## Abstract

Understanding signal fading effect is essential for the application of Rayleigh-scattering-based distributed acoustic fibre sensors (DASs) due to the nature of coherent beam interference within the pulse length. Statistical properties for the intensity of the Rayleigh backscattered light (i.e. intensity fading) and its impact on the sensitivity of DAS systems have been intensely studied over the last decades. Here we for the first time establish an analytical model for the phase signal retrieved from the dual-pulse heterodyne demodulated DAS system, which can be exploited to investigate the phase fading effect in this system. The developed model reveals that the phase fading phenomenon mainly originates from the randomness in the phase retardant of the Rayleigh scatters. The quantitatively resolved statistical features of the phase fading is confirmed by experimental results. Based on the analytical model, a noise figure is defined to characterize the global fading-induced noise level via taking into account contributions from all channels along the sensing fiber. The model also reproduces the anti-correlation relation between the power spectrum density of retrieved phase at the heterodyne frequency and the phase fading noise level. Following the analysis and the definition of the noise figure, an optimized real-time weighted-channel stack algorithm is developed to efficiently suppress the fading noise. Experimental results show that the algorithm can achieve a maximum noise figure reduction of 15.8 dB without increasing the system complexity.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Distributed optical fiber sensors are capable of sensing different physical quantities (e.g. temperature, strain and vibration) at positions along the optical fiber with well-defined spatial and measurement resolutions [1,2]. As an influential example, distributed acoustic sensor (DAS) can remotely retrieve information of local acoustic field via strain measurement, attracting enormous interests in recent years mainly due to its potential application in oil and gas industry as an advanced seismic geophone [3–6].

One important type of DAS system represents as phase-sensitive optical time-domain reflectometry ($\Phi$-OTDR), in which highly coherent pulsed lights are injected into the sensing fiber and Rayleigh backscattered lights are collected. Phase variations of the injected beam at different fiber positions can be retrieved by utilizing proper demodulation approaches, e.g. unbalanced interferometry with a 3$\times$3 coupler [7,8], phase-generated carrier algorithm [9], quadrature detection [10,11] and time-gated digital optical frequency domain reflectometry [12,13]. The retrieved phase signal is proportional to the strain induced by the external acoustic field [14].

Recently, we reported a dual-pulse heterodyne demodulated DAS (HD-DAS) technique being able to simultaneously resolve multiple acoustic-induced phase modulations over the fiber [15]. In this system, different from conventional single-pulse DAS systems, two pulses offset in both time and frequency domains are launched into the fiber, functioning similarly as the sensing and reference arms of an interferometer, thus each position on the fiber can be regarded as an independent sensor. In addition, as the two pulses propagate along the same fiber, the experienced environmental noises are self-referenced, allowing to achieve a noise floor as low as 8.91$\times$10$^{-4}$ rad/$\surd$Hz along with a high signal to noise ratio (SNR).

For the targeted applications of DAS (e.g. in downhole environment), in addition to the noise level of the sensor that ultimately limits the resolution of seismic wave detection, the performance is also restrained by the so-called signal fading effect. In DAS system, since the detected signal contains the interference among Rayleigh backscattered light from all positions within the pulse duration, plus the scattering cross section and the induced optical phase shift from those Rayleigh scatters are randomly dispersed over distance, the collected interference signal appears as intensity fluctuations leading to effects similar to noise. Signal fading can severely degrade the system’s ability on distributed detection and thus has drawn enormous attentions in recent years. Statistical properties of Rayleigh backscattered lights in single-mode fibers (SMFs) as well as the coherent fading in DAS systems have been analyzed [16–18]. Specifically, Staubli et al. studied the statistical properties of Rayleigh backscattered intensity in SMF [16]. Goldman et al. used Golay complementary codes in the direct detection and coherent OTDR scheme to mitigate the speckle-like fading noise of Rayleigh backscattering [17]. Liokumovich et al. developed a statistical model for the signals received in coherent OTDR which can be extended to examine the fading, transfer function and noise properties of retrieved phase [18]. Gabai et al. reported an approach to quantify the sensitivity of DAS system via considering the backscattered amplitude as a Rayleigh-distributed spatial random process [19]. Most of the previous analysis, however, focused on the intensity fading (rather than phase fading) effects, so far analytical formulation on the phase fading effects in DAS systems has not yet been reported.

