1. Introduction

Distributed optical fiber sensors based on spontaneous and stimulated Brillouin scattering have attracted extensive attention from both academia and industry over the past few decades [1–6]. The peak frequency of Brillouin gain spectrum (BGS), termed as the distributed Brillouin frequency shift (BFS), is widely used for measuring the temperature and strain along a fiber. Since the BGS theoretically satisfies a Lorentzian shape, Lorentz curve fitting (LCF) [7] is usually adopted to search the BFS with a limited sampling interval in frequency-domain. However, the LCF takes a large amount of time to process data owing to its iterative algorithm. In addition to the above-mentioned defect, it is also sensitive to the setting of initial parameters, which will lead to a large error when the signal-to-noise-ratio (SNR) is low. The cross-correlation method [8–11] works by calculating the frequency difference to obtain the BFS via convolving the ideal BGS with the measured BGS. Compared to LCF, such method is noniterative and free from the impact of initial parameters. Although its accuracy can be improved by combining the cross-correlation method with interpolation algorithm, it still suffers from a trade-off between the frequency sampling interval and accuracy in the practical applications.

To date, several machine learning based algorithms have been proposed and used for extracting the BFS. Clustering and classification algorithms, such as principal component analysis (PCA) [12], support vector machine (SVM) [13–15], have been proved to be effective. However, there’s a trade-off between the number of principal components or classes and the algorithm performance. The accuracy of the extracted BFSs depends on subdivided principal components or classes and the size of storage databases. Subsequently, multiple artificial neural networks (ANNs) have been proposed [16–23]. In consideration of the dimensionality of the measured BGSs along the sensing fiber, the ANNs could be divided into two categories. One is one-dimensional (1D) network in which data will be only analyzed in frequency-domain [18–23]. The other refers to two-dimensional (2D) network, where data will be processed jointly in time-frequency domain [16,17]. From another point of view, part of ANNs are to extract the BFSs or the temperature/strain information directly from the measured BGSs, so the processing time is reduced significantly [16–18,21–23]. The other part firstly denoise the measured BGSs and then extract the results [19,20]. However, a potential shift error of the actual BFS will be introduced inevitably as a result. In general, most of ANNs are trained by virtue of the specific data, so they require retraining or fine-tuning to satisfy different testing requirements. Besides, the noise in a sensing system is random so that the system noise model estimated by a certain ANN [16] is not so reliable.

The aforementioned methods have already shown the good performances in Brillouin optical time domain analyzer (BOTDA), because the BOTDA can achieve relatively high SNR. But in practice, Brillouin optical time domain reflectometry (BOTDR) [24–26] is much more widely employed for the merits of the single-end access, simple system architecture and easier implementation. The disadvantage of BOTDR system is that it would be affected by the relatively low SNR in principle. The reason is that the probe light is unidirectionally injected, which results in that only the low-intensity spontaneous Brillouin scattering (SpBS) can be stimulated. Therefore, a better signal processing method is required to meet the demands of the actual measurement in low-SNR BOTDR system.

In this paper, we have proposed a wavelet convolutional neural network (WNN) consisting of a unidimensional convolutional neural network and a self-adaptive wavelet neural network to implement the robust and fast temperature measurement in BOTDR. According to the detailed noise analysis in frequency-domain and time-domain, the noise along the time-related sensing distance satisfies the distribution similar to Gaussian white noise. Therefore, the temporal correlation could be ignored and the measured BGSs along the fiber are regarded as the 1D data before inputting into the proposed WNN. In view of the significance of local characteristics in frequency-domain, various self-adaptive wavelet activation functions are applied to each output of the full-connection layer for multi-scaled analysis and scale translation. Furthermore, a large number of ideal simulated BGSs with relative frequency ranges, random SNRs, BFSs and spectral widths (SWs) are generated as the training dataset in order to improve the universality and robustness of the WNN in different actual cases. Based on the above-mentioned parameters, a general model of the WNN is trained and finally built, which can be implemented in more different scenarios than the reported ANNs [16–23]. Based on the general model of the WNN, both the simulated and measured results show that it possesses better accuacy and slighter fluctuation of the extracted results in most cases, especailly in the cases with low SNR. Meanwhile, it verifies that the proposed WNN has better robustness and flexibility. Furthermore, the BFSs extracted by the WNN show a great linearity to the ambient temperature, and the processing time of the temperature measurement along the 18 km single-mode (SM) fiber can be reduced to around 0.54 s.

