Dynamic strain measurement in Brillouin optical correlation-domain sensing facilitated by dimensionality reduction and support vector machine

Yuguo Yao; Yuguo Yao; Yosuke Mizuno; Yosuke Mizuno

doi:10.1364/OE.455454

1. Introduction

Brillouin scattering in optical fibers is well known for its utilization in fully-distributed sensing of temperature and strain [1–10]. The Brillouin optical correlation-domain techniques measure the distribution of Brillouin gain spectrum (BGS) by manipulating the periodical coherence of the continuous lightwaves [11–13]. The principle allows the correlation-domain sensing to mitigate the limitation of phonon lifetime (∼10 ns) and achieve a relatively high spatial resolution compared with the peer techniques. Besides, the correlation-domain sensing techniques have the merits of random accessibility, which enables the successive measurement of one single random position at a high repetition rate [14–17].

The acceleration of the measurement speed in the distributed sensing is essential to the real-time monitoring of fast varying events along the optical fiber. To increase the measurement speed, the straightforward idea is abandoning the use of the lock-in amplifier (LIA) that restricts the acquisition speed of BGSs and using a voltage-controlled oscillator (VCO) that permits the pump-probe frequency offset to be scanned at a high speed [14,15]. The idea simply accelerates the acquisition speed of a single BGS, while the signal quality issues that come along with the speed-up need to be considered. For instance, the BGSs acquired without the LIA may exhibit a low signal-to-noise ratio (SNR), especially when the distributed sensing system works at a high spatial resolution and the weak real BGS signal can easily be buried by the random noise or the background noise. Increasing the averaging time may reduce the influence of random noise but also slow down the effective BGS acquisition speed.

Additionally, the Brillouin frequency shift (BFS) that truly reflects the distribution of strain or temperature under test needs to be extracted from the measured BGSs [18–28]. The post-processing can be implemented separately from the acquisition of the BGSs. However, when BFS extraction time is comparable to or even longer than the BGS acquisition time, the BFS extraction time cannot be neglected. Therefore, fast BFS extraction is essential to the real-time operation in Brillouin optical correlation-domain sensing.

Machine learning methods have been proposed to implement the BFS extraction in Brillouin correlation-domain sensing. Artificial neural networks (ANN) based on classification have been utilized to realize fast BFS extraction [27]. By treating each BGS as an n-dimensional vector and discretizing the BFS into a series of evenly spaced values, three-layer NNs have been trained with simulated signals and adopted to extract the BFS with high accuracy. Compared with the conventional signal processing based on Lorentzian curve fitting (LCF), the post-processing speed is increased by at least 1000 times and is promising to be used in online real-time distributed sensing.

The best BFS extraction time for each BGS achieved in Ref. [27] is 20 µs, which means that the BGS acquisition rate cannot exceed 50 kHz by using general ANN as the signal processing method under the assumption that the BFS extraction time cannot be longer than the BGS acquisition period. When a higher BGS acquisition rate is expected under the condition that real-time property is required, the BFS extraction time needs to be further compressed by employing faster machine learning methods, if common software and hardware platforms such as MATLAB and central processing unit (CPU) are used to implement the algorithms. For example, to realize a 100 kHz BGS acquisition rate, the BFS extraction time should be reduced to 10 µs or less.

Also, the three-layer NNs designed in Ref. [27] are inefficient. Training one signal NN with all the simulated data costs over one hour. In most situations, the well-trained NNs are not self-adapting and cannot be reused once the point number in one single BGS, the SNR, the dynamic range, or any other condition is changed. To further accelerate both the training time and BFS prediction time of the machine learning models, we try to refine our algorithms or explore new methods.

The BGS acquired in Brillouin optical correlation-domain sensing consists of the background and the real Brillouin spectrum [1,11,12]. Unlike the image processing-related machine learning applications, the BGS shape is relatively simple and intuitively, modeling one BGS with a 10000-dimensional vector is reluctant. It is reasonable to believe the BGS can be modeled with fewer features. Thus, reducing the feature space dimension is expected to significantly reduce the training time and BFS extraction time.

In this paper, we implement dimensionality reduction on BGS datasets using principal component analysis (PCA), which is a method commonly used to compress the data dimension as a pre-processing in many machine learning algorithms. Then we train fast BFS extraction models with support vector machines (SVMs) based on classification and regression. In the experiment, real-time dynamic strain sensing is demonstrated using the trained machine learning models.

2. Principle and simulation

In this section, the principle of PCA and SVMs based on classification and regression will be clarified and implemented to simulate BGS signals. The performances of different machine learning models will be compared.

2.1 Principle of PCA and SVMs in Brillouin optical correlation-domain sensing

Figure 1 shows the principle of BFS extraction in Brillouin optical correlation-domain sensing using SVMs, after the dimensionality reduction implemented by PCA [29,30]. The training dataset includes a number of BGSs labeled with different BFSs of k classes. Each BGS consists of n sampling points and can be modeled as a vector in n-dimensional linear space. In the paper, the training datasets are created by simulation. To reduce the dimensionality of the linear space from n to m, all the BGSs are firstly assembled into one matrix

(1)$$X = {\left[ {\begin{array}{c} { - \;(BG{S_1})\; - }\\ { - \;(BG{S_2})\; - }\\ { - \;(BG{S_3})\; - }\\ \vdots \\ { - \;(BG{S_p})\; - } \end{array}} \right]_{p \times n}}, $$

where p is the number of the BGSs in the training dataset, and each BGS occupies one row of the matrix X. The dataset is then centralized by subtracting the mean of all the BGSs as

(2)$${X_{center}} = X - \frac{1}{p}\sum\limits_{i = 1}^p {(BG{S_i})} = X - \overline {BG{S_i}}. $$

The eigenvalues and the corresponding eigenvectors of the covariance matrix $X_{center}^T{X_{center}}$ can be found by

(3)$$X_{center}^T{X_{center}}{{\mathbf u}_i} = \sigma _i^2{{\mathbf u}_i}, $$

where $\sigma _i^2$ is the i-th eigenvalue and ${{\mathbf u}_i}$ is the i-th eigenvector. The eigenvalues are in descending order, or $\sigma _1^2 > \sigma _2^2 > \sigma _3^2 > \ldots > \sigma _n^2$.

Fig. 1. Schematic of BFS extraction in Brillouin optical correlation-domain sensing with (a) support vector machine classification (SVC) and (b) support vector machine regression (SVR), after dimensionality reduction implemented by principal component analysis (PCA).

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The first m eigenvectors, namely the orthonormal principal components that describe the m largest variances, are combined into the projection matrix as

(4)$${U_{proj}} = {\left[ {\begin{array}{ccccc} |&|&|&{}&|\\ {{{\mathbf u}_1}}&{{{\mathbf u}_2}}&{{{\mathbf u}_3}}& \ldots &{{{\mathbf u}_m}}\\ |&|&|&{}&|\end{array}} \right]_{n \times m}}. $$

The BGS dataset is then projected to m-dimensional space by multiplying the projection matrix as

(5)$${X_m} = {X_{center}} \cdot {U_{proj}} = {\left[ {\begin{array}{c} { - \;{{\mathbf x}_1}\; - }\\ { - \;{{\mathbf x}_2}\; - }\\ { - \;{{\mathbf x}_3}\; - }\\ \vdots \\ { - \;{{\mathbf x}_p}\; - } \end{array}} \right]_{p \times m}}. $$

The projected dimensionality m is often selected to be much lower than the original dimensionality n to effectively reduce the dataset size. The ratio of the variance explained by the first m principal components to the total variance in the dataset can be represented by

(6)$$r = \frac{{\sum\limits_{i = 1}^m {\sigma _i^2} }}{{\sum\limits_{i = 1}^n {\sigma _i^2} }}. $$

The value of r is expected to be close to 1 with a small number of principal components.

