Spatio-temporal joint oversampling-downsampling technique for ultra-high resolution fiber optic distributed acoustic sensing

Hao Li; Cunzheng Fan; Zhengxuan Shi; Baoqiang Yan; Junfeng Chen; Zhijun Yan; Zhijun Yan; Deming Liu; Perry Shum; Qizhen Sun; Qizhen Sun; Qizhen Sun

doi:10.1364/OE.455747

1. Introduction

Fiber optic distributed acoustic sensing (DAS) is one of the most powerful techniques to obtain the internal property of the structure through detecting the transmission of the acoustic signals. Many engineering applications based on fiber DAS such as seismology [1], perimeter security [2], hydroacoustic detection [3] and pipeline health monitoring [4] have been reported. In particular, high-resolution and high SNR DAS system is an extremely important detection tool for seismic imaging and VSP exploration, whose sampling bandwidth only needs a few hundred Hertz. Due to the advantages of the long measurement distance, high-sensitivity and the ability of linear transformation, phase-sensitive optical time domain reflectometry (φ-OTDR) is the most popular technique of the fiber DAS [5].

Through injecting highly coherent probe pulses and then demodulating the optical phase of the Rayleigh backscattering signal (RBS), the external acoustic waves acting on the sensing fiber can be precisely recovered by several typical techniques for phase extracting such as 3*3 coupler method [6], phase generated carrier (PGC) method [7] and IQ demodulation [8]. However, these DAS systems are suffered from multiple noises caused by low reflected RBS intensity of the single mode fiber (SMF) [9], coherent fading [10], polarization fading [11], optical amplification noise [12], pulse modulated noise [13] and system hardware noise, leading to a deterioration of the acoustic sensing performance. In order to obstacle the influence of these noises, researchers have proposed many schemes. Considering that the phase demodulation noise is inversely correlated with the intensity of the backscattering light [9], the scattering enhanced fiber is used to increase the SNR of the RBS and further reduce the sensing noise [14–16]. For fading problems, the most common method is diversity technology. For example, frequency division multiplexing (FDM) [17–21] and wavelength division multiplexing (WDM) [22] are proposed to reduce the coherent fading of SMF, and polarization diversity is demonstrated to eliminate the polarization fading noise [23], in which the strain resolution of 0.4nε/√Hz and 40pε/√Hz are reported in Ref. [19] and [21], respectively. Since different probe components have different RBS patterns, the fading can be eliminated by combining the phase of different components. In addition, coding technique [24,25], orthogonal double pulse [26] and polarization maintaining fiber [27] are also introduced to solve the coherent or polarization fading problem, in which 75pε/√Hz strain resolution is realized [25]. Although the fading noise can be reduced, the complexity and cost of the methods mentioned above are greatly increased since multiple components with different frequencies or different modulation formats need to be generated. Besides, the fading noise can also be suppressed through the optimized algorithm such as weighted-channel stack algorithm for the dual-pulse heterodyne system [28]. However, the higher pulse repetition frequency is required and then the sensing distance is limited. Besides, the spatial resolution will be deteriorated due to the multiple channels averaging. For suppressing the system background noises, several conventional methods are employed. At the transmitter, probe pulse with high coherence and high extinction ratio generated by the combination of narrow linewidth laser, high performance modulator and ASE noise filter is injected to enhance the SNR of the RBS, and further reduce the demodulated phase noise. At the receiving end of the system, through decreasing the photodetector noise and optimizing the sampling resolution of the data acquisition card (DAQ), the acquisition and quantization noise can be greatly reduced. However, other noise such as out-of-band noise are still existed. Apart from the denoising methods mentioned above, the averaging techniques are proposed to suppress the noises of the SMF based DAS [29–31]. However, the severe crosstalk between channels will be introduced. For present coherent DAS system, the highest strain resolution reported by utilizing scattering enhanced fiber can reach hundreds of fε under the bandwidth of kHz level [32] and 166 pε at ultra-low frequency of 0.001Hz [33]. However, for the SMF based system, it can only reach tens of pε in the order of several kHz sampling bandwidth due to the low SNR of optical signal. Consequently, there still exits further space to optimize the noise floor and strain resolution of the SMF DAS.

In this paper, a spatio-temporal joint oversampling-downsampling technique is proposed to significantly reduce the noise floor of the SMF DAS, which is based on the heterodyne coherent detection and polarization diversity system. In the spatiotemporal joint dimension, dense oversampling is carried out first, and then downsampling is performed to compress the target spatial and temporal acquisition bandwidths simultaneously. Through the two-dimensional oversampling-downsampling process, the temporal out-of-band noise and the spatial interference fading noise can be greatly reduced without incurring penalties in the complexity and cost of the system. The fiber DAS system based on the proposed spatiotemporal oversampling-downsampling algorithm can be applied to the high-precision detection area, such as passive source seismic imaging and VSP exploration.

