Fiber-optic distributed sensors based on coherent Rayleigh scattering have been widely studied due to their ability to monitor temperature and strain measurements with ultrahigh sensitivity [1,2]. The temperature and strain dependence of the fiber refractive index allows obtaining a reliable distributed profile of these variables, being a suitable and reliable tool for several applications. In particular, phase-sensitive optical time-domain reflectometry (φ-OTDR) has undergone a relevant progress in the last decade, demonstrating ultrahigh quasi-static measurand resolution using a frequency-scanned φ-OTDR approach [3–6] and the capability of performing distributed acoustic sensing [7].

For quasi-static measurements, temperature and strain changes are retrieved by scanning the laser wavelength (or its optical frequency) and analyzing the local shift of the Rayleigh spectral response at each fiber position [3,4]. The spectral shift is commonly estimated by calculating the local spectral (amplitude-based) cross correlation (CC) between a given measurement spectrum and a reference spectrum obtained at a known temperature and strain condition. This CC is based on the sum of products obtained from the reference spectrum and the shifted version of a measured spectrum. Therefore, spectral samples with high Rayleigh intensity contribute more to the correlation amplitude than low intensity samples. Given the negative exponential distribution of the Rayleigh intensity probability density function [5], the CC can lead to high-amplitude correlation sidelobes, which can be even higher than the mainlobe indicating the true frequency shift. Unwanted correlation sidebands lead to large errors in the frequency shift estimation [8]. These large errors cannot only affect standard frequency-scanned φ-OTDR sensors, but also any Rayleigh sensor using standard (amplitude-based) CCs to estimate spectral shifts, as in orthogonally polarized φ-OTDR [9,10] and optical frequency-domain reflectometers [11]. Large errors represent a relevant limitation for all these distributed Rayleigh fiber sensors; and thus, some mitigating techniques have been already proposed in the literature [8,11]. These methods are essentially based on scanning a very large spectral range, resulting in a wide spectrum that is used as a reference. Then, measurements based on a narrower spectral scanning are compared to the reference using either a least mean square approach [8] or a standard CC calculation [11]. Unfortunately, this requires very long thermal and strain stability (needed to acquire a very large number of traces scanning the spectrum), which may not be always possible to realize.

In this Letter, the use of phase cross correlation (PCC) is proposed and experimentally validated as a robust method to estimate the Rayleigh spectral shift. Compared with the standard CC, the method correlates the instantaneous phase of the measured Rayleigh spectra, resulting in an amplitude-unbiased estimation of the spectral shifts. This instantaneous phase is associated with the artificially constructed spectra and not with the Rayleigh optical phase defining φ-OTDR traces along the fiber. Since the method is only based on the phase of the spectra, the impact of high-intensity spectral regions is mitigated, reducing large errors along the fiber. Comparing the proposed PCC approach with the conventional CC, experimental results demonstrate the reduction of large errors in distributed temperature measurements along a 5.63-km fiber.

In a frequency-scanned φ-OTDR sensor, the temperature- and strain-induced frequency shift of a locally measured Rayleigh spectrum ${s_k}({{\nu_i}} )$ is estimated by cross correlating this spectral shape with a local reference spectrum $s_k^0({{\nu_i}} )$ using a geometrically normalized cross correlation function (CCF), given as[3]

Note that this conventional CC process gives more weight to high-intensity spectral samples. The presence of low-probability high-intensity samples entering or exiting the scanned spectral region after spectral shifts leads to unwanted high peaks in the CCF. Such spurious peaks can be even higher than the one indicating the true frequency shift, so they can be wrongly selected as the main correlation lobe, resulting in large estimation errors [8].

To reduce the presence of large errors, an amplitude-unbiased CC approach must be used to estimate the spectral shift. This can be achieved by using PCC, as proposed here, which compares the similarity between the measured and reference spectra based on their phase coherence rather than on the Rayleigh intensity spectral shape [12,13].

