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Abstract
The development of optical fiber sensors has led to the possibility of accumulating vast, real-time databases of acoustic and other measurements throughout fiber networks, which brings even more widespread concern on improving the sampling effectiveness. In this paper, we present two kinds of sweep frequency methods based on using a neural network to extract temperature from the Brillouin gain spectrum (BGS). Gauss centralization and variance weight probability methods are proposed to compare with the uniform sweep frequency method. By analyzing formulas of the ideal BGS model, we find the gain near the peak of Brillouin gain spectrum has greater correlation with temperature extraction than other positions. Therefore, the Gaussian centralized sweep method is proposed. We further investigate the variation of the weights in the neural network and Brillouin data distribution in different positions and find that the variance is positively correlated with the weights in hidden layers. So, we propose the sweep frequency method based on variance weight probability and make a complement to interpret the rationality of this method in neural network. In all the aforementioned approaches, 281 points are obtained between the 9.07 GHz to 9.35 GHz range under the same condition. The data of each method is trained ten times and tested through the same neural network structure. All the RMSE of each test stage covers all data collecting the passage. The result shows that the RMSE of variance weight probability sweep frequency method is 0.5277, which is superior to the Gauss centralization sweep frequency method that was 0.6864 and the uniform sweep frequency method that was 0.9140.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The growing industry of Artificial intelligence of things (AIoT) has presented the ever-increasing need for ubiquitous, spatially distributed and low-cost sensors that operate with efficient intelligence management and reliable network monitor. As front-end AIoT-based hardware, Optical fiber sensors are highly desirable to support the requirements of the system due to their particular advantages such as multi-parameter sensing, wide dynamic range, electromagnetic interference immunity, and remote transmission [1]. Brillouin optical time domain analysis (BOTDA) method of the distributed optical fiber sensing technology is increasingly gaining popularity. BOTDA-based distributed optical fiber sensors have been developed for measured temperature and strain sensing, particularly in structural health monitoring and other engineering with high spatial resolution and long measurement distances. Many research works have been discussed in the field of BOTDA technique during the past two decades [2–7]. In the conventional BOTDA sensing system, a pump pulse light and a probe light counter-propagate in the fiber. Pump pulse signal transmits its power to amplify the probe signal by Stimulated Brillouin Scattering (SBS) effect when the frequency difference is equal to the Brillouin frequency shift (BFS) in the same fiber. As the temperature change of the fiber will influence BFS distribution linearly, and the BFS can be calculated from the BGS. We can process the BGS collected from the BOTDA-based sensing system to extract the temperature and other useful information.
As we know, conventional Lorentz curve fitting (LCF) technique will find the BFS by calculating the peak of the BGS to extract temperature [8–11]. It is a good understanding method for our cognition, but LCF contains tedious steps to calculate final temperature. And the accuracy of result is not better than ANN [12]. Recently, machine learning is effective in several areas, and especially ANN has been developed to process data whose relationship is challenging to comprehend in former methods. The ANN has been widely applied in data science and engineering estimation to process complicated input-output relationships for its non-linear mapping ability [13–21]. If using neural network methods, training of the pair of spectrum profile and temperature label pre-acquired to obtain a model which will be deployed in extracting temperature. Although the ANN method is more direct and efficient than the conventional LCF method, the time-consuming step-by-step sweeping frequency still takes up most of the total time.
To improve the efficiency of data collection in neural network process, we adjust the sweeping strategy to reduce sample points as few as possible. We hope to seek out more significant points to substitute the uniform sweep frequency method. This paper demonstrates two kinds of sweeping frequency strategies compared with the uniform sweep frequency method. In the beginning, this paper introduces the comparison of the test results of uniform sweep frequency in the neural network setting different step frequencies. It concludes that the uniform sweep frequency method has some limitations. Under the same structure conditions, we try to find a better sweep frequency strategy. Through the reference to the ideal BGS model formula, we find linewidth and peak value has a strong influence on temperature extraction. Then, experiment to sample the points of BGS by using the Gauss centralization sweep frequency method. To find the theoretical support and improve the accuracy, we analyzed the network weights processed by multiplied all layer weights and calculated through the sigmoid layer to reproduce the hidden layer data as a single matrix between input and output. Experiments show that a certain relationship weight data and input data, so we propose variance weight method. By comparing the corresponding performance with other methods, we choose more effective points directed by the weights distribution for training and deployment to replace uniform points.