The n-dimensional BGS dataset can be recovered by multiplying the projected m-dimensional data with the transpose of the projection matrix and adding the mean of all the BGSs as

(7)$${X_{re}} = {X_m} \cdot U_{proj}^T + \overline {BG{S_i}} = {\left[ {\begin{array}{c} { - \;(BG{S_1})^{\prime}\; - }\\ { - \;(BG{S_2})^{\prime}\; - }\\ { - \;(BG{S_3})^{\prime}\; - }\\ \vdots \\ { - \;(BG{S_p})^{\prime}\; - } \end{array}} \right]_{p \times n}}, $$

although some details of the original dataset may be lost during the PCA processing.

After the dimensionality reduction by PCA, the large-scale BGS dataset can be trained with various machine learning algorithms effectively using an ordinary computer effectively. Figure 1 shows two methods based on SVM. Figure 1(a) illustrates the basic principle of multi-class support vector machine classification (SVC), and Fig. 1(b) clarifies the principle of support vector machine regression (SVR). In this paper, the two algorithms will be testified and compared using both simulated and experimental datasets.

The SVC algorithm finds a hyperplane separator for every two distant BFS classes. The equation of the hyperplane can be represented by

(8)$${{\mathbf w}^T}{\mathbf x} + b = 0, $$

where w and b are the weights and the bias to be optimized by the algorithm. The projection into the m-dimensional space for each BGS in the two BFS classes needs to satisfy the constraint

(9)$$\left\{ {\begin{array}{ll} {{{\mathbf w}^T}{{\mathbf x}_i} + b \ge + 1,\quad {y_i} ={+} 1}\\ {{{\mathbf w}^T}{{\mathbf x}_i} + b \le - 1,\quad {y_i} ={-} 1} \end{array}} \right., $$

where y_i = +1 and -1 are the labels for the two different BGS classes. The equality sign stands when a BGS sample is a support vector determining the separator. All the support vectors in Fig. 1 are marked with golden backgrounds. The distances from the support vectors belonging to two different classes to the hyperplane need to be maximized under the condition

(10)$$\begin{aligned} &\mathop {\max }\limits_{{\mathbf w},b} \frac{2}{{||{\mathbf w} ||}}\\ &s.t.\quad {y_i}({{\mathbf w}^T}{{\mathbf x}_i} + b) \ge 1 \end{aligned}, $$

using the SVC algorithm. If there are k different BFS classes, the algorithm will need to find a separator for every two classes and the total number of hyperplanes will be k(k-1)/2. After the SVC model is well trained, each BGS sample contained in the test dataset without a BFS label is classified into one of k BFS classes, after being projected into the m-dimensional space constituted by the same principal components of the training dataset.

In the SVR algorithm, the BFS labels for the BGSs are treated as continuous values. The algorithm finds a linear equation to ensure all the BGS samples in m-dimensional space are as close to it as possible. The linear equation can be described by

(11)$$f(x) = {{\mathbf w}^T}{\mathbf x} + b, $$

where w and b are the parameters to be trained. The optimization problem can be described as

(12)$$\begin{aligned}& \mathop {\min }\limits_{{\mathbf w},b} \frac{1}{2}{||{\mathbf w} ||^2}\\ &s.t.\quad |{f({{\mathbf x}_i}) - BF{S_i}} |\le \varepsilon \end{aligned}, $$

in which the constraint condition allows a residual error less than ε for each BGS. To allow some of the BGS samples to fall outside the ±ε region, regularization is introduced into the optimization function as

(13)$$\mathop {\min }\limits_{{\mathbf w},b} \frac{1}{2}{||{\mathbf w} ||^2} + C\sum\limits_i {{l_\varepsilon }(f({{\mathbf x}_i}) - BF{S_i})}, $$

in which C is the regularization constant. The ε-insensitive loss function l_ε can be represented by

(14)$${l_\varepsilon }(z) = \left\{ {\begin{array}{ll} {0},& |z |\le \varepsilon \\ {|z |- \varepsilon} ,&|z |> \varepsilon \end{array}} \right., $$

which ignores the residual error within ±ε and counts the loss by measuring the distance between the BFS labels and the ±ε boundaries for the BGS samples outside the ±ε region. The BGS samples that fall outside the ±ε region are treated as support vectors and marked with golden backgrounds in Fig. 1(b). When the linear model described in Eq. (11) is well trained, the BFSs for the BGSs in the test datasets are then predicted as continuous quantities using the SVR model.

2.2 Visualizing BGS datasets in 2D and 3D

In this section, we implement dimensionality reduction to the BGS datasets by PCA and have a brief impression of the distribution of BGSs with different BFSs in 2D and 3D linear space. The simulation method used to generate the BGS datasets is similar to that in Ref. [27]; the laser frequency is assumed to be sinusoidally modulated. The parameters for one simulated BGS that need to be adjusted include the BFS, the full width at half maximum (FWHM), the SNR, and the strained section length in the middle of the optical fiber under test, while the modulation frequency f_m and the modulation depth Δf are fixed to 1 MHz and 4 GHz, and the BGS amplitudes are normalized.

To visualize each BGS sample in 2D and 3D clearly, a limited number of BGSs are created for one dataset. The simulation is performed by setting the BFS to the range from 0 MHz to 90 MHz with 10 MHz incremental step (10 classes). The FWHM is set to the range from 30 MHz to 39 MHz with 1 MHz step (10 cases). The strained section length is set to the range from 0.5 Δz to 1.0 Δz with 0.1 Δz step, where Δz is the nominal spatial resolution of Brillouin optical correlation-domain sensing that can be represented by [11,12]

(15)$$\Delta z = \frac{{{v_g}\Delta {\nu _B}}}{{2\pi {f_m}\Delta f}}, $$

where v_g is the group velocity of light in optical fiber, and Δν_B is the FWHM. Therefore, the total number of BGSs in one simulated dataset is 10×10×6 = 600. Four BGS datasets with the same structure are created by adding random noise of different amplitudes. The SNRs for the datasets are equal to ∞, 20 dB, 15 dB, and 10 dB.

