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Abstract
Compressive sampling theory asserts that certain signals can be recovered from far fewer samples than traditional methods use. We propose to enhance the performance of Brillouin sensing systems by improving the signal-to-noise ratio of the Brillouin spectra with random undersampled measurements of the original noisy Brillouin spectra. The number of acquisitions can be significantly reduced, and at the same time the measurement accuracy can be improved due to the increased signal-to-noise ratio of recovered Brillouin spectra measured based on compressive sampling principle compared to those measured directly by conventional methods. Experiments show that by performing ∼30% of the acquisitions that are required by conventional systems, over 7 dB signal-to-noise ratio enhancement can be obtained. Our proposal can be applied to any practical Brillouin sensing system whose performance can be enhanced by taking the advantages of recent advancements in computational methods without costly or sophisticated hardware modifications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Distributed optical fiber sensors are able to measure a spatially distributed profile of environmental quantities such as temperature, strain, pressure, etc., and this distributed-measurement capability offers unique advantage compared to conventional discrete sensing techniques, especially for the capability of continuous long-distance monitoring with a single unaltered optical fiber as the sensing element. Time-domain based Brillouin sensing technique, such as Brillouin optical time-domain analysis (BOTDA) or Brillouin optical time-domain reflectometry (BOTDR), has been proven to be one of the most popular distributed-sensing techniques that are able to measure temperature and strain over many tens of kilometers with moderate spatial resolution [1–4]. The sensing principle is to measure distributed Brillouin spectra along an optical fiber, and by determining the Brillouin spectral peak shifts, one can sense the ambient environmental variations. However, in order to obtain distributed Brillouin spectra, both the conventional BOTDA and BOTDR need to scan over a wide frequency range discretely around the Brillouin frequency of the optical fiber, so that a number of time-domain traces associated to each discrete frequency are measured. In addition, at each frequency, a large number of time-domain traces need to be averaged in order to achieve decent signal-to-noise ratio (SNR) since the backscattering signal is rather weak. Hence, it takes considerable time to complete one sensing measurement over rather long distance using Brillouin sensing systems. In recent years, much effort has been spent on developing fast BOTDA sensing techniques, such as slope-assisted method [5–7], frequency-comb-based sweep-free configuration [8–10], fast-frequency sweeping approach [11,12], and optical-chirp-chain scheme [13], etc. The aforementioned methods primarily focus on dynamic measurement with fast acquisition rate, and normally require expensive signal generation solutions to fulfill the objective [8–13], which also limits the sensing distance.
Inspired by recent advances in the computational and mathematical methods, which are having great impacts on areas where hardware design may have certain limitations, we propose to adopt compressive sampling (CS) principle to study the feasibility to reduce the number of frequency measurements for conventional Brillouin sensing systems without costly hardware modifications [14]. CS theory [15–17] asserts that certain signals can be recovered from far fewer samples or measurements than what the conventional methods use, thus it has the potential to be used to reconstruct Brillouin spectra at different locations along an optical fiber from much fewer number of measurements of the frequency components. We have shown in [14] that the number of measurements can be reduced by 70% compared to traditional method. However, that preliminary analysis only focuses on ideal Brillouin spectra without any measurement noise; therefore, this raises a question whether the method can be applied to practical systems with relatively large noises in the measurement because of the weak backscattering signals. Only reducing the number of acquisition without analyzing the measurement accuracy in practical cases does not provide sufficient evidence for this method to be widely adopted in reality.
In this work, we investigate the quality of CS-based Brillouin spectrum recovery with different noise levels, finding out that the performance of conventional Brillouin sensing systems can be significantly enhanced with regard to the acquisition amount reduction and the measurement accuracy improvement at the same time. There is a strong connection between CS and denoising; therefore, by random undersampling of the Brillouin spectral frequency components, the recovery of original Brillouin spectrum can be turned into a sparse signal denoising problem suggested by the CS theory. We use a fast iterative shrinkage-thresholding algorithm (FISTA) developed by A. Beck and M. Teboulle [18] for the Brillouin spectrum recovery. The algorithm was originally proposed as a class of iterative shrinkage-thresholding algorithm (ISTA) [19,20] for solving the linear inverse problems with a better convergence rate, and it was successfully applied to a wavelet-based image deblurring problem [18]. We investigate the feasibility to use this algorithm in the CS-based Brillouin sensing systems by recovering Brillouin spectra with a wide range SNR levels. The empirical and experimental results show that CS approach can be applied to practical Brillouin sensing systems, simultaneously realizing significant reduction of the acquisition amount and great enhancement of the measurement SNR. For the cases where the noise levels are moderate, by using only ∼30% of the frequency components which are used by traditional methods, the Brillouin spectrum can be recovered, and at the same time over 7 dB SNR enhancement can be achieved in the recovery process which can provide a much better measurement accuracy compared to that from direct measurements by conventional method. Even for the cases that the measurement has rather low SNR which may occur in the field test where unexpected large losses are present in the sensing fiber, this method is still robust enough to recover the spectra with acceptable accuracy. Moreover, our proposed method also has the potential to be applied to other fast BOTDA systems to further improve the performance. It is worth mentioning that there are many researches on improving SNR and measurement accuracy of Brillouin spectra [21–24], while our proposed method focuses on distributed fiber sensing applications where reducing the number of acquisitions and improving the SNR at the same time could have great potential in practical sensing applications.