2. Results and analyses
2.1 Experiment setup and fundamental theory
The BGS test system under different temperature conditions, as shown in Fig. 1. The output light is emitted by the 1550 nm laser device and enters the 50:50 coupler. It is divided into upper and lower branches. The upper branch light passes through the electro-optic modulator (EOM) and the erbium-doped fiber amplifier (EDFA1) into the optical isolator (OI) and then goes into the fiber under test (FUT) as the signal light of the SBS effect. The type of the FUT is the Highly Non-Linear Fiber (HNLF), and the length of the FUT is 1 km. The average power of the signal light is about 1 mW. The radio frequency (RF) signal of the electro-optic modulator is input by the vector network analyzer (VNA). The DC signal source is used to adjust the working point of the EOM. The lower branch light is amplified by EDFA2 and enters the optical fiber as a pump light. The pump light is a continuous light, its power is about 18 mW. The temperature information produced in the above steps is carried by signal light through the photodetector (PD) into the VNA. When adjusting the water bath (WB) temperature, the BGS under different temperature conditions can be obtained. In the experiment, 77 groups of BGS were collected between 16.5 ∼80℃ by changing the temperature of the WB. The temperature change interval is about 1℃. The frequency range is 9.07 GHz to 9.35 GHz, and the frequency interval is 0.175 MHz.
Fig. 1. BGS test system. The schematic diagram of the experiment setup with PC (Polarization Controller), EOM (Electro-Optic Modulator), DC (Direct Current), OI (Optical Isolator), VNA (Vector Network Analyzer), EDFA (Erbium-Doped Fiber Amplifier), PD (Photo Detector), WB (Water Bath), FUT (Fiber Under Test).
The obtained BGSs conform to the shape of the Lorentz curve in the ideal case, could be represented using a function:
In conventional methods, the BFS is obtained by the BGS, and then to be calculated to the corresponding temperature. In contrast to this approach, the ANN can learn the complex relationship between input and output. It can be used to associate BGS with temperature directly. Figure 2 shows a typical ANN structure with input and output layers together with one hidden layer, which is interconnected with different weights.
The artificial neural network is trained to acquire knowledge about the BGS patterns under different temperatures. It allows better accuracy even if the data points on the collected BGS become fewer. The typical neural network is composed of an input layer, a hidden layer and an output layer, named ${x_m}$, ${h_k}$, and ${y_n}$, respectively as shown in Fig. 2. The processing units are interconnected between layers. The weights from ${x_m}$ to ${h_k}$ and ${h_k}$ to ${y_n}$ are ${w_{mk}}\; $ and ${w_{kn}}$, respectively. The input layer is accepted as a vector matrix [${x_1},\; \; {x_2} \ldots \; {x_m}$]. The output layer is a vector matrix of [${y_1},\; \; {y_2} \ldots \; {y_n}$]. The ANN output layer labels have the same dimensions as the output vector matrix. The ANN estimates the errors of the results obtained in each iteration and corrects the weights by Back-Propagation (BP) algorithm to minimize the errors. After several iterations, the output results are as close as possible to the labels.
We use the ANN to extract temperature label from the BGSs test system. The whole process of each stage is shown in Fig. 3. In the process of data training and testing, the prediction accuracy of the neural network using multi-hidden layer structure does not have a remarkable superior to single-hidden layer network. And the multi-hidden layer training is slow and easy to over-fitting, so the single-hidden layer structure is adopted as the basic network [22–23]. For further optimizing the model structure, the number of hidden layer neurons was changed several times, and we make a final decision to use 50 instead of 100 or 10. Stochastic Gradient Descent (SGD) is used to reduce the loss value continuously. Therefore, the neural network model we proposed was constructed with three layers, as it’s shown in Fig. 3 including one input layer with shape (1, 281), one output layer with shape (1, 1), and a single-hidden layer with shape (1, 50).
Before entering the neural network model training, data preprocessing is carried out first. The BGS data is sampled as input data by different sweep frequency methods. Each sampled BGS data with shape (1, 281). The raw data of the BGS data with shape (1, 1601) are normalized in the range of 0 to 1. Sigmoid function as activation layer placed after the hidden layer. Then, the input data are trained in the same structure of the neural network.
To make full use of the data, we adopt the cross validation to evaluate the performance of the methods. With the help of the optimization loss function and the large numbers of iterations, the ANN model learns to map the temperature to the BGS. The cross validation to evaluate the method with the RMSEs is shown in the Fig. 4. In each composite training, 77 sets of Brillouin scattering spectrum data were divided into seven-folds, 11 groups of data each fold. One-fold of data was selected as the test set in every single training, and the other six-folds were 66 groups of data as the training set. Every composite training contains seven single training, and each training randomly selects 66 groups of data and their temperature label for training. The remaining 11 groups of data contains the corresponding label for testing to obtain the RMSE. And then taking the average of the RMSE of the composite training. The RMSE is considered an excellent general-purpose error metric for numerical predictions. The RMSE in this model is given in Eq. (3).
where ${T_{predicted}}$ is the ANN predicted temperature of the BGS in the sweep frequency method, The ${T_{observed}}$ is the true temperature label of the BGS. The performance of different sweep frequency methods can be evaluated by the difference between predicted value and corresponding observed value.2.2 Uniform sweep frequency
In data acquisition, sweep frequency affects the number of acquisition points and the acquisition time. It is beneficial that sampling few points to obtain a high accuracy result. We need to selectively pick out more effective points. Specifically, selecting the most significant fewer points in raw points acquired in the experiment above is the key to reducing sweep time. It is of great significance to select sweep step frequency by using the sweep frequency method.
Each original BGS profile has 1601 points, and the frequency range is 9.07 GHz to 9.35 GHz. We use the function to generates the evenly spaced vectors. It returns the evenly spaced 281 numbers as the serial numbers, and we select the serial number from the original BGS datasheet. The selected 281 data inside the frequency range with uniformly-spaced. This approach can be equated to uniform sweep frequency method with 1 MHz step frequency. The method for selecting points is shown below. Blue points present raw data points, with interval of 0.175 MHz, as reference points. Red points present the 281 points sampled in uniform sweep frequency method from the BGS raw data, with the interval of 1 MHz. It shows in Fig. 5(a) and the result is shown in Fig. 5(b). The average RMSE of ten composite training sessions was 0.9140.
Fig. 5. Uniform Sweep Frequency experiment result. (a) 281 points using Uniform Sweep Frequency. (b) RMSE of ten composite training sessions in uniform sweep frequency method.
For this neural network model, we use the same method to calculate RMSE by adjusting sweep frequency step, and then try to find out the relationship between the number of sampling points and the accuracy of temperature extraction. As shown in Fig. 6(a), the numbers of the uniformly-spaced sample points selected from the original BGS data are 141,281,421,561,701. It can be equated to the uniform sweep frequency with step frequency are 2 MHz, 1 MHz, 0.66 MHz, 0.5 MHz and 0.4 MHz. With the increase in the number of training sessions, the accuracy of extraction temperature changes. The stability and applicability is important for evaluating the sweep frequency method. The accuracy fluctuation of temperature extraction is relatively soft or regular, which is an ideal result. And that is also one of the reasons why we adopt ten training sessions. Through analysis, it is found that the temperature extraction accuracy with 281,421,561 sweep frequency points has better stability.
Fig. 6. RMSEs of five kinds of uniform interval sweep frequency method. (a) Line chart of the RMSEs. (b) Heatmap of the RMSEs.