2. Principle

For a coherent SMF DAS system, the traditional demodulation method is to divide the optical fiber into m sections, and the differential phase between two ends of gauge length (GL) is calculated for each section. Then, the n probe pulses transmitted according to the target acquisition bandwidth are calculated in the same way, corresponding to n samples for sensing fiber. Finally, the results are arranged and combined according to the spatial and temporal orders to obtain the demodulated m*n two-dimensional matrix. However, the demodulated SNR will be seriously affected by interfere fading and polarization fading, and large out-of-band acquisition noise will be introduced. Here, a two-dimensional processing method named spatiotemporal joint oversampling-downsampling is proposed to optimize the noise floor of the SMF DAS. As illustrated in Fig. 1, for the target spatiotemporal samples of s_m*t_n_, the oversampling is performed first, in which multiple sub-samples characterized as (s_ms,t_nt) are carried out within the adjacent target sampling intervals of spatial and temporal dimensions, respectively. Further, the sub-vectors of sub-sampling points are formed, and then the downsampling process is performed. The sub sampling points are compressed to the target spatiotemporal sampling rate with certain downsampling coefficients R_s and R_t, and the final output result (s_m’,t_n’) is served as the target matrix of m*n. The specific analysis is as follows.

Fig. 1. The schematic of the spatiotemporal joint oversampling-downsampling method.

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2.1 Spatial oversampling-downsampling

In the classical heterodyne coherent detection system, pulses with single carrier frequency are repeatedly injected into the fiber under test (FUT) to perform the acoustic sensing. The Rayleigh backscattered light collected by coherent detection is processed in digital terminal. As illustrated in Fig. 2(a), in the spatial dimension, the sensing fiber is spatial sampled based on the digital RBS data points sampled by DAQ with a fixed bandwidth and divided into multiple sections with the same channel spacing (CS). Different colored boxes represent different spatial sampling sections, in which the width W is less than the pulse width W_p and one single sensing channel is defined as fiber section between two adjacent sampling sections. The RBS data points obtained by DAQ are decorrelated with each other. The spatial oversampling-downsampling proposed in this work is not aimed at the fast time axis sampling determined by DAQ, but is an artificially sampling process based on the RBS digital data obtained by DAQ.

Fig. 2. The schematic of spatial sampling. (a) traditional spatial sampling; (b) ideal vector R(Z_i) and noise vector N(Z_i); (c) deviation between ADV and IDV; black line: original near fading vector, red line: keep the ideal vector while decrease the modulus of the noise vector, blue line: increase the modulus of ideal vector on the basis of weakened noise vector.

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Through multiplying with a pair of orthogonal reference signals [34], the phase vector of each sampling point can be obtained. Then, the spatial phase vector of each section with the width of W can be expressed as:

(1)$$\begin{aligned} \overrightarrow {V({{Z_i}} )} &= \frac{1}{M}\sum\limits_{m = {l_{(i,1)}}}^{{l_{(i,M)}}} {[{{A_m}\exp ({j{\varphi_m}} )+ {A_{n(m)}}\exp ({j{\varphi_{n(m)}}} )} ]} = |{\overrightarrow {r({{Z_i}} )} } |\exp ({j{\varphi_i}} )+ |{\overrightarrow {n({{Z_i}} )} } |\exp ({j{\varphi_{n(i)}}} )\\ &= {A_i}\exp ({j{\Phi _i}} ). \end{aligned}$$

where $\overrightarrow {V({{Z_i}} )}$, $\overrightarrow {r({{Z_i}} )}$ and $\overrightarrow {n({{Z_i}} )}$ are respectively the phase vector, ideal vector and noise vector of one sampling section, ${l_{(i,m)}}$ is the sampling point within ${Z_i}$ sampling section, ${Z_i}$ represents the first sampling point of ${Z_i}$ sampling section (${Z_i} = {l_{(i,1)}}$), and ${Z_{i + 1}} - {Z_i} = CS/\Delta z$, ${l_{(i,m + 1)}} - {l_{(i,m)}} = 2{n_{eff}}{f_{DAQ}}\Delta z/c = 1$, $M = {W / {\Delta z}}$ ($\Delta z$ is the sampling resolution determined by DAQ, ${n_{eff}}$ is the refractive index, ${f_{DAQ}}$ is the sampling rate of DAQ, and c is the light velocity) is the total sampling number in the width of W, ${A_m}$ and ${A_{n(m)}}$ are respectively the amplitude of ideal phase vector and noise vector of each sampling point, as well as ${\varphi _m}$ and ${\varphi _{n(m)}}$ are the phase of ideal phase vector and noise vector, respectively. Traditional demodulated method is to calculate the spatial differential vector between two ends of GL, in which one sensing channel (CH) can be characterized as:

(2)$$\begin{aligned} \overrightarrow {C{H_i}} &= \overrightarrow {V({{Z_i} + gl} )} \cdot conj[{\overrightarrow {V({{Z_i}} )} } ]= |{\overrightarrow {R({{Z_i}} )} } |\exp [{j({{\varphi_{i + gl}} - {\varphi_i}} )} ]+ |{\overrightarrow {N({{Z_i}} )} } |\exp [{j({{\varphi_{n(i + gl)}} - {\varphi_{n(i)}}} )} ]\\ &= {A_{i + gl}}{A_i}\exp [{j({{\Phi _{i + gl}} - {\Phi _i}} )} ]= {A_{i + gl}}{A_i}\exp [{j\Delta {\Phi _i}} ]. \end{aligned}$$

where $\overrightarrow {C{H_i}}$, $\overrightarrow {R({{Z_i}} )}$ and $\overrightarrow {N({{Z_i}} )}$ are respectively the actual differential vector (ADV), ideal differential vector (IDV) and noise differential vector (NDV), $\overrightarrow {V({{Z_i} + gl} )}$ represents the sampling section whose starting sampling point is ${Z_i} + gl$, and $gl = GL/\Delta z$. Then, the phase difference $\Delta {\Phi _i} = {\Phi _{i + gl}} - {\Phi _i}$ can be obtained through the arctangent and unwrapping calculation as shown in Fig. 2(b). However, there will be large noise due to the influence of interfere fading as illustrated in Fig. 2(c). For either side of the two ends of GL, if the coherent fading occurs, i. e. the intensity of $|{\overrightarrow {R({Z_i})} } |$ near to zero, the value of ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$ will become much smaller, making the actual phase vector $\overrightarrow {C{H_i}}$ deviating from ideal vector $\overrightarrow {R({{Z_i}} )}$ seriously, and further resulting in the fading sensing channels. Even there is no fading, but the RBS SNR (the value of ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$) of the sampling area is not very high, the demodulation noise will also be relatively large. Notably, by increasing $|{\overrightarrow {R({Z_i})} } |$ (blue line) or decreasing $|{\overrightarrow {N({Z_i})} } |$ (red line), the ADV can be closer to IDV, leading to a low-noise demodulation.

Here, a spatial oversampling-downsampling method is proposed to eliminate the demodulated noise mentioned above. First of all, the spatial oversampling is implemented to change the sparse channel spacing to dense channel spacing (DCS) artificially, in which CS is reduced several times. Hence, dense sampling sections are introduced as illustrated in Fig. 3(a), in which different colored boxes represent different spatial sampling sub-sections and the width of each sub-section (W) is less than the pulse width (W_p). Besides, the value of DCS is less than W, and thus the multiple sampling sub-sections are generated within one traditional sampling section determined by spatial resolution.

Fig. 3. The schematic of spatial oversampling. (a) spatial oversampling; (b) phase vector and noise vector; (c) average between multiple ADSVs.

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The spatial phase vector of kth sub-section in ${Z_i}$ section can be expressed as:

(3)$$\begin{aligned} \overrightarrow {V({{Z_{i,k}}} )} &= \frac{1}{M}\sum\limits_{m = {l_{(i,k,1)}}}^{{l_{(i,k,M)}}} {[{{A_m}\exp ({j{\varphi_m}} ){\,+\,} {A_{n(m)}}\exp ({j{\varphi_{n(m)}}} )} ]} {\,=\,} |{\overrightarrow {r({{Z_{i,k}}} )} } |\exp ({j{\varphi_{i,k}}} ){\,+\,} |{\overrightarrow {n({{Z_{i,k}}} )} } |\exp ({j{\varphi_{n(i,k)}}} )\\ &= {A_{i,k}}\exp ({j{\Phi _{i,k}}} ). \end{aligned}$$

where $\overrightarrow {V({{Z_{i,k}}} )}$, $\overrightarrow {r({{Z_{i,k}}} )}$ and $\overrightarrow {n({{Z_{i,k}}} )}$ are respectively the phase vector, ideal vector and noise vector of one sub-section, ${l_{(i,k,m)}}$ is the sampling point of ${Z_{i,k}}$ sub-section, ${Z_{i,k}}$ represents the first sampling point of ${Z_{i,k}}$ sampling section (${Z_{(i,k)}} = {l_{(i,k,1)}}$), and ${Z_{i,k + 1}} - {Z_{i,k}} = DCS$, ${l_{(i,k,m + 1)}} - {l_{(i,k,m)}} = 2{n_{eff}}{f_{DAQ}}\Delta z/c = 1$. Similarly, the spatial differential sub-vector between two ends of GL can be calculated as:

(4)$$\begin{aligned} \overrightarrow {C{H_{i,k}}} &= \overrightarrow {V({{Z_{(i,k)}} + gl} )} \cdot conj[{\overrightarrow {V({{Z_{i,k}}} )} } ]\\ &= |{\overrightarrow {R({{Z_{i,k}}} )} } |\exp [{j({{\varphi_{(i,k) + gl}} - {\varphi_{i,k}}} )} ]+ |{\overrightarrow {N({{Z_{i,k}}} )} } |\exp [{j({{\varphi_{n[(i,k) + gl]}} - {\varphi_{n(i,k)}}} )} ]\\ &= {A_{(i,k) + gl}}{A_{i,k}}\exp [{j({{\Phi _{(i,k) + gl}} - {\Phi _{i,k}}} )} ]. \end{aligned}$$