2. Principle
For conventional Brillouin sensing systems, either BOTDA or BOTDR, one usually measures Brillouin backscattering time-domain traces in a series of frequency components around the Brillouin frequency of the sensing fiber. The process is illustrated in Fig. 1 below. After all the traces represented by red dashed lines are measured, the Brillouin spectrum distribution along the entire sensing fiber can be obtained. The resonance frequency variations at each location can be calculated afterwards to determine ambient temperature or strain variations. CS approach can be applied in this acquisition process to reduce the number of time-domain traces that need to be measured. After the spectra with reduced frequency components are obtained, CS recovery algorithms are performed to reconstruct the original Brillouin spectra at each location, and at the same time, the SNR can be improved greatly to enhance the measurement accuracy. Therefore, in this entire process, conventional Brillouin sensing systems can gain a performance enhancement in terms of acquisition speed and measurement accuracy without costly or sophisticated hardware modifications.
This recovery procedure can be immediately recognized as a linear inverse problem of a linear system with the form of
where $x$ is a length $n$ vector representing the true and unknown Brillouin spectrum which needs to be recovered; $z$ is an unknown noise vector presenting in the measurement process; $A$ is an $m \times n\;(m < n)$ “sensing matrix”; $b$ is the observation vector with a length of m. An approach to the above linear inverse problem is to seek the solution based on ${l_1}$ regularization which has the form where ${||\cdots ||_2}$ and ${||\cdots ||_1}$ represent ${l_2}$ and ${l_1}$ norm, respectively; the regularization parameter $\lambda > 0$ provides a tradeoff between fidelity to the measurements and noise sensitivity [18], which can be properly optimized to obtain decent results; $W$ is an orthogonal linear transformation to ensure that $W(x)$ is a sparse transformation. In this work, we use wavelet transformation to transform Brillouin spectrum into a sparse domain. The presence of the ${l_1}$ term is used to introduce the sparsity in the optimal solution of Eq. (2), which meets the sparsity requirement of CS. The other requirement of CS is the incoherent sampling, so that the sensing matrix $A$ can be designed to perform random undersampling based on the theory that random matrices are largely incoherent with any fixed basis [17]. By random undersampling, the process in finding the solution of Eq. (2) can be considered as a sparse signal denoising process. Therefore, in principle, one is able to recover the original Brillouin spectrum, and at the same time, increase the SNR greatly. The solution to Eq. (2) can be found through FISTA algorithm which is well developed in [18], and we adopt the algorithm throughout the following work.3. Results and discussion
3.1 Simulation results
Before performing any experiment, we first run a series of simulations to verify our proposal. We generate a Lorentzian curve with $n = 1024$ data points that corresponds to a total measurement range of 1023 MHz with 1 MHz frequency increment. The linewidth (full width at half maximum) is chosen to be 50 MHz. We determine the number of data that needs to be randomly selected for the recovery by using the percentage of root-mean-square difference (PRD), $PRD =||\hat{x} - {x||_2}/||{x||_2} \times 100\%$, as the criterion, where $\hat{x}$ is the reconstructed spectrum and x is the original spectrum, respectively. We found that over 90% successful recovery rate (PRD is less than 2.5%) can be achieved when 30% of the data is chosen randomly. White Gaussian noises are added with controlled SNR values to the Lorentzian curve afterwards to simulate practical cases. By random selecting 30% of data points, and use the FISTA algorithm to perform the spectral recovery, we find out that the spectra can be recovered successfully, and at the same time the SNR can be improved greatly. Figure 2 shows a few examples to compare the reconstructed spectra with respect to the original ones. The red curves represent reconstructed spectra, all of which clearly show great noise reduction. The SNR levels of the reconstructed spectra are shown in the figures as well.
