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Abstract
Phase-sensitive optical time domain reflectometry becomes an effective tool to realize distributed sensing, and the optical phase of the received light is usually used to quantify the strain for both dynamic and static measurement. The analysis on the overall phase error has been improved by considering the proportionality of the detection noise to the local optical power. The estimation accuracy is greatly improved by using the proposed theory, the probability density of the estimation accuracy over 99% is > 0.6, ∼39 times larger than the previously reported method. The phase unwrapping may malfunction due to the noisy signal, causing large phase errors. Point break detection algorithms are used to locate the incorrect phase unwrapping points, so the temporal evolution of the phase retrieved at each position can be divided into several sections with different offset. The phase unwrapping error is then suppressed by removing the offset.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A phase-sensitive optical time domain reflectometry (φOTDR) is an effective tool to realize static or dynamic measurement in a spatially-resolved manner. It acquires the environmental information, usually strain, based on the interference of the Rayleigh backscattered light. The φOTDR exhibits an ultra-high sensitivity thanks to the interference process, for example, the strain sensitivity of the φOTDR is three-order of magnitudes higher than a distributed sensor based on Brillouin scattering [1]. The φOTDR becomes an attractive solution to acoustic/vibration in large-scale because of the high sensitivity and simple configuration; it has been applied to structural health monitoring, environmental monitoring, traffic monitoring and so on [2–4]. The φOTDR has been performed not only in the standard single mode fiber, but also in multimode fibers, Rayleigh scattering enhanced fibers and multicore fibers to achieve a better sensing performance [5–8].
External perturbations alter the local property of the sensing fiber, i.e. the refractive index and the length of the fiber. Thus, the perturbation leads to the change of the backscattered light in two aspects. On one hand, the light intensity varies due to the change of the interference condition. And the environmental information can be retrieved from the optical intensity change. This intensity-based measurement usually relies on the restorability of the φOTDR signal, which requires a frequency scanning [9]. The frequency response of the intensity-based φOTDR is thus compromised, so it is mostly used for static sensing. On the other hand, the fiber property change introduces a phase delay to the received light and the environmental information can be retrieved from this delay. The phase-based measurement is a good candidate for dynamic sensing because its maximum detectable frequency is only limited by the time of flight of the optical pulse in the fiber. Hence, this method draws more and more attention in recent years. The phase information can be retrieved by direct and coherent detection [10]. Interferometers need to be used to encode the phase information into optical intensity, which can be directly detected [11]. Coherent detection acquires the beat between Rayleigh backscattered light and a local oscillator, then the phase is obtained usually by the IQ demodulation [12]. Note that phase unwrapping is necessary for most phase-based methods to reconstruct the actual phase.
Noise is the main factor that limits the sensing performance of the φOTDR [13]. There are many noise sources in the sensing system. For example, the low extinction ratio of the optical pulse can introduce the coherent Rayleigh noise [14], the phase noise from the laser [15] and accumulative phase noise can cause a drift of the obtained phase [16]. Even in an optimized φOTDR system, there still exists noise from the photodetector, e.g. thermal noise and shot noise, and it is impossible to completely remove this type of noise. The relationship between the resultant phase error and the detection noise has been theoretically described based on the error propagation function [17], and current theoretical analysis usually assumes the noise is a constant along the fiber. Our recent research however demonstrates clearly that the detection noise, playing a dominant roll in most cases, varies randomly along the fiber because it is dependent on the local signal level [18]. Therefore, current analysis oversimplifies the situation and is supposed to provide inaccurate estimation of the phase error.
Currently, most φOTDR systems measure the optical phase to calibrate the environmental information and phase unwrapping is usually necessary during the data processing. The noise however can introduce false phase unwrapping and leads to a phase offset of ∼ 2π. Consequently, the φOTDR system is unable to provide reliable results due to such a large error, which may lead to hazardous consequences in practice. Many methods have been so far proposed to alleviate the phase error, such as using auxiliary weak reflection points in the sensing fiber [19], and employing pre-distorted pulse and processing the signal in both time and frequency domain [20], or relies on deep learning [21]. However, most of the methods require hardware modifications, increasing the system complexity and cost.
This paper addresses two points regarding the phase error in a φOTDR system. On one hand, the theoretical estimation of the phase error is improved by considering the detection noise as a position dependent variable, which is determined by the local optical power, instead of a constant along the fiber. In this way, the estimation accuracy is greatly enhanced, the probability density of the accuracy over 99% is improved by ∼39 times compared with the state of the art. On the other hand, the phase unwrapping error is highly suppressed by a pure data processing method, which identifies the unwrapping error by the change point detection algorithm and suppresses the error by DC removal. The working principle is straightforward, and this method requires no hardware modification. The error estimation and unwrapping error suppression have been experimentally demonstrated by a φOTDR system based on an imbalanced Mach-Zehnder interferometer (IMZI) with a 3×3 coupler. However, the methods proposed in this paper are believed to work for other types of φOTDR systems based on the optical phase for sensing.
