(Equation 1.6) shown in the left panel of the simulated signal of a ground velocity wavelet shown in the right panel. The signals’ cross‐section along the white line is shown in the bottom panels in radians.
Source: Based on Correa et al. (2017).
1.1.3. DAS Optical Phase Recovery
The randomness of the COTDR signal can be reduced through proper control of the external interferometer phase shift ψ0, which can be achieved in many ways. All these methods are based on the fact that COTDR intensity is random in distance but will vary harmonically depending on the phase, as follows from Equation 1.1 (see Figure 1.6). So, phase control can reveal phase information regardless of the random nature of the signal.
We will start our phase analysis with a simple, although not very practical, approach, where the phase shift ψ0 is locked onto a fringe sin(ψ0 + Φ) ≡ 1. Such an approach was used earlier to analyze the spatial resolution in phase microscopy (Rea et al., 1996). Then Equations 1.8 and 1.9 can be averaged over an ensemble of delta correlated backscattering coefficients 〈r(u)r(w)〉 = ρ2δ(u − w) as:
Equation 1.10 demonstrates that the sign of Doppler shift can be measured by DAS with proper phase control. The same data can be extracted directly from phase information, as is clear from Equation 1.11.
So far, we have analyzed the short pulse case, where the pulsewidth is significantly smaller than the external interferometer delay. In reality, such pulses cannot deliver significant optical power, which is necessary for precise measurements. Fortunately, Equations 1.10–1.11 can be generalized for a nonzero length optical pulse e(z) directly from Equation 1.5 in the same way that an optical incoherent image was obtained in Goodman (2005) using correlation averaging 〈(a ⊗ r1)(a ⊗ r2)〉 = 〈a2〉 ⊗ 〈r1r2〉. This expression is valid for an uncorrelated field, generated by random reflection points 〈r1(z1)r2(z2)〉 = δ(z1 − z2). This calculation confirms that Equation 1.11 remains the same, as it represents averaging over different harmonic signals, but Equation 1.10 will be reshaped to:
Equation 1.11 gives us the possibility to introduce a dimensionless signal as a phase change over a repetition or sampling frequency FS period A(z) = FS · ∂Φ/∂t, and so the DAS output A(z) can be represented for pulsewidth τ(z) = e(z)2 from Equations 1.3, 1.10, and 1.11 as:
Figure 1.6 Intensity changes are irregular along distance but harmonic along phase shift axis.
In Equation 1.14, the elongation corresponding to ΔΦ = 1 rad is A0 = 115nm, calculated for λ = 1550, neff = 1.468 and Kε = 0.73, which has been measured for conventional fiber (Kreger et al., 2006). The DAS signal is a convolution of pulse shape (as is typical for OTDR‐type distributed sensors) with a measured field, which is the spatial difference in fiber elongation speed of points separated by a gauge length.
Phase measurements can be made in a more practical way than locking the interferometer onto a fringe by using intensity trace Ij(z, t) j = 1, 2, ..P from P multiple interferometers with different phase shifts. Such data can be collected consequentially in P optical pulses, but it reduces sensor bandwidth by P times. Alternatively, the information can be collected for one pulse using a multi‐output optical component, such as a 3×3 coupler. In the general case, the phase shift Φ(z, t) can be represented (Todd, 2011) via the arctangent function ATAN of the ratio of imaginary Im Z to real part Re Z of linear combinations of intensities:
