1. Introduction

As a new monitoring technology, distributed acoustic sensor (DAS) based on phase sensitive optical time-domain reflection technology has received extensive attention in many fields such as structural health monitoring, pavement anomaly detection, perimeter security, oil and gas pipeline monitoring in recent years [1–4]. It has unique advantages such as high-density and large-scale monitoring, anti electromagnetic interference, and convenient deployment, and is considered a potential detection technology in the field of seismic exploration that can replace traditional seismic detectors [5,6]. The DAS system emits modulated pulse light and monitors abnormal acoustic or vibration events along the entire fiber by receiving Rayleigh backscattered signals generated in the fiber [7]. By performing orthogonal demodulation, phase unwrapping, and phase unwrapping on the obtained RBS signal, external vibration information can be restored. Generally, the DAS system requires stable narrow linewidth laser to ensure phase results at the output end [8]. However, currently commercial lasers have the problem of light source frequency drift (LSFD), which leads to the phase signal obtained by the DAS system drifting up and down with the frequency drift of the laser. In addition, due to the randomness of the external environment and the accumulation of phase noise generated during the transmission of light in the DAS system, both directly or indirectly affect the signal-to-noise ratio (SNR) of the phase signal and the subsequent extraction of effective phase information. The frequency drift of laser and phase noise can easily be misunderstood as external disturbances, leading to a sharp decline in the detection ability of DAS system for low-frequency information. Therefore, simultaneously suppressing phase noise and phase drift in phase signals is crucial for DAS system.

At present, there are two main solutions to suppress phase noise and phase drift interference on phase signals in DAS system. One solution is to optimize hardware devices such as lasers in the system, reduce phase interference introduced by hardware devices, or add reference structures, that is, improve the DAS system structure to improve the quality of phase signal acquisition; Another solution is to process the demodulated phase signal through software algorithms to improve the SNR of the phase signal and retain useful phase information. Experts and scholars have done a lot of valuable work in hardware device optimization and system structure improvement. In 2011, Pan et al. improved the structure of distributed Bragg reflector (DBR) fiber lasers by adding strong external cavity feedback to form a composite FP cavity with very narrow linewidth. The frequency noise in the 1kHz frequency domain has been reduced by 10dB [9]. In 2015, Zhu et al. proposed an active compensation method based on laser scanning and cross correlation calculation to suppress the impact of LSFD. By utilizing the proposed active compensation method, LSFD tracking can be achieved and inter trace distortion can be greatly suppressed. The experimental results show that the 1.68 Hz vibration event is successfully identified when the average frequency drift velocity is 0.5 MHz/s, which extends the application field of the Phi-OTDR sensor system to quasi-static measurement [10]. In 2020, Ma et al. proposed a digital differential coherent optical time-domain reflectometry based on heterogeneous dual-sideband linear frequency-modulated (LFM) pulse. Positive and negative LFM signals are loaded onto the upper and lower optical sidebands, respectively. By digitally differentiating the phase demodulation results of two sidebands, the sensitivity was at least doubled at 1 kHz, and 26.10 dB of low-frequency common mode phase drift noise was suppressed at 27 Hz [11]. In the same year, Wu et al. proposed bipolar Golay coded F-OTDR with heterodyne detection, which compensated for laser frequency drift through a new real-time compensation method and combined spectrum extraction and mixing methods to eliminate interference fading. Compared with monopole codes, it achieved an improvement of 7.1 dB SNR at the cost of expanding spatial resolution [12]. In 2021, Wang et al. demonstrated a simple and effective method for monitoring laser frequency changes using a reference random fiber grating. Within a maximum data acquisition time of 200 seconds, the frequency variation of a distributed feedback (DFB) laser with a MHz linewidth is obtained from the reference part of the sensing signal [13]. In 2022, Liu et al. injected modulated Golay encoded pulses into two optical fibers and subtracted the phase drift of the sensing and reference fibers to compensate for frequency drift. The demodulation disturbance of the two fibers increased by 2.8dB and 18.9dB, respectively [14]. In the same year, Mohammadmasoud Zabihi et al. added additional probe frequencies to the Phi-OTDR setup to provide reference frequencies, which provide information about changes in the laser source and other environmental noise such as humidity and temperature, helping to optimize the results extracted from low-frequency noise [15]. The above methods all use adding a reference optical path or improving the system structure to improve the phase signal SNR and reduce the impact of LSFD and phase noise on phase information. However, the increase in hardware equipment leads to an increase in system costs and limited effectiveness in improving phase signal quality.

