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Abstract
Distributed optical fiber Brillouin sensors can monitor the temperature and strain along a fiber by estimating the Brillouin frequency shift (BFS) according to the measured Brillouin spectrum. The system performance is highly dependent on the algorithm of BFS extraction. The well-established Lorentz curve fitting (LCF) method is generally employed because the Brillouin spectrum theoretically satisfies a Lorentz shape. Recently, machine-learning methods have been proposed for more effective BFS extraction, but they have some drawbacks and limitations. The machine-learning algorithms require a large amount of data and high computing power to find suitable extraction methods. However, with prior knowledge, Brillouin spectrum can be treated as a regular signal that requires only three degrees of freedom to define. The unique sparsity characteristics of Brillouin spectrum have not been well studied or exploited. In this paper, we propose a sparse representation method for Brillouin spectrum that extracts three sparse features of the Brillouin spectrum through the dictionary-learning algorithm (K-means singular value decomposition). The correlation between the sparse coefficient and the BFS is experimentally calibrated and verified. The accuracy of the proposed algorithm is comparable to that of LCF, and its processing is six times faster. This sparse representation method for Brillouin spectra is promising as an alternative universal BFS extraction method for distributed Brillouin sensors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Distributed optical-fiber Brillouin sensors have attracted great interest owing to their capability to sense distributed strain and temperature. They are widely used in the detection of oil and gas pipeline leakage, monitoring of bridge safety, and fire alarms [1–5]. Among various Brillouin sensing technologies, Brillouin optical time-domain analysis (BOTDA) is one of the most widely used. In a BOTDA system, the Brillouin gain spectrum (BGS) at each position of the fiber is obtained by scanning the frequency difference between probe and pump light. The frequency difference corresponding to the peak gain of BGS is equal to the Brillouin frequency shift (BFS), which is linearly related to the strain and temperature [6,7]. To obtain the BFS from the measured BGS, Lorentz curve fitting (LCF) is generally used because the BGS theoretically has a Lorentzian shape [8]. However, LCF takes significant time to process the data and has a large error when the signal-to-noise ratio is low [9]. Partial quadratic fitting method can reduce the processing time. However, the BFS’s accuracy using fitting method is highly sensitive to the setting of initial parameters [10]. The cross-correlation method improves upon LCF and calculates the frequency difference to obtain BFS by convolving the ideal BGS with the noisy BGS [11,12], which is more accurate, noniterative and free from initial values. Due to its great advantages, many novel methods based on cross-correlation have been proposed. By adding moving average before cross-correlation and replacing the reference curve with triangular pulse, 95% performance improvement has been achieved with the software and hardware solutions [13]. Interpolation algorithm based on cross-correlation technology is introduced to estimate BFS, which greatly reduce the computational cost [14]. However, these methods suffer from a trade-off between the scanning frequency spacing and accuracy, leading to higher requirements on the frequency-sweep interval than LCF.
Recently, many algorithms have been proposed to extract BFS directly without any curve-fitting process. Clustering and classification algorithms, such as principal component analysis and support vector machine, have been reported for temperature extraction with good performance [15–17]. However, a trade-off always exists between the number of classes or principal components and the algorithm performance. Accurate extraction depends on subdivided classes or principal components and large storage databases. On the other hand, artificial neural networks have been proven effective in extracting BFS [18–20]. However, machine-learning algorithms need an intensive training process in advance, which is relatively time-consuming. Moreover, retraining or fine-tuning is required to accommodate different types of actual data, which limits its application potential.
Based on the characteristics of the BOTDA system, it was found that BGS is sparse in a discrete-cosine-transform domain. Using this feature, a compressed sensing technique was proposed to recover BGS from far fewer measurements [21]. However, BGSs in a discrete cosine basis do not have the best sparsity. The physical characteristics of BGS indicate that it should have a limit of 3-sparseness under the appropriate dictionary, which implies that the spectrum can be uniquely determined by extracting the sparse parameters.