2. Theory
2.1 Working principle
In a φOTDR system, optical pulses from a coherent light source are launched into the sensing fiber. The pulse is continuously deflected at the inhomogeneities during its transmission along the fiber, and the backscattered light is acquired as a function of time. External stimuli can change the refractive index and the size of the inhomogeneity, resulting in a variation of the Rayleigh signal. For example, an applied strain $\varepsilon $ can cause a phase delay of the backscattered light by [22]
where n is the refractive index of the fiber, ${\nu _0}$ is the optical frequency of the incident pulse, l is the strained length, c is the light speed in vacuum and $\xi ={-} 1/2{n^2}[{({1 - \mu } ){p_{12}} - \mu {p_{11}}} ]$ with the Poisson’s ratio $\mu = 0.17$ and strain-optic coefficients ${p_{11}} = 0.121$ and ${p_{12}} \approx 0.27$. Most φOTDR sensing techniques aim to measure this phase delay in order to quantify the strain.One popular optical phase retrieval module for the φOTDR system is based on an IMZI and a symmetrical 3×3 coupler [11,17], as shown in Fig. 1. The Rayleigh backscattered light $E({t,z} )$ from position z and at measurement time t enters the module via a 50:50 splitter, so it is divided evenly into two light beams. The two beams pass through two arms in the interferometer, respectively. The beam in the upper arm is delayed compared to the other because the upper arm is $\Delta l$ longer. This length difference corresponds to the gauge length. Due to the imbalanced arms, the light backscattered at positions z and $z + \Delta l$ are actually combined at a 3×3 coupler. As a result, the output of the coupler is dependent not only on the amplitude of $E({t,z} )$ and $E({t,z + \Delta l} )$, but also on the optical phase difference between them. This phase difference carries the environmental information between the positions z and $z + \Delta l$. Three identical photodetectors are used to acquire the output of the coupler, and the obtained signal can be expressed as [17]:
Fig. 1. Diagram of a phase retrieval module in a φOTDR system based on imbalanced Mach-Zehnder interferometer with a 3×3 coupler.
The phase difference can be retrieved from the obtained signal by IQ demodulation and differentiation and cross-multiplication. The differentiation process in the latter method is essentially a high-pass filter, which is prone to noise. Therefore, IQ demodulation is used here. The in-phase (I) and quadrature (Q) components are expressed as [17]:
The phase unwrapping is applied to the temporal evolution of the retrieved phase at each position along the fiber. According to the classical unwrapping algorithm [23], the operation is dependent on the difference of two neighboring points:
- 1. when the difference between the two points $\Delta \varphi ({{t_{n + 1}}} )$ and $\Delta \varphi ({{t_n}} )$ is within [−π, π], it is believed that the phase $\Delta \varphi ({{t_{n + 1}}} )$ is not wrapped and no unwrapping operation is needed;
- 2. when the difference is larger than π, the phase $\Delta \varphi ({{t_{n + 1}}} )$ is considered as wrapped, and 2π is subtracted from the phase retrieved at ${t_{n + 1}}$ and later;
- 3. when the difference is smaller than ‒π, 2π is added to the phase retrieved at ${t_{n + 1}}$ and later.
2.2 Phase error analysis
Noise always exists in a φOTDR system and it results in a phase error that eventually deteriorates the measurement. Advanced optoelectronic components with high performance can reduce some types of the noise. For example, a semiconductor optical amplifier (SOA) or acousto-optic modulator can generate optical pulses with a high extinction ratio, so that the coherent noise can be reduced [24]. The photodetection noise is however inevitable even in an optimized φOTDR system and it is usually considered as the dominant noise source.
Due to the existence of the detection noise, the obtained signal at a given position varies randomly during the measurement time. Consequently, the phase difference $\Delta \varphi $ retrieved by Eq. (4) fluctuates as well during the repeated measurement and causes a variation in the strain determination, introducing large measurand errors. According to the propagation of the uncertainty, the variance of the phase difference $\Delta \varphi $ at a given position z is expressed as [17]
Equation (6) demonstrates that the variance of the phase difference is inversely proportional to the local value of ${\bar{I}^2} + {\bar{Q}^2}$. The value turns out to be dependent on the product of the received optical intensity at z and $z + \Delta l$ according to Eq. (2) and (3) [17]. The random nature of the received light makes the obtained phase variance $\sigma _{\Delta \varphi }^2$ fluctuate along the fiber. Equation (6) provides satisfactory estimation of the phase variance. As a random variable, the optical intensity of the Rayleigh backscattered light can be very low at some positions, which is called fading. At fading points, the ${\bar{I}^2} + {\bar{Q}^2}$ is small and the SNR is low, causing large phase errors.