2. BOTDR experimental setup

The experimental setup of the BOTDR sensing system is shown in Fig. 1(a), in which the SpBS spectra are measured by a self-heterodyne detection combined with an electrical coherent demodulator. A single optical source generated by a continuous wave (CW) tunable laser operating at 1550 nm is split into two branches. The CW signal from the upper branch is modulated by a pulse electro-optic modulator (EOM) to generate the pump pulses. An erbium-doper fiber amplifier (EDFA) is utilized to increase the peak pump power. Subsequently, a bandpass filter (BPF) is used for removing the amplified spontaneous emission caused by EDFA. After that, the optical pulses are then launched into the sensing fiber through a tri-port optical circulator (OC). The CW light from the lower branch is modulated by a microwave EOM to generate a carrier-suppressed dual-band signal, which can be regarded as a local reference optical oscillator. A radio frequency (RF) signal is used to drive the EOM, where the scanning range of the output signal frequency is around the BFS. Subsequently, a polarization scrambler (PS) is adopted to suppress the polarization-dependent noise. The SM optical fibers with different lengths are used as the fiber under test (FUT) for sensing. Part of that is placed in the temperature-controlled chamber (TCC) for the further temperature measurement.

 

Fig. 1. The BOTDR experimental setup for temperature measurement. EOM: electro-optic modulator, EDFA: erbium-doped fiber amplifier, BPF: bandpass filter, OC: optical circulator, FUT: fiber under test, TCC: temperature-controlled chamber, RF: radio frequency, PS: polarization scrambler, PD: photodetector, LPF: lowpass filter, A/D: analog-to-digital converter, WNN: wavelet convolutional neural network.

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At the receiver side, the backscattering light along the FUT is mixed with the depolarized CW optical local oscillation signal, whose power must be below the saturation point of the photodetector (PD). The electrical beating signal is selected by a BPF and then demodulated by the electrical coherent detection scheme. An electrical local oscillator (ELO) at a certain frequency close to the BFS of the sensing fiber is used to scan the Brillouin spectrum with a bandwidth of several hundreds of megahertz. Then a low-pass filter (LPF) is employed to get the baseband of the SpBS spectra. The beating signals recorded by an analog-to-digital (A/D) convertor pass through a BOXCAR [27] to improve the SNR level by the signal accumulation and averaging. Finally, the BFS and the actual temperature are going to be extracted by our proposed algorithm.

3. Principle and method

3.1 Noise analysis

To demonstrate the feasibility of the proposed WNN in the BOTDR experimental system, the noise analysis in frequency-domain and time-domain is significant. Here, a 4 km SM fiber is used to collect system noise by turning off the CW tunable laser. The scanning frequency ranges from 11.0 GHz to 11.198 GHz with a step of 2 MHz. The average time is set as 128 times, and the sampling rate of A/D converter is 100 MHz with the bit resolution of 9 bits.

In order to generalize the characteristics of the measured noise in time-domain, we analyze the noise distribution along the sensing distance and observe them at each frequency point within the frequency scanning range. Taking the specific frequency point at 11.08 GHz as an example, it clearly shows that the noise along the sensing distance changes randomly with a DC amplitude of 28.3797 mV, as shown in Fig. 2(a). As shown in Fig. 2(b), the mean value of the noise stays at zero in time-domain. Meanwhile, its variance varies randomly within a certain range from 3.54 to 4.32. It’s worth noting that the results are calculated by subtracting the DC amplitude of the noise. Besides, we calculate the autocorrelation function (ACF) of the noise to characterize its correlation in time-domain. Figure 2(c) shows the ACF results at 11.08 GHz. It can be seen that the correlation coefficients are constrained to the confidence interval whose range is within $\pm 0.0316$. The ACF results along the 4 km fiber at each frequency point are all depicted in Fig. 2(d), which proves that the noise in time-domain is white. The frequency histogram of the noise without its DC amplitude is further calculated to show its distribution, as shown in Fig. 2(e). In order to derive the probability density function (PDF) of the noise at 11.08 GHz, the kernel density estimation (KDE) [28,29] as the non-parametric method is used and its estimation result is depicted as the green line in Fig. 2(e). Meanwhile, we also use its corresponding mean value and variance value to generate the Gaussian distribution as the dotted red line. Obviously, the PDF $p_1(x)$ estimated by KDE is close to the Gaussian distribution $p_2(x)$. Furthermore, the difference between the estimated PDF and the Gaussian distribution is calculated by $\sqrt {(p_1(x)-p_2(x))^{2}}$ and it can be used to evaluate the performance of the estimated PDF. Figure 2(f) shows that the difference at each frequency scanning point is very little and the average of it is equal to 0.0036. The aforesaid characteristics all prove that the noise in time-domain satisfies the typical distribution of Gaussian white noise, which indicates that the measured noise along the sensing distance could be considered irrelevant.