The BGS datasets are firstly projected to 2-dimensional space by PCA using Eqs. (1)–(5), with BGSs labeled with different BFSs donated by dots with different colors, as shown in Fig. 2. In Fig. 2(a), it is clearly observed that each BFS class contains 60 BGS samples, except that the BFS = 0 MHz class (red dots) has only 10 samples, as the strained section length is invalid when the BFS is zero. We then take a close look at the BFS = 90 MHz class (pink dots) to analyze the distribution of the BGS samples in this class as an example. Six original BGS samples are shown in the inset of Fig. 2(a) for comparison, by setting BFS = 90 MHz, FWHM = 30 MHz, and the strained section length = 0.5 Δz, 0.6 Δz, 0.7 Δz, 0.8 Δz, 0.9 Δz, and 1.0 Δz. The 60 pink dots assemble as 6 clusters, and each cluster represents one case of strained section length, as the marks indicate in Fig. 2(a). In each cluster, there are 10 BGS samples, which stand for 10 cases of different FWHMs. The distances between the dots in one cluster are closer than the distances between different clusters, which means that the strained section length influences more on the BGS shapes than the FWHM in this simulation. It is also found that the samples in the 0.5 Δz cluster are more concentrated than in other clusters. The reason may be that the BGS with strained section length = 0.5 Δz is the only one in the BFS = 90 MHz class, whose real Brillouin signal power is lower than the background noise and the FWHM change is relatively insensitive in this case, as seen in the inset of Fig. 2(a). Figure 2(b), (c), and (d) show that, when the SNR gets lower, the dots in each cluster deviate from their original positions. The BGS samples belonging to different BFS classes start to merge and it becomes harder to identify clear boundaries.

Fig. 2. Simulated Brillouin gain spectra (BGSs) compressed by PCA to 2-dimensional space for visualization: (a) SNR = ∞, (b) SNR = 20 dB, (c) SNR = 15 dB, and (d) SNR = 10 dB. The simulated BGSs have FWHM = 30–39 MHz (1 MHz step), BFS = 0–90 MHz (10 MHz step), and strained section lengths = 0.5–1.0 times the nominal spatial resolution (0.1 time step). Inset in (a): original BGSs with FWHM = 30 MHz, BFS = 90 MHz, and strained section length = 0.5–1.0 times nominal spatial resolution (0.1 time step).

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To classify different BFS classes easily and visualize the BGS datasets in 3-dimensional space, the four BGS datasets with different SNRs are projected to 3D space by setting m = 3 in the PCA algorithm. The results are shown in Fig. 3. Compared with the projection in 2D space, the dots belonging to different BFS classes are separable, although the boundaries are still unclear when the SNR is low. Projecting the BGS datasets to higher dimensionality (m > 3) can further increase the separability, although at the cost of longer machine learning model training and prediction time. The datasets in 4 or higher dimensional space are also hard to be visualized by static figures.

Fig. 3. Simulated Brillouin gain spectra (BGSs) compressed by PCA to 3-dimensional space for visualization: (a) SNR = ∞, (b) SNR = 20 dB, (c) SNR = 15 dB, and (d) SNR = 10 dB. The simulated BGSs have the same parameters with those in Fig. 2.

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2.3 Determining the dimensionality for model training

The PCA is used to reduce the dimensionality of BGS datasets for faster training of machine learning models. However, details in the BGS samples are inevitably lost during the compression. It is essential to determine the dimensionality m for a particular dataset to a value that is as small as possible but still involves most of the information in the original dataset.

The BGS datasets used in this section are created by setting the BFS to the range from 0 MHz to 395 MHz with 5 MHz step (80 classes), the FWHM to the range from 30 MHz to 49 MHz with 1 MHz step (20 cases), and the strained section length to the range from 0.5 Δz to 1.4 Δz with 0.1 Δz step (10 cases). The overall number of BGSs in one dataset is 80×20×10 = 16,000. The SNR is also simulated to be ∞, 20 dB, 15 dB, and 10 dB for four different BGS datasets, as we did in Section 2.2.

The variances explained by each of the first 20 principal components, namely $\sigma _i^2$ (i = 1, 2, 3, …, 20) in Eq. (3), are presented in Fig. 4 (left y-axis, blue). The cumulative percentages in total variance explained by the first m principal components indicated by Eq. (6) are also illustrated in Fig. 4 (right y-axis, red). In Fig. 4(a), the first several principal components contribute to the most significant variances, which implies that the BGS dataset can be expressed using a limited number of dimensionality. Here, we assume that the BGS dataset can be fully described when the cumulative percentage reaches 99%, ignoring the trivial details lost during the compression. For the SNR = ∞ case, the critical number of m is 13. In other words, the BGS dataset can be represented using 13-dimensionality instead of the original 10,000- dimensionality. In Figs. 4 (b)–(d), the cumulative percentage curves ascend at a slower speed due to the noise. In the extreme case shown in Fig. 4(d), the first 20 principal components explain only around 60% of the total variance while the SNR is 10 dB. The random noise comprised of irregular variance leads to this phenomenon, although we know that the ratio curve will gradually increase to 100% with all the principal components are included. In the low SNR cases, we still select the dimensionality m = 13, which is equal to the number in the SNR = ∞ case, as the rest 40% of the total variance presents random noise that has no contribution to the BFS prediction.

Fig. 4. Variance explained by each of first 20 principal components (left y-axis) and cumulative percentage in total variance explained by first 20 principal components (right y-axis), for simulated BGS datasets processed by PCA: (a) SNR = ∞, (b) SNR = 20 dB, (c) SNR = 15 dB, and (d) SNR = 10 dB. The simulated BGS datasets have FWHM = 30–49 MHz (1 MHz step), BFS = 0–395 MHz (5 MHz step), and strained section lengths = 0.5–1.4 times the nominal spatial resolution (0.1 time step).

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To understand the effect of PCA on the BGS datasets better, BGS samples are recovered using Eq. (7) and compared with the original BGSs. Figure 5 illustrates four examples in the original BGS dataset (BFS = 0, 100, 200, and 300 MHz, FWHM = 30 MHz, strained section length = 1.0 Δz) and the corresponding BGSs recovered by 1, 3, 5, and 13 principal components under different SNRs. With the increase of the number of principal components used in the recovery, the recovered BGSs gradually approach the shape of the original ones. When m = 13, the recovered BGSs are approximately the same as the original ones while SNR = ∞. In the low SNR cases, the evolution of the recovered BGS follows the same pattern, although the noisy details cannot be reproduced.

Fig. 5. Original BGS and BGSs recovered by 1, 3, 5, and 13 principal components (PCs): (a1) – (a4) BFS = 0 MHz, (b1) – (b4) BFS = 100 MHz, (c1) – (c4) BFS= 200 MHz, and (d1) – (d4) BFS = 300 MHz. SNR = ∞, 20 dB, 15 dB, and (d) 10 dB. The BGSs has FWHM = 30 MHz, and strained section lengths = 1.0 times the nominal spatial resolution.

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It is notable that the selection of m we described above only applies to this particular BGS dataset. The selection may be different when the simulation conditions are adjusted.

To compare different datasets, we create two additional BGS datasets. One has the BFS ranging from 0 MHz to 79 MHz with 1 MHz step (80 classes), and the other has the BFS ranging from 0 MHz to 15.8 MHz with 0.2 MHz step (80 classes), while all the other parameters remain unchanged.

By using the same analysis method, it found that the critical number m is 6 and 4 for 0–79 MHz case and 0–15.8 MHz case, respectively. The results are reasonable as, when the BFS dynamic range is smaller, the variance of all the BGSs in one dataset also decreases, and fewer principal components are required to describe the whole dataset.

2.4 Training datasets and BFS extraction using SVC/ SVR

In this part, SVC and SVR models with six different kernel functions are implemented onto the three BGS datasets prepared in Section 2.3. The number of principal components used in the model training is 13, 6, and 4 for the three BGS datasets, as we discussed in Section 2.3. The performances of classification and regression methods with different kernel functions will be compared.

