of the two measurands with the conventional way in a standard optical fiber. This arises from the fact that standard optical fibers are characterized by a single Lorentzian-shaped Brillouin gain spectrum, whose Brillouin frequency shift (BFS) depends on both temperature and strain. So far, simultaneous temperature and strain sensing has been demonstrated by using many different approaches, including two optical fibers, specialty fibers, or even complex hardware systems and algorithms. Specifically, a simple solution is based on the use of the two-fiber configuration, where one fiber is mechanically isolated, and thus not affected by strain effects [7]. However, the requirement of two optical fibers cannot be always fulfilled. Strain and temperature discrimination in a single optical fiber has been demonstrated by using special fibers such as large effective area fibers (LEAF) [8–10], photonic crystal fibers [11], dispersion compensating fibers [12], few mode fibers [13], and polarization maintaining fibers [14]. Apart from using special fibers, strain and temperature effects can be decoupled by using hybrid systems based on more than one scattering effect [15–18].
Over the last years, machine learning has been used in BOTDA sensors to alleviate the cross-sensitivity problem. Artificial neural networks have been applied to LEAF fibers outperforming the equation solving method [19] and also to standard fibers classifying strain and temperature values [20]. Furthermore, machine learning has been applied to enhance the performance of the Brillouin-based distributed sensors either by extracting the BFS [21–29], extracting directly temperature from the spectra [30–37], or denoising the spectra [38–41]. In BOFDA, convolutional neural networks have been applied to either shorten the measurement time or enhance the performance of the system when a long sensing range (over 30 km) is needed [42,43].
To our knowledge, this is the first report on a BOFDA system for simultaneous temperature and strain sensing in a standard single-mode telecom optical fiber using machine learning. BOFDA, as mentioned previously, provides long measurement length and high spatial resolution, and most importantly, does not require ultra-fast sampling circuitries which has a positive impact on the system’s cost. We note that even though in the reported BOFDA system, we make use of an expensive VNA, this can be replaced by a digital setup that reduces the cost significantly [44]. The standard telecom optical fiber used in this paper contributes towards a cost-effective solution for distributed simultaneous temperature and strain sensing and enables the potential use of laid-out fiber optic networks for monitoring of geotechnical structures and energy transport infrastructure over long time periods.
The paper is structured as follows: In the theory section, we introduce the operating principle of a classic BOFDA system, the machine learning algorithms used to discriminate the two effects and the error estimation algorithm used to evaluate the discrimination performance. In the experimental section, we demonstrate a BOFDA system with narrow-band filtering that offers a high signal-to-noise ratio (SNR). This high SNR allows us to record the multipeak Brillouin gain spectrum (BGS) of the standard legacy SMF-28 (Corning) optical fiber. In total, four BFSs are extracted from the BGS which are used as features for the discrimination of temperature and strain effects. The measurements are conducted under different temperature and strain conditions using a fiber stretcher placed in a climate chamber. After obtaining the data and analyzing basic relations between features and target values (strain and temperature), simple frequentist and more sophisticated Bayesian-based algorithms are applied. The best performance in terms of temperature and strain mean absolute error is reached by using Gaussian process regression (GPR) which is a Bayesian-based and nonparametric machine learning algorithm. The errors obtained by the nonlinear machine learning algorithms are benchmarked against the errors obtained by the linear least-squares regression (equation solving method).
2. Theory
2.1 BOFDA sensing principle
Brillouin backscattering results from the scattering of the so-called pump waves with the acoustic phonons of the medium. The Brillouin scattering is inelastic with the frequency difference between the incident ${v_p}$ and the backscattered Stokes waves ${v_s}$, termed the BFS, being equal to the frequency of the acoustic waves. In order to strengthen the backscattered signal, the effect is stimulated by using counterpropagating Stokes waves with frequencies equal to those of the spontaneous Brillouin backscattered waves. However, a slight frequency detuning can still initiate Stokes amplification due to the damping rate of the medium. We note that the acoustic velocity of the acoustic modes of the medium affect the frequency of the backscattered Stokes waves, and thus the BFS. So, different acoustic modes result in different BFSs and thus a multipeak BGS can be obtained.
In contrast to the well-known Brillouin optical time-domain analysis (BOTDA), where the pulse response of the system $h(t)$ is directly recorded, BOFDA measures the system’s complex transfer function $H(j\omega )$ by applying RF modulation to the continuous pump wave and using a vector network analyzer (VNA). The transfer function $H(j\omega )$ can then be converted to time domain $h(t)$ through inverse fast Fourier transformations. Similar to BOTDA, the BGS of the BOFDA system is described by a sum of Lorentzian-shaped profiles:
2.2 Frequentist and Bayesian-based machine learning algorithms
Temperature and strain discrimination is performed by using frequentist and Bayesian-based machine learning algorithms. All the algorithms are trained and tested using experimental data with the inputs (features) being the four obtained BFSs ${v_{B1}},\ldots ,{v_{B4}}$ of the BGS (as we show in the experimental part) and the outputs (targets) being the temperature $T$ and strain $\varepsilon $. The inputs and outputs can be written in matrix form as $\mathbf{X} = [{v_{B1}},…,{v_{B4}}]$ and $\mathbf{Y} = [T,\varepsilon ]$, respectively. The algorithms used in this paper are linear and polynomial regression based on both frequentist and Bayesian statistics. Furthermore, gaussian process regression (GPR) which is a more sophisticated Bayesian-based machine learning algorithm is also employed. All the machine learning algorithms are applied using the Python’s scikit-learn library [52].