It has to be pointed out that the analysis in Ref. [17] and other similar studies neglect the dependence of the detection noise on the signal level and oversimplifies the situation. In practice, the Rayleigh backscattered light is very weak, so it usually needs pre-amplification before the photodetection. The amplified spontaneous emission (ASE) from the pre-amplifier will travel with the boosted signal and beats with it at the photodetector. Thus, the sig-ASE noise dominates the measurement and it is dependent on the detected signal level, which fluctuates along the fiber. As a result, the variance of the detection noise $\sigma _n^2$ is a position dependent variable instead of a constant. In addition, the signal obtained at the three photodetectors are different at a given position, as shown by Eq. (2), so the sig-ASE noise varies with the photodetector, which results in a non-zero covariance of the I and Q components.
The analysis above implies that the previous study provides an inaccurate estimation of the phase variance based on Eq. (6) because it assumes the detection noise is independent on the signal level, which is not the actual case. And Eq. (5) can actually provide a more accurate estimation. If the sig-ASE noise is taken into consideration, the variance of the I and Q components become position dependent variables, and they can be statistically obtained at each position based on the detected signal P1, P2 and P3 over the measurement time. Then the covariance can be determined based on the temporal series of the I and Q values at a given position. Although it is difficult to obtain an analytical expression of the covariance, it is very simple to calculate it based on the experimental results. In this way, the phase variance $\sigma _{\Delta \varphi }^2$ can be estimated with a high accuracy.
Note that phase unwrapping is necessary to retrieve the actual value, but the existence of noise can cause errors during the unwrapping process. Due to the noisy signal, the retrieved phase may be considered wrongly as wrapped, so the unwrapping process is performed. Consequently, ±2π is added to the fake unwrapping points, which introduces a large measurand error. Such an unwrapping error has been reported in [19,20,21,25,26] and usually happens at the fading points where the SNR is low [19,20]. The phase unwrapping error can also be observed at the unperturbed fiber section, and the unwrapping can fail several times at the same position, resulting in a phase variance over 103 as reported in [17,26] for different φOTDR systems. It is impossible to estimate the unwrapping error because it occurs randomly. However, it is of great significance to reduce and even correct the error.
3. Experiment and results
3.1 Experiment
A φOTDR system is built to analyze the phase error, the phase retrieval module is based on the IMZI and a 3×3 coupler. The whole setup is plotted in Fig. 2. The light source is a semiconductor laser (RIO ORION) with a very narrow linewidth. The phase noise and the intensity noise are very small so that their contribution to the phase error can be neglected. The continuous lightwave from the laser is converted into pulses by a semiconductor optical amplifier (SOA), the extinction ratio of the generated pulse is high enough so that the coherent Rayleigh noise can be neglected as well. The generated pulse is amplified by an Erbium-doped fiber amplifier (EDFA) and the ASE is suppressed by a narrowband optical filter. The peak power of the optical pulse is adjusted by a tunable attenuator to avoid any nonlinear effects. The pulse is guided into the sensing fiber via a circulator and the Rayleigh backscattered light is redirected to the phase retrieval module by the same circulator.
Fig. 2. Experimental setup of a φOTDR system based on imbalanced Mach-Zehnder interferometer with a 3×3 coupler.
Before entering the phase retrieval module, the backscattered light is pre-amplified by another EDFA and a narrow filter with 25 GHz bandwidth is used to suppress the ASE. After the phase retrieval module, the output of the coupler is connected to three identical photodetectors in order to acquire the signal P1, P2 and P3.
The sensing fiber is a ∼1442 m long standard single mode fiber, most part of the fiber is well isolated from external perturbations but the fiber section from 1408.1 m to 1418.6 m is wrapped around a PZT and subject to a dynamic strain at 1 kHz. The pulse width is set as 10 ns and the pulse repetition rate is 40 kHz. The bandwidth of the photodetector is 125 MHz, large enough for the pulse width [27,28]. The length difference of the two arms in the IMZI is 2 m, determining the gauge length. The output of the photodetectors is digitized and sent to a personal computer for data processing. The sampling rate of the analog-to-digital convertor is 500 MS/s and the sampling resolution is 14-bit for 500 mV.
3.2 Phase error
To analyze the phase error, the measurement is repeated 10,000 times under the same condition. The obtained phase should remain the same for each measurement in the ideal case. However, the signal P1, P2 and P3 obtained at a given position change randomly during the whole measurement due to the inevitable detection noise. Consequently, the retrieved phase varies slightly for each measurement and its variance is usually used to characterize the phase error.
