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)
The averaging function p(z) = 5m should optimally be chosen to be compatible with the pulsewidth τ(z) = 50ns, which should be around half the interferometer length L0 = 10m. With this width of the averaging function, it has no significant effect on the spatial resolution of the DAS. Modeling with and without weighted averaging is presented in Figure 1.9, which demonstrates that significant noise reduction can be achieved. It should be noted that this noise reduction is particularly marked in comparison with the coherent OTDR response, by contrasting with Figure 1.5. Nevertheless, weighted averaging suppresses rather than completely removes the effect of flicker noise, and some channels still demonstrate excessive noise (in addition to shot noise). Hence, the response over all depths at a given time for Figure 1.9 will contain spikes for faded channels.
As is explored in Section 1.3, the problem of flicker noise can be overcome by introducing engineered bright scatter zones along the fiber with constant spatial separation and uniform amplitude. Such scatter zones also reflect more photons, and so improve the shot noise detection limitation. In addition, the use of such engineered fiber allows the use of phase‐detection algorithms with improved sensitivity and extended dynamic range.
Figure 1.9 The left‐hand panel shows modeling of raw DAS acoustic data (Equation 1.12); the right‐hand panel shows the same shot with weighted averaging denoising (Equation 1.13) applied. The signals’ cross‐section along the white line is shown in the bottom panels in radians. The modeled
source is shown in the right panel of Figure 1.5.
1.1.6. Time Integration of DAS Signal
A DAS interrogator measures, in accordance with Equation 1.13, the speed difference between two sections of fiber that are separated by interferometer length L0 (referred to also as the gauge length), as presented in Figure 1.10. In pulse‐to‐pulse consideration, the DAS response is linearly proportional to the fiber elongation averaged over the gauge length in the nanometer scale, or strain rate in the nanostrain per second scale. The consideration can also be extended to multiple pulses by time integration of the DAS signal. So, if fiber rests initially and ground displacement equals to zero u(z, t1) = 0, then:
(1.22)
meaning a time integrated DAS signal can be considered as an output of a huge caliper that is measuring fiber elongation between two points with sub‐nanometer precision. This measuring principle is different from that of a geophone but is similar to an electromagnetic linear strain seismograph that can measure changes in distance between two points on the ground (Benioff, 1935).
Figure 1.10 Illustration of two time‐consecutive measurements when DAS output is proportional to fiber elongation