The six kernel functions used are linear, quadratic, cubic kernels, and Gaussian kernels with three distinct scaling factors [29]. With different kernels, the hyperplane separators of SVC and the regression function of SVR may exhibit sophisticated nonlinear models that fit the specific datasets better which cannot be described by a simple linear model, although the results are hard to predict before the models with different kernels are implemented.

Table 1 and Table 2 list the performance of BFS extraction by SVC and SVR with different kernel functions. Each model is implemented on three BGS datasets with SNR = ∞, 20 dB, 15 dB, and 10 dB. In Table 1, the four values in each box indicate (a) classification accuracy; (b) average BFS prediction time (the time cost of PCA is estimated to be 10⁻⁴ ms and negligible compared with that of SVM); (c) standard deviation (SD); (d) maximum error (the maximum difference between the BFS label and the predicted BFS should not be too large even when a model performs well on classification accuracy and SD). The average training times of each method for the three datasets are also listed in the table. The performance achieved by the best method for each dataset is marked in blue color. The abnormally large errors are marked with red, which means the corresponding method is inappropriate for the dataset. It is found that the choice of the best method depends on the specific dataset, and the SNR influences the performance significantly. The training of an SVC model usually costs several hundred seconds, and the prediction of a new BFS often requires several milliseconds.

Table 1. Performance of BFS extraction by SVC with different kernel functions^a

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Table 2. Performance of BFS extraction by SVR with different kernel functions^a

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In Table 2, the three values in each box indicate (a) SD; (b) average prediction time; (c) maximum error. The best performance for each dataset is marked with blue and the abnormal errors are marked with red. The time for training an SVR model is often short than training an SVC model, and the training time of some SVR models with Gaussian kernel may be down to several seconds, which is useful when the sensing system is highly flexible and many parameters need to be finely adjusted. The average prediction time using the SVR models is also clearly shorter than that using the SVC models. The fastest prediction time needs only several microseconds.

When considering the training and prediction speed, the regression-based methods are the better choices, as the classification-based methods require the calculation of weights involving many separators for multi-class problems. However, when high accuracy is required in the measurement, classification-based methods are the preferable choices. The SD and the maximum error obtained with SVC are lower than those obtained with SVR in most cases, especially for the large dynamic range cases (BFS = 0–395 MHz). The reason is believed to be that the BGS datasets with a larger dynamic range include more variation even when projected to lower dimensionality and that it is harder for the SVR methods to correctly fit a dataset that is not so “flat”.

3. Experiment

In this section, we experimentally verified our methods by performing distributed temperature measurement and dynamic strain measurement based on high-speed BOCDR with a high BGS sampling rate. Figure 6 shows the time sequence of BGS acquisition and BFS extraction for each BGS required by real-time measurement in high-speed BOCDR. To realize real-time measurement, the BFS extraction time for each BGS T_e needs to be shorter than the BGS acquisition period T_a. In other words, the post-processing for one BGS starts instantly when the corresponding BGS is completely acquired and finishes before the next BGS acquisition period ends. When the conventional method such as LCF is adopted, the BFS extraction often cannot be fast enough to satisfy the high repetition rate, and the post-signal processing is done off-line after all the necessary BGS data are acquired. Therefore, machine learning methods are proposed to accelerate the BFS extraction and motivate truly real-time successive monitoring at a high repetition rate.

Fig. 6. Time sequence of acquisition and BFS extraction for each BGS in real-time measurement. T_a is the period of BGS acquisition, and T_e is the BFS extraction time for each BGS.

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The first experiment was demonstrated with a polymer optical fiber (POF)-based high-speed BOCDR. The experimental setup was the same as the configuration in Ref. [8] and could realize a 42 Hz BGS repetition rate with no averaging. A 3.0 m perfluorinated graded-index POF was used as the sensing fiber [9], and a 1.0 m section in the middle of the fiber was heated to 55 °C, while the ambient temperature was 25 °C. The 30 °C temperature deviation corresponds to a -96 MHz BFS change in the POF. The modulation frequency f_m was scanned from 13.525 MHz to 13.590 MHz, and the frequency modulation depth Δf was set to ∼0.4 GHz, corresponding to a nominal spatial resolution of ∼0.65 m and a measurement range of ∼8.2 m.

Each original acquired BGS consists of 1371 sampling points and can be treated as 1371-dimensional vectors. The dimensionality can be reduced to 4 by implementing PCA before classification or regression-based algorithms. The SVC and SVR models are trained by datasets with BFS range = 2720–2880 MHz and BFS step = 5 MHz and 1 MHz, respectively. The BFS extracted by SVC and SVR without and with PCA for the distributed measurement is shown in Fig. 7. All four cases show correct results for the temperature distribution, although with slight differences. The BFS extracted by SVC for each BGS can only be one of the discrete values at 5 MHz or 1 MHz interval, while the BFS extracted by SVR can be any continuous value as expected. In this measurement, the difference in the BFS step does not have a significant impact on the results, and there is no need to further use a finer step below 1 MHz. The implementation of PCA has almost no influence on the SVC results but increases the maximum BFS fluctuation of the SVR results by approximately 5 MHz. However, the accuracy deterioration induced by the dimensionality reduction is acceptable considering the signal processing acceleration brought by PCA.

Fig. 7. The BFS extracted by (a) SVC without PCA, (b) SVR without PCA, (c) SVC with PCA, and (d) SVR with PCA for distributed measurement with a 42 Hz repetition rate. Blue curve: BFS step = 5 MHz, red curve: BFS step = 1 MHz.

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The average BFS extraction time for each method is also indicated in Fig. 7. The extraction time needed by SVC is longer than the time needed by SVR, as the multi-class classification algorithm is more computationally complex than the regression algorithm. The SVC with BFS step = 1 MHz is more time-consuming when compared with the BFS = 5 MHz case, as more classes need to be identified when a finer step is used. For SVR, the BFS step does not influence the extraction time notably, because the regression-based method optimizes the single target function regardless of the BFS label intervals.

The PCA reduces the dimensionality of each BGS from 1371 to 4 and shows remarkable improvements in the BFS extraction time for both SVC and SVR. By implementing PCA, the extraction speed is accelerated by 5.5–8.3 times for SVC, and 20.8–24.4 times for SVR. The BGS acquisition period T_a is 1/42 = 0.0238 s in this measurement, so both SVC and SVR can satisfy the requirement that BFS extraction time T_e should be shorter than T_a, even without PCA. However, if the BGS sampling rate is further increased, the situation will be different.

The POF-based system was then used to demonstrate the dynamic strain measurement. Instead of heating, a ±0.3% sinusoidal dynamic strain at 2 Hz was applied to the 1.0 m section of the POF. The correlation peak was fixed at the midpoint of the fiber by setting f_m to 13.552 MHz with the other setups remaining the same.

The SVC and SVR models are trained by datasets with BFS range = 2780–2860 MHz and BFS step = 5 MHz and 1 MHz, and the PCA also compresses the BGS dimensionality from 1371 to 4. The BFS extracted by SVC and SVR without and with PCA for the dynamic strain measurement is shown in Fig. 8. All four methods show correct curves for the 2 Hz dynamic strain. The analysis regarding the accuracy and the extraction time of SVC and SVR with or without PCA is similar to that of the distributed temperature measurement. The PCA accelerates the BFS extraction speed by 9.5–9.8 times for SVC and 36.8–37.1 times for SVR.