In terms of software algorithms, in 2013, Wu et al. proposed a signal separation method based on multi-scale wavelets, which can separate phase noise caused by frequency drift of fluctuations, as well as time-varying sound or other interference caused by air motion at different scale components. The experimental results show that compared with traditional differential and fast Fourier transform denoising methods, the SNR of the signal is always the best, and can be improved up to 35 dB under optimal conditions [16]. Although wavelet transform has good time-frequency localization characteristics, the limitation of wavelet transform length can cause signal leakage and other issues. In addition, the selection of wavelet basis, decomposition layers, threshold, and threshold function can all affect the final denoising effect. In 2019, Lv et al. proposed Empirical Mode Decomposition (EMD) for adaptive extraction and elimination of phase drift, which was eliminated under different vibration frequencies of 1 Hz, 5 Hz, and 10 Hz. 0.5 Hz and 0.3 Hz vibrations were detected, with SNR of 55.58 dB and 64.44 dB, respectively [17]. In the same year, Chen et al. proposed a method based on empirical mode decomposition and Pearson correlation coefficient fusion (EMD-PCC) to filter out phase noise in the $\phi$-OTDR system. The interference signal was accurately recovered at vibration frequencies of 200 Hz, 300 Hz, 400 Hz, and 500 Hz [18]. The EMD method does not require preselection of parameters and adaptively decomposes based on the signal itself. However, the inherent mode components (IMF) obtained using this method may contain feature components of different time scales, which is the problem of modal aliasing. In addition, there are also issues such as end effects and difficulty in determining iteration termination conditions, and the effectiveness of EMD is severely affected by noise. In subsequent research, some experts and scholars have proposed EMD improvement methods, such as EEMD, CEEMD, CEEMDAN, etc. Although EMD methods have improved their sensitivity to noise to some extent, residual noise still exists in the decomposition results and non adaptive parameters are introduced. In 2022, Mao et al. proposed a vibration signal denoising method based on Variational Mode Decomposition (VMD) and Pearson Correlation Coefficient (PCC), which performs K-layer VMD denoising on the phase signal after I/Q demodulation. It can significantly improve the SNR of the vibration signal under single frequency single point vibration, multi point double point vibration, and multi frequency single point vibration conditions [19]. VMD transfers the acquisition of signal components into a variational framework, effectively avoiding problems such as modal aliasing, underenvelope, and end effects. However, improper selection of penalty factors and decomposition layers in the VMD method will affect the denoising effect, and its robustness is poor when facing sudden signals and noise interference. Therefore, the above methods generally have problems such as artificially setting parameters and poor stability of denoising effects. In addition, most of the above methods only address phase noise or phase drift, without considering the impact of both types of interference on the phase signal, resulting in phase information loss and phase signal distortion in complex seismic exploration environments.

In order to simultaneously eliminate the impact of phase noise and LSFD on the phase signal in DAS system and improve the quality of the phase signal, this paper proposes a SEE-SGMD-PCC method. This method solves the end effect problem based on the symmetric continuation method of extreme points, maintains the measure and essential characteristics of the original phase signal based on symplectic geometry theory, and filters and reconstructs the dominant components of the signal based on Pearson correlation coefficient. The simulation results show that the proposed method can effectively recover the phase information of single frequency and multi frequency in the face of different levels of phase drift and Gaussian white noise interference. Detailed analysis of the sources of phase noise and causes of phase drift in the DAS system was conducted prior to conducting actual experimental testing. In actual DAS system experiments, compared with methods such as Wavelet, EMD-PCC, VMD-PCC, etc., the SNR of the phase signal processed by the SEE-SGMD-PCC method is significantly improved. The method proposed in this paper can accurately identify signal components, suppress phase noise and phase drift at the same time, and there is no loss of phase information. In two on-site experiments of perimeter security and buried optical cable, the proposed method effectively suppressed phase drift and phase noise, proving the feasibility of the proposed method. Overall, the SEE-SGMD-PCC method greatly improves the quality of phase signals without changing any system structure, providing a new technical means for the application of DAS in seismic exploration.