Note that the measured and reference spectra ${s_k}({{\nu_i}} )$ and $s_k^0({{\nu_i}} )$ are artificially constructed from samples of a set of independent φ-OTDR traces at a given fiber position [4]. The shape of the local spectral signal is inherently defined by the instantaneous phase of the specific artificial waveshape. Hence, the similarity between measured and reference spectra can be assessed by comparing their instantaneous phases. This can be performed calculating the phase cross correlation function (PCCF) [12,13]. It must be clarified that the instantaneous phase used in the PCCF does not correspond to a measurement of the Rayleigh scattering optical phase defining the φ-OTDR traces along the fiber, but it is the phase associated to the artificial local signals ${s_k}({{\nu_i}} )$ and $s_k^0({{\nu_i}} )$ which can be estimated by using Hilbert transform and analytic signal theory [14]. The analytic signal $S({{\nu_i}} )$ of a zero-mean real-valued Rayleigh intensity spectrum $s^{\prime}({{\nu_i}} )$ can be uniquely defined using its Hilbert transform $H[{s^{\prime}(\nu {_i} )} ]$ as $S({{\nu_i}} )= s^{\prime}({{\nu_i}} )+ jH[{s^{\prime}({{\nu_i}} )} ].$ The signal can also be represented in exponential form as $S({{\nu_i}} )= a({{\nu_i}} ){e^{j\phi ({v_i} )}}$, where $a({{\nu_i}} )$ and $\phi ({{\nu_i}} )$ are the amplitude (related to the Rayleigh intensity) and instantaneous phase of the analytic signal, respectively. The PCCF is a coherence functional that can be used to measure the similarity between ${s_k}({{\nu_i}} )$ and $s_k^0({{\nu_i}} )$, as a function of the frequency shift ${\delta \nu} $, and is mathematically defined as

 

Fig. 1. Flow diagram for the frequency shift estimation between spectra ${s_k}$ and $s_k^0$ using (a) standard amplitude CC and (b) proposed PCC.

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To validate the proposed method, the experimental setup shown in Fig. 2 has been used. The optical frequency of a highly coherent laser (<1-kHz linewidth), operating at 1550 nm, is scanned using 2-MHz steps in a 2-GHz spectral range. This is performed using a fast internal piezoelectric wavelength modulation (WL mod.) provided by the laser and controlled by a sawtooth signal with a period of 140 ms, supplied by an arbitrary waveform generator (AWG). Two cascaded electro-optical modulators (EOMs) create optical pulses of 10 ns with high extinction ratio and a period of 70 µs using a synchronized generator. The peak pulse power is boosted by an erbium-doped fiber amplifier (EDFA) to reach 20 dBm at the input of the fiber under test (FUT). At the detection stage, the Rayleigh scattering light is amplified by a second EDFA, while the amplified spontaneous emission (ASE) noise is filtered out by a bandpass filter (BPF). The acquisition of traces is performed with a 350-MHz photodetector (PD) and an oscilloscope (OSC). The acquisition sampling rate is set to 1 GS/s, providing a spatial sampling of 10 cm, while the spatial resolution is 1 m. The sensing fiber is a ∼5.63-km single-mode fiber, placed under thermal and mechanical isolation.

 

Fig. 2. Experimental setup of frequency-scanned φ-OTDR.

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The last 100 m of the fiber is loosely coiled and immersed into a water bath to apply temperature changes of −0.15 K and −0.735 K, which are also monitored by a thermocouple. The spectral shifts ${\widehat {\delta \nu }}$ induced by the applied temperature variations $\delta T$ are expected to be ∼200.9 MHz and ∼984.4 MHz, respectively, according to

During the experiment, two-time averaged φ-OTDR traces at 1000 different laser wavelengths are measured along the fiber. Each trace has a static signal-to-noise ratio of 10.0 dB at the far end of the fiber. The local Rayleigh spectral response at each fiber position is artificially constructed from the measured φ-OTDR traces [4]. This process is repeated for each fiber location. The total measurement time needed to obtain the averaged spectral responses is 140 ms.

Figure 3 illustrates the spectra measured before (black lines) and after (purple lines) a thermal change of −0.15 K is applied at two different fiber locations: ${z_1}$ = 5550 m [Fig. 3(a)] and ${z_2}$ = 5620 m [Fig. 3(b)]. The vertical axes on the left- and right-hand sides are displaced for a better visualization of the spectra. Note that the local spectral shape remains almost unchanged after the disturbance at both spatial locations, but a spectral shift ${\widehat {\delta \nu }}$ can be seen, as indicated by vertical dotted lines. Figure 3(a) shows the presence of a high intensity peak (left side of the figure, purple line), which is not present in the reference spectrum. However, Fig. 3(b) shows the opposite situation, where a high intensity peak (right side of the figure, black line) leaves the analyzed spectral window due to the induced shift. Thus, as will be verified hereafter, large errors are expected at these two fiber locations when using the amplitude CC approach.

 

Fig. 3. Scanned Rayleigh intensity spectra measured at different fiber locations: (a) ${z_1}$= 5550 m and (b) ${z_2}$ = 5620 m, before and after a temperature change of −0.15 K is applied to the sensing fiber. Black arrow indicates the temperature-induced spectral shift ${\widehat {\delta \nu }}$.