The previous study oversimplifies the phase error analysis by assuming the noise $\sigma _n^2$ is constant along the fiber and estimates the phase error as presented by Eq. (6). However, the dominant noise is usually sig-ASE noise for direct detection and shot noise for coherent detection. Both noise types are proportional to the Rayleigh signal, a random variable along the fiber, so the noise $\sigma _n^2$ is actually position dependent. Therefore, Eq. (5) is supposed to describe the phase error more accurately because it is the fundamental equation based on the error propagation and it considers the position dependence of the noise via the covariance term. The accuracy of the theoretical analysis on the phase error is defined as
A comparison of the phase error obtained experimentally and theoretically by Eqs. (5) and (6) is shown in Fig. 3. The longitudinal profiles of the phase error obtained by different methods are plotted in Fig. 3(a). In addition, the phase error obtained over the whole unperturbed fiber section is statistically analyzed and the corresponding probability density function (PDF) is plotted in Fig. 3(b). The PDF function presented here shows the probability for a given phase error range. The obtained error varies randomly around 10‒2 rad2 as shown in Fig. 3(a), this behavior agrees with the analysis in Section 2. The theoretically obtained error match well with the experimental result for most part of the fiber. Particularly, the red curve, computed by Eq. (5) overlaps with the experimentally obtained curve (blue) except some spikes at the fading positions. This overlap clearly shows that the phase error can be estimated with a higher accuracy if the detection noise is considered as position dependent. For the same reason, Eq. (5) also offers an accurate estimation on the statistical feature of the phase error, as Fig. 3(b) shows that the red PDF curve overlaps with the experimental result. The experimentally obtained PDF curve exhibits that there is a certain probability of large phase errors over 102 rad2 which is caused by the incorrect phase unwrapping.
Fig. 3. Comparison of the phase error obtained by experiment (blue) and theories assumes the noise is position dependent (red) and independent (black). (a) Longitudinal profile of the phase error for a short fiber section. (b) Probability density function of the phase error along the whole unperturbed fiber. (c) Probability density function of the estimation accuracy along the whole unperturbed fiber.
As presented by Fig. 3(a), the former analysis Eq. (6) underestimates the error at the curve dips and overestimates at the local peaks, because it assumes the detection noise is constant along the fiber. According to our recent research, the dominant noise is the sig-ASE noise if pre-amplification is used, so the detection noise is proportional to the received signal level [18]. At the fiber position with a very strong signal, the local ${\bar{I}^2} + {\bar{Q}^2}$ value is large, so the phase error is supposed to be very small (the curve dips) according to Eqs. (5) and (6). Due to the strong signal, the actual noise at these positions is larger than the noise value used in Eq. (6), thus the previous analysis provides an even smaller error than the experimental result. The situation is totally opposite at the error peaks where fading occurs. At the fading points, the signal level is very low, so the former analysis overestimates the noise in this case. Consequently, it provides a larger phase error than the actual value. The deviation at these two scenarios leads a broader PDF peak in Fig. 3(b).
The two theoretical methods are also compared by the estimation accuracy as defined by Eq. (7). The spatial profile of the accuracy can be calculated based on the theoretical analysis and the experimental results, and it is expected to vary along the fiber. As some positions, such as 1129.5 m, the results of Eq. (5) and (6) agree with the experimentally obtained phase error very well, so the corresponding accuracy can reach over 99% for both methods. However, the former model (Eq. (6)) can offer wrong results as discussed above, so its accuracy may hugely decrease in some cases. For example, the accuracy is only 23.5% at the position indicated by the arrow at 1145.5 m in Fig. 3(a). In comparison, the accuracy of the method proposed here is 99.8%, thus the accuracy is improved by 76%. At fading points, the experimentally obtained phase error can be over 10 rad2, much larger than the theoretical estimation, as shown in Fig. 3(a). In this case, the estimation n accuracy becomes very low. Figure 3(c) plots the PDFs of the accuracy for both theoretical methods. Compared with the published method (Eq. (6)), the theoretical analysis proposed in this paper demonstrates a much larger probability density at high accuracy range close to 100%. By considering the longitudinal variation of the noise, the new method achieves a probability density of ∼0.617 for the accuracy over 99%, whereas the previous analysis exhibits a density of just ∼0.016, demonstrating clearly the large improvement of this method. The accuracy for both methods can be very low due to the phase unwrapping error usually at the fading points. Generally speaking, the proposed method exhibits a higher accuracy than the former analysis.