2. Principle

2.1 Principle of SEE-SGMD-PCC algorithm

The phase drift and noise elimination method based on SEE-SGMD-PCC algorithm can be summarized into four steps: phase signal extension, symplectic geometry mode decomposition, screening of dominant components of signal and signal reconstruction. First, the extremal symmetric continuation (SEE) is used to extend the input time series to reduce the influence of end effects on symplectic geometry mode decomposition (SGMD). Then, the SGMD method decomposes the extended time series without any user-defined parameters to obtain symplectic geometry components (SGCs). Next, we use Pearson correlation coefficient (PCC) to distinguish between the signal dominant component and the noise dominant component in SGCs. Finally, the principal components of the signal are linearly superimposed to complete signal reconstruction, and then the extended parts at both ends of the signal are removed to obtain the processed signal. The specific processing flow of the SEE-SGMD-PCC algorithm is shown in Fig. 1.

 

Fig. 1. Processing flow of the SEE-SGMD-PCC algorithm.

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2.1.1 Phase signal extension

Due to the limited description of the signal near the two ends and the lack of constraints on the information outside the ends, although the analysis window scale or dimension of the signal can be adjusted during the signal decomposition process, the analysis window inevitably extends beyond the beginning and end of the signal. As a result, the features near the two ends will not be well expressed and some information will be lost, causing the values at both ends of the signal component in the decomposition result to deviate from the actual value, This leads to the issue of end effect, as shown by the red box mark in Fig. 2(b). The signal without phase signal extension exhibits end effects, so it is necessary to reduce the impact of end effects on SGCs through phase signal extension.

 

Fig. 2. Schematic diagram of end effects. (a)Original phase signal (b) Comparison of processing results.

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In order to reduce the impact of end effects on SGCs during the decomposition process, we use the SEE method to extend the signal $X = \{ {x_1},{x_2},{x_3}, \ldots,{x_N}\}$, where $N$ is the length of the original data. The specific expansion process is as follows:

For a signal sequence $X$ with a length of $N$, the index of its maximum sequence is $M(i)$, where $i = 1,2,3, \ldots,m$; The index of its minimum sequence is $L(j)$, where $j = 1,2,3, \ldots,l$. Perform data extension on the signal $X$ according to Eq. (1) and Eq. (2):

2.1.2 Symplectic geometry mode decomposition

Before using SGMD to decompose the signal $Y$, the relevant definitions, theorems and properties of symplectic geometry are as follows:

Definition 1. the real matrix is represented by $R$, the symbols ${A^T}$, ${A^{ - 1}}$ and ${A^{ - T}}$ respectively represent the transposition, inverse and inverse transposition of matrix $A$, the symbol ${I_b}$ represents $b$-order identity matrix, and ${J_1}$ represents the anti-symmetric matrix $\left [ {\begin {array}{cc} {} & {{I_n}}\\ { - {I_n}} & {} \end {array}} \right ]$.

Definition 2. Set the matrix $S \in {R^{2n \times 2n}}$, if $J_1^T{S_{}}{J_1} = {S^{ - T}}$ is satisfied, then matrix $S$ is a real symplectic matrix.

Definition 3. Set the matrix $A \in {R^{n \times n}}$, if $G = {G^T} \in {R^{n \times n}}$ and $F = {F^T} \in {R^{n \times n}}$ are satisfied, then matrix $H = \left ( {\begin {array}{*{20}{c}} A & F\\ G & { - {A^T}} \end {array}} \right )$ is a Hamilton matrix.

Definition 4. Set the matrix $\omega \in {R^{n \times n}}$, $P \in {R^{n \times n}}$, then matrix $P = I - 2\omega {\omega ^T}/{\omega ^T}\omega$ is a Householder matrix or elementary reflectivity.

