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Abstract
We propose to employ the image deconvolution technique for Brillouin optical time domain reflectometry (BOTDR) systems to achieve a flexible and enhanced spatial resolution with pump pulses longer than phonon lifetime. By taking the measured Brillouin gain spectrum (BGS) distribution as an image blurred by a point spread function (PSF), the image deconvolution algorithm based on the two-dimensional Wiener filtering can mitigate the ambiguity effect on the Brillouin response. The deconvoluted BGS distribution reveals detailed sensing information within shorter fiber segments, improving the inferior spatial resolution and simultaneously maintaining other sensing performance parameters. Thanks to the proposed technique, a typical BOTDR sensor with 40 ns pump pulses reaches a submetric spatial resolution as high as 10 cm. Compared to the differential-spectrum-based BOTDR retrieving the same spatial resolution, the image deconvolution technique shows advantages in system complexity and measurement uncertainty. Moreover, the proposed technique is promising to improve the spatial resolution of other distributed optical fiber sensing (DOFS) techniques such as BOTDR systems with complex pump modulation methods.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since distributed optical fiber sensing (DOFS) based on Brillouin scattering was first proposed [1], it has been widely researched and applied in many fields such as structural health monitoring [2–5]. It offers brilliant sensing features including high spatial resolution, long sensing range, and the capability of measuring temperature and strain simultaneously. Its essential sensing principle is to acquire the Brillouin gain spectrum (BGS) distribution, extract the distribution of Brillouin frequency shift (BFS) and thus obtain the distributed temperature and/or strain information along the fiber-under-test (FUT). Among the Brillouin-scattering-based sensing techniques, the Brillouin optical time domain analysis (BOTDA) [1] and the Brillouin optical time domain reflectometry (BOTDR) [6] are the most popular ones. The BOTDA systems employ the stimulated Brillouin scattering (SBS) of two counter-propagating light, which feature a high signal-noise ratio (SNR) and accuracy compared with the BOTDR systems utilizing the spontaneous Brillouin scattering (SpBS). However, the BOTDR systems perform the measurement by accessing only one end of the sensing fiber, providing good flexibility in embedding the sensing cable into structures [7].
Spatial resolution and sensing accuracy are two major performance parameters of Brillouin-scattering-based DOFS systems. The former is normally proportional to the pulse width for the two time-domain approaches, while the sensing accuracy is related to the SNR, the frequency interval, and the full width at half maximum (FWHM) of the measured spectrum [8]. Sadly, the trade-off between the two parameters has been commonly noticed due to the phenomenon of FWHM broadening and SNR degradation as shorter pulses are employed. It results from the pulse-based sensing principle, leading to ambiguity on the measured BGS distribution in both the time/spatial and frequency domain. Several techniques have been proposed and developed to overcome this trade-off, including pulse modulation and post-processing methods.
Pulse modulation techniques have been commonly used in BOTDA systems to obtain submetric spatial resolution while preventing the measured FWHM from broadening [9–12]. Similarly, differential-spectrum-based BOTDR [13] employs a pair of pulses with a slight width difference. 0.4 m spatial resolution is achieved on a 7.8 km sensing fiber. Based on a BOTDR analytical solution [14], the ambiguity of the measured BGS distribution is expressed as a two-dimensional convolution between a point spread function (PSF) and the intrinsic BGS. A further analysis in aspect of Time-Frequency localization shows that different pulse shape profiles offer BOTDR systems different responding features in the time and frequency domains for the FUT with heated sections shorter than the FWHM of the pulses [15]. Synthetic BOTDR [16] was proposed to realize a sub-meter spatial resolution. It injects a group of synthetic pulses consisting of two pulses with different pulse widths and different phase shifts into the FUT, and achieves a spatial resolution decided by the short-width pulse with the FWHM determined by the long-width pulse. Other methods include the differential cross spectrum technique [17] and the phase-shift pulse technique [18]. The constraints of these techniques are apparent in the increased system complexity and measurement time.
On the other hand, post-processing techniques allow the sensing systems to realize a higher spatial resolution without any systematic modifications. The subdivision technique has been proposed to obtain submetric spatial resolution parameters in both BOTDA and BOTDR systems [19,20]. However, the subdivision technique needs a reference fiber with a stable frequency shift in front of the temperature/strain change region. Researchers have also proposed to employ quadratic time-frequency transforms instead of commonly used short-time Fourier transform (STFT) to obtain better transition responsivity and frequency resolution at the same time [21]. Recently, deconvolution techniques are proposed to process the measured Brillouin signal traces of BOTDA systems to achieve submetric spatial resolution [22,23].