 

Fig. 2. The analysis of the system noise in time-domain. (a) Noise distribution of the sensing distance at the specific frequency point of 11.08 GHz. (b) Variance (bule, left axis) and mean value (orange, right axis) of the noise in time-domain at each frequency scanning point. (c) Autocorrelation function (ACF) of the noise in time-domain at 11.08 GHz. (d) ACF in time-domain at each frequency scanning point. (e) Probability density function (PDF) of the noise in time-domain at 11.08 GHz. (f) Difference between the PDF by kernel density estimation and Gaussian distribution at each frequency scanning point.

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In addition, we analyze the noise distribution in frequency-domain at each sensing position along the fiber. At the sensing distance of 3.8 km, the frequency distribution is shown in Fig. 3(a) and its DC amplitude is 28.3284 mV. Similar to Fig. 2(b), Fig. 3(b) shows that the mean value in frequency-domain along the transmission distance also stays at zero and the fluctuation range of variance is from 2.27 to 5.99. However, as shown in Fig. 3(c), the ACF results calculated in frequency-time shows that the correlation coefficients are constrained to the confidence interval whose range is within $\pm 0.2043$. The ACF results in frequency-domain at each position of the sensing distance are further shown in Fig. 3(d). It illustrates that the noise distribution in frequency-domain has strong correlation and it will immensely affect the measured BGSs distributions. Besides, the frequency histogram, the KDE results and the corresponding Gaussian distribution at 3.8 km are all depicted in Fig. 3(e), which shows that the estimated PDF are quite different from the typical Gaussian distribution. The difference at each point of the sensing distance in frequency-domain is also calculated in Fig. 3(f) and the average level is 0.0117. It can be clearly seen that the noise is frequency-related and it doesn’t satisfy the Gaussian distribution.

 

Fig. 3. The analysis of the system noise in frequency-domain. (a) Noise distribution of the frequency scanning range at the distance point of 3.8 km. (b) Variance (bule, left axis) and mean value (orange, right axis) of the noise in frequency-domain at each position of the sensing distance. (c) ACF of the noise in frequency-domain at 3.8 km. (d) ACF in frequency-domain at each position of the sensing distance. (e) PDF of the noise in frequency-domain at 3.8 km. (f) Difference between the PDF by kernel density estimation and Gaussian distribution at each position of the sensing distance.

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According to the above-mentioned characteristics of the measured system noise, the noise distribution along the sensing distance can be regarded as Gaussian white noise and its impact on the measured BGSs in time-domain can be neglected. Therefore, in the following parts, the measured BGSs are going to be analyzed as the 1D frequency sequences.

3.2 Proposed algorithm: WNN

As illustrated in Fig. 4, the proposed WNN consists of two networks, named as the convolutional network and the wavelet network, respectively. At first, the measured BGSs regarded as 1D frequency sequences are processed based on Z-score normalization [30], and then will be imported into the proposed WNN as the input.

 

Fig. 4. The architecture of the proposed algorithm: wavelet convolutional neural network (WNN).