Linear regression based on frequentist statistics, and particularly on ordinary least squares assumes linear relations between targets and features and the model can be written as:
where k denotes the target ($T$, $\varepsilon $). The goal of the frequentist regression learning is to estimate the coefficient matrix $\boldsymbol{\beta }$ that best describe the data by minimizing the residual sum of squares (cost function) as follows:After the optimization of the coefficient matrix $\boldsymbol{\beta }$ one can use this matrix to estimate the temperature and strain using the input matrix (BFSs) as follows:
where the matrix $\mathop {\mathbf{Y}}\limits^ \wedge $ denotes the estimated temperature and strain values. We note that in in the literature this method is also mentioned as equation solving method because it is based on statistical analysis and the goal is the estimation of the coefficient matrix $\boldsymbol{\beta }$ from a system of equations. However, in this paper we preferred the term “linear least squares regression” that is used in the field of statistics and machine learning [53,54]. In the results section, we use the linear least squares regression as a baseline to compare other more sophisticated and complex algorithms.Polynomial regression is also utilized to describe the nonlinearities in the dataset and works similarly to linear regression but by using features converted to their higher order terms. The complexity of the polynomial regression increases with the degree of polynomials, and thus the model becomes more prone to overfitting. We note that overfitting is a term that refers to the failure of a model to generalize on test data due to the complete or partial memorization of the training set [55]. The overfitting is addressed by applying L2 regularization, which adds a penalty term $\alpha$ to the cost function to force the weights to be close to zero [54]. The cost function including L2 regularization is written as follows:
Apart from frequentist regression, regression based on Bayes’ theorem [54] is also applied. In contrast to a classic frequentist approach, which requires a long run frequency of events, a Bayesian approach is based on an a-priori (domain) knowledge which is updated using the available data. The use of an a-priori knowledge limits the required amount of data rendering Bayesian regression more attractive when limited data is available [56], which, in turn, can have a positive impact on the time devoted on measurements in the lab. Furthermore, Bayesian regression provides probability distributions of the outputs instead of point estimates. This means that one can derive the mean prediction values associated with the confidence prediction interval which is widely used in active (sequential) learning tasks [54].
Linear regression can also be applied from a Bayesian viewpoint. The output Y is not treated as a single value but rather as a probability distribution and the Eq. (2) can be rewritten as:
where N denotes the Gaussian distribution with the mean being the coefficient matrix multiplied by the input matrix. The variance is given by the square of the standard deviation ($\sigma $). The prior (domain knowledge) on the weights is given by a spherical Gaussian and the model is estimated by maximizing the log-marginal likelihood as described in the Appendix A of [57]. We note that the prior acts in the same way as the penalty term in Ridge regression which explains the origin of the module’s name, BayesianRidge, used to implement this algorithm. The order of the Bayesian ridge regression can be increased by performing polynomial feature expansion.GPR is a powerful Bayesian based and nonparametric machine learning algorithm [58]. In the case of GPR, the prior belief is given by a distribution of functions which is updated using training data resulting in joint posterior probability distributions over all possible values. The prior of the gaussian process is specified by a mean and a covariance function (kernel). For the sake of simplicity, the mean can be assumed constant and zero [59]. The covariance function, in practice, controls among others, the smoothness, the periodicity and the stationarity of the predictions. In this paper, the well-known squared exponential covariance function [58] is used. In total, two hyperparameters need to be optimized, namely the length scale of the kernel and the so-called alpha value, which represents the noise level in the targets [52]. Both hyperparameters are specified as a scalar (isotropic kernels).
2.3 Error estimation algorithm based on leave-one-out cross-validation
An unbiased error estimation method is of great importance in machine learning to ensure that the trained models do not overfit and generalize well on new data. In this paper, an error estimation approach based on leave-one-out cross-validation [60] is applied. Leave-one-out cross-validation is a special case of cross-validation where the number of folds N is equal to the number of the different temperature and strain combinations. So, the model is trained and tested in total N times. We note that the temperature and strain values in the test set are never included in the training set. The final performance is estimated by averaging the strain and temperature errors resulting from the predictions on the test sets. We note that because every measurement is conducted at different temperature and strain levels, our approach provides an estimation of the models’ interpolation ability as well. Specifically, in this paper we make use of 50 different combinations of strain and temperature, which means that the training set consists always of 49 temperature and strain sets. All temperature and strain errors are estimated in terms of mean absolute error.