Regarding on dual-pulse HD-DAS system, the phase fading phenomenon has been experimentally observed [20,21]. Nevertheless detailed analysis of its mechanism and statistics has not been disclosed. The properties of the fading noise in HD-DAS system are expected to be distinct from those in single-pulse systems due to its self-reference feature and the heterodyne demodulation procedure. Here in this work, we for the first time establish an analytical framework for the demodulated phase of dual-pulse HD-DAS system, and further applies the formulas to study the phase fading effects. It is revealed that the random phase retardant of the local scatters dominates the observed phase fading effects. A dimensionless parameter, namely the noise figure, is defined to quantify the global level of phase fading. Using the noise figure and the mathematical model, the phase fading noise is suppressed using an optimized weighted-channel stack algorithm.

## 2. Analytical modelling of phase fading effect in HD-DAS system

#### 2.1 Interference model

We consider the optical fiber as composed of many reflectors, the reflectivity and phase shift of which are given by $r(z,\tau )$ and $\theta (z,\tau )$, where $z$ is the coordinate along the fiber and $\tau$ is the slow-time frame. The $z$ dependence of $r$ and $\theta$ can be modelled respectively as Rayleigh and uniform density distributions [20]. The time dependence is due to the fact that $r$ and $\theta$ may vary slowly with environmental perturbations. The electric field of the backscattered beam from the two pulses can then be expressed as:

#### 2.2 Heterodyne demodulation

When applying the heterodyne demodulation, Eq. (3b) is modulated to lower frequencies while Eq. (3a) and Eq. (3c) are shifted to higher frequencies. Following the established demodulation procedures [15], Eq. (3) is consequently mixed with $I_{r1} = sin(\Delta \omega n)$ and $I_{r2} = cos(\Delta \omega n)$ and filtered by a low-pass filter (LPF), giving:

#### 2.3 Phase fading effect and the noise figure

To study the phase fading effect, we consider a sinusoidal acoustic signal with angular frequency $\omega _s$ and with a uniform amplitude applied on the entire fiber length. The accumulated amplitude of the vibration-induced phase change per unit length is $\Psi$ rad and the amplitude of phase perturbation acted on each individual reflector is thus $\psi = \Psi /(1/\delta l)$. The applied phase signal on each reflector can be expressed as:

Substituting it into Eq. (6) gives:The analytical solution given in Eq. (6) was then used to reveal the origin of phase fading phenomenon. Specifically, four distribution combinations of $r$ and $\theta$ parameters were set in the simulation and the retrieved phase signal was calculated. The applied parameters are listed in Table 1, which were set similar to the experimental conditions (see Section 3.). In the simulation, the ADC sampling rate was 100 MS/s, such that the ADC spatial resolution was 1 m meaning there were 500 ADC channels along the fiber.

The simulation results of the demodulated phase $\widehat {\Phi }(m,n)$ are shown in Fig. 1. As can be seen, the distributions of $r$ and $\theta$ strongly affect the demodulated phase signal. When both $r$ and $\theta$ are constant, phase fading vanishes and the sinusoidal signal can be retrieved without distortion (Fig. 1(a)). When $r$ is Rayleigh distribution while $\theta$ is a constant, modest phase fading appears on top of the sinusoidal wave (Fig. 1(b)). However, when $\theta$ is randomly distributed (uniform distribution here), severe phase fading effect (including significant signal distortion and amplitude fluctuation over channels) appears regardless $r$ is a constant or randomly distribution (Fig. 1(c) and 1(d)). This indicates that the phase fading phenomenon is mainly induced by the randomness in the phase retardant $\theta$ of each Rayleigh scatter. Figure 1(e) plots the root-mean-square (RMS) values, i.e. the effective amplitude, of the retrieved time domain signal versus position for each channel (i.e. $\widehat {\Phi }_{rms}(m)$, where $m$ is the channel number) when $r$ and $\theta$ have distinct distributions. It can be seen that for a uniformly applied acoustic signal, when $r$ and $\theta$ are constants, the RMS value is a constant over position (red). While $r$ or $\theta$ are set random values, the RMS of the phase strongly fluctuates over position.

