equals ATAN StartFraction sigma-summation Underscript j equals 1 Overscript upper P Endscripts alpha Subscript j Baseline upper I Subscript j Baseline left-parenthesis z comma t right-parenthesis Over sigma-summation Underscript j equals 1 Overscript upper P Endscripts gamma Subscript j Baseline upper I Subscript j Baseline left-parenthesis z comma t right-parenthesis EndFraction"/>
(1.16)
where V is the visibility given by the ratio of peak‐to‐peak intensity variation to average intensity of the interference signal. In particular, for a symmetrical 3×3 coupler,
The theoretical expression for DAS resolution (Equation 1.13) was obtained from analysis of an interferometer locked onto a fringe, and it is necessary to test how this is applicable to practical phase measurement algorithms. Also, Equation 1.13 contains averaging over a statistical ensemble, and it is important to understand what it means in a real application. To answer the questions, we have compared theoretical values with a simulation based on a 3×3 coupler setup for 100 different random Rayleigh scattering patterns for a wide variety of parameters and found good comparison after averaging. To illustrate this analysis, three optical pulsewidth settings were used for interferometer delay (gauge length) of L0 = 10m and a ground velocity zone of 40 m (Figure 1.7a–c).
All traces (Figure 1.7a–c) correspond to strain measurements rather than to ground velocity profile measurements. If the pulsewidth is small, τ = 10ns, then averaging is not important, and the correspondence between different phase recovery algorithms are clear (Figure 1.7a). For a reasonable pulsewidth, τ = 50ns, only averaged simulation results correspond to theory (Figure 1.7b). If pulsewidth τ = 100ns becomes equal to L0 = 10m in the OTDR scale, then averaging is critical, but after it 100 times averaging correspondence is good (Figure 1.7c). It is important to mention that this simulation did not include photodetector noise, and noise‐like 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