3.3 Phase unwrapping error
Ideally, the obtained phase should be a constant if the fiber is isolated from environmental perturbations and the sensing system is free from any noise. The noise from different sources however can introduce a random change to the detected φOTDR signal, and finally cause a fluctuation of the retrieved phase over time. Therefore, the phase obtained from most part of the fiber demonstrates a small variance of ∼ 10‒2 rad2. However, large phase errors are obtained by the experiment as shown in Fig. 3. and they are supposed to originate from the poor SNR. The noisy phase retrieved at these positions can induce unnecessary unwrapping, further increasing the error. The temporal evolution of the phase obtained at some positions with large errors is analyzed. The phases obtained at around 1141 m (unperturbed) are plotted as a function of time in Fig. 4(a). It shows obviously that the temporal evolution consists of different segments with various DC values and the offset between two neighbor segments is about 2π, indicating the offset is caused by the incorrect phase unwrapping. The unwrapping failure also occurs at the vibrated section as shown in Fig. 4(b). The phase retrieved at 1410.5 m exhibits a sinusoidal waveform oscillating at 1 kHz as expected. However, the phase is unwrapped wrongly at ∼8.1 ms and ∼48.1 ms, introducing an offset to the phase retrieved in-between.
Fig. 4. Temporal evolutions of the retrieved phases at (a) unperturbed and (b) vibrated fiber sections. Phase unwrapping failures cause randomly and unexpected phase offset, resulting in a larger phase error.
A phase unwrapping failure results in a phase offset of 2π, equivalent to a strain change of ∼ 0.68 µɛ for a 1 m gauge length according to Eq. (1). Such a failure can occur many times during the measurement, leading to large phase errors as shown in Fig. 3. Owing to the phase unwrapping error, the φOTDR system is unable to provide unreliable measurement at some points and can provide false alarm in practice. It is therefore necessary to suppress or correct the phase unwrapping error.
According to the analysis in [17], the phase unwrapping error usually occurs at the fading position with a low ${\bar{I}^2} + {\bar{Q}^2}$ value for a φOTDR system based on coherent detection. The analysis above reveals the same dependence of the unwrapping error on the local ${\bar{I}^2} + {\bar{Q}^2}$ value for an IMZI based φOTDR system. The phase error and the corresponding averaged ${\bar{I}^2} + {\bar{Q}^2}$ value are shown in Fig. 5 for a short fiber section from 1120 m to 1160 m. The phase error demonstrates an obvious invers correlation with the local ${\bar{I}^2} + {\bar{Q}^2}$ value, and large errors (> 10 rad2) always occur at the places with a low ${\bar{I}^2} + {\bar{Q}^2}$ value. Figure 5 also illustrates that the incorrect phase unwrapping introduces large phase changes only at distinct points, thus the phase obtained at these points can be post-processed to remove the phase unwrapping error.
Fig. 5. Comparison of longitudinal profile of the phase error and ${\bar{I}^2} + {\bar{Q}^2}$ value for a short fiber section. Large errors occur at the position with small ${\bar{I}^2} + {\bar{Q}^2}$ value.
4. Phase unwrapping error prediction and suppression
It is investigated in the last section that the phase unwrapping error may occur randomly at positions with a low ${\bar{I}^2} + {\bar{Q}^2}$ value and introduce a fast phase change. Environmental stimuli, e.g. an abrupt stretch of the fiber, can also cause such a phase change. The large phase changes induced by the incorrect unwrapping and environmental variation are different in two aspects. One is that the phase unwrapping failure occurs at discrete points, whereas the abrupt environmental variation usually influences the retrieved phase over a certain range. The other is that the incorrect unwrapping always results in a phase change of ∼2π, but the external stimuli caused phase change is dependent on the environmental variation. Therefore, the phase unwrapping error can be distinguished easily from the large and sharp phase changes, then be suppressed or corrected.
The unwrapping failure divides the time series of the retrieved phase at a given position into several sections as shown in Fig. 4, and it is necessary to determine the boundary of each section in order to correct the unwrapping error, which is essentially a change point detection (CPD) problem. Change point detection finds its application in a variety of areas, such as finance, climatology, statistics and image processing [29]. The main task of the CPD is to quantify the homogeneity of the time series, and the employed measure is usually called cost function. Various cost functions have been proposed depending on the time series feature and the type of change points.
As shown in Fig. 4, the typical temporal evolution of the optical phase caused by unwrapping failure consists of multiple sections with different average or median. And optical phase can be considered as a Gaussian distributed variable with a fixed variance if the dominant noise is unchanged during the measurement. Thus, the mean-shift model can be applied to detect the change point, and the corresponding cost function can be written as [29]
The cost function can describe the homogeneity of the retrieved phase within a given time slot. A change point is believed to exist between two successive time windows if the results of the cost function are very different. The discrepancy of the two cost function results between two neighboring time windows $\Delta \varphi ({t - {t_w},t} )$ and $\Delta \varphi ({t,t + {t_w}} )$ is defined as [29]:
If there is no phase changing points between $t - {t_w}$ and $t + {t_w}$, the cost function results of the time slots [$t - {t_w},\; t + {t_w}$], [$t - {t_w},\; t$], and [$t,\; t + {t_w}$] are very similar, so the corresponding discrepancy is a small value. On the contrary, the cost function results are dissimilar if a phase unwrapping error occurs, and the discrepancy becomes large, forming a peak. To locate all the change points, e.g. unwrapping failures in this paper, a time window with a fixed width tw needs to slide over the whole measurement time to obtain the discrepancy. Then the discrepancy peak is located, and the position represents the time when the unwrapping error occurs. The time window should be large enough to represent the statistic feature of the phase; meanwhile, it should be also comparatively small so that each time window contains at most one changing point.
The false unwrapping points or the changing points at fiber position 1410.5 m are identified using the changing point method as described above. The red line in Fig. 6 represents the retrieved phase and corresponds to the y-axis on the left-hand side. It shows clearly that the unwrapping error occurs four times within a short time period of 125 ms, dividing the temporal phase series into 5 sections. The discrepancy value is obtained along the fiber based on the three types of cost functions with a window width of 12.5 ms, corresponding to 500 points. The discrepancy is calculated for each type and normalized to its maximum. According to Fig. 6 all the discrepancy curves exhibit four peaks, and they position at the time when unwrapping the error occurs, demonstrating that all the three cost functions are able to identify the change point in the temporal series of the phase retrieved by the φOTDR system. Due to the working principle, the peak shapes obtained by the cost functions are a bit different, however, the full width at half maximum of the discrepancy peak is very similar. It is can be concluded that the cost functions have the very similar performance in determining the phase unwrapping positions.
Fig. 6. Changing point detection for the temporal series of the phase retrieved at 1410.5 m by the φOTDR system. The normalized discrepancy curves are obtained by three cost functions.
As shown in Fig. 6, the phase retrieved within 125 ms can be divided into 5 sections with an offset of ∼2π due to the phase unwrapping error. Thus, it is impossible to quantify the environmental information faithfully based on the retrieved phase, and large errors can be expected. However, the offset can be easily removed by subtracting the mean value of each section. The result of the phase unwrapping error correction is shown in Fig. 7, which clearly demonstrates that the unwrapping correction method proposed in this paper applies to both perturbated and unperturbed fiber sections. The corrected result preserves all the characteristics of the original phase. For example, the spikes in the original phase remains the same shape after the error suppression. More spikes however may appear at the break points, as shown between 100 ms and 150 ms in Fig. 7(a), which are caused by the phase correction. These spikes can be removed by replacing the phase at the bread points by the interpolated value.
Fig. 7. Comparisons of phases obtained at position (a) 1141.4 m and (b) 1410.5 m with and without phase unwrapping error suppression.
The impact of unwrapping error suppression on the frequency domain is shown in Fig. 8, the power spectral densities (PSDs) of the obtained phases before and after the suppression are compared at the still and perturbated positions. According to Fig. 4, some temporal sections of the obtained phase exhibit an offset due to the phase unwrapping error, thus the SPDs demonstrate high power at DC and the low frequency range, as shown in Fig. 8. The offset is removed after the error suppression, so the power becomes smaller as presented by the red curves in Fig. 8. And the suppression has minimum influence on the dynamic measurement, because the frequency peak caused by vibration at 1 kHz remains almost the same after the suppression. The peaks at higher frequencies as shown in Fig. 8(b) are just the harmonics of the measured vibration.
Fig. 8. Power spectral density of the phases at position (a) 1141.4 m and (b) 1410.5 m with and without phase unwrapping error suppression.
The statistical feature of the phases is changed after the suppression of the unwrapping errors along the fiber and the change is shown in Fig. 9. The phase errors before and after the suppression are plotted as a function of distance in Fig. 9(a). The analysis in Section 2 and 3 reveals that the large error occurs at fading points where the ${\bar{I}^2} + {\bar{Q}^2}$ is low and the low SNR is likely to cause the phase unwrapping error, further increasing the total phase error at a given position. Figure 9(a) shows clearly that the phase unwrapping error suppression proposed here can reduce the overall phase error by at least one order of magnitude at the positions where the unwrapping error occurs. However, it is unable to reduce the error caused by the low SNR, particularly at the fading position. In this case, the unwrapping error can occur constantly over the measurement time, making the change points difficult and even impossible to detect. Therefore, only few points are detected and corrected at these positions, so that the error cannot be totally suppressed. Consequently, some error peaks remain high as shown in Fig. 9(a). It is possible to further reduce the peaks by repeating the error estimation and suppression process or by using a smaller window size. But both methods are at a cost of the processing time. The blue and red curves overlap for most fiber section, demonstrating that the unwrapping error suppression has no impact on the phase error at these places which is simply determined by the SNR. Figure 9(b) also shows that the PDF of the phase after the suppression is almost the same as the PDF of the original phase, however, the probability for large phase error decreases due to the suppression.