As shown in the heat map Fig. 6(b) below, specifically the warm-to-cool color scale characterizes the change in RMSE from high to low. Through the observation of the heat map, we can see that the number of sampling points from 141 to 562 decreases in warm color blocks and increases in cool color blocks, and RMSE shows a downward trend. When the number of sampling points is 703, the number of warm color blocks increases and the number of cool color blocks decreases. Therefore, the downward trend presents an inflection point. We further analysis the trends by calculating the mean of the RMSEs in different sampling points. The mean is 1.16, 0.9, 0.89, 0.84, and 1.01. It shows that the accuracy of temperature cannot be continuous improved by increasing the number of the sampling points. When the configuration of the neural network model is fixed, the accuracy of temperature extraction by uniform sweep method has some bottleneck.
Moreover, we need a better method for selecting sweep frequency besides the benchmark experiment by uniform interval sweep frequency. We selected 281 acquisition points of 1 MHz sweep step frequency as a reference. By analyzing the BGS and formulas, it is found that the weight of information carried by different position points was different. Then, we introduced the Gaussian centralization sweep frequency method and variance weight sweep frequency method, and have conducted further research on the model.
2.3 Gaussian central sweep frequency
By calculating the skewness and kurtosis of the spectrum data, it is further verified that the shape of BGS is a Gauss-like curve with single peak and symmetry. Considering the above situation, we assume that the point at the peak of BGS may have a noticeable effect on temperature prediction, the Gauss centralization sweep frequency strategy is proposed to optimize the sampling method in the experiment. Gaussian function to select the position of sweep points is shown as formulas below.
From the raw data, we can acquire estimated the value of the linewidth (FWHM). According to the formula (5), the standard deviation can be estimated at about 200. The property of a normal distribution also indicates that the peak of curve at the center. The points to select should accord with the characteristic above that the peak of BGS is around 800 in the x-axis, the result in the range 0 ∼ 1 is multiplied by 200 plus 800 to obtain points with the Gaussian sweep frequency method. In other words, we generate a gauss function as an index array $({\mu = 800,\sigma = 200} )$ to pick out 281 points from 1601 points. The selected point is shown in Fig. 7(a). These points can be considered as points of probability selection of Gauss distribution near the peak value.
Fig. 7. Gaussian Sweep Frequency experiment result. (a) 281 points using Gaussian Sweep Frequency. (b) RMSE of ten composite training sessions in the Gaussian sweep frequency method.
In this system, each pre-processed BGS obtained in the Gauss centralization sweep frequency method has the same start frequency, end frequency, and frequency interval. The upper limit for the sampling density is 0.175 MHz as same as the step frequency of the original BGS from the device. The selected points in the Gauss centralization sweep frequency method are among the original data. The corresponding amplitude to the frequency in each BGS is different for temperature difference. Then the data are trained in the ANN model to map the BGS to the temperature. Green points are 1601 points former, and 281 points are the red ones covering them. After ten composite training sessions as the same as the training and testing stage in section 2.2. The deviation of the temperature extraction is shown in Fig. 7(b) with an average of 0.6864.
2.4 Variance weight sweep frequency
The RMSE of the Gaussian centralization sweep frequency method in section 2.3 is less than that of the Uniform sweep frequency method in 2.2. Furthermore, we desire to further study the critical position of data for temperature extraction of BGS. Through analysis of the trained neural network model, and try to find out the relationship between the weights of hidden layer and the input data. We analyze the weights of 50 hidden layer neurons by using the matrix multiplication law. To multiply the weights matrix of the input layer of (281, 50) by (50,1). Vector of weights with the shape of (281, 1) can be obtained favorable by multiplying the weights of the hidden layer and output layer, but we cannot ignore that the structure with sigmoid function after the hidden layer. Input will be affected by sigmoid function, so the effect of the sigmoid function is essential.