In order to quantify the level of phase fading by considering all the channels along the fiber, a dimensionless parameter, termed as noise figure (NF), is defined as follows:

where $\widehat {\Phi }_{sd}$ and $\widehat {\Phi }_{avg}$ represent respectively the standard deviation and averaged value of $\widehat {\Phi }_{rms}(m)$ over all channels. Table 2 summaries the value of $\widehat {\Phi }_{sd}$, $\widehat {\Phi }_{avg}$ and the noise figure for the four cases considered in Fig. 1. It can be seen that $\widehat {\Phi }_{avg}$ barely changes with the distribution of $r$ or $\theta$ parameters while $\widehat {\Phi }_{sd}$ (and hence the noise figure) varies significantly.A larger value of NF indicates a higher level of signal fluctuation induced by the phase fading, meaning the system’s SNR is more sensitive to the fading. Using results in Table 2, the contribution of $\theta$ fluctuation to the phase fading can be estimated as about 5.5 times of $r$ fluctuation. This is also consistent with our finding that the phase fading is mainly originated by the phase retardant term [20]. Moreover, it’s interesting to note that when $\theta$ is uniform distribution and $r$ changes from constant to Rayleigh distribution, the degree of fading slightly decreases. This can be attributed by the self-reference feature of the dual-pulse HD-DAS system, in which case even though the two pulses experience slightly different values of the $r$ and $\theta$, the randomness may still be negated due to the relatively close distance between the two pulses.

To better visualize the NF factor, we generated 1000 sets of different $r$ and $\theta$ realizations (which can be regarded as 1000 pieces of fiber) with the same statistic distribution (i.e. $r$ is Rayleigh distribution and $\theta$ is uniform distribution) and the simulated results are given in Fig. 2(a) with Fig. 2(b) displaying the histogram of the noise figure. The red and magenta lines plot respectively $\widehat {\Phi }_{sd}$ and $\widehat {\Phi }_{avg}$. It can be seen that $\widehat {\Phi }_{avg}$ remains almost unchanged when the realizations of $r$ and $\theta$ are varied while $\widehat {\Phi }_{sd}$ is apparent to be more sensitive to the statistical fluctuation. Figure 2(b) displays the histogram of the noise figure, showing that the values are highly concentrated around the mode value ($\sim$0.42 in this case).

#### 2.4 Relation with the power spectral density at the heterodyne frequency

The heterodyne demodulation procedure also induces novel features in the phase fading noise in HD-DAS system. Figure 3 plots the 1000 sets of simulation results (with different $r$ and $\theta$ realizations) of the cross-correlation coefficient ($C_v$) between the power spectrum density (PSD) at the heterodyne frequency (noted as $PSD_{\Delta f}$) and the RMS of the retrieved phase for different channels. In this simulation, we consider the effect of external noises (e.g. laser noise, AOM modulator and ambient noises) on all channels as uniform, thus Gaussian-white noises were added to Eq. (2):

## 3. Experimental investigation

#### 3.1 Experimental setup

We conduct experiments to characterize the phase fading effect in HD-DAS system. The setup is shown in Fig. 4. A narrow-linewidth CW laser (NKT Koheras BasiK E15) was used as the light source. The laser beam was split into two paths by an optical coupler (OC1) and was modulated by two acousto-optic modulators (AOM1 and AOM2) to create the heterodyne pulse pair. The heterodyne frequency of the pulse pair was $\Delta f$ = 50 kHz and the pulse width was $w$ = 6 m. The pulse pair was generated at a repetition rate of $f_r$ = 200 kHz. An $L_d$ = 10 m long delay fiber was placed after AOM2 to separate the two heterodyne pulses in the time domain, and the spatial resolution can be calculated as $\delta z = (w+L_d)/2$ = 8 m. Then the pulse pair was amplified by an Erbium-doped fiber amplifier (EDFA1) before injecting into the sensing fiber though a circulator. The length of the fiber under test (FUT) was 500 m, placed inside an acoustic isolation box (filled with sponge around the fiber coil) to mostly suppress the environmental perturbations. The Rayleigh backscattered lights from the FUT were amplified by EDFA2 before reaching the photodetector (PD). The collected signal was sampled by a high-speed data acquisition card (DAQ) with a sampling rate of $f_{ADC}$ = 100 MS/s before signal processing.

