performance in Figure 1.7c can be explained by the COTDR signal, which will be overlaid on the DAS signal with nonzero pulsewidth. This is a natural limit for increasing SNR by extending pulsewidth; we have a compromise between SNR and signal quality at around L0 = 2τ. Finally, we can expect that the theoretical expression (Equation 1.13) can be used for spatial resolution analysis for different phase recovery algorithms after a proper averaging.
Figure 1.7 Comparison of DAS theoretical response (Equation 1.13) with simulation for a 3 × 3 coupler.
1.1.4. DAS Dynamic Range Algorithms
An acoustic algorithm (Equation 1.15) transforms the DAS intensity signal into a phase shift proportional to fiber elongation value; a question then is how large can this phase shift be? An algorithm based on such ambiguous function as ATAN(x) can give a result only inside a limited region. The classic approach to recover large phase changes is unwrapping: stitching together two consecutive points t and t + Δt from different branches of signal (Itoh, 1982):
This unwrapping, or phase tracking, concept works only if the phase difference is inside two quadrants:
Equation 1.17 makes it possible to measure significant fiber elongation, much longer than the wavelength. If the sampling rate FS = 1/Δt is higher than the acoustic frequency F, a larger acoustic amplitude can be integrated A0FS/2F ≈ 68μ over time for F = 50Hz and FS = 50kHz. Moreover, even this value has improved, and Equation 1.18 gives an idea of this. If the phase is a smooth function, we can differentiate in time Φ(t) before unwrapping. Then, the first differential linear term is removed, and condition becomes more relaxed:
So, the second order tracking algorithm can be obtained by differentiating the signal before unwrapping:
Equation 1.20 has an analog in classical optics, where, instead of the wavefront phase gradient, the wrapped curvature of the wavefront can be unwrapped to increase the dynamic range (Servin et al., 2017). A comparison of these algorithms is presented in Figure 1.8 using modeling for a harmonic signal with a linearly increasing amplitude. It is visible that both algorithms can recover a significant phase range, but the second order tracking algorithm can deliver in excess of a 10 times larger dynamic range.
Theoretically, even higher order algorithms can be designed by repeating this process using higher order derivatives, but they are noisier as more points are involved in the calculation—as can be seen by comparing Equations 1.18 and 1.19. From a practical point of view, the proposed 1D (in time) unwrapping algorithms are error‐free and simple enough to be implemented in real time. Potentially, noise immunity can be improved by transition to 2D (in time and distance) unwrapping, similar to that used in a synthetic aperture radar system (Ghiglia & Pritt, 1998). This solution can extract as much information about the phase as possible, but it is difficult to implement without post‐processing.
Figure 1.8 Comparison of first and second order tracking algorithms for DAS.
1.1.5. DAS Signal Processing and Denoising
In all phase‐detection schemes, the change in optical phase between the light scattered in two fiber segments is determined, meaning we are measuring the deterministic phase change between two random signals. The randomness of the amplitude of the scattered radiation imposes certain limitations on the accuracy of the sensor, through the introduction of phase flicker noise. The source of flicker noise is an ambiguity: when the fiber is stretched, the scattering coefficient varies, and can become zero. In this case, the differential phase detector generates a noise burst regardless of which optical setup is used. The amplitude of such noise increases with decreasing frequency (as is expected for flicker noise) when the phase difference is integrated into the displacement signal.
From a quantum point of view, we need, for successive phase measurements, a number of interfering photon pairs scattered from points separated by the gauge length distance. In some “bad” points, there are no such pairs, as one point of scattering is faded. A natural way to handle this problem is to reject “bad” unpaired photons by controlling the visibility of the interference pattern. As a result, the shot noise can increase slightly as the price for the dramatic reduction of flicker noise. The rejection of fading points can be practically implemented by assigning a weighting factor to each measurement result and performing a weighted averaging.
This averaging can be done over wavelength if a multi‐wavelength source is used. Alternatively, we can slightly sacrifice spatial resolution and solve the problem by denoising using weighted spatial averaging (Farhadiroushan et al., 2010). The maximum SNR is realized when the weighting factor of each channel is chosen to be inversely proportional to the mean square noise in that channel (Brennan, 1959), meaning the squared interference visibility, V2, can be used for the weighting factor as:
(1.21)