1. Introduction

Distributed optical fiber strain sensing systems, offering significant advantages over conventional sensors, including distributed measurement capabilities, passivity, and immunity to electromagnetic interference, have been widely employed in various fields such as structural health evaluation [1], invasion monitoring [2], etc. Distributed fiber strain sensing primarily relies on two types of scattering in optical fibers: Rayleigh scattering and Brillouin scattering. Generally, Rayleigh scattering-based strain sensors yield more accurate strain measurements due to their significantly higher strain sensitivity, approximately 150 MHz/$\mathrm {\mu }\mathrm {\varepsilon }$, in contrast to the Brillouin scattering-based counterparts with sensitivities around 0.05 MHz/$\mathrm {\mu }\mathrm {\varepsilon }$ [3].

When it comes to the implementation of Rayleigh scattering-based strain sensors, two primary methods are commonly utilized: Optical Time Domain Reflectometry (OTDR) and Optical Frequency Domain Reflectometry (OFDR). Typically, the OTDR-based approach achieves meter-level spatial resolution, which is insufficient for certain applications where localized information is of interest. Additionally, the process of step-by-step pulse wavelength changes for Rayleigh spectrum reconstruction in OTDR is time-consuming, and the measurable strain range is typically limited to several tens of $\mathrm {\mu }\mathrm {\varepsilon }$ due to a relatively small frequency coverage range [4]. Although it is feasible to expand the frequency-swept range in OTDR, the simultaneous achievement of a broad frequency range with a dense frequency interval poses practical challenges. An alternative approach was demonstrated by J. Pastor-Graells et al. involving chirped pulses-based OTDR, which enabled single-shot measurements to reduce time redundancy [5]. However, this method significantly increased the complexity of the hardware system, and the measurable strain range remained constrained by available hardware resources. As a result, OTDR-based sensors are typically optimized for small-amplitude and high-sensitivity strain sensing, while achieving both high spatial resolution and extensive strain range remains a challenge beyond the capabilities of OTDR. On the other hand, OFDR was first introduced as a distributed sensing method in [6]. OFDR demonstrates exceptional spatial resolution and a large strain range, provided it employs a wide-range frequency-swept probe lightwave in the OFDR configuration. Furthermore, the intrinsic de-chirping property in OFDR mitigates the burden on hardware resources, especially when compared to chirp-pulse-based OTDR, rendering OFDR more suitable for static and quasi-static sensing applications.

In a distributed strain sensing system, the most crucial performance indicators encompass the sensing range, spatial resolution, strain resolution, and measurable strain range. OFDR holds the potential to achieve a high-performance sensing system overall, benefiting from the aforementioned advantages. However, achieving a simultaneous and optimal balance between an extended sensing range, high spatial resolution, a large strain range, and fine strain resolution is an exceptionally challenging task. The limiting factor is the system noise inherent to OFDR, primarily the phase noise induced by the frequency sweep non-linearity of the probe lightwave [7–9]. The impact of the nonlinear frequency tuning of the probe lightwave on the performance metrics of an OFDR system is highly intricate, resulting in complex interdependencies among them. Trade-offs must be made among different performance aspects. For instance, in the pursuit of long-range sensing, an optical power spectrum analysis method combined with an external modulation technique was proposed, achieving a 100-km sensing range at the cost of spatial resolution, which was limited to just 25 m [8]. The use of a phase-noise-compensated algorithm led to 25-km strain sensing with improved spatial resolution down to 2.5 m [9]. Typically, external modulation-based OFDR systems suffer from a limited measurable strain range and modest spatial resolution due to the relatively narrow radiofrequency sweeping range, usually within several GHz. In the quest for high spatial resolution applications, Yang et al. introduced a periodic-phase-noise-estimated (PPNE) deskew filtering algorithm to eliminate phase noise and a 1-km sensing range with 1-cm spatial resolution was recently achieved [10]. However, a 1-cm spatial resolution may still fall short for high-performance OFDR systems. In a different context, Yu et al. achieved a minimum spatial resolution of 0.5 mm over a 104-meter sensing range [11], but at the expense of strain resolution, which deteriorated to 19.31 $\mathrm {\mu }\mathrm {\varepsilon }$. In applications necessitating high strain resolution, Song et al. achieved a remarkable strain resolution of 2.3 $\mathrm {\mu }\mathrm {\varepsilon }$ but at the cost of spatial resolution, which was limited to 7 cm [12]. It is also worth noting that this strain resolution was exclusively tested using the fiber section at the front end of the sensing fiber. Strain resolution at an extended sensing range, such as at the end of the fiber, was not demonstrated. In summary, there exists an inherent trade-off among key performance indicators, and further advancements in overall system performance, relying solely on current hardware and data analysis methods (e.g., the phase-noise compensation method), appear to be a formidable challenge if not unattainable.

To address these limitations, conventional signal processing algorithms have been employed to enhance OFDR system performance. These methods do not consider the physical mechanism of the noise source, relying solely on the stochastic characteristics of system noise as prior knowledge. Therefore, they differ from existing phase noise elimination-based approaches and hold the potential to overcome the limitations mentioned above, thus enhancing overall system performance. For instance, the wavelet transform is a well-established denoising technique that has been utilized to improve the strain resolution of an OFDR system to 1 $\mathrm {\mu }\mathrm {\varepsilon }$ along a 30-m sensing fiber with a spatial resolution of 5 mm [13]. However, the wavelet denoising approach requires the manual selection of wavelet basis and signal decomposition levels, increasing complexity and limiting versatility in diverse scenarios. Recently, several image processing methods have been applied to OFDR systems. Multiple sets of original distance-dependent one-dimensional cross-correlation signals derived from OFDR traces are first translated into a two-dimensional image. Conventional image processing techniques are then applied to denoise the image [14]. It’s worth noting that image processing denoising relies on the signal consistency between adjacent locations, and in cases of abrupt signal changes, there may be a loss of valid information. Image processing methods have also been employed to improve spatial resolution [15] and extend the strain measurement range [16]. In these approaches, fake peaks obtained from cross-correlation analysis caused by Rayleigh spectrum distortion are directly removed from the original images. Nevertheless, in specific cases involving localized changes, the problem of losing essential information remains.

In light of the limited performance improvements achieved by conventional signal and image processing algorithms, this paper introduces and experimentally demonstrates a high-performance distributed sensor based on OFDR empowered by deep learning. Deep learning algorithms have found widespread application in the field of fiber optic sensing to enhance performance, including applications in Raman-OTDR [17–19], phase-sensitive OTDR [20], Brillouin-Optical Time Domain Analysis [21], and more. Specifically, in the realm of artificial intelligence (AI) algorithms for reducing signal noise in fiber optic sensing, Zhang et al. introduced a noise reduction approach based on one-dimensional deep convolutional neural networks (1D-CNN) [18,19]. This approach emulates distributed optical fiber sensing scenarios by generating random numbers according to specific rules that simulate sensing signals occurring anywhere along the fiber. By simulating the actual noise in these signals, a substantial amount of signal noise is generated through these data simulations and is subsequently used as training data for a deep learning-based noise reduction model. This model is then applied to real-world fiber optic sensing signals, and its effectiveness has been demonstrated in several studies. However, this approach does have certain limitations. Specifically, when the noise reduction network is designed using a stack of small convolutional kernels, the model’s receptive field remains relatively small. Consequently, it can only process sensing information within a limited scope at a time, making it challenging to effectively evaluate random noise fluctuations over a broader signal range. This limitation results in modest performance in signal noise reduction. Previous research has indicated that increasing the network’s depth further does not significantly enhance its noise reduction capabilities [17]. CNNs were once the predominant algorithmic choice in modern computing [22]. However, the advent of Transformers [23] in recent years has presented substantial challenges to CNNs. Many researchers have expounded on the merits of Transformer models from various angles, including module construction [24], sparsity [25], multihead self-attention [26], and, notably, the establishment of extensive receptive fields, which stands as a pivotal factor contributing to model superiority. Presently, in the domain of CNN network design, architectures centered on large convolutional kernels have shown remarkable performance improvement with Vision Transformers (ViTs) [27] in tasks such as image recognition and object detection [28].

In view of these recent advancements, we embark on a fresh re-evaluation of CNN design for denoising applications. The critical question that arises is whether superior performance can be achieved by employing a modest number of large convolutional kernels or by resorting to the abundant use of smaller kernels. Existing research underscores the efficacy of networks designed around a limited number of large depth-wise (DW) convolutions compared with alternative approaches such as dilated convolutions, feature pyramids, or networks featuring an abundance of smaller kernels [28]. Within the scope of this paper, we advocate for a paradigm shift by utilizing a sparse ensemble of large convolutional kernels, as opposed to the traditional stack of smaller kernels, in constructing a neural network tailored for one-dimensional signal processing. The empirical findings corroborate that large-kernel CNNs possess a more expansive effective receptive field, coupled with a heightened proclivity towards waveform characteristics in contrast to the localized texture focus observed in small-kernel CNNs. This, in turn, results in superior denoising performance.

