when δ is the Dirac delta function. Then convolution can be removed from Equation 1.5 because δ(z) ⊗ a(z) = a(z), and the distance variation of Doppler shift ΔΩ(z) = Ω(z) − Ω(z − L0) can be represented via variation of intensity I(z, t) = E(z, t)E(z, t)*. The expression in braces in Equation 1.6 represents a two‐beam interference, so the intensity will vary harmonically depending on the phase. As we are interested in the intensity change, only the interference term needs be taken into consideration, which can be reshaped using the intensity derivative:
(1.7)
Then using convolution properties ∂[a ⊗ b(t)]/∂t = a ⊗ ∂b(t)/∂t, we can find intensity variation via phase shift Φ of backscattered light where there is argument of backscattering complex function:
The COTDR signal can be deduced from Equation 1.8 if we set L0 = 0 and ψ0 = 0. Even such a simple setup can deliver information on the Doppler shift and hence the ground speed v(z) through the intensity variation ∂I/∂t ∝ Δv in accordance with Equations 1.3, 1.8. Unfortunately, the proportionality factor contains an oscillation term, so we cannot distinguish positive speed from negative.
The result of computer modeling of a COTDR response on a differential Ricker wavelet for ground speed (Hartog, 2017) is presented in Figure 1.5. The right side shows 1D seismic wave moving in the z direction (in m) with a reflection from an interface with a positive reflection coefficient. Below the image is a time series of apparent velocity, when units are normalized to the expected optical phase shift in radians between points separated by gauge length 10 m. The left side of the figure corresponds to the relative pulse‐to‐pulse variation of the COTDR signal calculated in accordance with Equations 1.8–1.9. The sign of response changes randomly in accordance with an optical pulsewidth of 50 ns or 5 m. As a result, the signal cannot be effectively accumulated for multiple seismic pulselosityes because of the temperature drift between seismic shots. Temperature drift changes the phase constant of the fiber β0 and, in accordance with Equation 1.4, the effective reflection coefficient r(z) also changes. As a result of such drift, every seismic shot will have a unique, random, alternating, speckle‐like signature that cancels the averaging sum. Fortunately, this problem can be overcome by optical phase recovery, when, after similar averaging, average values appear. Thus, the actual DAS output will be a combination of fiber speed information and the unaveraged portion of the random COTDR signal.
Figure 1.5 COTDR response (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