Distributed Acoustic Sensing in Geophysics. Группа авторов

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Figure 1.26 Acoustic measurements using DAS with precision engineered fiber: The left panel represents strain rate measurement (Equation 1.39) and the right panel displays ground speed measurement (Equation 1.41) after filtering and integration. The signals’ cross‐section along the white line is shown in the bottom panels in radians. The modeled source is shown in the right panel of Figure 1.5.

      The spatial spectral response in the wavenumber domain K_z can be represented by Fourier transform ℑ:

(1.42)

where ℑ(K_z) is the spatial spectral response of the seismic wave. Comparisons of DAS with engineered fiber spectral response for spatial sampling equal to the gauge length and half of gauge length are presented in Figure 1.27 based on Equation 1.41. For the high spatial sampling, we have a gain in the frequency range, which is highlighted by the gray filling. Moreover, it is easy to filter out the aliased component for high sampling as the spectral density is zero for maximum frequency, seen by comparing the position of the black and gray vertical lines in Figure 1.27. This advantage can explain the absence of “staircasing” and the smooth output in Figure 1.26 right panel. An additional advantage of high sampling is that, for a typical L₀ = L_G = 5m, the sampling is twice or even three times smaller than the sensor separation in a geophone array. This spatial frequency margin is useful because DAS timing is different from analog geophones. For a geophone antenna, we can filter out high‐frequency space‐time components in the time domain by electrically filtering individual channels before sampling to prevent spatial aliasing. This approach is ineffective for DAS when the time sampling acts directly on the rapidly changing photocurrent. The problem can be solved for DAS by mechanical filtering in the acoustic area using a special design of the sensing cable, as in Carroll & Huber (1986). An alternative approach involves some oversampling in the spatial domain, and the result is not completely independent. Subsequent filtering then removes high spatial frequencies and prevents aliasing.

Finally, we can neglect the comb function in Equation 1.42, following which Equation 1.42 is exactly equivalent to the expression for a conventional fiber (Equation 1.30) with pulsewidth equal to the scattering period L_S = τ = 5m.