Ultimately, we experimentally demonstrate that the denoising model based on large-kernel CNNs significantly improves the performance of an OFDR-based distributed sensing system. This allows for the maintenance of a sub-microstrain (0.91 $\mathrm {\mu }\mathrm {\varepsilon }$) sensing resolution alongside a sharp spatial resolution of sub-millimeter (0.857 mm) over a 140-m sensing range. The introduced model is based on the fusion of data and physics-driven methodologies. This innovation is achieved by constructing a training dataset entirely through pure synthetic data and analytics engine, subsequently employing the well-trained model to perform denoising on real-world sensor signals. This strategic approach effectively eliminates the need for amassing extensive sensor data across diverse scenarios, resulting in a fast and cheap model training process. This model boasts a larger receptive field, enabling effective noise reduction and a reduction in random fluctuations in sensor signals. In contrast, traditional image denoising methods are often vulnerable to parameter sensitivity, resulting in a modest denoising performance. Furthermore, CNNs that rely on small convolutional kernels, despite their greater model depth, grapple with limited receptive fields, making it difficult to address the random fluctuations in sensor signals and, consequently, leading to only moderate denoising performance. In comparison, the algorithm proposed in this paper proves to be more adept at OFDR denoising. This work represents the inaugural experimental demonstration of OFDR-based distributed sensing, achieving sub-millimeter spatial resolution and sub-$\mathrm {\mu }\mathrm {\varepsilon }$ strain resolution across an extensive sensing range spanning over a hundred meters. In the proof of concept, we focus on strain sensing based on a standard single-mode fiber because strain sensors are widely used in diverse applications and single-mode fibers are easily accessible and low-cost. But the proposed strategy is applicable to any OFDR systems based on either single-mode fibers or specialty fibers for the measurement of any quantities.

This paper is organized as follows. In Section 2, we analyze the performance of a homebuilt OFDR-based distributed sensing system and focus on the limitations of the current system, providing the basis for the need for deep-learning-based approaches. In Section 3, we introduce our data and physics-driven model for improving system performance, and analyze its superiority compared to a deep learning model commonly employed in denoising applications. Section 4 includes the experimental results and discussion. Section 5 concludes this paper.

2. Performance evaluation of OFDR-based distributed sensing

A typical OFDR-based distributed sensing system is schematically depicted in Fig. 1. The fundamental operating principle of OFDR relies on swept-laser interferometry, and the strain demodulation process has been thoroughly detailed and described in prior research [6,8,9,29]. As mentioned earlier, for long-range sensing, the primary noise source that distorts the Rayleigh spectrum for sensing information extraction is the residual phase noise stemming from the non-linearity of the probe lightwave’s frequency sweep. While the intensity noise, including the detection shot noise, the thermal noise of the photodetector, and the quantization noise of the analog-to-digital conversion, plays a main role only for short-range sensing [30]. Nevertheless, the extent to which phase noise affects system performance is closely intertwined with the selection of system parameters. Among the four main system performance indicators, both spatial resolution and the measurable strain range benefit from a larger frequency-swept range. In essence, a broader frequency-swept range yields finer spatial resolution and an extended strain range [11]. In addition to a limited frequency-swept range, another factor limiting the measurable strain range is the position mismatch between the reference and measured spectra arising from the strain accumulation, especially for high-spatial resolution configuration. A position mismatch correction algorithm is generally employed to solve this problem [31,32]. The sensing range of OFDR is ultimately bounded by the accumulation of phase noise. In general, the phase-noise-compensated algorithms [9] or the PPNE-deskew filtering algorithm [10] can partially mitigate phase noise, yet residual phase noise continues to exert a detrimental impact on sensing performance. With a fixed residual phase noise, the strain resolution is found inversely proportional to the signal spectral width. A wider Rayleigh spectral range leads to a more irregular appearance, compressing the cross-correlation spectral peak to a narrower profile for a more precise spectrum shift calculation [33]. The dependence of strain measurement resolution on other performance metrics of an OFDR system can be expressed as:

 

Fig. 1. Typical OFDR experimental setup.

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To validate this relationship, we conducted tests to measure the strain distribution along a 140 m-long strain-free sensing fiber based on a homebuilt OFDR system. We used different frequency-swept-range probes of 20 nm, 40 nm, 60 nm, 80 nm and 100 nm while maintaining a fixed $N$ of 200, corresponding to spatial resolutions of 8 mm, 4 mm, 2.67 mm, 2 mm and 1.6 mm, respectively. The standard deviation of the strain fluctuation is calculated to determine the strain resolution [10,11], and the results are shown in Fig. 2(a). It is shown that the strain measurement resolution is approximately linearly proportional to the frequency-swept range. Subsequently, we conducted tests with a fixed frequency-swept range of 20 nm but varying the distance-domain window lengths from 200 to 400 points with intervals of 50 points. The evolution of strain resolution with respect to 1/$N$ is given in Fig. 2(b). Again, a linear relationship is revealed, as can be expected from Eq. (1).