3. Experimental
3.1 BOFDA experimental setup and data acquisition
The developed BOFDA experimental setup is shown in Fig. 1. A distributed feedback laser which emits at 1550 nm with a power of 18 dBm is utilized. The BOFDA system is based on stimulated Brillouin scattering, and thus apart from the pump signal, an additional probe (Stokes) signal is required to stimulate the backscattered Brillouin emission. In this setup, this is achieved by using a 20/80 optical splitter, where the 20% of the laser power goes to the pump path (lower branch) and the other 80% to the probe path (upper branch). The probe path performs frequency scanning to tune the BFS with the help of a Mach-Zehnder modulator (MZM). The MZM is driven by an RF signal generator and operates at its zero-transmission point creating two sidebands and suppressing the carrier [61]. The suppression of the carrier significantly decreases the probe power which after the modulator is at -1 dBm. The frequency offset (Δfb) between the carrier and the sidebands is determined by a radiofrequency (RF) signal generator (SG). A fiber Bragg grating (FBG) filter is used to discard the lower sideband as shown in Fig. 1. The probe power after the FBG filter is approximately -10 dBm. The isolator protects the components of the probe path from the transmitted pump signal. The pump path is responsible for the acquisition of the spatially resolved information. This is achieved using an additional MZM operating in the linear range of the transfer function to modulate the amplitude of the pump continuous wave. The modulation frequency step ($\Delta {f_m}$) and the maximum modulation frequency determine the measurement length and the spatial resolution, respectively [62]. An erbium-doped fiber amplifier (EDFA) that provides a constant output power is used to amplify the pump signal and set the power at 7 dBm. A polarization scrambler (PS) operating at 700 kHz is employed to reduce polarization fading [63]. The backscattered signal from the fiber-under-test travels through a circulator to a second FBG filter, which is used to filter out the Rayleigh scattered signal which is of high importance for a high SNR Brillouin spectrum as we show later. In the end, a VNA receives the electrically transformed signal from a photodiode which is given by the so-called transfer function $H(j\omega ,\Delta {f_b})$. The transfer function is then converted via inverse fast Fourier transformations into the time domain $H(t,\Delta {f_b})$. Given the refractive index of the optical fiber, the transfer function is finally converted into the spatially resolved BGS.
Fig. 1. BOFDA experimental setup and fiber configuration. The optical fiber is placed in a climate chamber with a segment of approximately 30 m coiled around a fiber stretcher. DFB laser: distributed feedback laser; MZM: Mach–Zehnder modulator; SG: signal generator; FBG filter: fiber Bragg grating filter; EDFA: erbium-doped fiber amplifier; PS: polarization scrambler; VNA: vector network analyzer.
Along with the BOFDA setup, Fig. 1 shows the configuration of the optical fiber. An approximately 450-m standard silica optical fiber is placed in a climate chamber with a segment of 30 m coiled around a fiber stretcher. This part of the fiber is exposed to both temperature and strain conditions. The fiber stretcher is based mainly on two rotatable cylinders with one of them placed on top of a remotely controlled translation stage to regulate the distance between them. The temperature in the climate chamber ranges from 20 °C to 40 °C, while the range of the applied strain is from 0 µɛ to 1380 µɛ. The total number of temperature and strain combinations is 50. Specifically, a step of 5 °C is set, while at each temperature 10 different strain levels are applied.
All measurements are conducted with the same BOFDA experimental settings. The modulation frequency step $\Delta {f_m}$ and the maximum modulation frequency ${f_{m,\max }}$ are set to 192 kHz and 16.32 MHz, respectively. We note that modulation frequency step is equal to the minimum modulation frequency $\Delta {f_{\min }}$, and thus a modulation frequency range of 16.128 MHz corresponding to 85 frequencies in total, is used. The spatial resolution $\Delta z$ and measurement length ${L_{\max }}$ are estimated using the following equations [44]:
where c and n are the speed of light in vacuum and the effective refractive index of the fiber. The refractive index of the commercially available standard SMF-28 (Corning) is 1.4682 at 1550 nm. So, using the Eqs. (6) and (7), the spatial resolution and the measurement length are calculated to be 6.3 m and 532 m, respectively.The Brillouin frequency sweep range is set from 10.84 GHz to 11.3 GHz. The VNA uses a very narrow bandwidth of 40 Hz, and no signal averaging is performed. The latter is attributed to the narrow bandwidth detection as well as the use of continuous waves. All these configurations result in a total measurement time of approximately 16 min for the optical fiber at hand.
A spatially resolved BGS along the optical fiber and a BGS at a random position are shown in Fig. 2(a) and Fig. 2(b), respectively. The BGS exhibits a high SNR which renders visible not only the fundamental peak but other three peaks as well Fig. 2(c) and Fig. 2(d) are retrieved using the same experimental conditions and settings but without the FBG filter, placed before the photodiode. Therefore, the use of the FBG filter which filters out the noise that arises from the Rayleigh backscattered signal is of great importance.
Fig. 2. Characteristic Brillouin gain spectrum retrieved from the experimental setup with (a, b) and without (c, d) an FBG filter before the detector. The left column shows a 2D representation of the BGS along the whole optical fiber while the right column depicts a BGS at a random position. The use of an FBG filter that filters out the Rayleigh backscattered signal improves the SNR significantly.
3.2 Feature extraction of the multipeak Brillouin gain spectrum
A multipeak BGS of high SNR allows a relatively fast and of high quality multipeak Lorentzian curve fitting (LCF). As mentioned in section 2.1, an LCF is performed to extract the BFSs which serve as features for the machine learning models. Figure 3(a) shows a characteristic example of such a fit on a normalized BGS at a random position along the optical fiber. Apart from the BGS (black dots) and its multipeak LCF (purple curve), the Lorentzian components of each peak are depicted with colored dashed curves as well. Due to the low amplitude of the secondary peaks, the BGS is plotted in logarithmic scale.
Fig. 3. a) Multipeak Lorentzian curve fitting (LCF; purple curve) of the Brillouin spectrum (black dots) at a random position along the fiber. The individual Lorentzian components (LC) are depicted with colored dashed curves. b) 2D representation of the BGS along the optical fiber including the extracted BFSs of the four Brillouin peaks.
Figure 3(b) shows a BGS along the whole optical fiber including the extracted BFSs illustrated with the same colors as the corresponding Lorentzian components in Fig. 3(a). The approximately 30-m segment of the optical fiber wound on the stretcher is clearly distinguished at 300 m. The BFSs of the other parts of the fiber are of the same value which is attributed to the same temperature conditions.