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The convolutional network begins with an input layer with the size of $1 \times 100$. The number of 100 refers to the number of frequency scanning points. Here, such input data are sent into two convolutional paths and these convolution operations are used to filter the sequences by using different convolutional kernels [31]. As for the upper path, the data are processed by 16 convolutional kernels in one-dimension (Conv1D) with the size of $1 \times 3$ and stride $1$ to aim at generating 16 filtered sequences with the length of $98$. The batch normalization (BN) [32,33] is introduced to make the training faster and more stable through normalization of the layers’ inputs by re-centering and re-scaling. The rectified linear unit (ReLU) [34] activation function is used for adding nonlinear factors. After that, a max pooling layer with stride $2$ is used to reduce the length of filtered sequences by down-sampling with the maximum value, where the size of its output is $49\times 16$. Next, the data are filtered by 16 Conv1D with the size of $1 \times 4$ and stride $1$. The length of each filter sequence is 46. The BN, ReLU and max pooling are adopted subsequently, and the output is with the size of $23 \times 16$. Finally, filtered by 16 Conv1D of size $1 \times 2$ and stride $1$ together with the BN, the output data from the upper path is with the size of $22 \times 16$ and can be denoted as $Conv1$. Meanwhile, as for the lower path, we utilize 16 Conv1D with the size of $1 \times 7$ and stride $4$ to realize the generation of the filtered sequences with the size of $24 \times 16$. Furthermore, getting through BN, ReLU and max pooling, the data are filtered by 16 Conv1D with the size of $1 \times 3$ and stride $1$. The output from BN termed as $Conv2$ is with the size of $22 \times 16$, which is equal to $Conv1$. The above-mentioned two paths of the convolutional network are used to build a residual connection [35], in which the output of the connection denoted as $Conv$ is calculated by the formula $Conv = ReLU(Conv1+ReLU(Conv2))$. The residual operation could effectively avoid the problem of vanishing gradients and mitigate the degradation problem of the trained network.

The wavelet network [36–38] is an improved full-connection neural network and it contains an input layer, four hidden layers and an output layer. The size of the input layer is 352, corresponding to the size of the matrix $Conv$. Four hidden layers are with the size of 128, 84, 64 and 20, respectively. The output of the wavelet network gives the extracted BFS. In order to obtain more local characteristics in frequency-domain, the wavelet transform method [39] is used to realize the multi-scaled analysis and the scale translation. Therefore, all activation functions of the output neurons are various wavelet functions instead of the classic sigmoid functions. In the network, each wavelet activation function needs to be trained for extracting different local characteristics from the sequences adaptively. The equation of the wavelet activation function $f$ could be expressed as Eq. (1):

3.3 Model training

Owing to that the measured BGSs meet the Lorentzian curves theoretically, we can generate ideal BGSs $g(v)$ for training the model according to the following Eq. (2):

In the common Brillouin sensing system, the frequency scanning ranges of the measured BGSs are not the same, which mainly attributes to the physical properties of sensing fibers and the different operating wavelength. To improve the flexibility of the proposed WNN, the length $N$ of generated BGSs is set as 100. Besides, the frequency scanning range could be unified from $(Base+0)$ MHz to $(Base+2\times (100-1))$ MHz with a fixed step of 2 MHz. Although the value of $Base$ would be changed according to different experimental settings, it could not affect the relative values of the extracted results. In this way, the WNN could handle different BGS traces with different frequency ranges by converting an absolute frequency range to a relative frequency range [0 MHz, 2$\times$(100-1) MHz], hence, its universality could be improved. In order to further enhance the universality and robustness of the model, the BGSs are generated with the random BFS and SW, where the random range of the BFS is set from 20 MHz to 178 MHz with a step of 0.5 MHz and the range of the SW is from 20 MHz to 70 MHz with a step of 2 MHz. Meanwhile, the peak amplitude $g_B$ ranges randomly from 0.8 to 1.2. According to the Eq. (3), the Gaussian white noise $n(v)$ will be added to the ideal BGSs with a random SNR, where the range of SNR is from 1 to 21 dB with a step of 2 dB.

Moreover, the Adam [40] is adopted to optimize the model parameters according to the training data. The training process is based on the Python environment running on a computer with an AMD Ryzen 9 3950X 16-core processor and a Nvidia TITAN RTX GPU with TU102-core.

4. Results and discussion

To fully demonstrate the performance superiority of the proposed WNN, the data are going to be processed by three different methods, respectively. Other two methods are the LCF based on Levenberg-Marquardt (LM) algorithm [41] and the basic NN [18] containing input and output layers together with two hidden layers. For LM algorithm, the parameters are continuously evolved throughout the iterative process until iteration step is less than the stopping criteria, which is set as $10^{-8}$. The layout of the optimized NN is finally trained to be 100-40-10-1, where the training environment, the training dataset and the testing dataset are all same as the WNN.