where $\overrightarrow {C{H_{i,k}}}$, $\overrightarrow {R({{Z_{i,k}}} )}$ and $\overrightarrow {N({{Z_{i,k}}} )}$ are respectively the actual differential sub-vector (ADSV), ideal differential sub-vector (IDSV) and noise differential sub-vector (NDSV), $\overrightarrow {V({{Z_{(i,k)}} + gl} )}$ represents the sub-section whose starting sampling point is ${Z_{(i,k)}} + gl$, which can be shown in Fig. 3(b). After the oversampling process, the downsampling is proposed to recover the original channel spacing with the spatial downsampling coefficient R_s, which means that the oversampling interval DCS should be returned to CS. In the process of downsampling, each ADV’s starting point is selected with the interval of CS, and then ${R_s}$ $\overrightarrow {C{H_{i,k}}}$ are averaged to calculate the actual differential vector as follows:

(5)$$\begin{aligned} \overrightarrow {C{H_i}^{\prime}} &= \frac{1}{{{R_s}}}\sum\limits_{k = 1}^{{R_s}} {\overrightarrow {C{H_{i,k}}} } = \frac{1}{{{R_s}}}\sum\limits_{k = 1}^{{R_s}} {|{\overrightarrow {R({{Z_{i,k}}} )} } |\exp [{j({{\varphi_{(i,k) + gl}} - {\varphi_{i,k}}} )} ]} + \frac{1}{{{R_s}}}\sum\limits_{k = 1}^{{R_s}} {|{\overrightarrow {N({{Z_{i,k}}} )} } |\exp [{j({{\varphi_{n[(i,k) + gl]}} - {\varphi_{n(i,k)}}} )} ]} \\ &= |{\overrightarrow {R({{Z_i}} )^{\prime}} } |\exp [{j({{\varphi_{i + gl}}^{\prime} - {\varphi_i}^{\prime}} )} ]+ |{\overrightarrow {N({{Z_i}} )^{\prime}} } |\exp [{j({{\varphi_{n(i + gl)}}^{\prime} - {\varphi_{n(i)}}^{\prime}} )} ]\\ &= {A_{i + gl}}^{\prime}{A_i}^{\prime}\exp [{j({{\Phi _{i + gl}}^{\prime} - {\Phi _i}^{\prime}} )} ]= {A_{i + gl}}^{\prime}{A_i}^{\prime}\exp [{j\Delta \Phi {^{\prime}_i}} ]. \end{aligned}$$

Without any moving process, multiple ADSVs generated by the oversampling within one target sampling interval are averaged to realize spatial downsampling. Compared with the traditional demodulated method whose SNR is determined by ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$, the average of multiple ADSVs can completely eliminate the interfere fading, since there always exist ADSVs whose IDSV’s modulus is much larger than noise modulus, as well as the average process greatly weakens the influence of the NDSVs with random modulus and angles. The modulus of $R{({{Z_i}} )^\prime }$ will increase due to the small acute angle among multiple IDSVs, while the modulus of $N{({{Z_i}} )^\prime }$ will decrease greatly since the angle among multiple NDSVs is random. Hence, the SNR of the proposed phase demodulation method is determined by ${{|{\overrightarrow {R({{Z_i}} )^{\prime}} } |} / {|{\overrightarrow {N({{Z_i}} )^{\prime}} } |}}$, which is much larger than ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$ in the traditional method, and the signal vector modulus will be much larger than the noise vector modulus. Unlike the traditional moving average method which is performed by average among multiple adjacent sensing channels, the proposed downsampling process is carried out in the phase vector domain before phase calculation. Therefore, the proposed method has no interfere fading spots and can greatly reduce the noise floor of the DAS system without any crosstalk.

In order to verify the performance of the proposed method, a simulation is carried out. A sensing fiber with the length of 3500 m is modeled, in which the Rayleigh scattering points with sub-wavelength scale are aggregated into millimeter scattering points to reduce the pressure of redundant data calculation. The amplitude of each random Rayleigh scattering point with sub-wavelength scale follows Rayleigh distribution, as well as the phase follows (-π, π) uniform distribution. The carrier frequency of the heterodyne coherent DAS system is set to 200 MHz under the condition that the sampling frequency is set to 1GSa/s. Besides, the pulse width is 40 ns, and the transmission attenuation of the fiber is 0.2 dB/km. Then the optical fiber is sampled 2000 times, in which each sampling is mixed with a simulated acquisition noise that is caused by external environment, and the noise amplitude and phase follow Gaussian distribution. Figure 4 shows the simulation results with increasing R_s. It is obvious that the noise level characterized by average modulus is decreased with the increase of R_s, as well as the probability that IDV is less than the average noise modulus also decreased, of which both contribute to the effective demodulated phase noise reduction.

Fig. 4. Effects with increase of R_s. (a) mean value of noise vector’s modulus; (b) probability that IDVs are less than average noise modulus.