 

Fig. 2. (a) Relation between the strain resolution and the frequency-swept range. (b) Relation between the strain resolution and the reciprocal of the distance-domain window length (i.e., 1/$N$).

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Hence, it can be inferred that there is a trade-off relationship to some extent between spatial resolution and strain resolution [11]. Increasing the frequency-swept range and decreasing the distance-domain window length contribute to improved spatial resolution but can degrade the strain resolution. In Appendix A, we briefly introduce the demodulation principle of Rayleigh sprectrum-based sensing scheme. The trade-off relation between the sensing resolution and the spatial resolution can be qualitatively explained. This serves as a general guideline for developing a data-driven neural network to facilitate the construction of a high-performance OFDR system with high spatial resolution, high strain resolution, large strain range, and an extensive sensing range. In the proof-of-concept demonstration, we first opt for the largest available frequency-swept range of the light source in a homebuilt OFDR system to obtain the ultimate spatial resolution and ensure a full coverage of the measurable strain range, and then a deep learning model is well-trained to improve the strain resolution. Eventually, an overall high-performance strain sensing system can be achieved.

3. Data and physics-driven neural network

Noise in OFDR sensing systems is typically characterized as random [15]. As mentioned earlier, traditional noise reduction methods, including wavelet denoising [13] and image processing-based approaches [14,15], are often sensitive to parameters and generally exhibit moderate denoising efficiency. In the field of AI algorithms, inspired by the success of 1D-CNN in Raman-OTDR, we have also applied the 1D-CNN to signals collected from a homebuilt OFDR system [34]. The 1D-CNN denoising model has shown a substantial performance improvement for OFDR-based sensing system (the details are presented in Appendix B). However, practical experience has revealed limitations in the denoising capabilities of neural networks relying on small convolutional kernels, mainly due to their limited receptive fields. Existing research [28,35] indicates that increasing the model’s receptive field can significantly enhance performance by capturing a broader data context. Nevertheless, expanding the model’s receptive field introduces challenges, such as increased computational complexity and difficulties in preserving local signal details. Therefore, the key challenge lies in designing the network intelligently to strike a balance between these issues.

3.1. Large kernel denoising network

In our research, we have implemented a network architecture utilizing shallow layers with large convolutional kernels to formulate the large kernel denoising network (LKDNet). The schematic representation of the LKDNet’s network structure is shown in Fig. 3. The network consists of three core modules: Stem module, Large-Kernel convolutional module, and Transitional module.

 

Fig. 3. The structural diagram of the introduced LKDNet.

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The Stem module, designed to capture intricate signal details, is composed of a sequential arrangement of layers: a one-dimensional convolutional layer with a 3-unit kernel, followed by a one-dimensional DW convolutional layer with a 3-unit kernel, a 1$\times$1 convolutional layer, and another one-dimensional DW convolutional layer with a 3-unit kernel. The initial convolutional layer performs a down-sampling of the signal by a factor of two, followed by a DW convolutional layer to extract granular information. Subsequently, a 1$\times$1 convolutional layer is introduced to facilitate inter-channel information propagation and enhance non-linearity. Lastly, a DW convolutional layer concludes this initial phase, effectively extracting detailed signal features.

The Large-Kernel convolutional module is tasked with the extraction of signal features across an expansive receptive field. This module is composed of two primary types of network layers: LK Block and ConvFFN, with the module’s depth being highly customizable. Specifically, the LK Block layer is structured sequentially, encompassing normalization layers (BN) [36], 1$ \times$1 convolutions, one-dimensional DW convolutional layers employing large kernels, and additional 1$ \times$1 convolutions. Notably, shortcut connections have been incorporated into the network design. The design rationale for the LK Block layer adheres to the following guiding principles: 1) The adoption of DW separable convolutions to mitigate the computational overhead associated with large kernels [37]. 2) The integration of 1$ \times$1 convolutional layers both preceding and following the DW convolutional layer to augment the model’s non-linearity. 3) The implementation of shortcut connections [38]. Particularly in the context of large convolutional networks, the absence of shortcut connections can lead to significant performance degradation in handling signal details [39]. ConvFFN is made up of BN, 1$ \times$1 convolutions and GELU activations [40], with shortcut connections integrated into the network architecture. Inspired by the feedforward networks extensively utilized in Transformers [27,41], the ConvFFN block projects data into a higher-dimensional channel space and subsequently compresses it back to the original dimensionality. The inclusion of shortcut connections enhances the network’s capacity for effective nonlinear operations. The LK Block and ConvFNN are stacked alternately to form the large convolutional processing module, and the module’s network depth can be flexibly designed based on the complexity of data processing.