Fig. 9. Comparison of the phase error obtained before and after the phase unwrapping error suppression. (a) Longitudinal profile of the phase error for a short fiber section. (b) Probability density function of the phase error along the whole unperturbed fiber.
It has to be noted that data processing speed is a big concern because the φOTDR system usually acquires a large amount of data. As a result, the computer used here is with 32 Intel CPUs and 251.6 GB RAM. It takes in total ∼ 10 min to calculate the phase from the raw data and suppress the phase unwrapping error for 10,000 traces over 7056 sampling points. Several factors have impact on the processing speed, such as the sliding window size and the cost function type. The speed can certainly be improved if the algorithm is further optimized and if GPU is used.
5. Conclusion
The phase error in a φOTDR system is investigated theoretically and experimentally in this paper. At first, the phase error is analyzed in a new way by considering the longitudinal variation of the detection noise for the first time to the best of our knowledge. As a result, the theoretical calculation of the phase error is much closer to the experimentally obtained value, outperforming the previous research, which treats the noise as a constant over the fiber length. For example, the new method has demonstrated a probability density that is ∼ 39 times larger than the previous research for the accuracy over 99%. A phase error theory with a higher estimation accuracy helps researcher deepen the understanding of the whole system and find the limiting factor.
In addition, some break point detection algorithms have been introduced to suppress the phase error caused by improper unwrapping. The CPD methods can locate the incorrect unwrapping points and divided the temporal phase variation at one position into several sections. The unwrapping error is highly suppressed by removing the offset within each section. Compared with other reported methods, the proposed method delivers a simple and economic solution that relies purely on data processing, requiring no hardware modifications. This phase unwrapping error suppression method is therefore believed as a universal solution to all the φOTDR systems based on phase measurement. It needs to be pointed out that the break point detection algorithms may have more applications in distributed acoustic sensing because it can also detect the phase or frequency change of a signal.
Funding
Zentrales Innovationsprogramm Mittelstand (ZF4044230RH9).
Acknowledgements
The research project was carried out in the framework of the Zentrales Innovationsprogramm Mittelstand (ZF4044230RH9, project acronym “FoLO”). It was supported by the Federal Ministry for Economic Affairs and Energy (BMWi) on the basis of a decision by the German Bundestag. The authors thank Marcus Schukar and Sven Münzenberger at BAM for their technical support and all the partners in the FoLO project for helpful discussion
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.
References
1. X. Lu, M. A. Soto, and L. Thévenaz, “Temperature-strain discrimination in distributed optical fiber sensing using phase-sensitive optical time-domain reflectometry,” Opt. Express 25(14), 16059–16071 (2017). [CrossRef]
2. M. M. Molenaar, D. Hill, P. Webster, E. Fidan, and B. Birch, “First downhole application of distributed acoustic sensing for hydraulic-fracturing monitoring and diagnostics,” SPE Drill. & Complet. 27(01), 32–38 (2012). [CrossRef]
3. F. Walter, D. Gräff, F. Lindner, P. Paitz, M. Köpfli, M. Chmiel, and A. Fichtner, “Distributed acoustic sensing of microseismic sources and wave propagation in glaciated terrain,” Nat. Commun. 11(1), 2436 (2020). [CrossRef]
4. F. Peng, N. Duan, Y.-J. Rao, and J. Li, “Real-time position and speed monitoring of trains using phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014). [CrossRef]
5. M. J. Murray, A. Davis, and B. Redding, “Multimode fiber Φ-OTDR with holographic demodulation,” Opt. Express 26(18), 23019–23030 (2018). [CrossRef]
6. G. Cedilnik, G. Lees, P. E. Schmidt, S. Herstrøm, and T. Geisler, “Pushing the reach of fiber distributed acoustic sensing to 125 km without the use of amplification,” IEEE Sens. Lett. 3(3), 1–4 (2019). [CrossRef]
7. Z. Zhao, H. Wu, J. Hu, K. Zhu, Y. Dang, Y. Yan, M. Tang, and C. Lu, “Interference fading suppression in φ-OTDR using space-division multiplexed probes,” Opt. Express 29(10), 15452–15462 (2021). [CrossRef]
8. Z. Zhao, Y. Dang, M. Tang, L. Wang, L. Gan, S. Fu, C. Yang, W. Tong, and C. Lv, “Enabling simultaneous DAS and DTS through space-division multiplexing based on multicore fiber,” J. Lightwave Technol. 36(24), 5707–5713 (2018). [CrossRef]
9. Y. Koyamada, M. Imahama, K. Kubota, and K. Hogari, “Fiber-optic distributed strain and temperature sensing with very high measurand resolution over long range using coherent OTDR,” J. Lightwave Technol. 27(9), 1142–1146 (2009). [CrossRef]