Lemma 1. Set the matrix $H \in {R^{2n \times 2n}}$ be a Hamiltonian matrix, for the matrix $J_1$ defined above, there is ${J_1}H = {({J_1}H)^T}$.

Property 1. Set the matrix $H \in \Phi$, $\Phi = \{ H \in {R^{2n \times 2n}},|{J_1}H = {({J_1}H)^T}\}$ is the set composed of all Hamiltonian matrices; Set the matrix $S \in \Psi$, $\Psi = \{ S \in {R^{2n \times 2n}},|J_1^T{S_{}}{J_1} = {S^{ - T}}\}$ is the set composed of all real symplectic matrices, then $SH{S^{ - 1}} \in \Phi$, that is, the symplectic similarity transformation preserves the Hamiltonian matrix structure.

Theorem 1. Set the trajectory matrix to $Z$, let ${C_{n \times n}} = {Z^T}Z$, construct a new matrix $H = \left [ {\begin {array}{*{20}{c}} C & {}\\ {} & { - {C^T}} \end {array}} \right ]$, which is also a Hamiltonian matrix.

Theorem 2. There must exist a Householder matrix $H_1$ such that ${H_1}{H_{}}H_1^T$ is an upper Hessenberg matrix:

(a) Construct trajectory matrix ${Z_{p \times q}}$

Based on the Takens embedding theorem, setting the embedding dimension as $q$, the time delay embedding method can be used to map a one-dimensional signal $Y$ into a trajectory matrix ${Z_{p \times q}}$, which contains all the dynamic information of the time series.

According to reference [20], the selection idea of embedding dimension is to calculate the power Spectral density (PSD) of time series $Y$, and then estimate the maximum peak frequency ${f_{\max }}$ in PSD. When the normalization frequency ${f_{\max }/{f_s}}$ ($f_s$ is the sampling rate) is less than the given threshold ${10^{ - 3}}$, the embedding dimension $q$ is set to $(N+2r)/3$. When the normalization frequency ${f_{\max }/{f_s}}$ ($f_s$ is greater than the given threshold ${10^{ - 3}}$, the embedding dimension $q$ is set to $({f_s}/{f_{\max }}) \times 1.2$. The main purpose of selecting this dimension is to increase the time span of the embedding dimension by 20% compared to the average period of the required component sequence, in order to improve the decomposition effect of SGMD during the decomposition process.

(b) Symplectic geometry transformation

According to Theorem 1, in order to obtain the characteristic information of the signal $Y$, it is necessary to construct a Hamilton matrix related to its trajectory matrix $Z$. The trajectory matrix is subjected to autocorrelation processing to obtain the matrix $C$:

The size of the initial single component matrix obtained through symplectic geometry transformation is $p \times q$. In order to obtain the reconstructed signal expression with the same dimension as the one-dimensional signal $Y$, the initial single component matrix mentioned above is subjected to diagonal averaging processing. Diagonal averaging can transform the initial single-component matrix $D_i$ into a new time series with a length of $n=N+2r$, and $q$ new time series can be obtained by the specific transformation Eq. (10):

After processing through the above process, $q$ new time series were obtained, but they may have the same characteristics, period, and frequency components and are not completely independent. Time series with the same characteristics can be combined and reconstructed using time series correlation and frequency similarity. The specific processing process is as follows:

Due to the fact that the time series with the highest degree of correlation with the original signal is located at the front end of ${Y'_i}$, the correlation ranking with the original signal is ${Y'_1} > {Y'_2} > \cdots > {Y'_q}$. Therefore, ${Y'_1}$ is selected for feature comparison with other components, and several components with high similarity are summed to get the first symplectic geometry component $SG{C_1}$. Furthermore, by removing $SG{C_1}$ from $Y'$, the remaining signals can be expressed as $re{s_1}$:

2.1.3 Select dominant components of signal

The correlation between the symplectic geometry component and the original signal is calculated based on Pearson correlation coefficient (PCC), which is defined as Eq. (15). SGC with PCC greater than the set threshold is determined as the dominant component of the signal, while the remaining SGC is determined as the dominant component of the noise.

2.1.4 Signal reconstruction

The dominant component $SG{C_i},1 \le i \le r$ of the signal is obtained, where $r$ is the number of dominant components of the signal. Accumulate the dominant components of the signal to obtain the denoised signal:

2.2 Performance of simulation experiment

In order to test the phase drift and noise removal effects of the proposed algorithm under different conditions, four different types of simulation experiments were designed, as shown in Table 1, where $randn()$ represents Gaussian white noise with mean value of 0 and variance of 1, and $size()$ represents the size of the array.

Table 1. Composition of simulated experimental signals

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The Figs. 3–6 show the original time-domain signals and their corresponding power spectra of four different types of experiments, as well as the processed time-domain signals and their corresponding power spectra. There are different degrees of phase drift and Gaussian white noise in the four groups of original phase curves. It is difficult to observe effective phase signals in the time domain, and even there may be situations where noise can overwhelm the effective signal, which seriously affects the quality of the phase signal, making it difficult to extract the effective information.

 

Fig. 3. Time domain and spectrum diagrams.(a)the original signal 1 (b)the processed signal1.

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Fig. 4. Time domain and spectrum diagrams.(a)the original signal 2 (b)the processed signal2.

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Fig. 5. Time domain and spectrum diagrams.(a)the original signal 3 (b)the processed signal3.

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Fig. 6. Time domain and spectrum diagrams.(a)the original signal 4 (b)the processed signal4.

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From the results of several groups of experiments, the SEE-SGMD-PCC method can suppress the interference of phase drift and Gaussian white noise on phase signals of different frequencies, and restore the phase curve well. Therefore, the effectiveness of the proposed method has been preliminarily demonstrated through simulation experiments, reducing the impact of phase drift and phase noise on the phase signal, and possessing strong noise robustness.

3. Experimental verification

3.1 Phase signal acquisition

We adopts a distributed acoustic sensor based on digital heterodyne coherent detection technology, as shown in Fig. 7. The light from the narrow linewidth laser (NLL) is divided into two parts by a 90:10 fiber coupler 1 (OC1), where 90% of the light is used as the detection light and 10% is used as the local oscillator light (LO); The detection light is modulated into pulsed light by AOM. The pulsed light is amplified by the front-end erbium-doped fiber amplifier (EDFA) and then enters the fiber under test through a circulator (Cir). The piezoelectric transducer (PZT) connected to the tail end of the optical fiber under test is driven by an arbitrary function generator (AFG) to simulate the external vibration signals with different frequencies. After passing through the circulator, the backscattered Rayleigh scattering (RBS) signal light from the optical fiber under test is divided into the signal light in X and Y polarization directions by the polarization beam splitter (PBS). After passing through a polarization maintaining fiber coupler (PMOC), the local oscillator beats the signal light in two polarization directions at fiber couplers 2 and 3 (OC2, OC3), respectively. The beat signal is converted into an electrical signal by a photodetector (PD), and finally the electrical signal is collected by a data acquisition card (DAQ) and sent to a computer for analysis and processing. The specific parameter configuration of the DAS system is shown in Table 2.

 

Fig. 7. Experimental platform. NLL: Narrow Linewidth Laser; OC: Optical Coupler; AOM: Acousto-Optic Modulator; EDFA: Erbium-Doped Fiber Amplifier; Cir: Circulator; PBS: Polarization Beam Splitter; PMOC: Polarization Maintaining Optical Coupler; PD: Photodetector; DAQ: Data Acquisition; PC: Personal Computer; AFG: Arbitrary Function Generator; PZT: Piezoelectric Transducer; FUT: Fiber Under Test.

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Table 2. Parameter configuration of the DAS system

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The original signal collected by the DAS system is the Rayleigh backscatter beat signal, which needs to be demodulated to obtain the object processed by the method proposed in this paper, namely phase information. The specific processing steps are as follows:

Firstly, the two beat electrical signals collected by DAS can be expressed as:

3.2 Phase signal analysis

The theoretical model of traditional DAS assumes that the detection light pulse in the optical fiber is an ideal single frequency light, and there is no frequency drift problem. The collected signal is shown in Eq. (26). However, in actual DAS system, when there are no external vibration events, the received signal will also produce distortion. As shown in the Fig. 8, as the acquisition time increases, more severe signal distortion will occur.