After extracting the features and before applying machine learning, it is of great importance to briefly analyze the relation between the features and the targets. Figure 4(a) shows two BGS (along with the LCF) resulted from measurements conducted at 20 °C (blue) and 30 °C (orange), when strain is zero. It is obvious that the temperature increase results in a spectrum shift towards higher frequencies. The relation between the fundamental BFS and the temperature is linear and is shown in the inset of Fig. 4(a). The BFS temperature sensitivity of the fundamental peak, which is conventionally called temperature coefficient is 1.094 (± 0.004) MHz/°C. The second, third and fourth peak shift also linearly with temperature with their temperature sensitivities being 1.106 (± 0.004) MHz/°C, 1.176 (± 0.011) MHz/°C and 1.078 (± 0.013) MHz/°C, respectively. We observe that the coefficients’ errors are one order of magnitude larger for the last two peaks which is most likely attributed to their lower amplitude. The different temperature sensitivities indicate that temperature and strain discrimination is partially feasible even with a simple linear regression model.
Fig. 4. a) Experimental BGS (dots) along with Lorentzian curve fitting (curves) at 20 °C (blue) and 30 °C (orange), when no strain is applied. The inset plot depicts the linear relation between the BFS of the fundamental peak and the temperature. b) BFS vs. applied tensile strain for the fourth peak at 20 °C (blue), 30 °C (orange) and 40 °C (purple). The legend includes the strain sensitivities Cɛ at each temperature.
Figure 4(b) shows, as an example, how the BFS of the last (fourth) peak changes when strain is applied to the fiber under different temperature conditions. For the sake of clarity, only data from the measurements conducted at 20 °C (blue), 30 °C (orange) and 40 °C (purple) are shown. We observe that the BFS changes linearly with the strain. The strain coefficient slightly increases with the temperature as shown in the legend of the Fig. 4(b). On average, the strain coefficient of the fourth peak is 51 kHz/µɛ. Although the fourth peak has a two-orders of magnitude lower amplitude and it is broader than the fundamental peak, the quality of the linear fit is very good. The fit quality is evaluated by estimating the coefficient determination R-squared which, on average is 0.999. For the sake of completeness, we mention that the strain coefficient for the first three peaks is 50 kHz/µɛ. Furthermore, we observe that the strain range changes with temperature which is attributed to the thermal expansion of the stretcher’s material (aluminum).
4. Results
4.1 Temperature and strain discrimination using frequentist-based machine learning
The well-established and simple linear least-squares regression is employed as described in section 2.2. Due to the model’s simplicity, basic trends that can be used to discriminate the two parameters are captured. As already mentioned in the theory section, this algorithm corresponds to the conventional equation solving method, and thus it works as a baseline model for the rest of the paper. The temperature and strain mean absolute errors are calculated using the previously described cross-correlation method and are found to be 5 °C and 114 µɛ, respectively.
Due to the non-linearities in the data that arise from the dependencies of strain coefficients on temperature and vice versa, polynomial regression is also utilized. Although the degree of the polynomials increases the model’s complexity and facilitates a perfect fit, the model’s ability to generalize on new data drops significantly. As an example, we mention that while the 4th order polynomial model provides a strain error of 81 µɛ, the 5th order polynomials increase the error to 186 µɛ. This is worse than the error of the simple linear least-squares regression model.
To prevent overfitting, ridge regression is applied. Ridge regression, as already mentioned, is a regression algorithm (linear or polynomial) in combination with L2 regularization. Thanks to the penalty term in Eq. (4), the error of the 5th order polynomial model decreases from 186 µɛ to 58 µɛ, which is the lowest achieved by using polynomial regression. Figure 5 summarizes the strain errors using all the aforementioned algorithms and applying different penalty terms. In general, we observe that when complexity increases (degree of polynomial), a higher penalty value is required to reach the minimum strain error. We note that the temperature error is highly correlated with the strain error and therefore, the lowest strain error (58 µɛ) corresponds to the lowest temperature error which is 2.6 °C.
Fig. 5. Overview of the frequentist-based models’ performance in terms of strain errors. The orange and red solid frames correspond to the linear and polynomial models without regularization (α=0), respectively. On the other hand, the dashed frames indicate that L2 regularization is applied (ridge regression).
4.2 Temperature and strain discrimination using Bayesian-based machine learning
After applying simple frequentist-based regression, the same algorithms are applied from a Bayesian viewpoint. As already discussed in the section 2.2, the BayesianRidge module from the sklearn python library is used. Although the results are similar with those obtained using frequentist algorithms, Bayesian ridge regression proves to be advantageous because no optimization of the penalty term is required [52], and no overfitting is observed. This also makes the training procedure faster.
To further improve the performance in terms of strain and temperature errors, we make use of the Gaussian process regression (GPR), which, in contrast to the previously used algorithms, it is not restricted to a specific parametric equation. However, the kernel function, which describes the similarity between two neighboring points, has to be specified. After trying all the kernel functions included in the sklearn library, we conclude that the radial-basis (squared exponential) function kernel delivers the lowest strain and temperature errors. Furthermore, because the data are supposed to include some noise, the optimization of a hyperparameter that describes the variance of the noise on the labels is of great importance. A systematic study of the impact of this hyperparameter on the discrimination performance results in an optimized value equal to 10−5. The total strain and temperature error achieved using GPR is 45 µɛ and 2 °C, respectively. This means that GPR provides approximately 60% and 22% lower errors than the linear and polynomial (ridge) regression, respectively.