In this work, a novel image deconvolution technique based on two-dimensional Wiener filtering is proposed and demonstrated to realize a flexible submetric spatial resolution for BOTDR systems. Image denoising techniques have been widely researched to decrease the noise level of the measured BGS distributions [24–26]. Regarded as an image blurred by a PSF, the BGS distribution from BOTDR systems can be deblurred via specially designed image deconvolution algorithm in the time domain and frequency domain simultaneously. Results show that the technique offers a flexible and enhanced spatial resolution while maintaining other sensing performance. Distributed sensing with a flexible and submetric spatial resolution down to 10 cm over 1.8 km FUT is demonstrated with a BOTDR system employing 40 ns pump pulses while retaining a low BFS uncertainty. We also compare the proposed technique with the differential-spectrum-based BOTDR. Finally, the further improvement of the spatial resolution in the differential technique BOTDR is also investigated by using the proposed image deconvolution technique based on Wiener filtering.
2. Principles
2.1 Mathematical model of BOTDR systems based on PSF
In BOTDR systems, the temperature or strain is demodulated from the BFS of the SpBS light. Frequency-sweep process and STFT are two common techniques to obtain the BGS via BOTDR. In the STFT-based BOTDR systems, the measured Brillouin gain spectrum k(t, v) can be expressed as [14]:
The integral of PSF in the frequency and time domain expressed as:
Fig. 1. (a) Simulated point spread functions ψ of the 10 ns pump pulse; (b) Simulated point spread functions ψ of the 40 ns pump pulse; (c) The energy density distribution of the PSF in the time domain (blue line: 10 ns, red line: 40 ns); (d) The energy density distribution of the PSF in the frequency domain (blue line: 10 ns, red line: 40 ns).
According to Eq. (1a), the expectation of the measured BGS k(t,v) is the two-dimensional convolution of the time-varying Lorentzian spectrum l(t,v) and PSF ψ(t,v), indicating that the intrinsic BGS l(t,v) is blurred to k(t,v) by the PSF ψ(t,v) in the time domain and the frequency domain. When the PSF of a short pulse is narrow in the time domain and wide in the frequency domain, the spectrum will be broadened after convolution. On the contrary, when the PSF of a long pulse is wide in the time domain and narrow in the frequency domain, the spatial resolution will be reduced after convolution.
A typical BOTDR system is illustrated in Fig. 2. A 1550.12 nm laser (NKT, Coheras BasiK) with a linewidth of 200 Hz is split into two branches with a 3 dB coupler. In the upper branch, the continuous light is modulated by an electro-optic modulator (EOM) and an acoustic-optic modulator (AOM, Gooch & Housego, 200 MHz) to form a 40 ns rectangular pulse light with an extinction ratio of over 50 dB and a rising/falling edge time of less than 1 ns. An erbium doped fiber amplifier (EDFA, Beogold, PEDFA) is employed to boost he peak power of the pulse to 17.6 dBm, which is low enough to avoid small gain SBS effect [27]. Subsequently, the pulse is guided into the FUT via an optical circulator. In the lower branch, the reference light is modulated by another EOM with a fixed driving frequency of 10 GHz. The lower-sideband of the frequency-modulated light is chosen by a bandpass filter (BPF) and sent to a polarization scrambler (PS). The SpBS signals backscattered from the FUT are mixed with the reference light by a 3 dB coupler, received with a balanced photodetector (PD, Thorlabs, PDB480C-AC,1.6 GHz), and converted to intermediate frequency (IF) signals of several hundred MHz. The output IF signals are acquired by a data acquisition card (DAQ, Gage, CSE24G8) with sampling rate of 2 GSa/s.
Fig. 2. Experimental setup. NL Laser, narrow linewidth laser; EOM, electro-optic modulator; AOM, acoustic optical modulator; EDFA, erbium-doped fiber amplifier; BPF, bandpass filter; PS, polarization scrambler; PD, photodetector; DAQ, data acquisition card; VOA, variable optical attenuator; AFG, Arbitrary Function Generator; RF, Radio Frequency; FUT, fiber under test.