 

Fig. 8. DAS signal distortion.(a)20-bar curves (b)100-bar curves.

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This is because lasers are susceptible to changes in external environment and electromagnetic fields. When external vibrations occur in the laser, such as building vibrations, personnel activities, and other events that can cause the vibration of the instrument itself, it can cause changes in the internal resonant cavity of the laser, such as changes in the length of the resonant cavity or displacement of the mirror inside the resonant cavity, thereby leading to the frequency stability of the laser. In addition, the random noise caused by the Spontaneous emission of laser atoms also has an uncontrollable impact on the laser, so the commercial narrow linewidth lasers commonly used in DAS systems cannot guarantee long-term stability.

When using the DAS system to collect signals, the phase change term caused by laser frequency drift will be directly superimposed on the phase term, and the beat frequency electrical signal we collect will become:

3.3 Experimental results

3.3.1 Single frequency vibration signal

In this experiment, AFG is used to generate six different frequencies of sine wave driving PZT in the range of 0.5Hz-1kHz to simulate external vibration events. The Figs. 9–14 show the original phase signal after IQ orthogonal demodulation, with the red part of each graph representing the spectrum, and the horizontal and vertical coordinates representing the frequency and amplitude, respectively; The black part displays the phase curve, with horizontal and vertical coordinates representing time and phase, respectively; The blue part shows the magnified red curve, and the presence of phase drift affects the observation of other frequency components. We amplified the red curve near the effective frequency component, so that we can observe the amplitude of the effective frequency component well. The green box marks the phase drift. It can be seen that the original phase signal exhibits varying degrees of phase drift, and phase noise causes serious interference to the original phase signal, resulting in the inability to obtain accurate external vibration information. For the original phase signals with different frequencies mentioned above, the actual data denoising effect of the proposed method is tested, and methods such as Wavelet, EMD-PCC, and VMD-PCC are used as comparative methods for the proposed method in this paper.

 

Fig. 9. Original phase signal of 0.5Hz.

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Fig. 10. Original phase signal of 1Hz.

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Fig. 11. Original phase signal of 5Hz.

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Fig. 12. Original phase signal of 20Hz.

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Fig. 13. Original phase signal of 700Hz.

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Fig. 14. Original phase signal of 1kHz.

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We use signal-to-noise ratio (SNR) to evaluate the suppression effect of different methods on phase drift and phase noise. The definition of SNR is shown in Eq. (32):

The Figs. 15–20 show the denoising results of different methods at 0.5Hz, 1Hz, 5Hz, 20Hz, 700Hz, and 1kHz, respectively. In Figs. 15–20(a), the wavelet method to some extent suppresses phase noise, but its SNR is negative, indicating that the wavelet method cannot effectively remove phase drift. In Figs. 15–20(b), although the EMD-PCC method can eliminate the impact of phase drift on the phase signal, there is a certain degree of distortion observed in the time-domain signal processed by this method. In Figs. 15–20(c), VMD-PCC seems to provide better processing results compared to the first two methods, and the SNR has been improved to a certain extent. However, both EMD and VMD methods have end effects and mode mixing issues. When complex waveforms appear, they will be decomposed into several inaccurate components, resulting in other noise components in the phase signal, thereby affecting the SNR and effective information of the phase signal. Compared to the other three methods, the method proposed in this paper does not require any user-defined parameters and can adaptively select parameters to decompose and process the signal in the face of varying degrees of noise interference and phase drift. After processing using the SEE-SGMD-PCC method, a pure spectrum graph as shown in the Figs. 15–20(d) was obtained, and the SNR of the phase curve was greatly improved.