In this paper, we propose a novel analytical model for sparse representation in BOTDA systems. Considering that three parameters can describe a BGS as a Lorentz curve, the signal is deemed to be represented as 3-sparse under the transformation dictionary. We use a dictionary-learning algorithm called K-means singular value decomposition (K-SVD) to search for the best dictionary for the sparse representation of BGSs. After two-step linear dictionary learning, one sparse parameter corresponding to BFS can be extracted to monitor fiber-temperature changes. The uncertainties in average temperature obtained using LCF and K-SVD along the fiber are found to be 0.3134 °C and 0.3211 °C, respectively. The extraction accuracy of the proposed method is almost the same as that of LCF, but it requires only one-sixth of the processing time. Unlike other machine-learning methods, the proposed dictionary-learning method uses prior information on BGS and obtains sparse feature parameters without a training set. Therefore, it is a universal extraction method based on BGS’s sparseness, and is applicable to any Brillouin sensors.
2. Principle
2.1 Sparse representation of BGS
Sparse representation is a method of representing signals with as few atoms as possible in a given hyper-complete dictionary. It is widely used in image denoising [22], pattern recognition [23,24], target tracking [25], and compressed sensing [26]. Signal representation is generally based on orthogonal linear transformation. However, most natural signals are superpositions of multiple types of data, which cannot be expressed by a single orthogonal basis. Recently, a series of overcomplete redundant functions named as dictionaries were shown to surpass the fixed orthogonal basis for sparse representation [27]. The atomic structure of a dictionary is designed to match the characteristics of a signal. The best linear combination of certain atoms is used to represent the signal. Mathematically, its solution is [28]:
where Y denotes the signal to be represented; X is the coefficient matrix to match Y as sparsely as possible; and D is the designed over-complete dictionary matrix; ${||X ||_0}$ is the l0 norm, which is the number of non-zero components in the X.In a BOTDA system, if the attenuation of an acoustic wave is considered, the Brillouin gain spectrum has a Lorentz spectrum profile and can be expressed as [29]:
On the other hand, the Lorentzian shape of BGS implies that it can be expressed in three degrees of freedom in a certain space. Generally, the sparsity is related to the number of independent variables of the signal. Sparsity representation can be understood as finding a new non-redundant coordinate system to redefine the original signal. Considering the independence of the three parameters in Eq. (2), we assume that the optimal sparsity of the signal is 3. Then, the goal is to solve three characteristic parameters at each position. The transformation from a BGS to the three characteristic parameters can be considered as a limit-sparseness processing of the signal. The objective function of this sparse representation for BGSs is
2.2 Sparse solution based on K-SVD dictionary-learning
To find the sparse representation of BGS, designing an appropriate over-complete dictionary D is essential. In general, D can be chosen from a prespecified set of linear transforms or designed by adapting its atoms to fit signals. Linear transformations using a fixed basis, such as the discrete cosine transform mentioned above, are concise and convenient. However, BGS cannot be transformed into a 3-sparse signal under any fixed basis. We choose the dictionary-learning algorithm K-SVD to design a dictionary. K-SVD is similar to the K-means method. K-means clustering is an unsupervised learning method based on the nearest-neighbor principle [30,31], which has no requirements for the type of data. This method arbitrarily finds K points and sets these data points as the initial cluster center, following which the Euclidean distance between other sample points and the center point is calculated. The non-clustered center points are categorized as a class in which the nearest cluster center is located. The object of K-means processing is a series of points, while K-SVD processes signals with sparsity. When the K-SVD algorithm approximately represents each signal sampling as a 1-sparse signal, it degenerates into a K-means clustering algorithm. As shown in Eq. (5), if we set a penalty term with the l2 norm for the sparse representation of BGSs, the problem is transformed into a problem of solving the shortest Euclidean distance:
Our K-SVD algorithm involves the following three steps. After these three steps, K-SVD can seek a proper dictionary to represent BGSs that strictly adhere to the sparse constraints.
- 1) Initialize dictionary D randomly. Meanwhile, initialize coefficient matrix X as an all-zero matrix.
- 2) Fix dictionary D and then calculate the sparse coding of all samples. Eq. (4) is split into N parts as follows:$$\mathop {\min }\limits_{{x_i}} ||{{g_i} - D{x_i}} ||_2^2 \quad subject \; to \; \forall i,{||{{x_i}} ||_0} = 3 ( i = 1,2,\ldots k).$$Eq. (7) can be considered as a typical compressive sensing problem, and the sparse solution X can be solved using the orthogonal matching pursuit algorithm [34].