The shape of layers is shown in Fig. 3. Assuming that the input layer and hidden layer connect weights is ${w_1}$ (281, 50), input dimension is (1, 281), output weights is ${w_2}$ (50, 1), sigmoid function shape is as same as the output of hidden layer. The primary purpose of Sigmoid is to compress the hidden layer results into the range of 0-1, to add non-linear factors. Due to the formula:
where x is the output of hidden layer but in front of the sigmoid. When analyzing weights data, the function we used as the following formula: using a mean of input ${x_{mean}}$ to represent input data, which satisfies the requirement of dimension. Sigmoid results of weights contain the shape (1, 50) since the shape of ${x_{mean}}\ast {w_1}$ is (1, 50), and it is regarded as ${w_s}$. A temporary multiplier is still necessary as a sigmoid matrix between the hidden and output layers. $\frac{{{w_s}}}{{{x_{mean}}}}$ is proposed in place of the multiplier. Then multiply hidden layer ${w_1}$, sigmoid matrix $\frac{{{w_s}}}{{{x_{mean}}}}$, and output layer weights ${w_2}$ to obtain the final weights with the shape (281, 1) will be analyzed. And we show the weights do not contain a step to multiply the sigmoid matrix as Fig. 8(a). The full version with sigmoid shows in Fig. 8(b) expands the x-axis to 1601 to get a more convenient analysis.Fig. 8. Integration of weights (a) without sigmoid matrix in 281 points, (b) using sigmoid matrix in 281 points.
Figure 8(b) shows that the weights are near zero and disorderly in the left and right edge positions. The weights between 600 and 1200 in the x-axis fluctuate greatly, which is close to the center of the Brillouin spectrum. To prove universality, we trained a 1601-points neural network model and analyzed the weights according to the above process. In this experiment, the shape of the final integration weights is (1601, 1) as shown in Fig. 9(a).
Figure 9(b) shows that the weights of this measurement still reveal the law mentioned above. The weight is related to the characteristic distribution of input data. The Gauss centralization sweep frequency method in 2.3 also demonstrates that the center of BGS has a higher value, but the weight value is close to 0 near the peak of the mean value in Fig. 9(b). We consider finding other data characteristics have a greater impact on the prediction of temperature. To Analyze the effectiveness of the points from the input data, the fluctuation in every point is calculated by the value of 66 group data in the same position. The variance of the 66 group normalized input data is calculated as shown in Fig. 10, the shape is (1, 1601).
Contrasting Fig. 9(b) and Fig. 10, it shows that the absolute weights of hidden layer neurons are more significant where the variance vibration is strong. To verify the validity of the sweep points of the BGS according to the weight of variance. Each point has a variance, and the variance determines the probability of sampling the point. These points are selected according to the Variance Weight (V-W) method shown in Fig. 11(a).
Fig. 11. Variance weight sweep frequency experiment result. (a) 281 points using variance weight sweep frequency. (b) RMSE of ten composite training sessions in V-W Selected method
According to this method, 281 points are being processed like section 2.2 and used the same way to verify the prediction effect. The RMSE of ten composite training sessions is shown in Fig. 11(b) with an average of 0.5277.
3. Discussion
3.1 Comparison of the deviation
Comparing the three sweep frequency methods mentioned above, RMSEs are shown in Fig. 12(a). It can be seen the differences in the trends among the three types of sweep frequency methods. Red line represents the RMSE of the Uniform sweep frequency method, which is the standard strategy adopted widely. The green line represents the RMSE of the Gaussian centralization sweep frequency method, which is better than the red line. Furthermore, the blue line is the RMSE of the V-W sweep frequency method, whose RMSE is the minimum in three methods. The Mean, Max, Min of the RMSEs illustrate the deviation between the temperature extraction value and the temperature label. The standard deviations (SD) of the RMSEs can describe the uncertainty of the temperature in different methods. The above statistical characteristic in the different sweep frequency methods are shown in the bar chart Fig. 12(b). It shows that the V-W method is better than the other two methods. To further illustrate the broad applicability of the sweep frequency method we proposed, we re-evaluate the ANN model using Mean Absolute Error (MAE). As it is shown in Fig. 12(c) and Fig. 12(d), the performance of the V-W method remains well. For the reason that the RMSE is more sensitive to outliers than MAE. Therefore, the RMSE is preferred to us to evaluate the method. It is a credible conclusion due to the similarity between integration weights and all input data variance.