 

Fig. 15. 0.5Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 16. 1Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 17. 5Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 18. 20Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 19. 700Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 20. 1kHz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Overall, in response to the phase drift caused by laser frequency drift under external disturbances ranging from 0.5Hz to 1kHz, the amplitude at 0Hz in the spectrum results processed by the wavelet method remained basically unchanged, indicating that the wavelet method judged the phase drift as useful information of the signal itself. The phase drift was not effectively eliminated, and there was a certain end effect problem, as shown in Figs. 15–20(a); The EMD-PCC method itself has problems such as modal aliasing, and there is a certain residual phase drift at 0Hz, as shown in Fig. 15(b)); The amplitude at 0Hz in the spectrum results processed by VMD-PCC and the method proposed in this paper is basically suppressed to around 0. Both methods fully utilize the correlation between signal components and the original signal, and can remove the influence of phase drift on useful signals. In response to the phase noise problem in the phase curve, compared to the other three methods, the proposed method can effectively suppress phase noise in the DAS system even when facing complex waveforms and strong noise interference, improve the SNR of the phase signal, and enhance the low-frequency detection ability of the DAS system.

3.3.2 Multi-frequency vibration signal

In order to further evaluate the applicability and denoising effect of the proposed algorithm in the DAS system, two sets of multi frequency vibration event experiments were designed. The two sets of experiments included frequencies of 100Hz, 300Hz, 500Hz and 200Hz, 400Hz, 600Hz, respectively. The above two signals generated by AFG drove PZT to simulate multi-frequency external vibration events. The two sets of phase curves obtained through phase demodulation are shown in the Fig. 21, and obvious external disturbance events can be seen from the enlarged spectrum. However, there are also serious phase drift and phase noise issues. Similarly, Wavelet, EMD-PCC, VMD-PCC, and other methods were used as comparative methods to test the noise suppression and drift elimination effects of SEE-SGMD-PCC under multi-frequency disturbances.

 

Fig. 21. Original phase signal.(a)signal_1: 100Hz+300Hz+500Hz (b)spectrum of signal_1 (c)signal_2: 200Hz+400Hz+600Hz (d)spectrum of signal_2.

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Figures 22–23 show the processing results of two sets of multi frequency vibration experiments using different methods. Observing Figs. 22–23(a), it can be observed that although the main frequency phase information processed by the Wavelet method is not lost, it still cannot solve the problem of phase drift, and the noise suppression effect is poor. In Figs. 22–23(b), although the EMD-PCC method effectively suppresses phase drift, the phase signal processed by it experiences phase information loss and severe phase signal distortion. When applying DAS system to on-site environments, this issue may lead to the loss of useful information, missed reporting of external disturbance events, and so on. For the denoising results shown in Figs. 22–23(c), the VMD-PCC method has achieved a certain degree of improvement in SNR after processing, without any loss of phase information. However, due to the fact that the decomposition effect of this algorithm depends on the setting of the number of modal decompositions, if the setting is not accurate, the ability to extract effective phase information will sharply decrease. Therefore, when facing complex external environments and multi frequency disturbance events, the VMD-PCC method cannot completely and effectively remove noise components and has poor stability. In Figs. 22–23(d), the SEE-SGMD-PCC method adaptively selects denoising parameters to increase the SNR of the 100-600Hz signal to 26.25dB, 45.94dB, 42.75dB, 48.38dB, 38.57dB, and 40.75dB, maximizing the recovery of the original phase signal and fully demonstrating the algorithm’s ability to suppress phase drift and phase noise when dealing with external multi-frequency disturbance information, thus facilitating the subsequent extraction of effective information.

 

Fig. 22. 100Hz+300Hz+500Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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Fig. 23. 200Hz+400Hz+600Hz phase signal processing results. (a)Wavelet (b)EMD-PCC (c)VMD-PCC (d)SEE-SGMD-PCC.

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3.3.3 Test of stability

In order to further test the stability of the proposed method in this paper, multiple external vibration experiments were repeated. Due to the need to manually determine the wavelet basis and decomposition levels in the Wavelet method, as well as the decomposition level $k$ and penalty parameter $\alpha$ in the VMD-PCC method, four sets of optimal custom parameters were selected for stability testing. In the wavelet method, db4, db3, sym6 and coif4 were selected, while in the VMD-PCC method, $(k,\alpha )$ was selected for (9,1600), (9,1000), (7,600) and (11,1400) respectively. The final test results are shown in Fig. 24. The test results show that the Wavelet and VMD-PCC methods have poor stability under the influence of artificially selected parameters, and the Wavelet method cannot solve the phase drift problem, resulting in little improvement in the SNR after processing; Although the EMD-PCC method does not require manual parameter setting, it is susceptible to end effects, modal aliasing, and other issues, and cannot guarantee stable suppression effects in environments with strong background noise; The average SNR improvement of the method proposed in this paper can reach over 80dB. Compared with the other three methods, the SEE-SGMD-PCC method proposed in this paper has good stability, and has good suppression effect on phase drift and phase noise that exist simultaneously in the phase curve, effectively improving the recognition ability of the DAS system for low-frequency vibration signals.