The Transition layer, situated within the heart of the large convolutional processing module, comprises a systematic arrangement of consecutive components, including 1$ \times$1 convolutions followed by a one-dimensional DW convolution with a 3-unit kernel.

When neural networks handle a substantial volume of sensor data, their performance can be constrained. Designing an effective data normalization method becomes a pivotal challenge. In this context, our output layer is configured to consist of a convolutional layer with a 3-unit kernel size along with a BN layer. Experimental findings underscore that the absence of the BN layer leads to performance deterioration when processing sensor signals exhibiting substantial variability. This paper employs two Large-Kernel convolutional modules for signal processing, hence LKDNet can be defined by parameters $[D1; D2], [C1; C2], [K1; K2]$, where $D$ represents the number of LK blocks, $C$ signifies the number of network channels, and $K$ denotes the size of the large convolutional kernels.

3.2. Data and physics-driven strategy

For different OFDR systems and application scenarios, the characteristics of sensor data vary depending on hardware parameters and sensing environments. How to obtain as comprehensive training data as possible for all scenarios is a significant challenge in the implementation of AI noise reduction for optical fiber sensing. Therefore, to replicate real-world sensing scenarios as authentically as possible, we employ random strength values as stochastic variables to simulate the sensor data that may spontaneously manifest on the optical fiber. Assuming that an OFDR system measures strains within a range spanning both positive and negative 10,000 $\mathrm {\mu }\mathrm {\varepsilon }$, we generate random sensor strain trajectories encompassing strengths ranging from −10,000 $\mathrm {\mu }\mathrm {\varepsilon }$ to +10,000 $\mathrm {\mu }\mathrm {\varepsilon }$ based on a data-driven strategy. The maximum consistent length for each strength is established in accordance with the receptive field of a reference network, and for this study, a consistent length of 250 is designated. Essentially, this entails that each strength value will persist within a specific range, and this range is defined as a random number ranging from 1 to 250.

Subsequently, we introduce the corresponding simulated noises to the sensor trajectories. The generation of noise that aligns with physical significance and accommodates a comprehensive range of scenarios poses a significant challenge in the AI denoising process. To tackle this challenge, we have undertaken the simulation of signal noise based on physics-driven OFDR noise data. In the demodulation process of an actual OFDR sensing system, the manifestation of noise is essentially discrete, as the result of the cross-correlation spectral analysis always falls on integer values. Hence, the noise intensity is expressed as a random integer multiple of a base value $\mathrm {\alpha }$, which is defined as follows:

 

Fig. 4. Data and physics-driven training set. (a) Simulated noisy sensor signals. (b) Signal strength consistency detail. (c) Signal zone switching detail.

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4. Results and discussion

4.1. Experiments

The schematic of the experimental setup utilized in the demonstration is given in Fig. 1. A brief introduction to the experiments is given below. The sensing fiber extends over approximately 140 meters, and a delay fiber, measuring 288 meters in length, is integrated into the auxiliary reference interferometer to facilitate phase noise compensation. A tunable narrow-linewidth (<10 kHz) laser (Santec, TSL-770) is employed as the probe source. As mentioned in Section 2, for the experiments described below, the wavelength-swept range is configured at 140 nm, representing the maximum available wavelength range. This choice of a 140-nm tuning range ensures an ultimate two-point spatial resolution of 5.7 $\mathrm {\mu }$m assuming no influence of phase noise, and a large-enough frequency-swept range also ensures a full coverage for an extended strain sensing range. The distance-domain window length $N$ is set to 150, indicating a sensing spatial resolution of 0.857 mm. Smaller values of $N$ can introduce outlier peaks in the cross-correlation spectra, leading to dead zones along the sensing fiber. Three different strains (0 $\mathrm {\mu }\mathrm {\varepsilon }$, 1000 $\mathrm {\mu }\mathrm {\varepsilon }$, and 2000 $\mathrm {\mu }\mathrm {\varepsilon }$) are applied to a 1-m fiber under test (FUT) segment at the end of the test fiber.

4.2. Strain information extraction

In the process of strain information extraction, position mismatch correction is performed to address the position deviation issue arising from the strain accumulation along the sensing fiber [31,32]. Position mismatch correction is a process that relies on accurate strain measurements at the front positions for correcting the rear positions. However, it’s important to note that a random drift in the initial wavelength of the probe lightwave between the reference signal and the measurement signal can introduce a fixed measurement deviation along the entire fiber [9], rendering the position mismatch correction algorithm ineffective. To counteract this, we isolate a short section of the sensing fiber in a sound-proof temperature-controlled box to serve as a reference. This reference fiber helps eliminate the noise floor caused by the random drift of the probe laser frequency [9], ensuring accurate strain demodulation and position mismatch correction. The demodulated strain distributions along the FUT before and after position mismatch correction are presented in Fig. 5.