Figure 6 presents a few distributed temperature and strain predictions using GPR corresponding to the section coiled on the fiber stretcher so that both temperature and strain changes are simultaneous extracted. We note that the performance is estimated using the leave-one-out cross validation method, as described in section 2.3, and hence the predictions are made on data that are not included in the training set. Every distributed measurement is illustrated with colored dots, as shown in the legend above the plots. The dashed and colored lines represent the set temperature and strain values, which facilitate the comparison between the set values and the GPR predictions. We observe that on average, the depicted GPR predictions along the stretched section of the fiber are close to the total strain and temperature errors (45 µɛ and 2 °C, respectively). Furthermore, the temperature predictions seem to fluctuate around the set values apart from the those corresponding to 40 °C, which tend to be mostly below the set value.
Fig. 6. Strain (left) and temperature (right) GPR predictions illustrated with colored dots. Every color corresponds to a single measurement along the optical fiber with the temperature and strain conditions shown in the legend above the plots.
Figure 7 provides a more compact overview of the temperature and strain predictions including data from all temperature conditions and strain levels applied to the examined optical fiber. Figure 7(a) and Fig. 7(b) show the mean temperature and strain predictions (orange dots) vs. the set temperature and strain values, respectively. We observe the GPR predictions follow very well the trend of the black dashed lines, which represent the set values. Besides the mean predictions, the prediction intervals are also illustrated (blue areas). These intervals correspond to the standard deviation of the predicted values and show the range of values in which most outcomes are expected to fall.
Fig. 7. GPR extracted mean temperature (a) and strain (b) vs. set values. The blue areas indicate the prediction intervals while the dashed black line represents the best possible outcome values.
We note that all the features (BFSs) used in this paper contribute to the system’s performance and none of them can be excluded from the training procedure without experiencing a significant increase in temperature and strain errors. Table 1 provides a comparison of the GPR performance in terms of mean temperature and strain errors when fewer peaks are used.
Table 1. GPR temperature and strain mean absolute errors when fewer peaks are used.
5. Discussion
In this paper, we applied simple machine learning algorithms to decouple temperature and strain effects in a standard telecom optical fiber. Linear least-squares regression assumes only linear relations between features and targets, and thus cannot capture nonlinearities in the data. Nevertheless, a temperature and strain error of 5 °C and 114 µɛ, respectively, was achieved. This model that corresponds to the equation solving method works as a baseline. Simple polynomial regression and ridge regression improved the performance by almost 30% and 50%, respectively. Ridge regression applied from a Bayesian viewpoint did not improve the performance further. However, no optimization of a penalty term was needed [52], which makes the Bayesian ridge regression more favourable than the frequentist-based algorithm. GPR, which is also based on Bayes theorem, reduced further the temperature and strain error by approximately 22% but at the cost of the required hyperparameter tuning procedure as described in the previous sections. This, however, is not time-consuming because the training lasts less than a second. Table 2 provides an overview of the algorithms’ performance.
Table 2. Performance comparison of the applied machine learning algorithms in terms of temperature and strain mean absolute error. Linear-least squares regression corresponds to the equation solving method used as a baseline.
We have shown that all the BFSs are necessary to achieve temperature and strain errors of 2 °C and 45 µɛ, respectively. Even though the measurement time is reduced when fewer peaks are recorded, this has a negative impact on the temperature and strain errors as shown in the Table 1. We note that the measurement time depends linearly on the total number of frequency sweep points. So, in the future we will try to reduce the number of frequency points by increasing the frequency sweep step. Furthermore, the measurement time could be potentially reduced by optimizing the number of averages and the bandwidth in the VNA.
We applied algorithms and followed a methodology for the models’ training but also for the performance evaluation which facilitates the transition from the laboratory to the field. All the algorithms are trained using only the BFSs of the multipeak BGS, without including spectral properties (like linewidth and gain) which cannot be reproduced easily. Moreover, the machine learning models are not that complex (like artificial neural networks) and their training is easy and fast. To ensure that the models generalize and interpolate well, the system’s performance is estimated using leave-one-out cross-validation.
We believe that the algorithms used in this paper can also be implemented in other Brillouin-based distributed fiber optic sensors (e.g., BOTDA and BOCDA) providing that those sensors are capable of recording a high SNR multipeak BGS similar to the one shown in Fig. 3(a). This results from the fact, that we made use of only the BFSs, which are the most conventional way to extract temperature and strain in all Brillouin sensor technologies. Therefore, we believe that our approach can have a significant impact on all the available Brillouin-based sensors.
In this paper, we made use of the SMF-28 Corning standard telecom optical fiber. However, we note that our methods can also be applied to other specialty fibers but the performance of the algorithms is expected to differ. The performance would depend, for example, on the number of the resolved peaks, the amplitude (SNR) of the peaks and their sensitivity on temperature and strain. In general, specialty fibers are more expensive than the standard fibers that we used in this paper. So, the use of standard fibers contributes towards a cost-effective solution for distributed simultaneous temperature and strain sensing and enables the potential use of laid-out fiber optic networks for monitoring of geotechnical structures and energy transport infrastructure over long time periods.