Fig. 2. Original and reconstructed Lorentzian line shape with 1024 data points and a linewidth of 50 for different SNR levels of the original spectra: (a) 6 dB, (b) 14 dB, (c) 20 dB, and (d) 28 dB. For the reconstructed curves, only 30% of the data is used.
In order to further investigate the measurement accuracy enhancement, we calculate the standard deviation of the peak frequencies obtained by fitting a number of different reconstructions, and compare to those from the original spectra. We use a robust least-square parabolic fit after discarding all data points below 50% of the peak value which can be obtained through a first round estimation of the peak frequency and the peak value as explained in [2]. The results are shown in Fig. 3. At each SNR level, we generate 100 different spectra with the same noise level, and then calculate the standard deviation of the peak frequencies shown by the blue curve to simulate the measurement error in a system. For each spectrum, 30% of the data is selected to perform the recovery, and the standard deviation of the peak frequencies of the recovered spectra is calculated as shown by the red curve. The averaged SNRs of the reconstructed spectra are also illustrated. For all the cases, the SNR has gained obvious improvements. However, we find out that when the SNR is below ∼10 dB, the standard deviations of the reconstructed spectra are larger than those of the original spectra indicating deteriorate measurement accuracies even though the SNRs are improved. Even when the number of selected data is increased, the measurement errors do not decrease as shown in the inset in Fig. 3. The reason is that minimizing ${l_1}$ norm often leads to sparse solution, and many small coefficients tend to carry a larger penalty than the large coefficients; therefore, small coefficients are suppressed and solutions are usually sparse. When the measurement has large noise, in the sparse domain, the noise may represent as large coefficients which can introduce errors during the minimizing process, even for the cases that large amount of data is selected. However, in a practical sensing system, one needs to obtain SNR as high as possible in order to maintain acceptable measurement accuracy. In the above simulations, for a 10 dB SNR, the standard deviation is about 0.7 MHz which means that the error between two repeated measurements could be several times larger than this value, so the accuracy may not be good enough for many applications. Hence, one may need to achieve a better SNR in practice in order to reduce the measurement uncertainty. However, the results confirm the robustness of the CS principle which can still be used in the situation where excess losses exist in the sensing cable, and limited work could be done to increase the SNR. In addition, the recovery algorithm could be further optimized or modified, and other algorithms could be developed specifically to the cases that large noises are presented in a Brillouin spectrum to improve the measurement accuracy.
Fig. 3. Error (standard deviation) of the peak frequencies obtained from 100 original Lorentzian curves (blue) and the corresponding reconstructed spectra (red) with different SNRs. The mean value of SNRs of the reconstructed spectra are also indicated. Inset shows the error versus the ratio of number of selected data to that of the total data.
For the original spectrum with a SNR greater than ∼10 dB, the reconstructed spectra not only show an improved SNR, but also a reduced fitting error (standard deviation) as shown in Fig. 3. For SNR higher than 14 dB, over 5 dB SNR enhancement can be achieved for all the cases, and the enhancement can be as high as 7 dB. At the same time, the measurement accuracy is improved as expected. This provides solid evidence that CS principle can enhance the performance of Brillouin sensing systems in terms of acquisition speed and measurement accuracy without any hardware modification. In order to further investigate the feasibility of the proposed method, we need to perform experiments as in reality the Brillouin spectra measured by a sensing system are not exactly Lorentzian.
3.2 Experimental results
We setup a conventional BOTDA system to acquire distributed Brillouin spectra to verify our proposal. The experiment setup is shown in Fig. 4. A laser beam from a semiconductor laser is split by an optical coupler; one path goes directly to the fiber under test (FUT) via a polarization scrambler (PS) and an optical isolator (ISO) and the other path is frequency shifted by a carrier-suppressed optical single sideband modulator (OSSB) which is driven by a synthesizer. The power of the resultant beam is boosted by an erbium-doped fiber amplifier (EDFA), and then is modulated into optical pulses by a pulse generator which in our case is a semiconductor optical amplifier driven by a function generator. The pulse is delivered into the FUT through an optical circulator (CIR). The signal out of the photodetector (PD) is acquired by a digitizer and results are analyzed on a computer.
Fig. 4. A conventional BOTDA system: PS: polarization scramble; ISO: optical isolator; FUT: fiber under test; CIR: optical circulator; PD: photodetector; EDFA: erbium-doped fiber amplifier; OSSB: carrier-suppressed single sideband modulator.