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2.2 Temporal oversampling-downsampling

Based on the spatial dimension process, the temporal oversampling-downsampling technique is proposed to furtherly reduce the out-of-band noise. The sampling function can be expressed as:

(6)$$p(t) = \sum\limits_{n ={-} \infty }^{n = \infty } {\delta ({t - nT} ).}$$

Then, a continuous signal $x(t )$ with the frequency domain of $X({j2\pi f} )$ is multiplied by $p(t )$ to perform time sampling:

(7)$${x_p}(t) = \sum\limits_{n ={-} \infty }^{n ={+} \infty } {x(nT)\delta ({t - nT} ).}$$

(8)$${X_p}({j2\pi f} )= \frac{1}{T}\sum\limits_{k ={-} \infty }^{k ={+} \infty } {X[{j2\pi ({f - k{f_s}} )} ].}$$

where ${x_p}(t )$ and ${X_p}({j2\pi f} )$ are time domain and frequency domain form of the discrete sampling signal, respectively, and ${f_s} = {1 / T}$ is the sampling rate. It is obvious that the ${X_p}({j2\pi f} )$ is a periodic function with frequency of ${f_s}$, which is composed of a group of shifted frequency domain function $X({j2\pi f} )$. Hence, for the time sampling of a continuous signal, the classical Nyquist theorem illustrates that if the rate for sampling is at least twice of its maximum frequency, the signal can be reconstructed without distortion, otherwise the spectrum aliasing will occur, which can be illustrated in Figs. 5(a) and 5(b). Here, we take a single frequency signal $x(t )= A\cos ({2\pi {f_n}t + {\theta_n}} )$ as an example. As demonstrated in Figs. 5(c)–5(d), the cosine spectrum is shifted and superposed with the interval of ${f_s}$. For the case of Fig. 5(c), the sampling rate ${f_s}$ is greater than twice of ${f_n}$, and thus the signal can be accurately recovered through a low-pass filter since it is oversampling without frequency aliasing. Besides, for the case of Fig. 5(d), ${f_s}$ is less than twice of ${f_n}$, then the aliasing occurs when the spectrum moved at the interval of ${f_s}$, which is called undersampling. However, even it is undersampling, its component will also appear in the in-band of the acquisition bandwidth ($DC - {{{f_s}} / 2}$), in which the in-band frequency component is ${f_s} - {f_n}$. For the in-band frequency component ${f_p}$ introduced by oversampling or undersampling, the more universal expression is as follows:

(9)$${f_p} = \left\{ {\begin{array}{cc} {{f_n},}&{{f_s} \ge 2{f_n}}\\ {|{{f_s} - m{f_n}} |,}&{0 < {f_s} < 2{f_n},\exists m \in Z,|{{f_s} - m{f_n}} |\le {f_s}/2} \end{array}} \right..$$

Fig. 5. The schematic of single frequency signal time sampling and the simulated sampling results of the single frequency signal; (a) oversampling; (b) undersampling;(c) in-band frequency component with oversampling; (d) in-band frequency component with undersampling; (e) fix signal frequency and change the sampling rate; (f) fix sampling rate and change the signal frequency.

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Based on the theorem mentioned above, a simulation is made to prove the process of oversampling and undersampling. A continuous function is generated artificially, and it is sampled by the sampling function described by Eq. (6). First, the frequency of signal to be collected is fixed (take ${f_n} = 1kHz$ as an example), and the sampling rate is a variable. As demonstrated in Fig. 5(e), the in-band frequency is located at 1kHz when the sampling rate is larger than 2kHz, while appears an oscillated wave at the condition of undersampling. Extend to all signals, each zero point is located at ${{{f_n}} / m}$, which is consistent with Eq. (9). Then, the sampling rate is fixed (take ${f_s} = 500Hz$ as an example) and change the frequency of signal to be collected. As demonstrated in Fig. 5(f), the in-band frequency changed with actual frequency in the form of folding triangle wave, in which the peak and vally are located at ${{{f_s}} / 2} + m{f_s}$ and $m{f_s}$, respectively, which is also consistent with Eq. (9).

In an acquisition system such as fiber DAS, the white noise is distributed over a wide frequency band, which can be regarded as the superposition of numerous single frequency signals mentioned above. Consequently, the out-of-band noise will be introduced into the in-band through the process of undersampling which is expressed as $|{{f_s} - m{f_n}} |$ for each single frequency noise signal, leading to a deterioration of noise floor of the system. As shown in Fig. 6(a), the target sampling bandwidth is $DC - {f / 2}$, and thus the sampling rate is set to ${f_s} = f$ according to the traditional method. However, the out-of-band noise which located at ${f_n} > {f / 2}$ will be introduced to in-band through undersampling, then the noise power of the system is distributed in the range of the whole in-band $DC - {f / 2}$. Here, a time oversampling-downsampling method is proposed to reduce the out-of-band noise. As shown in Fig. 6(b), the oversampling is first carried out to expand the noise distribution bandwidth with the sampling rate of ${f_s} = {R_t}f$, in which ${R_t}$ is the temporal downsampling coefficient, so that some out-of-band noise can be correctly introduced into the in-band according to Eq. (9) and the overall noise distribution of the system is changed to $DC - {{{R_t}f} / 2}$. Then, the downsampling is performed to recover the target bandwidth as illustrated in Fig. 6(c). First, the frequency ${f / 2}$ is used as the cut-off frequency for low-pass filter to compress the bandwidth. Then, the length of time domain data is extracted with the interval of ${R_t}$ to achieve spectrum compression. Finally, through the process of “oversampling-bandwidth compression-downsampling”, the out-of-band noise and the noise floor of the DAS can be greatly reduced.