Fig. 8. The BFS extracted by (a) SVC without PCA, (b) SVR without PCA, (c) SVC with PCA, and (d) SVR with PCA for a 2 Hz dynamic strain measurement with a 42 Hz repetition rate. Blue curve: BFS step = 5 MHz, red curve: BFS step = 1 MHz.

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In the second experiment, dynamic strain measurement was demonstrated with a higher repetition rate. The setup was basically the same as the previously developed configuration (see Fig. 1 in Ref. [17]), except that the microwave generator was replaced with a VCO, which led to a BGS sampling rate of 8 kHz. A laser diode (power: 6 dBm, wavelength: 1550 nm, linewidth: ∼1 MHz) was used as a light source, and its output was amplified to 23 dBm and injected into a fiber under test, which was a 3-m-long polarization-maintaining fiber [31] (BFS: 10.79 GHz; employed to mitigate the effect of polarization fluctuations caused by the vibration). A dynamic strain of ±0.25% sinusoidally vibrating at 40 Hz was applied to a static strain of 0.45% along a 64-cm-long segment in the sensing fiber. By setting the modulation frequency to 429.3 kHz, the correlation peak (4th order) was generated in the midpoint of the strained section. The modulation depth was 4.9 GHz, corresponding to a nominal spatial resolution of ∼50 cm.

Figure 9 is the time-varying BGSs with an 8 kHz repetition rate. The corresponding BGS acquisition period T_a is 1/8 kHz = 0.125 ms. As the frequency of the applied dynamic sinusoidal strain is 40 Hz, each period of sinusoidal strain consists of 200 BGSs. The peak-peak value of BFS induced by the dynamic strain is around 200 MHz. To extract the BFS using SVC and SVR, the training datasets are created with BFS = 10860–11050 MHz (step = 5 MHz, 40 classes) and SNR = 10 dB. Each BGS has 625 sampling points and is treated as a vector with 625 features.

Fig. 9. Time-varying BGSs with 8 kHz repetition rate. The frequency of applied dynamic strain is 40 Hz.

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The BFS extracted by SVC and SVR without and with PCA applied is shown in Fig. 10. The resultant BFS fluctuates significantly due to the low SNR arising from the fast frequency scanning speed. To compare the effect of different methods, the actual extracted BFS and the ideal sinusoidal BFS are indicated by dots and solid lines respectively, and root mean square errors (RMSEs) are calculated to evaluate the accuracy of the machine learning methods. In Fig. 10(a), the BFS extracted by the classification-based method for each BGS belongs to one of the 40 discrete given BFS values; while in Fig. 10(b), the extracted BFS extracted by the regression-based method can be any analog value, and some BFS may even fall beyond the preset dynamic range. The RMSEs for the two methods are 26.6 MHz and 29.5 MHz, respectively. The average BFS extraction time is 4.369 ms and 0.2842 ms. Although the RMSE of the results given by SVR is a bit larger than that given by SVM, the BFS extraction time of SVR is over 15 times faster than that of SVM. However, a BFS extraction time of 0.2842 ms is still longer than the BGS acquisition period of 0.125 ms, and both machine learning methods are not efficient enough and cannot provide a real-time solution for this measurement. As the RMSE is much larger than the BFS step (5 MHz) in the training dataset, there is no need to further apply a finer BFS step.

Fig. 10. The BFS extracted by (a) SVC without PCA, (b) SVR without PCA, (c) SVC with PCA, and (d) SVR with PCA for the dynamic strain measurement with an 8 kHz repetition rate. The dots and the solid lines indicate the actual and the ideal BFS respectively.

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By implementing PCA, each BGS is projected from 625 dimensions to 13 dimensions, and the computation time needed to train the models and predict the new BFS is significantly reduced. The actual BFS and the ideal sinusoidal BFS presented in Fig. 10 (c) and (d) are similar to those in Fig. 10 (a) and (b). The RMSEs of the results given by SVC and SVR with PCA are 29.6 MHz and 35.2 MHz, which are 11.3% and 19.3% larger than the results given by methods without PCA due to the information loss in the dimensionality reduction. The average BFS extraction time is 0.4287 ms and 0.0104 ms for SVC and SVR with PCA, which is 10.2 times and 27.3 times faster than the methods without PCA. The SVR with PCA provides the best BFS extraction time that is much shorter than the BGS acquisition period (0.125 ms) and can be used in real-time implementation. The time issue may be largely influenced by the platform and software used to implement the algorithm, whereas our results verify that using the regression-based method with dimensionality reduction can realize the real-time sensing at an 8 kHz repetition rate on a common CPU while the accuracy does not deteriorate severely.

4. Conclusion

In this paper, PCA is proposed in dimensionality reduction aiming at shortening the machine learning model training and BFS extraction time in Brillouin optical correlation-domain sensing, before SVM methods based on classification and regression are adopted in processing the compressed BGS signals. The results of a dynamic strain measurement show that the dimensionality reduction based on PCA proposed in the paper accelerates the BFS extraction speed by 10.2–27.3 times compared with the methods with no PCA, which enables the real-time strain measurement at an 8 kHz repetition rate without outstanding accuracy deterioration. In addition, the comparison between the SVC and SVR illustrates that the regression-based method is more efficient in model training and BFS prediction, although at the cost of a slight degrading of measurement accuracy.

Funding

Changshu Institute of Technology (KYZ2019056Q); Natural Science Research of Jiangsu Higher Education Institutions of China (17KJB510002, 21KJD510002); Japan Society for the Promotion of Science (21H04555); Murata Science Foundation; Telecommunications Advancement Foundation; Yazaki Memorial Foundation for Science and Technology; Takahashi Industrial and Economic Research Foundation; Konica Minolta Science and Technology Foundation.

Disclosures

The authors declare no conflicts of interest.

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.