 

Fig. 5. Demodulated strain distribution with different strains (a) before and (b) after position mismatch correction.

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4.3. Deep learning denoising

To denoise the demodulated OFDR traces, represented as strain-distance signals (used as the input of the model), we utilized the data and physics-driven LKDNet. For the experiment, we generated 50,000 noisy sensor traces following the description in Fig. 4, with each dataset containing ten thousand sample points. These synthetic 50,000 traces were used to train and test the model. It’s noteworthy that in cases of extensive signal coverage, it may be advantageous to incrementally augment the volume of training data, aiming to comprehensively simulate a wide array of sensing scenarios. The training dataset and the testing dataset were allocated in an 8:2 proportion, leading to 40,000 entries for the training dataset and 10,000 for the testing dataset. In order to bolster the model’s capability to handle data spanning a wide range of strain values—bearing in mind that deep learning models typically excel within the numerical range of −1 to 1—we normalized the input sample data to the model within the −1 to 1 range. The label data corresponds to the authentic data of the signal trajectory. Experimental findings have demonstrated that neglecting this data normalization approach results in reduced denoising performance for signals characterized by significant strain values. In our experiments with LKDNet, we obtained the best results by setting the parameters $D$, $C$, and $K$ to [2,2], [64,128], and [51,47] respectively, through empirical trails. The neural network underwent training for 300 epochs with a batch size of 32, commencing with an initial learning rate of 0.01. Every 200 training steps, the learning rate was dynamically reduced to one-tenth of its current value. During each epoch, the input data was initially propagated forward, followed by the computation of the mean squared error (MSE) between the output and the target, which was subsequently backpropagated. The optimization process employed the Adam algorithm to iteratively update the network parameters [43]. These experiments were conducted utilizing the PyTorch library on a Linux platform, leveraging a 128-core AMD EPYC 7742 processor. In the context of identical sensing signal traces, we conducted a comparative analysis of model training under two distinct conditions: the application of white Gaussian noise and the physics-driven noise as described in Section 3. Our experimental results revealed that the denoising performance of the models is nearly equivalent in both scenarios. Notably, when utilizing physics-derived noise, the models exhibit accelerated convergence during the training process, resulting in fast training, as illustrated in Fig. 6.

 

Fig. 6. MSE loss during the entire training process when trained using two different types of training data.

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The ultimate experimental results are shown in Fig. 7. The raw strain-distance profiles along the entire test fiber and the profiles that are processed by the LKDNet are compared in Fig. 7(a). It can be observed that the raw strain traces show significant fluctuations and the amplitude increases with increasing fiber lengths due to the accumulation of phase noise. On the other hand, the denoised signals have much smaller fluctuations in all three different settings of tensile strains applied to the FUT, indicating significant improvement in strain resolution after denoising by the LKDNet model. Enlarged views of the strain distribution along the strain-applied FUT segment and the strain-free fiber length are given in Fig. 7(b) and 7(c), respectively. The demodulated strains on the stretched fiber section match the applied values, without an accuracy degradation in the denoising process. Also, for the strain-free fiber section, the demodulated values after denoising lies close to the 0-$\mathrm {\mu }\mathrm {\varepsilon }$ axis. It is noteworthy that at the right-hand end of the stressed FUT, the polymer coating layer of the fiber was removed to ensure a stable connection between the FUT and the translation stage, which was moved outwards to stretch the FUT during the test. The schematic is shown in Fig. 7(d). Interestingly, as can be observed in an enlarged view of Fig. 7(e), the demodulated strains from this localized fiber section are slightly larger than those from the other fiber lengths. This is due to the fact that the removal of the polymer coating of this localized fiber segment changed the strain response coefficient of the fiber itself. The length of this "outlier" is found to be 43 mm, matching well with the experimental settings. This interesting feature is perfectly preserved after denoising, proving the effectiveness of the simultaneous achievement of denoising the raw signal and preserving the important signal details. We further evaluate the strain resolution by calculating the standard deviation of the random strain fluctuation along the test fiber [10,11]. It is found that the resolution is improved to 0.91 $\mathrm {\mu }\mathrm {\varepsilon }$ compared to 24.16 $\mathrm {\mu }\mathrm {\varepsilon }$ before denoising, with an improvement factor of $\sim$26. In the cases of applying large strains of 1000 $\mathrm {\mu }\mathrm {\varepsilon }$ and 2000 $\mathrm {\mu }\mathrm {\varepsilon }$, the LKDNet model shows a similar extent of performance improvement, as demonstrated in Fig. 7(f). This confirms the strain amplitude-independent characteristics of the model for denoising. Meanwhile, twelve independent measurements over ten minutes at an unstrained state of the FUT were performed, and the averaged strain resolution after denoising over different spatial locations is calculated to be 0.88 $\mathrm {\mu }\mathrm {\varepsilon }$, compared to 23.42 $\mathrm {\mu }\mathrm {\varepsilon }$ before denoising, again verifying the improved strain resolution.