#### 3.2 Measurement of the noise figure

Figure 5(a) displays the measured NF of the retrieved phase signal from 1000 sets of repeat measurements with the duration of each data set as 10 ms. Since the same piece of single mode fiber was used in the experiment, this measurement corresponds to a certain realization of $r$ and $\theta$ overlapped with external random noises. Since the external noises can be viewed as Gaussian and white, the measured histogram of the noise figure (Fig. 5(b)) appears as normal distribution. As a direct comparison, Fig. 5(c) and 5(d) plots respectively the simulated NF (the simulation parameters are listed in Tab. 1) and its histogram for 1000 sets of the random Gaussian noises (see Eq. (11)) and with one specific realization of $r$ and $\theta$ distribution. It can be seen that the features of simulated $\Phi _{sd}$ and $\Phi _{avg}$ can well-reproduce the experimental results and the simulated NF also fits very well to the Gaussian distribution with a similar average value and standard deviation.

Figure 6(a) plots a typical set of collected intensity signal over position. The red line represents the averaged intensity (namely the DC component) corresponding to Eq. (3a), while the green area denotes the amplitude of the interference signal (i.e. AC component) given by Eq. (3b) and Eq. (3c). Figure 6(b) and 6(c) plots respectively the $PSD_{\Delta f}$ and the root-mean-square error (RMSE) of the demodulated phase signal at different positions. As can be seen there is a clear anti-correlation relation (measured as $C_v \sim 0.83$) between $PSD_{\Delta f}$ and the RMSE: the lower PSD, the higher RMSE. Note that $PSD_{\Delta f}$ values in linear scale were used in the calculation of $C_v$. Interestingly, we note that the positions for large RMSE does not necessarily coincident with the minimum intensity point. This is because the terms in Eq. (3a) are filtered out in the heterodyne demodulation procedure, remaining only Eq. (3b) and Eq. (3c) in the demodulated phase. Equation (3b) and Eq. (3c) is the heterodyne oscillation part which is directly linked to the interference visibility and the PSD at the heterodyne frequency. A smaller visibility leads to a lower level of $PSD_{\Delta f}$ and thus a larger phase fading noise (i.e. higher RMS) for the retrieved phase. Equation (3b) and and Eq. (3c) is therefore more important for the quality of the demodulated phase rather than the minimum intensity (related to Eq. (3a)).

#### 3.3 Phase noise suppression using optimized weighted-channel stack algorithm

Regarding on the suppression of the fading noises in single-pulse DAS system, several approaches have been proposed in recent years. Hartog et al. used multiple optical interrogation frequencies to overcome the fading of detected intensity [23]. Chen et al. used an intensity modulator in the time-gated optical frequency domain reflectometry configuration, which induces harmonics that can be used to suppress fading noise via a matched filter and rotated-vector-sum method [4]. Those approaches to some extent increase the complexity of either the laser source or detection end, which is undesirable for on-site sensing applications.