The root-mean-square error (RMSE), standard deviation (SD) and processing time are considered as the vital evaluation parameters. According to Eq. (4), the RMSE characterizes the difference between the extracted BFS or temperature and its true value. Besides, the SD characterizes the fluctuation of the extracted BFS or temperature as Eq. (5):

In the following subsections, we generate the ideal BGSs with different parameters including various SNRs, BFSs and SWs to test the trained model as shown in Table 1, and the frequency ranges of the ideal BGSs are set as a relative range from 0 MHz to 198 MHz with the fixed step of 2 MHz. The value of $Base$ could be measured in the actual experiment according to the different sensing fibers. In the subsection of simulation results, the performances of the BFSs extraction using the LM, NN and WNN are demonstrated. Furthermore, the actual BGSs distributed throughout the 18 km SM optical fiber are measured in the BOTDR system. The BFSs extracted from the measured BGSs by the LM, NN and WNN are linearly fitted to the real temperatures, respectively. Finally, the performances of the temperature measurement by these three methods are also discussed.

Table 1. Test parameters for simulation

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4.1 Simulation results

Considering that the SNRs, BFSs and SWs have different effects on the extracted results, we first evaluate the performances of the extracted BFSs from each parameter. Figure 5(a)–5(c) show the performances of three methods at different SNR levels, where the BFS of simulated BGSs and the SW of those are set as 90 MHz and 40 MHz, respectively. For each SNR in Table 1, 10,000 BGSs are generated by adding Gaussian white noise. As shown in Fig. 5(a) and 5(b), the RMSE and SD decrease as the SNR increases. The RMSE and SD of the BFS extracted by the WNN are better than those extracted by the LM and the NN, when the SNR is lower than 11 dB. As for the situations of the SNR higher than 11 dB, the accuracy and the fluctuation level of the BFS extracted by the LM and the NN are closer to that by the WNN as the SNR increases. It shows that the BFSs extracted by the proposed WNN have better accuracy and slighter fluctuation than other two methods when the SNR is less than 11 dB. Figure 5(c) shows that the processing time of the LM decreases with the increase of the SNR, which means that the iterations of the LM will be decreased due to the growth of SNR. However, it still takes at least 156,766 ms to run the calculation. In comparison to the LM, the processing time of the WNN is going to be greatly reduced by three orders of magnitude and its handling time can be less than 221 ms. Besides, the calculating time of the NN is around 9 ms due to its simpler architecture than the WNN. The time difference at one order of magnitude with the unit $ms$ between the NN and the WNN is still acceptable, considering the improvement of accuracy and fluctuation level.

 

Fig. 5. The RMSE and SD of the BFSs extracted by the LM, the NN and the WNN methods with the simulation data, together with the processing time of these three methods. (a) RMSE, (b) SD and (c) processing time at different SNR levels; (d) RMSE, (e) SD and (f) processing time with different BFS data; (g) RMSE, (h) SD and (i) processing time with different SW data.

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Besides, to verify the impact of different BFS data on the performances of three different methods, the SNR and SW are set as 11 dB and 40 MHz, respectively. For each BFS in Table 1, 10,000 simulated BGSs are further generated for testing. Figure 5(d) and 5(e) illustrate that the corresponding processing results of the WNN are better than those of the LM for all cases. Compared to the NN, the processing results of the WNN are better under most cases. Additionally, the NN and the WNN shows better superiority in the aspect of the tolerance than the LM at several certain points. Meanwhile, the RMSE and the SD results of the BFS extracted by three methods all nearly maintain good consistency for different BFS data. As illustrated in Fig. 5(f), the processing time of the LM increases as the BFS increases, and its values remain at a level above 121,511 ms. By contrast, similar to Fig. 5(c), it merely takes about 209 ms for the WNN to process the same amount of data and the NN requires about 7 ms.

We also evaluate the impact of various SWs on the performances of the LM, NN and WNN, where the SNR and BFS are set as 11 dB and 90 MHz, respectively. For each SW in Table 1, we generate 10,000 simulated BGSs to show such performances of three methods. As shown in Fig. 5(g) and 5(h), the accuracy of the BFS extracted by these three methods all decreases with the increasing of SW. The RMSE and SD of the extracted BFS by the WNN are smaller than those by the LM and the NN, which verifies that the proposed WNN has superiority in terms of the computational accuracy and the fluctuation level. The results shown in Fig. 5(i) indicate that the LM takes more processing time as the SW increases, which is owing to that the LM has the smaller gradient to search for the BFS when the value of SW is larger. Similarly, the WNN still takes about 210 ms and the NN requires around 9 ms. In conclusion, to process 10,000 simulated BGSs under different circumstances, the time consumption of the WNN would be all reduced by three orders of magnitude compared with the LM. Meanwhile, its calculating time would be comparable to the NN, which can be seen from Fig. 5(c), 5(f) and 5(i).