 

Fig. 7. Experimental results. (a) Strain profiles as a function of fiber length before and after denoising by LKDNet. (b)(c) The enlarged views of the strain distribution along the strain-applied FUT segment and the strain-free segment. (d) The schematic diagram of the strain-applied scheme. (e) The enlarged view of the demodulated strain along the coating-removed segment. (f) The strain resolutions before and after denoising by LKDNet with different applied strains.

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The LKDNet has shown significant abilities for denoising the OFDR-based distributed sensing system and improving the strain resolution. The advantage of the LKDNet in denoising performance over the traditional 1D-CNN model derives from a more expansive receptive field. For experimentally verifing the superiority of LKDNet in denoising performance, we carry on 1D-CNN based denoising experiments based on equivalent hardware resources and identical raw data to compare the denoising performance of two approaches (the experimental details are shown in Appendix B). Experiments have shown that the 1D-CNN based denoising model achieves a strain resolution of 3.52 $\mathrm {\mu }\mathrm {\varepsilon }$, while the LKDNet achieves a strain resolution of 0.91 $\mathrm {\mu }\mathrm {\varepsilon }$, thus proving the advantages of the LKDNet based approach. We believe the proposed model in this paper could also be a useful tool for performance improvement in extensive optical fiber sensing systems such as Raman-OTDR, phase-sensitive OTDR, Brillouin-OTDA and etc.

5. Conclusion

In this paper, a data and physics-driven neural network is introduced to enhance the performance of OFDR-based distributed sensing systems. By analyzing the limitations of the current system and the shortcomings of existing signal demodulation approaches, a large-kernel convolutional neural network is innovatively employed to overcome these limitations, leading to a significant improvement in overall system performance. The model design not only focuses on denoising but also leverages the underlying physical characteristics of the sensing system to enhance algorithm efficiency, making it more practical. As a result, assisted with the deep learning model analysis, the system achieves a sub-microstrain (0.91 $\mathrm {\mu }\mathrm {\varepsilon }$) sensing resolution with a sharp spatial resolution of sub-millimeter (0.857 mm) over a 140-m sensing range. To the best of our knowledge, this marks the first experimental demonstration of OFDR-based distributed sensing with both sub-millimeter spatial resolution and sub-$\mathrm {\mu }\mathrm {\varepsilon }$ strain resolution over a long sensing range beyond a hundred meters.

In this proof-of-concept study, the focus was on distributed strain sensing using a long length of standard communication single-mode fiber. However, the demonstrated technique can easily be extended to other distributed fiber sensors based on both OFDR and OTDR modalities, using different types of optical fibers, such as core-exposed fibers. It’s important to note that previous efforts in OFDR systems primarily focused on increasing the complexity of the system hardware to improve performance. In contrast, the introduced technique is entirely software-based, capitalizing on recent advances in data analysis. This approach opens up new possibilities for the development of cost-effective, ultra-high-performance intelligent sensing systems for a wide range of industrial applications. The combination of synthetic training data and a data-and-physics-driven training strategy provides a promising way to leverage machine learning for enhanced sensor performance without requiring extensive hardware modifications.

Appendix A: demodulation principle of OFDR-based distributed sensing by the Rayleigh spectrum

For a linear sweep with a slope of $\gamma$, the optical field of the probe lightwave injected into the FUT can be described as:

(3)$$E(t)= E_0\exp\{j[\omega_0t+\pi\gamma t^2+\theta(t)]\}$$

where $\omega _0$ is the initial optical angular frequency and $ \theta (t)$ represents the phase noise stemming from the non-linearity of the probe lightwave’s frequency sweep. The interference signal of the local lightwave with the Rayleigh backscattered lightwave from the FUT can be written as:

(4)$$I(t)=\sum_{i}\sqrt {R_i} E_0^2 \cos[2\pi\gamma\tau_it +\omega_0\tau_i-\pi\gamma\tau_i^2+\theta(t)-\theta(t-\tau_i)]$$

where $i$ denotes the number of scatterers along the FUT; $R_i$ and $ \tau _i$ are the corresponding reflectivity and the round-trip time, respectively. After Fourier transforming the signal to the distance domain, the data corresponding to the fiber section of interest is selected, followed by the inverse Fourier transform. Then, the complex localized Rayleigh spectrum of the selected fiber section can be obtained:

(5)$$I(t)=\sum_{i=a}^{b}\sqrt R_i E_0^2 \exp\{j[2\pi\gamma\tau_it+\omega_0\tau_i-\pi\gamma\tau_i^2+\theta(t)-\theta(t-\tau_i)]\}$$

where $a$ and $b$ represent the boundary values of the selected fiber section. When there is a strain applied to this section of fiber, the signal is phase-modulated and can be described as:

(6)$$I(t)=\sum_{i=a}^{b}\sqrt R_i E_0^2 \exp\{j[2\pi\gamma\tau_it+\omega_0\tau_i-\pi\gamma\tau_i^2\\ +\theta(t)-\theta(t-\tau_i)+0.78\varepsilon\omega_0\tau_i-0.78\varepsilon\beta l_0]\}$$

where $\varepsilon$ is the applied strain; $\beta$ is the propagation constant; and, $l_0$ is the round-trip distance of the start location where the strain is applied. The linear phase term $0.78\varepsilon \omega _0 \tau _i$ induces a time-domain shift $\Delta t$ of the Rayleigh spectrum, which can be demodulated by a cross-correlation operation. Thus, the applied strain can be derived as [9]:

(7)$$\varepsilon=\frac{2\pi\gamma\Delta t}{0.78\omega_0}$$

Generally, the method of using the optical power spectrum of Rayleigh scattering is employed to address the issue of phase noise and eliminate the effect of the constant phase [8]. However, the residual phase noise is still the main noise source. For a phase noise-free OFDR system, a reflection event is manifested as a delta function in the distance domain. When there exists phase noise, the amplitude of the reflection peak drops and the energy of the peak spreads. The widened peak could cause interference between adjacent locations [44]. For strain sensing applications, if the selected spatial resolution (determined by the selected boundaries of the fiber section as shown in Eq. (5)) is too narrow to integrate the energy of a widened reflection peak, the Rayleigh spectrum would distort, leading to a deteriorated strain resolution. Therefore, the trade-off relation between the sensing resolution and spatial resolution can be qualitatively inferred, as illustrated in Fig. 8.

 

Fig. 8. Comparison of the spatial resolution selection between with and without phase noise.

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Appendix B: the experiment of sensing resolution improvement by 1D-CNN and wavelet denoising

A classical 1D-CNN for signal denoising, which offers a suitable computational complexity and facilitates practical application implementation. Its model architecture is characterized by its simplicity, comprising of convolutional layers and BN layers. The network consists of a total of 20 layers, employing a convolution kernel size of 3 and a step size of 1. The ReLU activation function is utilized. The first and last layers have 1 channel each, representing the strain-distance signal of OFDR. The remaining convolution layers have 64 channels. The denoised signal and the raw signal are compared in Fig. 9. The random fluctuation is much smaller after 1D-CNN denoising with a standard deviation of 3.52 $\mathrm {\mu }\mathrm {\varepsilon }$, which proves the effectiveness of the denoising model. However, compared to the denoising performance of LKDNet demonstrated in Fig. 7, 1D-CNN merely achieves a moderate denoising result while the strain resolution can be raised to 0.91 $\mathrm {\mu }\mathrm {\varepsilon }$ by LKDNet, which proves the superiority of the proposed model.

 

Fig. 9. Strain profiles as a function of fiber length before and after denoising by 1D-CNN.

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As a comparison, a classical wavelet denoising (WD) method is employed to improve the sensing resolution and the results are shown in Fig. 10. The db3 wavelet function is employed to decompose the strain-distance signal in three layers, and then the soft threshold method is utilized to process wavelet coefficients. A suitable threshold needs to be carefully adjusted to improve the sensing resolution as much as possible without deteriorating spatial resolution. In comparison, the denoising method based on deep learning models does not require manual adjustment of algorithm parameters, which is a significant advantage. From Fig. 10 we can observe that the improvement of sensing resolution achieved by the wavelet denoising is quite limited. The standard deviation along the sensing fiber is calaulated to be 8.48 $\mathrm {\mu }\mathrm {\varepsilon }$.

 

Fig. 10. Strain profiles as a function of fiber length before and after denoising by wavelet.

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Funding

Research Initiation Project of Zhejiang Lab (2022ME0PI01).

Acknowledgement

Researchers Supporting Project number (RSPD2024R654), King Saud University, Riyadh, Saudi Arabia.

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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