 

Fig. 24. Stability test result.

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3.3.4 Application

In order to further verify the feasibility of the proposed method in practical application scenarios, we conducted perimeter security experiment and buried optical cable experiment, as shown in Figs. 25–26. In the application of complex environments such as perimeter security and seismic exploration, obtaining data without phase drift and low phase noise is necessary to distinguish or invert perimeter intrusion events and underground structures. Therefore, higher requirements are put forward for DAS systems, and suppressing phase drift and phase noise is particularly crucial. In the perimeter security experiment, as shown in Fig25.(a) and (b), we spiral lay the optical cable on the perimeter fence and use hand tapping as the active source; In the experiment of buried optical cable, as shown in Fig26.(a) and (b), a length of approximately 19m of the cable was laid in soil with a depth of approximately 20cm, with stamping feet used as an active source near the buried optical cables. The DAS system serves as a collection device for signal acquisition in the above two application scenarios.

 

Fig. 25. Perimeter security experiment. (a)Schematic diagram of perimeter security (b)Experimental site map.

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Fig. 26. The experiment of buried optical cable. (a)Schematic diagram of perimeter security (b)Experimental site map.

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The phase signals collected from two different sites were separately extracted for 5 seconds as the input signals for the proposed method in this study. The tapping signal and stomping signal are shown in Fig. 27(a) and (c) respectively, revealing severe interference from phase drift and phase noise. These signals cannot be directly used for effective identification and extraction of useful information. The SEE-SGMD-PCC method proposed in this paper was used to process the original tapping signal and stomping signal. The processed results are shown in Fig. 27(b) and (d), and the phase drift phenomenon was well suppressed, effectively recovering the phase signal and fully preserving the information carried by the phase signal. This proves the feasibility of the proposed method in practical application scenarios and further improves the performance of the DAS system.

 

Fig. 27. The experiment results of practical application. (a)Original tapping signal (b)Processed tapping signal (c)original stomping signal (d)Processed stomping signal.

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4. Conclusion

In response to the problem of phase information being severely affected by phase drift and phase noise interference in DAS system, which cannot effectively monitor low-frequency information, this paper proposes an algorithm based on SEE-SGMD-PCC to simultaneously suppress phase noise and LSFD, and provides a process flow for using this algorithm to process phase signals. Several simulation experiments show that the proposed method can effectively restore phase information in the face of different drifts and Gaussian white noise. Then, the DAS based on digital heterodyne coherent detection technology was used as the experimental platform to analyze the sources of phase noise and phase drift, as well as their impact on phase information. Finally, single frequency vibration, multi frequency vibration experiments, and two on-site application experiments were designed. The single frequency experimental results show that the maximum SNR of the phase signal can reach 79.23dB, which greatly improves the low-frequency detection performance of the DAS system compared to methods such as Wavelet, EMD-PCC, and VMD-PCC. The multi-frequency experimental results show that the proposed method improves the SNR of the 100-600Hz signal to 26.25dB, 45.94dB, 42.75dB, 48.38dB, 38.57dB, and 40.75dB, respectively, without any problems such as phase information loss or phase curve distortion. It has the ability to identify external multi-frequency disturbance information and restores the original phase signal to the maximum extent possible. The results of two on-site application experiments also demonstrate the feasibility of this article in practical application environments. The method proposed in this paper adaptively improves the phase signal quality and low-frequency detection performance of the DAS system without adding any other hardware devices such as reference devices and changing the system structure, providing a feasible means for the application of the DAS system in complex seismic exploration environments such as cities, oceans, and mountains.