To obtain the accurate PSF of the system for the better filter design, the pump pulse envelope profile is also detected with another PD (Thorlabs, PD430C) and then collected by the DAQ. Figure 3(a) shows the detected pump pulse envelope f(t) with the pulse width of 40 ns, which is not a standard rectangular pulse due to the imperfect modulation hardware of the system. Figure 3(b) shows that h(t) in the STFT is selected as a 40 ns rectangular window, correspondingly Fig. 3(c) shows the original PSF of the system calculated by Eq. (1c). It indicates the measured BGS distribution will be strongly ambiguous in the time domain, resulting in a poor spatial resolution. To reduce the ambiguity effect of the measured BGS caused by the two-dimensional convolution, researchers have designed specific pulse formats to construct a δ-like PSF which is narrow in both the time domain and frequency domain. The synthetic BOTDR [14], differential cross spectrum technique [17] and the phase-shift pulse technique [18] can construct δ-like PSFs by using modulated pulses and corresponding filters. However, this will enlarge the system complexity and the measurement time.
Fig. 3. (a) The 40 ns pump pulse envelope f(t); (b) the Fourier transform window function h(t); (c) the corresponding Point spread functions ψ (top view).
2.2 Image deconvolution for enhancing spatial resolution of BOTDR systems
Since the ambiguity effect of the measured BGS distributions in BOTDR systems is very similar to the degradation of a blurred image [28], image deconvolution is here proposed to process the BGS distributions. Classic two-dimensional image deconvolution techniques include regularized filtering, Lucy-Richardson method, blind deconvolution, and Wiener filtering [29]. Different methods offer their own benefits for image deconvolution problems, Wiener filtering is adopted here considering its advantage in better time efficiency compared to other nonlinear filters [29], which is important for BOTDR applications.
Wiener filtering has been widely used in image deconvolution applications for blurred images with additive noise. The filtering will minimize the mean square error between the deblurred image and the original sharp image. For our application, the recovered BGS is calculated by
According to the analysis in the last section, the FWHM of the equivalent PSF energy density distribution in the time domain decides the system spatial resolution after the filtering. Firstly, the value of parameter γ is adjusted to restore the original PSF to an equivalent δ-like one using Eq. (4). Then the recovered BGS distribution can be obtained by substituting the optimized γ into Eq. (3) to deblur the measured BGS distribution. In this way, an enhanced and flexible spatial resolution can be achieved by carefully adjusting the value of parameter γ. The value of γ is initiated as 1 and minified by 1e0.1 times for each step. For each γ value, the equivalent PSF is obtained via Eq. (4) with original normalized PSF as shown in Fig. 3(c). The corresponding time-domain energy density distribution and its FWHM are also calculated. Figure 4(a) illustrates the FWHM evolution versus γ value.
Fig. 4. (a) FWHM evolution versus γ value, (b) The energy density distributions of equivalent PSFs with different γ values.
Figure 5 shows the equivalent PSFs when the values of γ are 3.3e-4, 2.5e-5, 6.3e-7, and 6.3e-8. The resulting FWHM values of the energy density distributions as shown in Fig. 4(b) are around 10 ns, 5 ns, 2 ns, and 1 ns, respectively, which means that the corresponding spatial resolution parameters are 1 m, 50 cm, 20 cm, and 10 cm. When the original PSF is normalized, the resulting spatial resolution is independent of the pulse light power and is enhanced with the decrease of parameter γ.
3. Experimental results and discussions
3.1 Verification of spatial resolution enhancement
In this experimental demonstration, the experimental setup in Fig. 2 is adopted with pump pulse width set as 40 ns, and the average times for the received original BGS is 100000. The demodulation process of the proposed technique includes three steps. Firstly, as the same as conventional STFT-based BOTDR systems, the acquired IF signals are applied STFT and then averaged to obtain the original time-varying BGS. Secondly, the Wiener filtering algorithm is designed based on the pulse and STFT window profiles and then applied to the spectrum. Finally, the BFS is extracted from the filtered spectrum by Lorentz curve fitting (LCF).
To verify the spatial resolution enhancement performance of the Wiener filtering algorithms designed above, the FUT composed of two types of fibers with different frequency shifts is arranged as Fig. 6. The BFS values of type I fiber (blue line) and type II fiber (green line) are 10716 MHz and 10635 MHz, respectively. Type II fiber segments of 10 cm, 20 cm, 50 cm, and 1 m are sequentially placed at the end of an 1865 m type I fiber, and each segment is separated by a span of 10 m type I fiber.
