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The theoretical concept of DAS is based on the assumption that the Rayleigh centers have no microscopic motion, but they are “frozen” inside glass during manufacture. In this case, the positions of the centers depend on the macroscopic motion of fiber and can coincide with the ground speed around a buried fiber (v). There are two time scales of relevance to DAS: (1) as optical pulse travels with speed c, significantly faster than ground motion, this dictates the spatial resolution; (2) seismic motion is responsible for interference changes pulse to pulse, which can be used to recover the seismic signal. All parameters for both fast and slow motions are summarized in the table of variables at the end of the chapter.

Let us calculate how the intensity of backscattered light changes when a section of fiber is moving with speed v(z) under a seismic wave (Figure 1.4). The Rayleigh centers will move with the fiber, so the frequency of the backscattered light will experience a Doppler shift Ω(z) proportional to its speed, like for Brillouin scattering (Hartog, 2017). The aim of DAS can be considered as the measurement of Doppler shift for Rayleigh scattering derived from the detected photocurrent. The phase shift can be measured between two separate points in space, and then the resultant Doppler shift can be recovered with spatial integration, as will be shown later in the text. The first step is to analyze changes in intensity between different optical pulses to derive the fiber speed information, which will be equal to the ground speed in a seismic wave.

Consider a coherent optical pulse e(t′) that is launched into a single‐mode optical fiber. The backscattered optical field E(t′) at time t′ for light reemerging from the launch end can be expressed as a superimposition of delayed partial fields backscattered with a reflection coefficient r₀(z) along the fiber axis z (Shatalin et al., 1998). This amplitude coefficient represents coupling between the forward and backward modes. For a speed of light in the fiber c ≈ 2 10⁸m/c, and wave propagation constant β, we can use group and phase delays 2z/c and , respectively. So, the emerging field will depend on interferometer optical delay, or gauge length, L₀ as:

(1.1)

For a regular fiber, the phase shift term in Equation 1.1 can be separated into a constant part and a part changing with “slow” time t, representing pulse‐to‐pulse parameter variation with Doppler shift frequency Ω(z), which is proportional to scattering particle velocity v(z) and wavelength frequency ω.

(1.2)

(1.3)

Here the strain coefficient K_ε relates the physical and optical length of fiber, n_eff is fiber effective refractive index, and λ is the laser wavelength. Equations 1.2–1.3 represent a well‐known dualism, when a change in interference can be considered not only as a result of a change in phase, but also as a beating of a frequency due to a Doppler shift. The concept finds application in Doppler lidars, where Rayleigh scattering light contains wind speed information, so the height distribution of the speed can be detected using OTDR (Garnier & Chanin, 1992). The DAS conception is somewhat different: we do not measure the absolute velocity of Rayleigh scatterers, but the difference in such velocity along the gauge length. Another difference is that Rayleigh centers are frozen in a glass of fiber at a melting point of about 800°. Their movement follows the movement of the fiber, and hence very low Doppler frequencies (down to mHz) can be measured.

For simplicity of further calculations, the reflective coefficient r₀(z) can be redefined as the effective reflective coefficient r(z):

(1.4)

Then, to extract the Doppler shift from the intensity equation, we need to control the phase shift ψ₀ between delayed optical fields in the interferometer. So Equation 1.1 can be rearranged using Equations 1.2–1.4 to:

(1.5)

Here the convolution symbol ⊗ is used to simplify the expression, and the OTDR scale z = 2ct′ for the “fast” time t′ is used. The convolution commutes with translations (Goodman, 2005), meaning that Equation 1.5 can be converted using a(z₁ − z₂) ⊗ b(z₁) = a(z₁) ⊗ b(z₁ − z₂) to:

(1.6)

Let us consider first the simple case of short pulse e(z) = δ(z) 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:

(1.8)

(1.9)

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).