The methods reported in this paper can be combined with those reported in our previously published work [64] and contribute towards multiparameter distributed fiber optic sensing including simultaneous temperature, strain and humidity sensing. In [64], we demonstrated a BOFDA system for simultaneous temperature and humidity sensing using a polyimide (PI)-coated optical fiber. In this paper, we made use of a standard SMF-28 Corning optical fiber with acrylate coating. We are confident that in the future all the three measurands will be discriminated using the two-optical fiber configuration. In our previous paper [64], we separated temperature and humidity effects using linear regression, but we believe that the results could be improved using the proposed GPR. We note that GPR, predicts apart from the mean predictions, also the confidence intervals which make the algorithm very attractive for application in field of active learning.
6. Conclusions
We demonstrated a BOFDA for simultaneous distributed strain and temperature sensing using a single low-cost standard telecom optical fiber. This was achieved by developing a BOFDA setup, which provides a multipeak Brillouin gain spectrum with very high SNR and implementing simple machine learning algorithms. The best performance in terms of temperature and strain mean absolute error was achieved by using GPR and estimated to be 2 °C and 45 µɛ, respectively. The model was trained using only the BFSs, which are the most reliable and established parameters to monitor strain and temperature changes in all Brillouin-based distributed fiber optic sensors. Thus, our methodology is not restricted to BOFDA sensors, but can also contribute towards simultaneous temperature and strain sensing in standard fibers for all the available Brillouin-based sensors. Therefore, we are confident that our approach will find applications in the field of structural health monitoring (e.g., of geotechnical and energy transport infrastructures) where the discrimination of temperature and strain effects is essential.
Funding
PhD program of Bundesanstalt für Materialforschung und -Prüfung (BAM).
Acknowledgments
The authors would like thank Harald Kohlhoff, Jessica Erdmann and Frank Basedau for the conceptualization and manufacturing of the fiber optic stretcher. C.K. would also like to thank Christoph Völker for the fruitful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. C. Y. Hong, Y. F. Zhang, G. W. Li, M. X. Zhang, and Z. X. Liu, “Recent progress of using Brillouin distributed fiber optic sensors for geotechnical health monitoring,” Sens. Actuators, A 258, 131–145 (2017). [CrossRef]
2. A. Minardo, A. Coscetta, E. Catalano, R. Bernini, and L. Zeni, High spatial resolution physical and chemical sensing based on BOFDA, SPIE Optics + Optoelectronics (SPIE, 2019), Vol. 11028.
3. R. Bernini, A. Minardo, and L. Zeni, “Distributed Sensing at Centimeter-Scale Spatial Resolution by BOFDA: Measurements and Signal Processing,” IEEE Photonics J. 4(1), 48–56 (2012). [CrossRef]
4. T. Kapa, A. Schreier, and K. Krebber, “A 100-km BOFDA Assisted by First-Order Bi-Directional Raman Amplification,” Sensors 19(7), 1527 (2019). [CrossRef]
5. A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016). [CrossRef]
6. K. Hotate and T. Hasegawa, “Measurement of Brillouin Gain Spectrum Distribution along an Optical Fiber Using a Correlation-Based Technique—Proposal, Experiment and Simulation,” IEICE Trans Electron E83, 1 (2002). [CrossRef]
7. X. Bao, D. J. Webb, and D. A. Jackson, “Combined Distributed Temperature and Strain Sensor-Based on Brillouin Loss in an Optical-Fiber,” Opt. Lett. 19(2), 141–143 (1994). [CrossRef]
8. X. Zhang, S. Liu, J. Zhang, L. Qiao, T. Wang, S. Gao, and M. Zhang, “Simultaneous Strain and Temperature Measurement Based on Chaotic Brillouin Optical Correlation-Domain Analysis in Large-Effective-Area Fibers,” Photonic Sens. 11(4), 377–386 (2021). [CrossRef]
9. X. Liu and X. Bao, “Brillouin Spectrum in LEAF and Simultaneous Temperature and Strain Measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012). [CrossRef]
10. J. Q. Peng, Y. G. Lu, Z. L. Zhang, Z. N. Wu, and Y. Y. Zhang, “Distributed Temperature and Strain Measurement Based on Brillouin Gain Spectrum and Brillouin Beat Spectrum,” IEEE Photonics Technol. Lett. 33(21), 1217–1220 (2021). [CrossRef]
11. L. F. Zou, X. Y. Bao, V. S. Afshar, and L. Chen, “Dependence of the Brillouin frequency shift on strain and temperature in a photonic crystal fiber,” Opt. Lett. 29(13), 1485–1487 (2004). [CrossRef]
12. Z. Li, L. Yan, X. Zhang, and W. Pan, “Temperature and Strain Discrimination in BOTDA Fiber Sensor by Utilizing Dispersion Compensating Fiber,” IEEE Sens. J. 18(17), 7100–7105 (2018). [CrossRef]
13. A. Li, Y. F. Wang, J. Fang, M. J. Li, B. Y. Kim, and W. Shieh, “Few-mode fiber multi-parameter sensor with distributed temperature and strain discrimination,” Opt. Lett. 40(7), 1488–1491 (2015). [CrossRef]
14. W. W. Zou, Z. Y. He, and K. Z. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17(3), 1248–1255 (2009). [CrossRef]
15. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “Simultaneous temperature and strain measurement with combined spontaneous Raman and Brillouin scattering,” Opt. Lett. 30(11), 1276–1278 (2005). [CrossRef]