10. A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors (CRC Press, 2017), Chap. 6.
11. R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000). [CrossRef]
12. Z. Pan, K. Liang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “Phase-sensitive OTDR system based on digital coherent detection,” Proc. SPIE 8311, 83110S (2011). [CrossRef]
13. H. Gabai and A. Eyal, “On the sensitivity of distributed acoustic sensing,” Opt. Lett. 41(24), 5648–5651 (2016). [CrossRef]
14. A. Morsali, S. Lecler, P. M. Pelletier, and P. Pfeiffer, “Significance of modulator extinction ratio in long distance coherent optical time domain reflectometry,” Opt. Eng. 60(03), 036106 (2021). [CrossRef]
15. G. Yang, X. Fan, S. Wang, B. Wang, Q. Liu, and Z. He, “Long-range distributed vibration sensing based on phase extraction from phase-sensitive OTDR,” IEEE Photonics J. 8(3), 1–12 (2016). [CrossRef]
16. Y. Lv, P. Wang, Y. Wang, X. Liu, Q. Bai, P. Li, H. Zhang, Y. Gao, and B. Jin, “Eliminating phase drift for distributed optical fiber acoustic sensing system with empirical mode decomposition,” Sensors 19(24), 5392 (2019). [CrossRef]
17. X. Lu, M. A. Soto, P. J. Thomas, and E. Kolltveit, “Evaluating phase errors in phase-sensitive optical time-domain reflectometry based on I/Q demodulation,” J. Lightwave Technol. 38(15), 4133–4141 (2020). [CrossRef]
18. X. Lu and K. Krebber, “Characterizing detection noise in phase-sensitive optical time domain reflectometry,” Opt. Express 29(12), 18791–18806 (2021). [CrossRef]
19. B. Redding, M. J. Murray, A. Donko, M. Beresna, A. Masoudi, and G. Brambilla, “Low-noise distributed acoustic sensing using enhanced backscattering fiber with ultra-low-loss point reflectors,” Opt. Express 28(10), 14638–14647 (2020). [CrossRef]
20. Y. Wakisaka, D. Iida, H. Oshida, and N. Honda, “Fading suppression of Φ-OTDR with the new signal processing methodology of complex vectors across time and frequency domains,” J. Lightwave Technol. 39(13), 4279–4293 (2021). [CrossRef]
21. F. Jiang, Z. Zhang, Z. Lu, H. Li, Y. Tian, Y. Zhang, and X. Zhang, “High-fidelity acoustic signal enhancement for phase-OTDR using supervised learning,” Opt. Express 29(21), 33467–33480 (2021). [CrossRef]
22. X. Lu and P. J. Thomas, “Numerical modeling of Fcy OTDR sensing using a refractive index perturbation approach,” J. Lightwave Technol. 38(4), 974–980 (2020). [CrossRef]
23. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982). [CrossRef]
24. H. F. Martins, S. Martin-Lopez, P. Corredera, M. L. Filograno, O. Frazão, and M. González-Herráez, “Coherent noise reduction in high visibility phase-sensitive optical time domain reflectometer for distributed sensing of ultrasonic waves,” J. Lightwave Technol. 31(23), 3631–3637 (2013). [CrossRef]
25. Z. Wang, J. Jiang, Z. Wang, J. Xiong, Z. Qiu, C. Liu, and Y. Rao, “Quasi-distributed acoustic sensing with interleaved identical chirped pulses for multiplying the measurement slew-rate,” Opt. Express 28(26), 38465–38479 (2020). [CrossRef]
26. X. Lu and K. Krebber, “Direct detection based φOTDR using the Kramers-Kronig receiver,” Opt. Express 28(24), 37058–37068 (2020). [CrossRef]
27. X. Lu, M. A. Soto, L. Zhang, and L. Thévenaz, “Spectral properties of the signal in phase-sensitive optical time-domain reflectometry with direct detection,” J. Lightwave Technol. 38(6), 1513–1521 (2020). [CrossRef]
28. X. Lu, M. A. Soto, and L. Thévenaz, “Optimal detection bandwidth for phase-sensitive optical time-domain reflectometry,” Proc. SPIE 9916, 99162N (2016). [CrossRef]
29. C. Truong, L. Oudre, and N. Vayatis, “Selective review of offline change point detection methods,” Signal Process. 167, 107299 (2020). [CrossRef]