- 3) Update dictionary D. Only one column of D is updated in each iteration while assuming that the other columns of D and the coefficient matrix X are fixed. The objective function for updating the k-th atom is rewritten as follows:$$||{g(\upsilon ,z) - DX} ||_2^2 = \left|\left|{g(\upsilon ,z) - \sum\limits_{j = 1}^k {{\alpha_j} \cdot {x_j}} } \right|\right|_2^2 = \left|\left|{(g(\upsilon ,z) - \sum\limits_{j \ne k} {{\alpha_j} \cdot {x_j}) - {\alpha_k} \cdot {x_k}} } \right|\right|_2^2 = ||{{E_k} - {\alpha_k} \cdot {x_k}} ||_2^2,$$where αk is the k-th atom of D; xk is the corresponding k-th row of X that needs to be updated; and Ek is the error produced by the other k - 1 atoms sparsely decomposing Y without using the k-th atom. This equation suggests that the matrix DX can be divided into fixed k - 1 terms and one unknown term αk xk. Here, we use the SVD of Ek to obtain these k - 1 matrices and solute αk and xk. The SVD of matrices is widely used in the identification of linear dynamic systems as the optimal approximation [35,36]. The SVD of an $m \times n$ matrix Ek with rank r is defined as: where V is an orthogonal $n \times n$ matrix, U is an orthogonal $m \times m$ matrix, and Σ is an $m \times n$ diagonal matrix with singular diagonal elements. Further, the SVD of Ek can be represented by the following sum: where ${\sigma _i}$ is the i-th singular value arranged by its numerical value and ui and vi are the i-th columns of the matrices U and V, respectively. As mentioned above, the least-squares solution of $||{{E_k} - {\alpha_k} \cdot {x_k}} ||_2^2$ can be obtained using the SVD of Ek, the first main vector of the left singular matrix U is used as the updated dictionary atom αk, and the product of the first main vector of the right singular matrix V and the largest singular value is taken as the updated sparse vector xk. Steps 2) and 3) are repeated until all the atoms and the sparse coding of all the signals are updated. The final solution D is the dictionary of sparse representations, and X is the sparse coefficient to be addressed.
2.3 Signal processing and feature extraction
To extract the BFS, the BGSs obtained from the BOTDA system are processed as shown in Fig. 1. Since SVD is a linear decomposition, K-SVD is more effective for extracting linear and dependent data. However, the BGS is non-linear with respect to the linewidth and BFS of the Lorentz curve; hence, BGSs cannot be separated in one dictionary-learning step. Therefore, we propose a two-step linear dictionary-learning process.
Fig. 1. Schematic diagram of two-step signal processing for measured BGSs: the first dictionary-learning output Gain-ksvd and the second dictionary-learning output BFS-ksvd.
The input of the first dictionary-learning step is the measured BGSs, which can be split into one-dimensional matrices corresponding to the BGS at each sampling point. The curve at each sampling point has its own sparsity. Therefore, it can be represented by the product of a dictionary and a column of the sparse coefficient matrix. According to Eq. (2), BGSs have a linear relationship with the peak gain. The measured distributed BGSs can be written in the following matrix form:
In Eq. (12), ${\upsilon _i}$ is the i-th scanning frequency; ${\upsilon _{Bj}}$ and $\Delta {\upsilon _{Bj}}$ are the BFS and linewidth at the j-th fiber position, respectively. In Eq. (11), the gain coefficient gB is a diagonal matrix, while the changes of the other two variables are merged into matrix A. The linewidth and BFS at each position simultaneously affect the value aij. In the first step, the sparsity should be set to 3 because there are three variables. Subsequently, the gain coefficient and two cross terms between the BFS and linewidth are obtained.
Based on the above procedure, the gain coefficient can be separated in the first step, which leaves only matrix A to be further considered in the second step. To effectively extract BFS, we need to process the raw BGSs in advance to ensure the linear relationship between the BFS and input. For this purpose, the gain coefficient obtained in the first step is used as the known value gB:
As expressed in Eq. (13), the second input signal is represented by three parts. The scanning frequency v in the first part changes equally at each sampling point. Therefore, it can be regarded as a constant in the process of updating the dictionary. The BFS has an exact linear relationship with the input, which means the learning algorithm can easily search for it. The reciprocal of the linewidth is the third part, which also needs updating. Therefore, the sparsity is set to 2 to reflect two variables in the second step. In Eq. (13), the BFS has a linear relationship with the processed data, while the linewidth is inversely related to the data. Therefore, the other sparse parameter cannot straightforward represent the linewidth. The linewidth can be obtained by further linear processing, but that will increase the computational complexity. As the BFS is what we are most concerned about, we will not further process signal to extract linewidth here.
3. Experimental verifications
Figure 2 shows the setup of the experimental BOTDA system. A narrow-linewidth laser with a center wavelength of approximately 1550 nm is used as the light source. The laser light is split into two branches by a 3-dB optical coupler. One branch is modulated with an EOM to generate carrier-suppressed dual-band probe light. A MS is used to drive the EOM with the output frequency sweeping from 10.5 GHz to 10.9 GHz in steps of 2 MHz. A PS is used to eliminate the effects of polarization-dependent gain fluctuation. The other branch passes through a SOA driven by an AFG to produce 20-ns-width pump pulses. The optical pulses are amplified by an EDFA followed by a BPF to remove amplified spontaneous emission before propagating into the fiber through an optical circulator. The longer wavelength of the probe light is obtained by an FBG and is converted into electrical signals by a 125-MHz PD. Finally, the electrical signals are acquired by an oscilloscope at 500 MSa/s and averaged 512 times. A single-mode-fiber spool of approximately 10 km is used for test. The 100-m section at the fiber end is placed in a TCC, which is set from 30 °C to 60 °C in steps of 5 °C, while the rest of the fiber is kept at a room temperature of 22 °C. In the experiments, both the LCF and K-SVD algorithms run on Matlab with an Intel Xeon Gold 6136 CPU to process 50000 BGSs.
Fig. 2. BOTDA experimental setup. EOM: electro-optic modulator, MS: microwave synthesizer, PS: polarization switch, TCC: temperature-controlled chamber, SOA: semiconductor optical amplifier, AFG: arbitrary function generator, EDFA: erbium-doped fiber amplifier, BPF: band-pass filter, FBG: fiber Bragg grating, PD: photodiode.
First, the TCC is set to 30 °C. The distributed BGSs along the FUT are measured and processed by LCF and the first dictionary-learning phase, respectively. Figure 3 shows the three sparse-representation parameters of BGSs obtained in the first dictionary-learning phase. The sparse-representation parameter of gain-ksvd along the fiber shows a trend remarkably similar to the gain variation obtained through LCF in Fig. 3(a) and 3(d). However, the other two parameters are inherently interacting; that is, the jitter of the linewidth in (c) and the BFS changes in (b) affect both parameters in (e) and (f), which require further separation.
Fig. 3. (a) Peak gain, (b) BFS, and (c) linewidth obtained by LCF fitting and (d) gain-ksvd and (e, f) two cross terms obtained in the first K-SVD dictionary-learning phase.
Subsequently, we measured BGSs at different temperatures and performed the first dictionary-learning phase. Figure 4 shows a detailed comparison of the gain obtained using LCF and the parameter gain-ksvd obtained using K-SVD dictionary learning at different temperatures. It should be noted that only three traces of the results at different temperatures are presented in the figure to compare the two methods clearly. It is clear that the variation trend of the gain coefficient at different temperatures can be reflected by the sparse parameter gain-ksvd extracted using K-SVD. Note that the numerical values of the gain coefficient and gain-ksvd obtained using the two methods are in different orders of magnitude because SVD is a universal method to solve the problem of mathematical extremes, and its results may not directly correspond to exact physical values. Therefore, the gain-ksvd should be treated as an indicator of the actual Brillouin gain. Here, we introduce a correlation coefficient r to quantify the similarity of the two obtained gains:
Fig. 4. Peak-gain distribution along the 10-km fiber obtained using LCF and the first K-SVD dictionary-learning phase for the BGSs when the 100-m section at the fiber end was heated to 30 °C, 35 °C, and 40 °C.
Table 1. Correlation coefficient between obtained gain using LCF and K-SVD
Figure 5(b) shows the sparse representation of BFS-ksvd along the fiber obtained in the second dictionary-learning step at different temperatures. The BFS results obtained using LCF are shown in Fig. 5(a) for comparison. Similar to the BFSs obtained using LCF, the sparse parameter obtained after the second K-SVD processing step shows a clear trend of increase with the temperature at the end of the FUT, which indicates its potential to reveal ambient temperature changes.