Here, the Wiener filtering algorithms with the γ values of 3.3e-4, 2.5e-5, 6.3e-7, and 6.3e-8 as designed in Section 2.2 are employed to retrieve spatial resolutions of 1 m, 50 cm, 20 cm, and 10 cm, respectively. To illustrate the effect of the filtering algorithms, comparisons have been conducted among the original results without applying any filtering, and those after applying filters retrieving 1 m, 50 cm, 20 cm, and 10 cm spatial resolution. Figure 7 shows the original BGS distribution without applying Wiener filtering, and those after applying the filtering. The normalized BGSs at the two FUT locations around 1865 m inside and outside the 10 cm type II fiber segment are depicted respectively in Fig. 8(a) and (b), where the main lobe of each curve remains as a Lorentzian-like profile. Figure 9 illustrates the corresponding BFS distributions extracted from the distributions as shown in Fig. 7 via Lorentz curve fitting. As shown in Fig. 7(a) and Fig. 9(a), all segments of type II fiber cannot be distinguished in either the spectrum or BFS distribution. Whereas, in Fig. 7(b) to 7(e), they can be recognized gradually with the decrease of parameter γ. This process can also be observed from Fig. 8(a) that the main peak of the BGS inside the 10 cm segment is gradually shifted to its correct frequency. The spatial resolution enhancement abilities of the filters can be confirmed according to the BFS distributions in Fig. 9(b) to 9(e). The shortest length of the recognized type II fiber segment equals the FWHM of ψt in the corresponding value of γ. The BFS uncertainty of the last 10 m sensing fiber in each condition is also calculated to evaluate the sensing performance, as 0.13, 0.23, 0.29, 0.67, and 0.93 MHz, respectively. The degradation of the BFS uncertainty is due to the rising background noise level of the BGS distribution with the γ value decreasing, as shown in Fig. 7 and 8.
Fig. 7. (a) Original measured BGS distribution using 40 ns pulses, and Wiener filtering recovered BGS distributions to retrieve (b) 1 m spatial resolution, (c) 50 cm spatial resolution, (d) 20 cm spatial resolution, (e) 10 cm spatial resolution.
Fig. 8. Normalized BGS curves at FUT locations around 1865 m (a) inside the 10 cm type II fiber segment and (b) outside the 10 cm type II fiber segment.
Fig. 9. BFS distributions extracted from (a) original measured BGS distribution using 40 ns pulses, and Wiener filtering recovered BGS distributions to retrieve (b) 1 m spatial resolution, (c) 50 cm spatial resolution, (d) 20 cm spatial resolution, (e) 10 cm spatial resolution.
Temperature distribution measurements are also conducted with the BOTDR system in Fig. 2 to assess the measurement accuracy after the deconvolution. The pump pulse width is also set as 40 ns. The Wiener filtering as mentioned above is applied to the received SpBS signals, where the parameter γ is set as 6.3e-8 aiming at the spatial resolution of 10 cm. The FUT configured as Fig. 10(a) is stored in room temperature of 16 °C and 20 cm fiber at its end is heated to 30, 40, 50, and 60 °C. Given the temperature coefficient of the FUT calibrated in advance as 1.1 MHz/°C, the temperature distributions depicted in Fig. 10(b) can be obtained according to the resolved BFS distributions. The temperature uncertainty for each measurement at the FUT end is calculated as 0.55, 0.88, 0.89, 0.80, and 0.82 °C, which agrees well with the above BFS uncertainty of 0.93 MHz. There are 3 resolved points at 1837.48, 1837.53, and 1837.58 m of the temperature distribution curves in the heated segment. The maximum measured temperature of the heated segment for each measurement locates at 1837.53 m, which is 15.4, 30.4, 40.9, 51.8, and 61.0 °C, where the maximum absolute error is 1.8 °C. The average temperature values of the three spots are calculated as 15.3, 29.7, 40.2, 51.1, and 60.4 °C. The lengths of the rising edge and the falling edge of the heated fiber segment in each measurement are both around 10 cm, which implies that the spatial resolution has attained 10 cm.
Fig. 10. (a) The FUT configuration of temperature distribution measurement, (b) The measured temperature distributions.
3.2 Comparison and compatibility with differential spectrum based BOTDR
To compare the sensing performance, the differential-spectrum-based BOTDR [13] is conducted using the setup in Fig. 2 and the FUT configuration in Fig. 6. The system utilizes a pair of pulses with a slight width difference and performs a two-step subtraction on the measured BGS distribution pairs, achieving sub-meter spatial resolution.
In this demonstration, the pulse pair widths are set as 45/40 ns, the STFT window widths are 60/65 ns, and 50000 averages are taken for each measured BGS distribution. In Fig. 11, the so extracted BFS distribution by the LCF method is depicted in the blue curve. The BFS distribution obtained in Section 3.1 retrieving 50 cm spatial resolution is simultaneously given in the red curve. Both results can reveal the 50 cm and 1 m segments of type II fiber, showing a good agreement with each other. However, the image deconvolution method offers a better BFS uncertainty based on 0.23 MHz over that of 0.45 MHz based on the differential-spectrum-based method.