Fig. 12. Results in three sweep frequency methods. (a) Line chart of the RMSEs. (b) Statistical characteristic of RMSEs. (c) Line chart of the MAEs. (d) Statistical characteristic of MAEs.
3.2 Comparison of effectiveness
All the code and data are running in the web-based interactive development environment. We built a private cloud server, and the data was processed in a virtualized environment. The hardware platform configuration of the server as follows: The CPU model is Intel Core i7-4790 CPU @ 3.60 GHz. The CPU cache size is 8192KB. The virtual address size is 48 bits. And the total memory size is 4048284 kB. By comparing the extracted temperature indexes of V-W sweep number 281 and uniform sweep number 561 in Table 1, it is found that the V-W method has a higher accuracy of extracting temperature with less sampling times. Meanwhile, the BGS file corresponding to each temperature label contains 561 points in uniform sweep with a file size of about 16KB, while the V-W method contains 281points with a file size of 8KB.The superiority of the V-W method is further manifested as the amount of the collected data increases.
Table 1. Network training time and data size of the different sweep frequency method
3.3 Robustness and limitation of the model
To make the model more robust, the size of the dataset is increased by adding noise. The original BGSs corresponding to the temperature has been added different noise. Then each temperature has 100 simulated BGSs with different SNRs from 19 dB to 72 dB. The 7700 sets of BGS-temperature dataset have been preprocessed with the above three sweep frequency method. Then they have been trained and tested in the ANN model. Comparing the result in three sweep frequency methods as shown in Fig. 13. It further verified that the V-W method in the model is better than the others.
Fig. 13. Results in three sweep frequency methods with 7700 sets of dataset. (a) Line chart of the RMSEs. (b) Statistical characteristic of RMSEs.
The ANN model we designed is to extract the temperature inside the training temperature range 16.5 ∼80℃. If the corresponding temperature of the BGS for extraction exceeds the known temperature range, the accuracy of the output in the former model would be affected. And the variance weight should be adjusted when expanding the temperature extraction range.
To sum up, we can use the variance weight method as a sweep frequency strategy when extracting the temperature of BGS by using ANN. When selecting the same number of sampling points, this method will improve temperature measurement accuracy. On the other hand, to solve the problem of frequency sweeping time-consuming, reducing the sampling times while maintaining a certain precision, and using this method to obtain more valuable and fewer points is a better choice.
4. Conclusion
To reduce the time of sweeping frequency in the BGS experiment system and increase the accuracy of temperature extraction at the same time, we proposed two kinds of sweeping methods that choose more valuable points to compare to uniform sweep frequency strategies. The weights of the training model are fully analyzed. Vector is used as the factor as a linear transformation. 281 points extracted by different methods are trained and tested. The results show that the sweeping method based on variance weight probability can extract temperature more quickly and accurately, with the average of RMSE is 0.5277, better than the other two methods. Looked from the effect that, the variance weight sweep frequency method carries on the extracting temperature from BGS which has the better performance, showed this method is effective feasible. In addition, this method not only overcomes the obstacle of time-consuming drawbacks in the neural network which is used to train the model, but also improves the stability and accuracy of temperature extraction. This work opens up a way of BGS compressed sampling and provides an innovative method to realize high-precision temperature extraction. Meanwhile, benefiting from the efficiency, stability, and practicability, the presented variance weight probability sweep frequency method has the potential in various applications of the distributed optical fiber sensing technique.
Funding
Science and Technology Project of Education Department of Jilin Province (JJKH20190110KJ); Jilin Scientific and Technological Development Program (20180201032GX); National Natural Science Foundation of China (61875070).
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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