Fig. 5. BFS distributions along the fiber obtained using (a) LCF with the measured BGSs and (b) K-SVD with the second dictionary-learning step when the fiber end was heated to 30 °C, 35 °C, 40 °C, 45 °C, 50 °C, 55 °C, and 60 °C; insets: local BFS distributions in the section where the fiber was heated.
In order to validate the feasibility of sparse representation for Brillouin sensing, we linearly fit the original traces of the BFS obtained using LCF and the sparse representation parameter BFS-ksvd with respect to temperature, as shown in Fig. 6. The temperature coefficient CT is calculated as 1.03 MHz /°C and 0.0848 /°C with linear fitting for the LCF and K-SVD results, respectively. Therefore, the BFSs obtained with sparse representation has excellent linearity with temperature, similar to those obtained from the LCF.
Fig. 6. BFS and BFS-ksvd as functions of temperature and the results of their linear fitting for LCF (blue, left axis) and K-SVD (orange, right axis), respectively.
Furthermore, we use the coefficient of determination R2 to quantify the degree of linear fitting:
Subsequently, the temperature distributions along the fiber are extracted for both methods using the fitted linear relationships with temperature. Figure 7 shows the detailed temperature distributions along the heated fiber section. This figure demonstrates that sparse representation for BGSs can successfully extract the temperature distribution with the same performance as LCF. To objectively evaluate the accuracy of the temperature measurement, we calculate the standard deviation (SD) and uncertainty of the estimated temperature along the unheated fiber segments for repeated measurements. The uncertainty is computed through quadratic fitting of the SD. As shown in Fig. 8, the SD along the fiber fluctuates within the range of 0.1 °C to 0.7 °C, and the maximum SD can be controlled below 0.7 °C for both methods. The average uncertainties of the LCF and K-SVD results along the fiber are 0.3134 °C and 0.3211 °C respectively. Therefore, the performance of the K-SVD sparse representation of BGS for temperature sensing is comparable to that of LCF.
Fig. 7. Temperature distributions obtained using LCF (dotted lines) and K-SVD (solid lines) when the fiber is heated to 30 °C, 35 °C, 40 °C, 45 °C, 50 °C, 55 °C, and 60 °C.
Compared with LCF, the proposed K-SVD method is advantageous in terms of computational complexity and time consumption. The two-step K-SVD converges within only one iteration with a total time of 34.1845 s (first dictionary-learning step: 24.0735 s; second dictionary-learning step: 10.1110 s), which is much less than the time of 182.9979 s consumed when using the LCF method on the same computing platform. For a new Brillouin sensor with different BGS characteristics, different sparse dictionary is learned. The dictionary and coefficients are obtained at the same time, which means the dictionary is not learned in advance. And the processing time includes the time spent on the dictionary learning. Although, there are many advanced methods that have obvious advantages in processing time. The algorithm proposed here is a new attempt. It does not need to be initialized and has no special requirements for data. And we believe that the proposed algorithm has much room for improvement.
4. Conclusion
We propose a method for the sparse representation of BGSs to extract information along the fiber in BOTDA systems. With prior knowledge of the Lorentzian shape of the BGSs, the proposed method can convert the three characteristic parameters of BGSs into sparse singular values through dictionary learning. Two sparse coefficients corresponding to gain and BFS can be extracted through a two-step linear K-SVD algorithm. Consequently, the temperature distribution of the fiber can be successfully resolved from the sparse coefficient. The maximum uncertainty of the temperature extracted using K-SVD is 0.3211 °C along the fiber, while the maximum uncertainty with LCF for the same data is 0.3134 °C. The K-SVD method can achieve a temperature-monitoring performance comparable to that of LCF with much faster processing. Therefore, the sparse coefficient based on the BGS’s sparse representation could be exploited as alternative sensing metrics like BFS. With prior knowledge of the BGS, information can be extracted more quickly and efficiently. It is worth mentioning that the proposed method is different from other machine-learning methods in that it does not need the manual adjustment of parameters or a time-consuming offline training phase before use. We believe that it is a universal and efficient method to obtain the BFS. Furthermore, the results indicate that the sparse nature of BGS is valuable and deserves more attention for distributed Brillouin sensing.
Funding
National Key Research and Development Program of China (2018YFB1801205); National Natural Science Foundation of China (61722108, 61931010); Innovation Fund of WNLO.
Disclosures
The authors declare no conflicts of interest.
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