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		Linear Kernel	Quadratic Kernel	Cubic Kernel	Gaussian Kernel (σ = 0.9)	Gaussian Kernel (σ = 3.6)	Gaussian Kernel (σ = 14)
BFS = 0∼395 MHz; Step = 5 MHz (80classes) 13 PCs	SNR = ∞	99.2%	100%	100%	99.4%	98.5%	95.4%
		2.276 ms	2.395 ms	2.576 ms	4.755 ms	2.577 ms	3.293 ms
		0.3532 MHz	0.0 MHz	0.0 MHz	0.4331 MHz	0.7056 MHz	1.243 MHz
		5 MHz	0 MHz	0 MHz	10 MHz	10 MHz	10 MHz
	SNR = 20dB	97.0%	97.7%	97.5%	97.4%	97.4%	94.6%
		2.227 ms	2.562 ms	2.769 ms	4.453 ms	3.029 ms	3.744 ms
		0.8381 MHz	0.7601 MHz	0.8750 MHz	0.8386 MHz	0.8447 MHz	1.280 MHz
		10 MHz	10 MHz	10 MHz	10 MHz	10 MHz	15 MHz
	SNR = 15dB	91.1%	91.2%	90.3%	87.7%	90.2%	87.9%
		2.066 ms	2.767 ms	2.323 ms	6.455 ms	2.743 ms	2.689 ms
		1.663 MHz	1.672 MHz	1.586 MHz	8.890 MHz	1.662 MHz	1.932 MHz
		15 MHz	15 MHz	10 MHz	195 MHz	15 MHz	15 MHz
	SNR = 10dB	55.2%	51.7%	48.0%	33.9%	52.2%	52.6%
		2.151 ms	2.667 ms	2.859 ms	7.805 ms	3.289 ms	4.049 ms
		4.137 MHz	4.468 MHz	4.910 MHz	57.37 MHz	4.413 MHz	4.988 MHz
		30 MHz	30 MHz	25 MHz	265 MHz	25 MHz	90 MHz
Average Training Time		437.4 s	492.8 s	487.9 s	775.0 s	536.4 s	581.1 s
BFS = 0∼79 MHz; Step = 1 MHz (80classes) 6 PCs	SNR = ∞	97.0%	100%	99.8%	90.5%	94.9%	66.3%
		1.975 ms	2.370 ms	2.775 ms	5.812 ms	3.204 ms	3.876 ms
		0.1365 MHz	0.0 MHz	0.0353 MHz	0.2248 MHz	0.1765 MHz	0.5655 MHz
		1 MHz	0 MHz	1 MHz	1 MHz	2 MHz	4MHz
	SNR = 20dB	60.0%	64.0%	59.2%	38.9%	52.7%	41.5%
		1.763 ms	2.477 ms	2.441 ms	5.071 ms	2.959 ms	3.230 ms
		0.7798 MHz	0.7261 MHz	0.7939 MHz	2.024 MHz	0.8143 MHz	0.9984 MHz
		4 MHz	4 MHz	5 MHz	71 MHz	4 MHz	5 MHz
	SNR = 15dB	25.3%	26.0%	23.9%	19.3%	22.9%	21.5%
		1.719 ms	2.227 ms	2.129 ms	6.593 ms	3.665 ms	3.989 ms
		1.801 MHz	1.829 MHz	1.907 MHz	4.826 MHz	1.871 MHz	1.970 MHz
		9 MHz	9 MHz	9 MHz	78 MHz	9 MHz	9 MHz
	SNR = 10dB	9.9%	9.1%	8.4%	8.0%	9.2%	9.5%
		1.769 ms	2.585 ms	2.953 ms	7.011 ms	3.839 ms	3.426 ms
		4.773 MHz	4.800 MHz	5.205 MHz	10.13 MHz	4.857 MHz	4.797 MHz
		25 MHz	26 MHz	26 MHz	74 MHz	29 MHz	29 MHz
Average Training Time		379.2 s	805.1 s	546.9 s	872.0 s	327.0 s	374.3 s
BFS = 0∼15.8 MHz; Step = 0.2 MHz (80classes) 4 PCs	SNR = ∞	55.1%	98.9%	99.2%	72.4%	78.4%	43.4%
		1.753 ms	2.466 ms	2.341 ms	4.559 ms	3.280 ms	3.133 ms
		0.2401 MHz	0.0087 MHz	0.0132 MHz	0.0976 MHz	0.1482 MHz	0.3897 MHz
		1.2 MHz	0.2 MHz	0.2 MHz	0.6 MHz	1.4 MHz	2.4 MHz
	SNR = 20dB	12%	13.9%	13.1%	9.7%	12.0%	10.9%
		1.604 ms	2.280 ms	2.366 ms	4.481 ms	3.104 ms	3.575 ms
		0.8020 MHz	0.7468 MHz	0.7533 MHz	0.8625 MHz	0.7426 MHz	0.8265 MHz
		3.6 MHz	4.2 MHz	3.8 MHz	4.6 MHz	4.0 MHz	4.2 MHz
	SNR = 15dB	5.7%	5.8%	5.7%	4.8%	5.9%	5.7%
		1.610 ms	2.312 ms	2.261 ms	4.815 ms	4.121 ms	3.911 ms
		1.613 MHz	1.626 MHz	1.688 MHz	1.915 MHz	1.620 MHz	1.607 MHz
		6.6 MHz	6.6 MHz	7.6 MHz	13.6 MHz	6.6 MHz	6.6 MHz
	SNR = 10dB	2.9%	2.9%	2.8%	2.5%	2.7%	2.9%
		1.619 ms	2.758 ms	2.771 ms	5.224 ms	4.587 ms	5.218 ms
		3.537 MHz	3.576 MHz	3.580 MHz	4.013 MHz	3.555 MHz	3.451 MHz
		13.4 MHz	15.2 MHz	15.0 MHz	13.6 MHz	14.2 MHz	13.4 MHz
Average Training Time		211.3 s	732.6 s	975.6 s	550.3 s	263.8 s	302.9 s

		Linear Kernel	Quadratic Kernel	Cubic Kernel	Gaussian Kernel (σ = 0.9)	Gaussian Kernel (σ = 3.6)	Gaussian Kernel (σ = 14)
BFS = 0∼395 MHz; Step = 5 MHz (80classes) 13 PCs	SNR = ∞	53.67 MHz	12.58 MHz	8.412 MHz	12.14 MHz	9.044 MHz	21.42 MHz
		0.0146 ms	0.0066 ms	0.0017 ms	0.0106 ms	0.0045 ms	0.0154 ms
		227.9 MHz	70.90 MHz	16.81 MHz	24.43 MHz	21.67 MHz	108.6 MHz
	SNR = 20dB	53.66 MHz	16.05 MHz	8.504 MHz	12.36 MHz	8.116 MHz	21.81 MHz
		0.0148 ms	0.0093 ms	0.0032 ms	0.0118 ms	0.0049 ms	0.0179 ms
		224.5 MHz	78.36 MHz	36.27 MHz	55.22 MHz	27.68 MHz	99.76 MHz
	SNR = 15dB	53.28 MHz	20.10 MHz	12.39 MHz	19.85 MHz	9.262 MHz	24.49 MHz
		0.0144 ms	0.0109 ms	0.0069 ms	0.0216 ms	0.0073 ms	0.0211 ms
		209.3 MHz	126.0 MHz	109.9 MHz	126.0 MHz	68.33 MHz	120.5 MHz
	SNR = 10dB	54.62 MHz	31.64 MHz	23.99 MHz	59.93 MHz	18.54 MHz	34.31 MHz
		0.0155 ms	0.0152 ms	0.0137 ms	0.0477 ms	0.0175 ms	0.0324 ms
		191.7 MHz	149.8 MHz	123.6 MHz	205.9 MHz	89.40 MHz	159.0 MHz
Average Training Time		215.2 s	555. 0 s	517.5 s	40.50 s	18.01 s	28.05 s
BFS = 0∼79 MHz; Step = 1 MHz (80classes) 6 PCs	SNR = ∞	3.199 MHz	1.694 MHz	1.889 MHz	2.574 MHz	1.560 MHz	1.547 MHz
		0.0074ms	0.0015 ms	0.0013 ms	0.0136 ms	0.0015 ms	0.0025 ms
		15.06 MHz	3.046 MHz	3.080 MHz	7.052 MHz	3.7124 MHz	8.423 MHz
	SNR = 20dB	3.370 MHz	1.501 MHz	1.088 MHz	2.425 MHz	1.132 MHz	1.546 MHz
		0.0086 ms	0.0022 ms	0.0013 ms	0.0117 ms	0.0019 ms	0.0036 ms
		13.73 MHz	7.740 MHz	4.538 MHz	15.47 MHz	3.986 MHz	8.873 MHz
	SNR = 15dB	4.460 MHz	2.609 MHz	1.826 MHz	4.405 MHz	1.718 MHz	2.504 MHz
		0.0101 ms	0.0082 ms	0.0033 ms	0.0197 ms	0.0046 ms	0.0147 ms
		18.21 MHz	14.21 MHz	7.017 MHz	30.43 MHz	7.534 MHz	12.23 MHz
	SNR = 10dB	7.267 MHz	5.788 MHz	4.793 MHz	8.802 MHz	4.457 MHz	5.568 MHz
		0.0126 ms	0.0144 ms	0.0144 ms	0.0388 ms	0.0294 ms	0.0353 ms
		24.95 MHz	22.00 MHz	21.23 MHz	44.93 MHz	21.83 MHz	21.03 MHz
Average Training Time		267.9 s	37.96 s	35.46 s	192.5 s	4.509 s	5.528 s
BFS = 0∼15.8 MHz; Step = 0.2 MHz (80classes) 4 PCs	SNR = ∞	1.100 MHz	0.4046 MHz	0.3583 MHz	0.4822 MHz	0.3154 MHz	0.3848 MHz
		0.0120 ms	0.0051 ms	0.0034 ms	0.0059 ms	0.0034 ms	0.0062 ms
		5.049 MHz	2.186 MHz	0.6086 MHz	1.502 MHz	0.6647 MHz	2.057 MHz
	SNR = 20dB	1.232 MHz	0.8100 MHz	0.7346 MHz	0.8547 MHz	0.6928 MHz	0.7749 MHz
		0.0105 ms	0.0093 ms	0.0095 ms	0.0235 ms	0.0179 ms	0.0197 ms
		5.330 MHz	3.978 MHz	3.475 MHz	4.865 MHz	3.293 MHz	3.918 MHz
	SNR = 15dB	1.747 MHz	1.610 MHz	1.551 MHz	1.798 MHz	1.525 MHz	1.576 MHz
		0.0114 ms	0.0154 ms	0.0167 ms	0.0391 ms	0.0362 ms	0.0368 ms
		6.508 MHz	6.382 MHz	6.192 MHz	7.355 MHz	5.843 MHz	6.288 MHz
	SNR = 10dB	3.128 MHz	3.127 MHz	3.101 MHz	3.461 MHz	3.120 MHz	3.098 MHz
		0.0190 ms	0.0208 ms	0.0227 ms	0.0558 ms	0.0498 ms	0.0505 ms
		10.92 MHz	11.11 MHz	11.56 MHz	11.45 MHz	11.96 MHz	11.23 MHz
Average Training Time		150.6 s	152.6 s	28.87 s	93.43 s	3.666 s	6.156 s