Furthermore, in order to analyze the joint effects of all parameters given by Table 1, the RMSE results of the extracted BFS by three different methods are depicted as the heat maps in Fig. 6(a), 6(b) and 6(c), respectively. Likewise, the SD results of that are also shown in Fig. 7(a), 7(b) and 7(c), respectively. The darkest red patch represents that the value of RMSE or SD is greater than 3.2, which indicates that the existing RMSE or SD performance of the extracted BFSs is not enough to support its practical applications. Additionally, the darkest blue patch means that the value of RMSE or SD is zero, indicating that the results are with excellent accuracy. In addition, the median value 1.6 is drawn as a white patch, which can be regarded as the performance threshold. Considering all the RMSE results in Fig. 6, it can be found that the BFSs extracted by the WNN possess better accuracy in most cases than NN. In the cases of the SNR less than 11 dB, the accuracy of the BFSs extraction by the WNN is better than that by the LM. Meanwhile, the accuracy of WNN is similar to that of the LM when the SNR is higher than 11 dB. As shown in Fig. 7, nearly all the SD results of the BFSs extracted by the WNN are better than other two methods. It means that the fluctuations of the BFSs extracted by the WNN are slighter than other two methods in all cases. Besides, the SD results of the BFSs extracted by the NN are better than LM when the SNR is less than 11dB. In conclusion, the simulated results verify that the WNN is with better accuracy and slighter fluctuation, and can be used for many more extreme cases, especially for the low-SNR scenarios.

 

Fig. 6. The comparison between the RMSE of the BFSs extracted by (a) the LM, (b) the NN and (c) the WNN.

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Fig. 7. The comparison between the SD of the BFSs extracted by (a) the LM, (b) the NN and (c) the WNN.

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4.2 Experimental results

In order to realize the temperature extraction experimentally, we collect the distributed BGSs at different temperatures and then use the LM, the NN and the proposed WNN methods to run the calculations for comparison. The system parameters of BOTDR sensing system have been given by Table 2. The pulse width is set as 100 ns corresponding to a spatial resolution of 10 m, and the average time is 128 times. The sampling rate of the A/D converter is set as 100 MHz with the bit resolution of 9 bits. The frequency scanning range is from 11 GHz to 11.198 GHz with a step of 2 MHz. The 18 km SM fiber is used for carrying out the experiments, which is constructed by several FC/APC connectors, three 4 km optical SM fibers and two 3 km optical SM fibers. The measured BGSs distribution along 18 km fiber are normalized as shown in Fig. 8(a). At the connecting positions, the signal glitches could be observed due to the edge reflection. The last 300 m optical fiber is placed in the TCC to measure the temperature in real-time, and the corresponding shifted BGSs caused by the change of temperature are enlarged in Fig. 8(b). Besides, considering the existed DC amplitude of the measured BGSs, the SNR of the actual BOTDR system could be calculated by the Eq. (6):

 

Fig. 8. (a) Normalized BGSs distributions along the 18 km SM fiber in BOTDR. (b) Enlarged view of the normalized BGSs distributions from 16.5 km to 17.9 km. (c) The SNRs along the sensing distance.

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Table 2. System parameters of the experimental setup

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As shown in Fig. 9(a), 9(b) and 9(c), the BFS results along the fiber calculated by three methods at different temperatures are depicted, respectively. It’s worth noting that the local distributions of the 300 m heated part are also shown clearly in the sub-graphs placed in Fig. 9(a), 9(b) and 9(c). Similar to the LM and the NN, the results processed by the WNN show a clear increasing trend with the increase of the temperature in the TCC, which shows its potential in monitoring the change of ambient temperature. Notably, the distribution of the BFSs extracted by the WNN is much smoother than other two methods.

 

Fig. 9. The BFS distributions along the fiber obtained by using (a) the LM, (b) the NN and (c) the WNN with the measured BGSs when the fiber end was heated to 47.80 $^{\circ }$C, 52.44 $^{\circ }$C, 56.80 $^{\circ }$C, 61.62 $^{\circ }$C, 65.82 $^{\circ }$C, 70.10 $^{\circ }$C, respectively; insets: local BFS distributions of the 300 m heated part.