In Section 2.4 it is revealed that $PSD_{\Delta f}$ shows a significant negative correlation with the RMS value of the retrieved phase. Following this feature, an optimized weighted-channel stack algorithm (WCSA) can be proposed as:

where $M$ is the number of stacking channels and $w(k)$ is the weight. $\mathcal {F}\{* \}_{\Delta f}$ is the amplitude of Fourier transform at the heterodyne frequency $\Delta f$. Since a higher PSD value at the heterodyne frequency is favorable to suppress the fading noise, the weighted average using the calculated signal power at the heterodyne frequency in Eq. (12b) can enhance the contribution from channels with a higher PSD. Furthermore, compared to the method proposed previously [21], this algorithm is also efficient in computation since only the Fourier transform at the heterodyne frequency (rather than the full band) is required, thus:The theoretical model allows a explicit optimization of the stacking number ($M$ parameter) in terms of NF for the WCSA algorithm. A larger $M$ parameter increases the effect of moving average and thus suppress the random fading noise, while it also leads to signal distortion caused by the aliasing between adjacent channels. In the simulation, we scan the value of $M$ parameter and calculate the corresponding NF. Figure 7(a) plots the obtained optimal $M$ parameter as a function of pulse duration. It can be seen that in the cases of higher spatial resolution when the difference in signal between channels is more pronounced, a stacking number of 4 is enough to optimally suppress the fading noise. In the cases of low spatial resolution, however, more stacking numbers are required.

We then exploited the optimized WCSA to re-demodulate the experimental signal in Fig. 5. We scanned the number of stacking channels $M$ from 1 to 7 and the averaged NF of those 1000 experiment data is shown in Fig. 7(b) (magenta line). As can be seen, in this configuration NF was suppressed the most when $M$ = 4, agreeing very well with the predicted optimal value (in our experiment the spatial resolution is 8 m). The averaged NF is decreased from 7.24 to 1.18 (a decline of 15.8 dB). As a comparison, we also plot the averaged NF values when the regular moving average algorithm was used (green line, proposed in [20]). In the moving average algorithm, the signal on all the stacks were simply averaged without weighting. It can be seen that for all the numbers of stacked channels, the suppression of the phase fading noise are more effective when using the optimized WCSA, demonstrating the advantage of channel weighting using Eq. (12). The red lines in Fig. 6(b) and 6(c) show the obtained $PSD_{\Delta f}$ and RMSE after applying optimized WCSA ($M$ was set to 4). It can be seen that the averaged $PSD_{\Delta f}$ can be improved from 69.0 dB to 73.1 dB, accompanied by a decrease of averaged RMSE from 0.029 rad to 0.017 rad.

Finally we conducted an experiment to validate the effect of optimized WCSA on the acoustic perturbation affected by fading. The experimental configuration is illustrated in Fig. 8(a). 17.5-meter-long sensing fiber was coiled on one piece of piezoelectric ceramic transducer (PZT). The fiber length between the PZT and the DAS interrogator was about 250 m. Sinusoidal perturbation at 1 kHz frequency was applied on the PZT. The red and blue curves in Fig. 8(b) plot respectively the RMS of the demodulated phase signal over position with and without applying WCSA. As can be seen, although an uniform acoustic signal was applied on the PZT, the demodulated phase appears evident fluctuation due to phase fading. Such a fluctuation can be greatly suppressed when optimized WCSA ($M$ = 4) was used. Figure 8(c) and (d) display the contour plot of the demodulated phase signal without and with WCSA. It can be seen the retrieved phase signal is more identical to sinusoidal wave when WCSA was applied.

## 4. Conclusion

In summary, we establish an analytical expression for the heterodyne demodulated phase signal of HD-DAS system and further demonstrates its capability to analysis the phase fading effect originated from the superposition of a vast number of Rayleigh scattering signals. The revealed statistics of the phase fading noise and the anti-correlation relation between $PSD_{\Delta f}$ and RSME of the retrieved phase signal are confirmed by experimental results. The reported analysis is also applicable to other DAS systems using heterodyne demodulation (e.g. single-pulse DAS system with coherent detection) or phase-generated carrier algorithm [9] for phase recovery. The developed analytical model may further be used to promote novel demodulation algorithm that can completely overcome the fading effect.

## Appendix: derivation of Eq. (9)

The numerator and denominator of Eq. (9) can be expanded as:

A simple expression of the demodulated phase can therefore be obtained:

## Funding

China Geological Survey (DD20190234).

## Disclosures

The authors declare no conflicts of interest.

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