Fig. 6. The schematic and the simulated results of temporal oversampling-downsampling; (a) traditional time sampling in which sampling rate is equal to twice of target bandwidth; (b) oversampling; (c) downsampling; (d) PSD comparison between R_t= 1 and R_t= 6; (e) the trend of noise floor and SNR.

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In order to demonstrate the effectiveness of the proposed method, a simulation is carried out. A frequency sweeping signal with the time length of 1s which contains a white Gaussian noise is presented as:

(10)$$x(t )= \cos ({2\pi {f_0}t + \pi \kappa {t^2}} )+ {N_w}.$$

where ${f_0} = 200Hz$ is the starting frequency of sweeping, $\kappa = 300Hz/s$ is the sweeping velocity, and ${N_w}$ is the noise whose power is -23dBW. Considering that the maximum frequency of the signal is 500 Hz calculated by the starting frequency and sweeping velocity, the target sampling rate is set as 2kHz to ensure a relatively good waveform recovery. Then, it is sampled according to Eq. (7) with the target bandwidth of $DC - 1kHz$. The power spectrum density (PSD) of traditional sampling and an example of oversampling-downsampling with R_t= 6 are shown in Fig. 6(d), in which the 200Hz∼500 Hz wide-band signal is recovered correctly. It can be seen that the signal power is unchanged while the noise floor is reduced by 6 dB, indicating that the SNR of this frequency sweeping signal is increased by 6 dB. Further, the dependences of noise floor and SNR on ${R_t}$ are depicted in Fig. 6(e). It can be seen that through the oversampling-downsampling method in time domain, the noise floor of the system can be effectively reduced, leading to a symmetric improvement of the SNR. Notably, the noise floor decreases exponentially with the increase of the R_t.

Further, combined with the spatial domain, the noise floor of the SMF DAS can be greatly reduced. For the heterodyne coherent DAS system, in order to adjust the initial phase to zero and further avoid phase unwrapping errors as much as possible, the time domain differential phase vectors are demodulated to characterize the external acoustic signals in our previous work [35]. The vector differential calculation is conducted between two adjacent sub-sampling points within one target sampling spacing, which can avoid the phase unwrapping errors in the traditional demodulation. After the bandwidth compression process, the downsampling and the differential-phase-sum reconstruction method within one target time window can be expressed as:

(11)$$\Delta {\Phi _i}^{\prime}({{t_j}^{\prime}} )= \sum\limits_{n = ({j - 1} )\times {R_t} + 1}^{j \times {R_t}} {\Delta {\Phi _i}^{\prime}({{t_n}} )} .$$

The accumulation reconstruction operation can stack the differential phase of all sub-sampling points within one target sampling spacing, so as to restore the real phase amplitude of the target sampling rate, which will not lead to the amplitude compression of the real acoustic signal. Finally, through the spatio-temporal joint oversamling-downsampling process mentioned above, the fading noise and out-of-band noise ca be greatly eliminated and suppressed.

3. Experimental setup

In order to verify the denoising performance of the spatiotemporal oversampling-downsampling method, a coherent heterodyne polarization diversity DAS system is set up as shown in Fig. 7. A high coherent narrow linewidth (NLL, linewidth < 0.1kHz) is employed as the light source. The continuous light emitted by NLL is split into two beams by the optical coupler (OC), which are respectively served as probe signal and local oscillator (LO). Then the probe light is injected into the acousto-optic modulator (AOM) to generate probe pulses with a 200 MHz frequency shift and 40 ns pulse width. After amplified by the Er-doped fiber amplifier (EDFA), the probe pulses are transmitted to the fiber under test (FUT) through the circulator (Cir). To generate external acoustic signals, a section of fiber is wrapped on the piezoelectric transducer (PZT). At the receiving end, the polarization diversity scheme is utilized to eliminate the influence of polarization fading. The probe pulse train with a certain time interval is sent into the sensing fiber for time dimension sampling. Similarly, the spatial sampling is realized by artificially dividing the sensing fiber with a certain spacing. Although the pulse number of measurements is increased to perform the time dimension oversampling of the sensing fiber, the real-time response performance of the system will not be affected, since the downsampling process of the spatial and time dimension is further conducted in the high-speed FPGA chip, whose processing speed is basically the same as the traditional detection method. Besides, the fiber length is 3520 m, the gauge length is set up to 10 m, and the PZT located at 1490 m is used to generate a standard sinusoidal acoustic signal, in which the length of the fiber wrapped on PZT is 13 m. The 200 Hz sine acoustic signal applied by PZT and the static environment signal of other areas of sensing fiber are collected by DAS system, in which the final target bandwidth and the CS are set to DC∼500 Hz and 10 m, respectively. Besides, the W and DCS mentioned in the principle part are set as 2 m and 1 m, respectively.