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Figure 4 shows a comparison between the CCF obtained using the amplitude CC (blue lines) and the proposed PCC approach (orange lines) applied to the same example spectra shown in Fig. 3. Results clearly show that the conventional amplitude-based method leads to a CCF with several high-amplitude correlation sidelobes. In addition, the correlation lobe with the highest amplitude does not correspond to the correct peak, as expected by the applied temperature change according to Eq. (5). Therefore, the estimation of the frequency shift leads to a large error (LE) as indicated by the blue stars and vertical dashed lines in Fig. 4. However, the use of PCC leads to a PCCF with an enhanced main correlation peak and decreased unwanted sidelobes. Thus, the method leads to a better identification of the correct correlation peak, while the use of polynomial fitting allows improving the accuracy on the frequency shift estimation. Since PCC removes the effect of the amplitude from the correlation calculation, the main CC peak occurs where the greatest number of sampled spectral points exhibit high coherence and similarity, making it an ideal amplitude-unbiased method to estimate spectral shifts with reduced impact of high intensity spectral samples.

 

Fig. 4. Amplitude and PCCFs of the spectra shown in Fig. 3, at (a) ${z_1}$= 5550 m and (b) ${z_2}$ = 5620 m. The vertical dashed lines and stars show the estimated frequency shifts ${\widehat {\delta \nu }}$.

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Figure 5 shows the estimated frequency shift profiles for the last 500 m of fiber, obtained by both the standard amplitude CC (blue line) and the proposed PCC (orange line), when thermal changes of −0.15 K [Fig. 5(a)] and −0.735 K [Fig. 5(b)] are applied. A localized frequency shift can be observed between 5510 m and 5630 m, which is consistent with the length of the fiber spool placed inside the water bath. The average frequency shifts in the disturbed region are 195.5 MHz and 980.4 MHz, for each case, corresponding to temperature changes of −0.146 K and −0.732 K according to Eq. (5). These values well agree with the temperature changes applied and measured using a thermocouple. In addition, note that both amplitude CC and PCC methods lead to very low frequency shift uncertainty (standard deviation), being ∼1 MHz for both methods, at 5.5 km where the fiber is thermally and mechanically isolated.

 

Fig. 5. Distributed frequency shift profile obtained along the last 500 m of a 5.63-km sensing fiber, using both the CC (blue curves) and PCC (orange curves) methods, when temperature changes of (a) −0.15 K and (b) −0.735 K are applied to the last 100 m. Insets show an enlarged view of the frequency shift profile in the perturbed fiber section.

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Note that when applying the smallest temperature change, the standard CC method leads to seven large errors within the perturbed fiber section (i.e., being larger than the half spectral width of the CC lobe, being 50 MHz), as depicted by the spikes in the blue line of Fig. 5(a). Two of these sampled positions correspond to the examples labeled as ${z_1}$ and ${z_2}$ in Figs. 3 and 4. In the case of the largest temperature change, the standard CC method leads to 59 large errors within the perturbed section. In contrast, when using the proposed PCC method, the frequency shift at all the spatial locations can be correctly estimated. The insets in Fig. 5 show a zoom-in on the disturbed area, in which fine details can be observed. The variability in this area can be attributed to different exposures to water temperature along the immersed fiber section presumably due to spatially uneven cooling distribution inside the water bath. The estimations performed by both methods are quite similar, except for the spatial locations where large errors occur. This way, results verify that the proposed PCC method leads to more reliable results compared with the standard amplitude CC approach, allowing for a reduction in the number of large errors along the sensing fiber.

In summary, a method based on PCC has been proposed and experimentally validated to estimate the frequency shift of the Rayleigh spectral response of frequency-scanned Rayleigh distributed sensors. Being amplitude unbiased, the technique mitigates the impact of high-intensity spectral samples in the CCF, thus reducing the presence of large errors in the frequency shift estimation, while keeping a very low frequency uncertainty. The method only needs an additional computing time to obtain the instantaneous phase of the analytic signal; however, in our Matlab implementation, this time remained below 2% of the CC computing time, being negligible compared with the overall processing time. While the method has shown no issues with phase recovery, an exhaustive analysis of the method under different scenarios may be still needed. A full statistical analysis is out of the scope of this Letter, but this could also be performed to estimate the probability of large errors and compare it with that reported for the classical amplitude CC [8]. In addition, different potential implementations of the PCC could still be investigated. In addition to its use in classical frequency-scanned φ-OTDR sensors, the technique can also be employed to estimate the distributed profile of Rayleigh spectral shifts in orthogonally polarized φ-OTDR systems, as well as in optical frequency-domain reflectometers.