Our approach can be used regardless of length of fiber in the measurement, and we choose the FUT to be 1 km for testing purpose in the experiment. By varying the number of averages when acquiring time-domain traces, we are able to obtain Brillouin spectra with different SNR levels. We first average each trace for 100 times, so that relatively low SNR Brillouin spectrum distribution can be obtained as shown in Fig. 5(a) which is acquired at 1 MHz frequency increment for a total of 1024 acquisitions. By randomly selecting 30% of the traces, Brillouin spectrum distribution can be reconstructed as shown in Fig. 5(b), which clearly shows improved SNRs compared to Fig. 5(a). We calculate average SNRs from 100 original Brillouin spectra and 100 reconstructed ones, respectively, confirming a 4.6 dB improvement from 11.5 dB to 16.1 dB is achieved on average. This improvement is consistent with the simulation results shown in Fig. 3, so that the accuracy improvement can be estimated accordingly. One of example spectra and its corresponding reconstruction are shown in Fig. 5(c), and the SNR improvement can be seen clearly. Power spectral densities of the two spectra from Fig. 5(c) are shown in Fig. 5(d) showing an obvious denoising effect based on CS principle.
Fig. 5. Measurement of 1 km fiber with 1 MHz frequency increment and 1024 MHz total frequency range using (a) conventional BOTDA method with 1024 frequency acquisitions, and (b) CS method to reconstruct the results with 307 random frequency acquisitions which is ∼30% of the number of acquisitions used in (a). (c) Original Brillouin spectrum and its corresponding reconstruction at the location of 450 m, and the SNR values at this location for original and reconstructed spectra are given. (d) Power spectral densities of the original and reconstructed spectra shown in (c). The average number of each trace is 100 to obtain relatively low SNR in the measurement.
Next, we average each traces for 4000 times, and the distributed Brillouin spectra are shown in Fig. 6(a). Similarly, we calculate average SNRs from 100 original Brillouin spectra and 100 reconstructed ones, respectively, confirming a 6.5 dB improvement from 27.3 dB to 33. 8 dB is achieved on average. This improvement is also consistent with the simulation results, so that the accuracy improvement can be estimated according to Fig. 3. The spectrum at 450 m and its corresponding reconstruction are shown in Fig. 6(c), and the SNR improvement can be seen clearly. It is worth mentioning that the measured spectrum is clearly deviated from a Lorentzian curve, but the reconstructed result can successfully retrieve the original spectrum with suppressed noises. Power spectral densities of the two spectra from Fig. 6(c) are shown in Fig. 6(d) illustrating an obvious denoising effect.
Fig. 6. Measurement of 1 km fiber with 1 MHz frequency increment and 1023 MHz total frequency range using (a) conventional BOTDA method with 1024 frequency acquisitions, and (b) CS method to reconstruct the results with 307 random frequency acquisitions which is ∼30% of the number of acquisitions used in (a). (c) Original Brillouin spectrum and its corresponding reconstructed one at the location of 450 m, and the SNR values at this location for original and reconstructed spectra are given. (d) Power spectral densities of the original and reconstructed spectra shown in (c). The average number of each trace is 4000 to obtain decent SNR in the measurement.
4. Conclusion
In conclusion, we demonstrate significant performance enhancement of conventional Brillouin sensing systems based on the compressive sampling principle. By random undersampling of the measurement in a conventional BOTDA or BOTDR system, the recovery of the original Brillouin spectrum of an optical fiber can be turned into a sparse signal denoising problem. We conduct both simulations and experiments confirming that only 30% of the acquisitions compared to those used by the conventional methods are sufficient to retrieve the Brillouin spectrum, and at the same time, the recovered spectral SNR can be greatly enhanced compared to that obtained by direct measurement. Over 7 dB enhancement is demonstrated, and the sensing accuracy can be improved as well. Note that when the SNR is rather low, less than 10 dB in the measurement, the current algorithm could not result in measurement accuracy improvement. However, the recovery algorithm could be further optimized or modified, and other algorithms may be developed to the cases that large noises are presented in a Brillouin spectrum to improve the measurement accuracy.
Funding
National Natural Science Foundation of China (61520106013, 61727816); Fundamental Research Funds for the Central Universities (DUT20LK47).
Acknowledgments
D.–P. Zhou would also like to thank the financial support of the Academic Program Development Funds from Dalian University of Technology, and the Research Fund for International Cooperation (ICR1907).
Disclosures
The authors declare no conflicts of interest.
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