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In order to validate the feasibility of the WNN for the temperature measurement, we linearly fit the extracted BFS obtained by three methods relative to the temperature value, as shown in Fig. 10. The temperature coefficient $C_T$ is calculated to be 1.11033 MHz/$^{\circ }$C, 1.21061 MHz/$^{\circ }$C and 1.10592 MHz/$^{\circ }$C by linear fitting for the LM, the NN and the WNN methods, respectively. As a result, the BFSs obtained by the WNN have the good temperature linearity, which is similar to those obtained by the LM. Besides, the temperature coefficient calculated by the NN is larger than that by the LM and WNN. Furthermore, we use the determination coefficient $R^{2}$ to evaluate the degree of linear fitting quantificationally, according to the Eq. (7):

 

Fig. 10. The BFS as the functions of the temperature and the results of the linear fitting for LM, NN and WNN, respectively.

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Subsequently, the temperature distributions along the fiber can be measured by the linearly-mapping of the extracted BFSs. Figure 11(a) shows the detailed temperature distributions along the heated part. The RMSE and the SD results of the temperature measured by three different methods are further calculated, as shown in Fig. 11(b) and 11(c), respectively. The RMSE results of the temperature measured by the WNN are smaller than that of other two methods, which proves that the accuracy of the WNN is the best of the three. Meanwhile, the RMSE results calculated by the NN are also less than LM, which means it also has better accuracy than the LM. Correspondingly, the SD results of the temperature measured by the WNN are also slighter than the LM and the NN, which verifies that the WNN has the slighter fluctuation than other two methods. In addition, considering the time consumption in the actual implementation, the processing time of the 18,000 collected BGSs along the 18 km fiber by three different methods are compared, which run on the same computing platform. The LM requires at least 402.898 s while the NN takes only 9 ms. Since the architecture of the proposed WNN is more complex than the basic NN, the WNN requires about 0.54s to complete the measurement. Although more time cost is introduced into the experiment, the accuracy and the fluctuation level of the WNN are well improved, which are also the benefits brought about by its more complex network structure.

 

Fig. 11. The extracted temperature in BOTDR sensing system. (a) The temperature distributions obtained by using LM (dotted circle lines), NN (dotted square lines) and WNN (solid lines) when the fiber is heated to 47.80 $^{\circ }$C, 52.44 $^{\circ }$C, 56.80 $^{\circ }$C, 61.62 $^{\circ }$C, 65.82 $^{\circ }$C, 70.10 $^{\circ }$C, respectively. (b) The RMSE of the temperature extracted by LM, NN and WNN. (b) The SD of the temperature extracted by LM, NN and WNN.

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5. Conclusion

In this paper, we analyze the characteristics of the noise distribution in time-frequency domain. The noise along the time-related sensing distance meets the typical distribution of Gaussian white noise. In addition, the noise in frequency-domain has the strong correlation. Hence, the impact of the noise in time-domain on the measured BGSs could be neglected and the measured BGSs along the fiber can be regarded as 1D input data of the proposed WNN.

Moreover, a signal processing method based on the proposed WNN containing a 1D convolutional neural network and a self-adaptive wavelet neural network are demonstrated. The 1D convolutional neural network is used to filter the noisy BGS traces and capture their features. Besides, the self-adaptive wavelet activation functions are used at each output of the full connection network in order to achieve the covering of more local characteristics in frequency-domain. Hence, the multi-scaled analysis and scale translation in frequency-domain could be realized adaptively in the hidden layers. In further, we generate a large number of ideal simulated BGSs with random SNRs, BFSs, SWs and relative frequency ranges, in order to train and finally build a general model for the SM fiber sensing in the BOTDR system.

The performances of the optimized WNN have been compared with the existing LM and basic NN method comprehensively. Both the simulated and experimental results show that the WNN has better robustness and universality to handle more extreme cases, and especially possesses the superiorities in the cases of lower SNR. The WNN only takes approximately 0.54 s to measure the temperature with 18,000 BGS traces along the 18 km SM sensing fiber. Compared with the time consumption of the LM, that of the WNN can be reduced significantly by three orders of magnitude. Meanwhile, its time cost would be increased than the basic NN due to the more complex network structure, but still be comparable to the NN. Nevertheless, the accuracy and the fluctuation level of the extracted results utilizing the WNN can be well improved than other two methods.

In conclusion, this work verifies that the proposed WNN might be an advisable solution to be utilized in the robust and fast temperature monitoring scenarios, which benefits from its advantages in terms of robustness and flexibility, high accuracy in the low SNR cases, and relatively-high processing speed.