Fig. 11. Extracted 50 cm spatial resolution BFS distributions based on differential spectrum based technique (blue curve) and image deconvolution (red curve).
The deconvolution technique is also capable of improving the spatial resolution of other BOTDR with complex pump modulation methods, including the differential-spectrum-based technique. The modulation and demodulation process of the technique can also be described as a PSF, which is expressed as
Fig. 12. (a) Pulse pair envelopes of 45/40 ns, and (b) the corresponding PSF for differential-spectrum-based BOTDR system.
With parameter γ carefully set as 1e-3, the equivalent PSF after filtering is given as Fig. 13(a). While the energy density distributions of the equivalent PSFs without and with deconvolution are illustrated in Fig. 13(b) as blue and red curves, respectively. As shown in Fig. 14, the type II fiber segments of 20 cm and 10 cm are clearly recognized after filtering. The result indicates that the Wiener filtering enhances the spatial resolution of the above differential-spectrum-based BOTDR system from 50 cm to 10 cm.
Fig. 13. (a) The equivalent PSF of the differential-spectrum-based BOTDR system after deconvolution, (b) The energy density distributions of the equivalent PSF in the time domain without deconvolution (blue curve) and with deconvolution (red curve).
Fig. 14. BFS distributions retrieved from differential-spectrum-based BOTDR system without deconvolution (blue curve) and with deconvolution (red curve).
3.3 Discussions
One may concern about the speed of the Wiener filtering algorithm. On the one hand, there is no need to change the pulse modulation method, which will increase the cost of experiment setups. The pulse envelop is sampled along with the sensing of Brillouin scattering light to construct a more accurate PSF, so the sensing error from the rising edge and fall edge of the pump pulse is reduced on the deconvolution process. Therefore, the pulse does not need to be modulated into an ideal rectangular. On the other hand, as a post-processing method, two-dimensional Fourier transform and inverse Fourier transform in the Wiener filtering can be conducted through fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT), accelerated by graphics processing unit (GPU). The time of the Wiener filtering algorithm is 1.74s. Compared with the time of STFT, this time is negligible. Thus, the proposed deconvolution algorithm is fast enough for long-distance sensing.
Another concern is the highest achievable spatial resolution based on the two-dimensional Wiener filtering. The two-dimensional PSF is affected by the response characteristics of the PD and acquisition card. A broader bandwidth of the PD and a higher sampling rate of the acquisition card allow a higher information bandwidth and more data points for STFT. As shown in Section 3.1, the decrease of parameter γ will improve the achieved spatial resolution, but simultaneously degrades the BFS uncertainty due to the increase of the resulting noise level after filtering. A lower noise level shall allow a smaller γ value and a better spatial resolution while retaining the BFS uncertainty. This effect fundamentally limits further enhancement of the resolved spatial resolution using Wiener filtering. Therefore, the highest spatial resolution is decided by the system bandwidth and the noise level. Limited by the 1.6 GHz bandwidth of PD and the noise in the BGS, the spatial resolution obtained in the experiment is 10 cm to maintain the BFS uncertainty under 1 MHz.
4. Conclusion
In this paper, the image deconvolution technique based on two-dimensional Wiener filtering is proposed to improve the spatial resolution of the BOTDR system. The measured BGS distribution modeled as the convolution of the intrinsic distribution and a PSF can be regarded as a blurred image. The ambiguity effect of the PSF on the measured distribution is the main reason for the trade-off between spatial resolution and BFS uncertainty for the BOTDR technique. A novel image deconvolution technique based on two-dimensional Wiener filtering is proposed to reduce the ambiguity effect. Experimental results show that the proposed technique can effectively improve the spatial resolution without sacrificing other sensing performance parameters. In the experimental demonstrations, a spatial resolution as high as 10 cm over 1.8 km sensing fiber is achieved using a typical BOTDR system with the Wiener filtering. The proposed technique is promising to enhance the resolution for other DOFS techniques on the premise of constructing the PSF accurately. For example, an investigation shows it can further improve the spatial resolution of the differential-spectrum-based BOTDR technique from 50 cm to 10 cm.
Funding
National Natural Science Foundation of China (61905029, 62105045); National Science Fund for Distinguished Young Scholars (61825501); Chongqing Natural Science Foundation of Innovative Research Groups (cstc2020jcyj-cxttX0005); Chongqing Natural Science Foundation of Postdoctoral Fellowship (cstc2021jcyj-bshX0128).
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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