16. K. Kishida, Y. Yamauchi, and A. Guzik, “Study of Optical Fibers Strain-Temperature Sensitivities Using Hybrid Brillouin-Rayleigh System,” Photonic Sens. 4(1), 1–11 (2014). [CrossRef]
17. A. Coscetta, E. Catalano, E. Cerri, N. Cennamo, L. Zeni, and A. Minardo, “Hybrid Brillouin/Rayleigh sensor for multiparameter measurements in optical fibers,” Opt. Express 29(15), 24025–24031 (2021). [CrossRef]
18. H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695 (2000). [CrossRef]
19. B. W. Wang, L. Wang, N. Guo, Z. Y. Zhao, C. Y. Yu, and C. Lu, “Deep neural networks assisted BOTDA for simultaneous temperature and strain measurement with enhanced accuracy,” Opt. Express 27(3), 2530–2543 (2019). [CrossRef]
20. R. Ruiz-Lombera, A. Fuentes, L. Rodriguez-Cobo, J. M. Lopez-Higuera, and J. Mirapeix, “Simultaneous Temperature and Strain Discrimination in a Conventional BOTDA via Artificial Neural Networks,” J. Lightwave Technol. 36(11), 2114–2121 (2018). [CrossRef]
21. Y. Yao and Y. Mizuno, “Neural network-assisted signal processing in Brillouin optical correlation-domain sensing for potential high-speed implementation,” Opt. Express 29(22), 35474–35489 (2021). [CrossRef]
22. Z. Y. Cao, N. Guo, M. H. Li, K. L. Yu, and K. Q. Gao, “Back propagation neutral network based signal acquisition for Brillouin distributed optical fiber sensors,” Opt. Express 27(4), 4549–4561 (2019). [CrossRef]
23. Y. Q. Chang, H. Wu, C. Zhao, L. Shen, S. N. Fu, and M. Tang, “Distributed Brillouin frequency shift extraction via a convolutional neural network,” Photonics Res. 8(5), 690–697 (2020). [CrossRef]
24. Y. Liang, J. Jiang, Y. Chen, R. Zhu, C. Lu, and Z. Wang, “Optimized Feedforward Neural Network Training for Efficient Brillouin Frequency Shift Retrieval in Fiber,” IEEE Access 7, 68034–68042 (2019). [CrossRef]
25. C. Lu, Y. Liang, X. Jia, Y. Fu, J. Liang, and Z. Wang, “Artificial Neural Network for Accurate Retrieval of Fiber Brillouin Frequency Shift With Non-Local Effects,” IEEE Sens. J. 20(15), 8559–8569 (2020). [CrossRef]
26. A. Madaschi, J. Morosi, M. Brunero, and P. Boffi, “Enhanced Neural Network Implementation for Temperature Profile Extraction in Distributed Brillouin Scattering-Based Sensors,” IEEE Sens. J. 22(7), 6871–6878 (2022). [CrossRef]
27. N. D. Nordin, F. Abdullah, M. S. D. Zan, A. A. A Bakar, A. I. Krivosheev, F. L. Barkov, and Y. A. Konstantinov, “Improving Prediction Accuracy and Extraction Precision of Frequency Shift from Low-SNR Brillouin Gain Spectra in Distributed Structural Health Monitoring,” Sensors 22(7), 2677 (2022). [CrossRef]
28. H. Zheng, S. Sun, Y. Qin, F. Xiao, and C. Dai, “Extraction of Brillouin frequency shift from Brillouin gain spectrum in Brillouin distributed fiber sensors using K nearest neighbor algorithm,” Opt. Fiber Technol. 71, 102903 (2022). [CrossRef]
29. H. Zheng, F. Xiao, S. Sun, and Y. Qin, “Brillouin Frequency Shift Extraction Based on AdaBoost Algorithm,” Sensors 22(24), C1 (2022). [CrossRef]
30. A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016). [CrossRef]
31. B. Wang, N. Guo, L. Wang, C. Yu, and C. Lu, “Robust and Fast Temperature Extraction for Brillouin Optical Time-Domain Analyzer by Using Denoising Autoencoder-Based Deep Neural Networks,” IEEE Sens. J. 20(7), 3614–3620 (2020). [CrossRef]
32. M. H. Wang, Y. Sui, W. N. Zhou, X. An, and W. Dong, “AIoT enabled resampling filter for temperature extraction of the Brillouin gain spectrum,” Opt. Express 30(1), 1 (2022). [CrossRef]
33. Y. Zhang, L. Yu, Z. Hu, L. Cheng, H. Sui, H. Zhu, G. Li, B. Luo, X. Zou, and L. Yan, “Ultrafast and Accurate Temperature Extraction via Kernel Extreme Learning Machine for BOTDA Sensors,” J. Lightwave Technol. 39(5), 1537–1543 (2021). [CrossRef]
34. M. H. Wang, Y. Sui, W. N. Zhou, W. Dong, and X. D. Zhang, “Sweep frequency method with variance weight probability for temperature extraction of the Brillouin gain spectrum based on an artificial neural network,” Opt. Express 29(1), 1 (2021). [CrossRef]
35. N. D. Nordin, M. S. D. Zan, and F. Abdullah, “Generalized linear model for enhancing the temperature measurement performance in Brillouin optical time domain analysis fiber sensor,” Opt. Fiber Technol. 58, 102298 (2020). [CrossRef]
36. N. D. Nordin, M. S. D. Zan, and F. Abdullah, “Comparative Analysis on the Deployment of Machine Learning Algorithms in the Distributed Brillouin Optical Time Domain Analysis (BOTDA) Fiber Sensor,” Photonics 7(4), 79 (2020). [CrossRef]
37. H. Wu, L. Wang, N. Guo, C. Shu, and C. Lu, “Brillouin Optical Time-Domain Analyzer Assisted by Support Vector Machine for Ultrafast Temperature Extraction,” J. Lightwave Technol. 35(19), 4159–4167 (2017). [CrossRef]
38. H. Wu, Y. Wan, M. Tang, Y. Chen, C. Zhao, R. Liao, Y. Chang, S. Fu, P. P. Shum, and D. Liu, “Real-Time Denoising of Brillouin Optical Time Domain Analyzer With High Data Fidelity Using Convolutional Neural Networks,” J. Lightwave Technol. 37(11), 2648–2653 (2019). [CrossRef]
39. H. Zheng, Y. Yan, Y. Wang, X. Shen, and C. Lu, “Deep Learning Enhanced Long-Range Fast BOTDA for Vibration Measurement,” J. Lightwave Technol. 40(1), 262–268 (2022). [CrossRef]
40. Y.-N. Yang, Y. Dong, and K. Yu, “SNR Improvement based on Attention-DNet for Brillouin Distributed Optical Fiber Sensors,” in 2022 27th OptoElectronics and Communications Conference (OECC) and 2022 International Conference on Photonics in Switching and Computing (PSC), (2022), pp. 1–3.
