pulses: the fiber response to strain is highly nonlinear, and therefore the changes in amplitude and phase cannot be directly matched to the original strain affecting the fiber. The next section discusses ways of addressing this. Therefore, COTDR systems are not that useful for seismic applications.
With the phase DAS technique, the method for optical phase analysis is a key feature of system design. All techniques rely on phase modulation between the beginning and end of a pulse, which can be considered as a double pulse. Such modulation can be performed before or after light propagation over optical fiber, as indicated in Figure 1.3. We have limited our discussion to schemas that have been patented and implemented in practice. In one scheme, which is similar to that used for multiplexed interferometer sensors (Dakin, 1990), two laser pulses with different frequencies may be sent down the fiber (Figure 1.3a). In this case, the acoustic phase shift will be transferred to a frequency difference and can be measured in the photocurrent radio frequency domain.
Figure 1.2 COTDR.
Figure 1.3 DAS schemas: MOD—intensity and frequency modulator; AOM—acousto‐optic modulator.
Figure 1.4 DAS optical setup. Distance is proportional to flytime.
Other solutions, such as that shown in Figure 1.3b, contain an embedded delay line that defines the spatial resolution. We will focus our analysis on this class of systems. Another configuration uses optical heterodyne, as shown in Figure 1.3c, where the backscatter signal is continuously mixed with a slightly frequency shifted local oscillator laser. In this case, the elongation along the fiber is measured by computing the difference of the accumulated optical phase between two sections of fiber, and the measurement is carried out at differential frequency f1 − f2. Although this technique offers a flexible spatial resolution, it requires a laser source with extremely high coherence to achieve reasonable signal‐to‐noise ratio (SNR) performance over several tens of kilometers of fiber. The details of the heterodyne concept are thoroughly covered elsewhere (Hartog, 2017). Another method involves sending multiple pulses of different frequencies, either in series or from pulse to pulse, and then computing the phase of the backscatter signal, as indicated in Figure 1.3d. The phase calculation in this case is similar to first case (Figure 1.3a).
1.1.2. DAS Interferometric Optical Response
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
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.
Here the strain coefficient Kε relates the physical and optical length of fiber, neff is fiber effective refractive index, and λ is the laser wavelength. Equations 1.2–1.3 represent a well‐known
