Distributed Acoustic Sensing in Geophysics. Группа авторов

Чтение книги онлайн.

СКАЧАТЬ

1.2. DAS SYSTEM PARAMETERS AND COMPARISON WITH GEOPHONES

In this section, we consider how DAS parameters (such as spatial resolution), gauge length, frequency response, and SNR enable DAS to become an effective tool for seismic measurements. Field data are also presented, with DAS output compared to geophone data.

      1.2.1. DAS Optimization for Seismic Applications

      Distributed fiber sensors measure physical parameters of an external environment continuously through the integration properties of light traveling along a lengthy optical path. This is quite different from point sensors, such as geophones, which make an inertial measurement of ground speed at fixed positions (SEAFOM, 2018). The DAS records a local strain rate, which can be converted into particle velocity to allow direct comparison with geophone data. Following Jousset et al. (2018), we can approximately represent DAS signal A(z, t) via ground displacement u(z, t), where F_S is the DAS sampling frequency and L₀ is the gauge length.

(1.23)

      If F_S → ∞, L₀ → 0, then the DAS signal can be presented in a double differential form:

(1.24)

These simplified expressions (Equations 1.23–1.24) give us a qualitative sense of the DAS algorithm output. For a subsequent quantitative analysis, we shall need the detailed expression that was obtained in the previous section. Namely, for a nonzero interferometer gauge length L₀ and optical pulsewidth τ, averaged over random scattering DAS output, A(z) can be represented by Equation 1.15 in expanded form:

(1.25)

      where F_S is sampling frequency and A₀ = 115nm is a scale constant (Equation 1.14). So, the velocity field can be recovered by spatial integration starting from a motionless point as:

(1.26)

Then DAS signal (Equation 1.25) can be transformed using shift invariant a(z₁) ⊗ b(z₁ + z₂) = a(z₁ + z₂) ⊗ b(z₁) to:

(1.27)

      where θ(z) is the Heaviside step function, whose value is zero for a negative argument. As expected, the DAS signal is represented (Equation 1.5) as a convolution of a point spread function with v(z).

Spatially integrated signal (Equation 1.27) was modeled for 10 m gauge length and 50 ns pulsewidth, as shown in Figure 1.5 (right panel). The results of modeling (Equation 1.25) are presented in Figure 1.11 (left panel), and the result is converted to geophone‐style data (Equation 1.26) in the right panel. From a practical point of view, low temporal frequencies, out of the range of interest, can be filtered out, and also spatial antialiasing filtering can be used. It is worth mentioning that the right panel of Figure 1.11 demonstrates the real change in polarity of the reflected seismic pulse. Also, spatial integration (Equation 1.26) acts as statistical averaging, which eliminates the randomness of the “staircasing” in Figure 1.5 left panel.

      The most valuable geophysical information is delivered by sound waves with frequencies below F_MAX = 150Hz, as higher frequencies are attenuated by the ground. For a speed of soundC = 3000m/s, this corresponds to an acoustic wavelength C/F_MAX = 20m, so Nyquist’s limit dictates that L_G ≤ C/2F_MAX = 10m is the maximum spacing of conventional sensors. Formally, the linear spline approximation G(z) of conventional antenna velocity v(z) output can be represented using expressions from (Unser, 1999), as:

      (1.28)СКАЧАТЬ