41. B. Wang, N. Guo, L. Wang, C. Yu, and C. Lu, “Denoising and Robust Temperature Extraction for BOTDA Systems based on Denoising Autoencoder and DNN,” in 26th International Conference on Optical Fiber Sensors, (2018). [CrossRef]
42. C. Karapanagiotis, A. Wosniok, K. Hicke, and K. Krebber, “Time-Efficient Convolutional Neural Network-Assisted Brillouin Optical Frequency Domain Analysis,” Sensors 21(8), 2724 (2021). [CrossRef]
43. C. Karapanagiotis, “Evaluation of the generalization performance of a CNN-assisted BOFDA system,” in Sensors and Measuring Systems; 21th ITG/GMA-Symposium, (2022), 1–4.
44. A. Wosniok, “Distributed Brillouin Sensing: Frequency-Domain Techniques,” in Handbook of Optical Fibers, G. D. Peng, ed. (Springer Nature, Singapore, 2018), pp. 1–25.
45. M. Newville, T. Stensitzki, D. B. Allen, M. Rawlik, A. Ingargiola, and A. Nelson, “Lmfit: Non-Linear Least-Square Minimization and Curve-Fitting for Python,” Astrophysics Source Code Library (2016).
46. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]
47. “BOTDR Measurement Techniques and Brillouin Backscatter Characteristics of Corning Single-Mode Optical Fibers” White paper, Corning Incorporated WP4259, 1–9 (2015).
48. F. Bastianini, R. Di Sante, F. Falcetelli, D. Marini, and G. Bolognini, “Optical Fiber Sensing Cables for Brillouin-Based Distributed Measurements,” Sensors 19(23), 5172 (2019). [CrossRef]
49. X. Bao, S. Xie, X. Liu, and L. Chen, “The non-uniformity and dispersion in SBS-based fiber sensors,” Proc. SPIE 8351, 83512Y (2012). [CrossRef]
50. W. Zou, Z. He, and K. Hotate, “Two-Dimensional Finite-Element Modal Analysis of Brillouin Gain Spectra in Optical Fibers,” IEEE Photonics Technol. Lett. 18(23), 2487–2489 (2006). [CrossRef]
51. F. Gyger, Z. Yang, M. A. Soto, F. Yang, K. H. Tow, and L. Thévenaz, “High Signal-to-Noise Ratio Stimulated Brillouin Scattering Gain Spectrum Measurement,” in 26th International Conference on Optical Fiber Sensors, (2018).
52. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and É. Duchesnay, “Scikit-learn: Machine Learning in Python,” J. Mach. Learn. Res. 12, 2825–2830 (2011).
53. J. Fox, Applied Regression Analysis and Generalized Linear Models, (Sage Publications Inc, 2008).
54. K. P. Murphy, Machine learning : a probabilistic perspective, (MIT Press, 2012).
55. X. Ying, “An Overview of Overfitting and its Solutions,” J Phys Conf Ser1168(2), 022022 (2019). [CrossRef]
56. D. McNeish, “On Using Bayesian Methods to Address Small Sample Problems,” Structural Equation Modeling A Multidisciplinary Journal 23(5), 750–773 (2016). [CrossRef]
57. M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” Appl. Phys. Lett. 1, 211–244 (2001). [CrossRef]
58. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2005).
59. C. M. Bishop, Pattern recognition and machine learning (Springer-Verlag New York, Inc., 2006).
60. Y. L. Zhang and Y. H. Yang, “Cross-validation for selecting a model selection procedure,” J Econometrics 187(1), 95–112 (2015). [CrossRef]
61. M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997). [CrossRef]
62. D. Garus, K. Krebber, F. Schliep, and T. Gogolla, “Distributed sensing technique based on Brillouin optical-fiber frequency-domain analysis,” Opt. Lett. 21(17), 1402–1404 (1996). [CrossRef]
63. A. Zadok, E. Zilka, A. Eyal, L. Thévenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express 16(26), 21692–21707 (2008). [CrossRef]
64. C. Karapanagiotis, K. Hicke, A. Wosniok, and K. Krebber, “Distributed humidity fiber-optic sensor based on BOFDA using a simple machine learning approach,” Opt. Express 30(8), 12484–12494 (2022). [CrossRef]