Fig. 7. System configuration of coherent heterodyne polarization diversity DAS system.

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4. Results and discussion

4.1 Relationship between noise floor and downsampling factor

The relationship between noise floor and spatial or temporal downsampling factor was firstly investigated. Modified the value of spatial downsampling factor R_s while the temporal downsampling factor R_t was set to a constant as 1. The same sensing area was selected and the average value of PSD noise floor along 780 m sensing fiber was calculated. As demonstrated in Fig. 8(a), the noise floor decreases with the increase of the spatial downsampling factor R_s, in which the trend is the same as the single exponential function in the simulation. Then, modified the value of temporal downsampling factor R_t while the spatial downsampling factor R_s was set to a constant. It should be pointed out that in order to obtain a more accurate influence of R_t on noise floor, R_s should be set to a relatively larger value to exclude the influence of fading noise, and hence R_s was set to 5 here. Considering the influence of fading noise on the actual variation trend of noise floor with R_t value increasement, the one-dimensional variation trend is illustrated instead of two-dimensional change trend. As illustrated in Fig. 8(b), the same average PSD noise floor was calculated, which decreases along with the increase of R_t in the same regularity with the simulation result as the sum of two exponential functions. The reason why the coefficient of the exponential function is different is that the noise of the actual DAS system is not exactly the same as that of simulation.

Fig. 8. The relationship between noise floor and (a) R_s or (b) R_t.

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4.2 Noise suppression of the static signal

Firstly, the spatial and temporal downsampling factors were both set to 1, corresponding to the traditional demodulation method. The results obtained under this condition are shown in Figs. 9(a) and 9(b), in which many large noise sensing channels caused by fading and out-of-band noise are existed. The average noise floor of the flat area between 100Hz-500 Hz is calculated as -54.19 dB (re rad²/Hz). Then, the spatial oversampling-downsampling with ${R_s} = 6$ was implemented, whose results can be shown in Figs. 9(c) and 9(d). Furthermore, the demodulated result of the spatiotemporal oversampling-downsampling with ${R_s} = 6$ and ${R_t} = 6$ is illustrated in Figs. 9(e) and 9(f). It can be seen that there are almost no sensing channels containing large noise and the fading noise is completely eliminated. Besides, the PSD result shows that the average noise floor is 10.62 dB lower than the traditional demodulated method. Moreover, the strain resolution was estimated by the multiple channels’ averaging PSD noise floor as shown in Fig. 9(g), which is obtained by setting both R_s and R_t equal to 10 according to the relationship illustrated in Fig. 8. Under the condition of DC-500 Hz target bandwidth, the noise floor is -72.31 dB (re rad²/Hz) of the 100Hz-500 Hz flat area and -61 dB (re rad²/Hz) at 10 Hz, and then the strain resolution of the DAS system is respectively calculated as 2.58pε/√Hz@100Hz∼500 Hz and 9.47pε/√Hz@10 Hz according to the phase strain sensitivity [36].It can be proved that the proposed algorithm can eliminate the fading noise and out-of-band noise effectively and hence realize the pε level measurement under the bandwidth of 10Hz∼500 Hz for standard SMF DAS.

Fig. 9. The time domain and frequency domain demodulated results; (a)(b) R_t= 1 and R_s= 1; (c)(d) R_t= 1 and R_s= 6; (e)(f) R_t= 6 and R_s= 6; (g) the noise floor under the condition of R_t= 10 and R_s= 10.

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4.3 Optimization of large noise sensing channels

Moreover, the probability of large noise and fading sensing channels was investigated, in which the sensing channels whose root-mean-square (RMS) value is greater than 0.06 are defined as large noise and fading channels. According to the target SS of 10 m, 3520 m sensing fiber is divided into 352 sensing channels, and the RMS value of each sensing channel is calculated. Further the RMS normalized probability distribution of different demodulated methods is studied, which is illustrated in Fig. 10. Without any denoising algorithm, the large-noise channels probability of the traditional demodulated method is up to 44.32%, which limits the sensing performance. After one dimensional processing, the large-noise channels probability can be effectively reduced which can be demonstrated in Figs. 10(b). Significantly, through joint processing, the large-noise and fading sensing channels can be completely eliminated as shown in Fig. 10(c).

Fig. 10. The normalized probability distribution of the large-noise sensing channels; (a) R_t= 1 and R_s= 1; (b) R_t= 1 and R_s= 6; (c) R_t= 6 and R_s= 6.