		Linear Kernel	Quadratic Kernel	Cubic Kernel	Gaussian Kernel (σ = 0.9)	Gaussian Kernel (σ = 3.6)	Gaussian Kernel (σ = 14)
BFS = 0∼395 MHz; Step = 5 MHz (80classes) 13 PCs	SNR = ∞	99.2%	100%	100%	99.4%	98.5%	95.4%
		2.276 ms	2.395 ms	2.576 ms	4.755 ms	2.577 ms	3.293 ms
		0.3532 MHz	0.0 MHz	0.0 MHz	0.4331 MHz	0.7056 MHz	1.243 MHz
		5 MHz	0 MHz	0 MHz	10 MHz	10 MHz	10 MHz
	SNR = 20dB	97.0%	97.7%	97.5%	97.4%	97.4%	94.6%
		2.227 ms	2.562 ms	2.769 ms	4.453 ms	3.029 ms	3.744 ms
		0.8381 MHz	0.7601 MHz	0.8750 MHz	0.8386 MHz	0.8447 MHz	1.280 MHz
		10 MHz	10 MHz	10 MHz	10 MHz	10 MHz	15 MHz
	SNR = 15dB	91.1%	91.2%	90.3%	87.7%	90.2%	87.9%
		2.066 ms	2.767 ms	2.323 ms	6.455 ms	2.743 ms	2.689 ms
		1.663 MHz	1.672 MHz	1.586 MHz	8.890 MHz	1.662 MHz	1.932 MHz
		15 MHz	15 MHz	10 MHz	195 MHz	15 MHz	15 MHz
	SNR = 10dB	55.2%	51.7%	48.0%	33.9%	52.2%	52.6%
		2.151 ms	2.667 ms	2.859 ms	7.805 ms	3.289 ms	4.049 ms
		4.137 MHz	4.468 MHz	4.910 MHz	57.37 MHz	4.413 MHz	4.988 MHz
		30 MHz	30 MHz	25 MHz	265 MHz	25 MHz	90 MHz
Average Training Time		437.4 s	492.8 s	487.9 s	775.0 s	536.4 s	581.1 s
BFS = 0∼79 MHz; Step = 1 MHz (80classes) 6 PCs	SNR = ∞	97.0%	100%	99.8%	90.5%	94.9%	66.3%
		1.975 ms	2.370 ms	2.775 ms	5.812 ms	3.204 ms	3.876 ms
		0.1365 MHz	0.0 MHz	0.0353 MHz	0.2248 MHz	0.1765 MHz	0.5655 MHz
		1 MHz	0 MHz	1 MHz	1 MHz	2 MHz	4MHz
	SNR = 20dB	60.0%	64.0%	59.2%	38.9%	52.7%	41.5%
		1.763 ms	2.477 ms	2.441 ms	5.071 ms	2.959 ms	3.230 ms
		0.7798 MHz	0.7261 MHz	0.7939 MHz	2.024 MHz	0.8143 MHz	0.9984 MHz
		4 MHz	4 MHz	5 MHz	71 MHz	4 MHz	5 MHz
	SNR = 15dB	25.3%	26.0%	23.9%	19.3%	22.9%	21.5%
		1.719 ms	2.227 ms	2.129 ms	6.593 ms	3.665 ms	3.989 ms
		1.801 MHz	1.829 MHz	1.907 MHz	4.826 MHz	1.871 MHz	1.970 MHz
		9 MHz	9 MHz	9 MHz	78 MHz	9 MHz	9 MHz
	SNR = 10dB	9.9%	9.1%	8.4%	8.0%	9.2%	9.5%
		1.769 ms	2.585 ms	2.953 ms	7.011 ms	3.839 ms	3.426 ms
		4.773 MHz	4.800 MHz	5.205 MHz	10.13 MHz	4.857 MHz	4.797 MHz
		25 MHz	26 MHz	26 MHz	74 MHz	29 MHz	29 MHz
Average Training Time		379.2 s	805.1 s	546.9 s	872.0 s	327.0 s	374.3 s
BFS = 0∼15.8 MHz; Step = 0.2 MHz (80classes) 4 PCs	SNR = ∞	55.1%	98.9%	99.2%	72.4%	78.4%	43.4%
		1.753 ms	2.466 ms	2.341 ms	4.559 ms	3.280 ms	3.133 ms
		0.2401 MHz	0.0087 MHz	0.0132 MHz	0.0976 MHz	0.1482 MHz	0.3897 MHz
		1.2 MHz	0.2 MHz	0.2 MHz	0.6 MHz	1.4 MHz	2.4 MHz
	SNR = 20dB	12%	13.9%	13.1%	9.7%	12.0%	10.9%
		1.604 ms	2.280 ms	2.366 ms	4.481 ms	3.104 ms	3.575 ms
		0.8020 MHz	0.7468 MHz	0.7533 MHz	0.8625 MHz	0.7426 MHz	0.8265 MHz
		3.6 MHz	4.2 MHz	3.8 MHz	4.6 MHz	4.0 MHz	4.2 MHz
	SNR = 15dB	5.7%	5.8%	5.7%	4.8%	5.9%	5.7%
		1.610 ms	2.312 ms	2.261 ms	4.815 ms	4.121 ms	3.911 ms
		1.613 MHz	1.626 MHz	1.688 MHz	1.915 MHz	1.620 MHz	1.607 MHz
		6.6 MHz	6.6 MHz	7.6 MHz	13.6 MHz	6.6 MHz	6.6 MHz
	SNR = 10dB	2.9%	2.9%	2.8%	2.5%	2.7%	2.9%
		1.619 ms	2.758 ms	2.771 ms	5.224 ms	4.587 ms	5.218 ms
		3.537 MHz	3.576 MHz	3.580 MHz	4.013 MHz	3.555 MHz	3.451 MHz
		13.4 MHz	15.2 MHz	15.0 MHz	13.6 MHz	14.2 MHz	13.4 MHz
Average Training Time		211.3 s	732.6 s	975.6 s	550.3 s	263.8 s	302.9 s