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4.4 SNR improvement of the dynamic signal

For demonstrating the SNR improvement of the dynamic signals, a 200 Hz sinusoidal acoustic signal is applied on the sensing fiber by PZT located at 1490 m. As illustrated in Figs. 11(a) and 11(b), the fidelity of the demodulated signal is relatively poor with the SNR of 35.6 dB. The amplitude of the recovered wave is not very uniform, which is mainly caused by the fading noise or small value of ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$. After the one dimensional process of spatial oversampling-downsampling with the downsampling factor of ${R_s} = 6$, the sine waveform has a relatively better fidelity with the SNR of 42.5 dB because of the increase of ${{|{\overrightarrow {R({{Z_i}} )^{\prime}} } |} / {|{\overrightarrow {N({{Z_i}} )^{\prime}} } |}}$ compared with ${{|{\overrightarrow {R({{Z_i}} )} } |} / {|{\overrightarrow {N({{Z_i}} )} } |}}$, which can be shown in Figs. 11(c) and 11(d). Furthermore, as illustrated in Figs. 11(e) and 11(f), the quality of the demodulated signal is improved with an excellent fidelity and a high SNR of 56.4 dB through joint spatiotemporal oversampling-downsampling process, which is 20.8 dB higher than traditional demodulated method. Besides, the signal amplitude will not be compressed with the increasement of the R_t value. Hence, the proposed method can effectively enhance the SNR of the dynamic acoustic signals.

Fig. 11. The time domain and frequency domain demodulated results of the 200 Hz sine wave; (a)(b) R_t= 1 and R_s= 1; (c)(d) R_t= 1 and R_s= 6; (e)(f) R_t= 6 and R_s= 6.

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4.5 Performance comparison with the averaging technique

Further, the proposed spatiotemporal joint oversampling-downsampling scheme is compared with the denoising method of slide averaging among multiple sensing channels. A 20 cm long section of 4 km optical fiber is pasted on the table, and multiple gently knocks at the position of the pasted section are conducted to simulate the external signals. Both of two denoising methods are performed with the channel spacing of 10 m, the spatial downsampling factor Rs and the channel slide averaging window are both set to 6. As shown in Fig. 12(a), for the proposed algorithm in this work, there is not any crosstalk since only the channel 123 contains the multiple knocking signals. While for the averaging technique illustrated in Fig. 12(b), although the noise floor is optimized, the severe crosstalk is introduced, in which there are signals in channel 123 to 128. As shown in Figs. 12(c) and 12(d), the PSD noise floor present that the proposed method is 8 dB lower than the averaging technique, demonstrating that the sensing performance of the spatiotemporal joint oversampling-downsampling method is better than the traditional averaging method without any crosstalk.

Fig. 12. (a)(c) The time/frequency domain demodulated results of the spatiotemporal oversampling-downsampling with the spatial downsampling factor of 6; (b)(d) the time/frequency domain denoising results of the averaging technique with the channel slide window of 6.

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5. Conclusion

In this work, we proposed and demonstrated an effective denoising algorithm named spatiotemporal oversampling-downsampling technique for heterodyne coherent DAS system. In the spatiotemporal joint dimension, dense oversampling is carried out first, and then downsampling is performed to compress the spatiotemporal bandwidth to achieve noise suppression. The spatial oversampling-downsampling is demonstrated to completely eliminate the interfere fading noise without increasing any system complexity and introducing any crosstalk. Besides, the temporal oversampling-downsampling is proposed to reduce the out-of-band noise. The experimental results show that the noise floor is inversely correlated with the downsampling factors. The strain resolution of the DAS system with the proposed scheme can reach 2.58pε/√Hz within the flat area of 100Hz-500 Hz and 9.47pε/√Hz@10 Hz in the experimental demonstration. In addition, the probability of the large-noise sensing channels is reduced from 44.32% to 0% by the spatiotemporal oversampling-downsampling method, demonstrating the basically completely elimination of large-noise sensing channels. For the dynamic acoustic signal, the demodulated SNR is improved by 20.8 dB. Compared with the channels slide averaging technique, the proposed method owns a better noise floor without any crosstalk. Based on the excellent denoising performance, our scheme can be applied to the high-precision detected fields such as passive seismic imaging and VSP exploration.

Funding

National Natural Science Foundation of China (61775072, 61922033); National Key Research and Development Program of China (2018YFB2100902); Innovation Fund of WNLO; Fundamental Research Funds for the Central Universities (HUST:2021JYCXJJ036).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 61922033, 61775072), the National Key Research and Development Program of China (Grant No. 2018YFB2100902), the Innovation Fund of WNLO, and the Fundamental Research Funds for the Central Universities, HUST:2021JYCXJJ036.

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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Spatio-temporal joint oversampling-downsampling technique for ultra-high resolution fiber optic distributed acoustic sensing

Abstract

1. Introduction

2. Principle

2.1 Spatial oversampling-downsampling

2.2 Temporal oversampling-downsampling

3. Experimental setup

4. Results and discussion

4.1 Relationship between noise floor and downsampling factor

4.2 Noise suppression of the static signal

4.3 Optimization of large noise sensing channels

4.4 SNR improvement of the dynamic signal

4.5 Performance comparison with the averaging technique

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (11)

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