		Linear Kernel	Quadratic Kernel	Cubic Kernel	Gaussian Kernel (σ = 0.9)	Gaussian Kernel (σ = 3.6)	Gaussian Kernel (σ = 14)
BFS = 0∼395 MHz; Step = 5 MHz (80classes) 13 PCs	SNR = ∞	53.67 MHz	12.58 MHz	8.412 MHz	12.14 MHz	9.044 MHz	21.42 MHz
		0.0146 ms	0.0066 ms	0.0017 ms	0.0106 ms	0.0045 ms	0.0154 ms
		227.9 MHz	70.90 MHz	16.81 MHz	24.43 MHz	21.67 MHz	108.6 MHz
	SNR = 20dB	53.66 MHz	16.05 MHz	8.504 MHz	12.36 MHz	8.116 MHz	21.81 MHz
		0.0148 ms	0.0093 ms	0.0032 ms	0.0118 ms	0.0049 ms	0.0179 ms
		224.5 MHz	78.36 MHz	36.27 MHz	55.22 MHz	27.68 MHz	99.76 MHz
	SNR = 15dB	53.28 MHz	20.10 MHz	12.39 MHz	19.85 MHz	9.262 MHz	24.49 MHz
		0.0144 ms	0.0109 ms	0.0069 ms	0.0216 ms	0.0073 ms	0.0211 ms
		209.3 MHz	126.0 MHz	109.9 MHz	126.0 MHz	68.33 MHz	120.5 MHz
	SNR = 10dB	54.62 MHz	31.64 MHz	23.99 MHz	59.93 MHz	18.54 MHz	34.31 MHz
		0.0155 ms	0.0152 ms	0.0137 ms	0.0477 ms	0.0175 ms	0.0324 ms
		191.7 MHz	149.8 MHz	123.6 MHz	205.9 MHz	89.40 MHz	159.0 MHz
Average Training Time		215.2 s	555. 0 s	517.5 s	40.50 s	18.01 s	28.05 s
BFS = 0∼79 MHz; Step = 1 MHz (80classes) 6 PCs	SNR = ∞	3.199 MHz	1.694 MHz	1.889 MHz	2.574 MHz	1.560 MHz	1.547 MHz
		0.0074ms	0.0015 ms	0.0013 ms	0.0136 ms	0.0015 ms	0.0025 ms
		15.06 MHz	3.046 MHz	3.080 MHz	7.052 MHz	3.7124 MHz	8.423 MHz
	SNR = 20dB	3.370 MHz	1.501 MHz	1.088 MHz	2.425 MHz	1.132 MHz	1.546 MHz
		0.0086 ms	0.0022 ms	0.0013 ms	0.0117 ms	0.0019 ms	0.0036 ms
		13.73 MHz	7.740 MHz	4.538 MHz	15.47 MHz	3.986 MHz	8.873 MHz
	SNR = 15dB	4.460 MHz	2.609 MHz	1.826 MHz	4.405 MHz	1.718 MHz	2.504 MHz
		0.0101 ms	0.0082 ms	0.0033 ms	0.0197 ms	0.0046 ms	0.0147 ms
		18.21 MHz	14.21 MHz	7.017 MHz	30.43 MHz	7.534 MHz	12.23 MHz
	SNR = 10dB	7.267 MHz	5.788 MHz	4.793 MHz	8.802 MHz	4.457 MHz	5.568 MHz
		0.0126 ms	0.0144 ms	0.0144 ms	0.0388 ms	0.0294 ms	0.0353 ms
		24.95 MHz	22.00 MHz	21.23 MHz	44.93 MHz	21.83 MHz	21.03 MHz
Average Training Time		267.9 s	37.96 s	35.46 s	192.5 s	4.509 s	5.528 s
BFS = 0∼15.8 MHz; Step = 0.2 MHz (80classes) 4 PCs	SNR = ∞	1.100 MHz	0.4046 MHz	0.3583 MHz	0.4822 MHz	0.3154 MHz	0.3848 MHz
		0.0120 ms	0.0051 ms	0.0034 ms	0.0059 ms	0.0034 ms	0.0062 ms
		5.049 MHz	2.186 MHz	0.6086 MHz	1.502 MHz	0.6647 MHz	2.057 MHz
	SNR = 20dB	1.232 MHz	0.8100 MHz	0.7346 MHz	0.8547 MHz	0.6928 MHz	0.7749 MHz
		0.0105 ms	0.0093 ms	0.0095 ms	0.0235 ms	0.0179 ms	0.0197 ms
		5.330 MHz	3.978 MHz	3.475 MHz	4.865 MHz	3.293 MHz	3.918 MHz
	SNR = 15dB	1.747 MHz	1.610 MHz	1.551 MHz	1.798 MHz	1.525 MHz	1.576 MHz
		0.0114 ms	0.0154 ms	0.0167 ms	0.0391 ms	0.0362 ms	0.0368 ms
		6.508 MHz	6.382 MHz	6.192 MHz	7.355 MHz	5.843 MHz	6.288 MHz
	SNR = 10dB	3.128 MHz	3.127 MHz	3.101 MHz	3.461 MHz	3.120 MHz	3.098 MHz
		0.0190 ms	0.0208 ms	0.0227 ms	0.0558 ms	0.0498 ms	0.0505 ms
		10.92 MHz	11.11 MHz	11.56 MHz	11.45 MHz	11.96 MHz	11.23 MHz
Average Training Time		150.6 s	152.6 s	28.87 s	93.43 s	3.666 s	6.156 s

Dynamic strain measurement in Brillouin optical correlation-domain sensing facilitated by dimensionality reduction and support vector machine

Abstract

1. Introduction

2. Principle and simulation

2.1 Principle of PCA and SVMs in Brillouin optical correlation-domain sensing

2.2 Visualizing BGS datasets in 2D and 3D

2.3 Determining the dimensionality for model training

2.4 Training datasets and BFS extraction using SVC/ SVR

3. Experiment

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (2)

Equations (15)

Optics Express