href="#ulink_a09dbb4d-9b4f-5a15-a79a-b087d12a64a6">Equation 1.41) looks similar to that for standard fiber signal (Equation 1.27). If, say, L0 = LS, then, in Equation 1.41, the curly expression { } represents a chapeau function for linear spline interpolation (Unser, 1999), In other words, v(z) is sampled and linearly interpolated with LS period in Equation 1.41 without any smearing, as it was for the case of conventional fiber (Equation 1.27).
The results of modeling (Equation 1.39) are presented in Figure 1.26, left panel. The spatially integrated version of this signal (Equation 1.41) was modeled for L0 = 2LS, and is shown in Figure 1.26, right panel. 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 1.26 is very similar to the original pulse (Figure 1.5), which demonstrates the real change of polarity of the reflected seismic pulse. Compared with Figure 1.10 (conventional fiber), Figure 1.26 shows better SNR and signal amplitude stability than with conventional fiber, and a more uniform size of the step in the “staircase” in the left panel, which can be easily filtered out.
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 Kz can be represented by Fourier transform ℑ:
where ℑ(Kz) 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 = LG = 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 LS = τ = 5m.
Figure 1.27 Comparison of DAS with engineered fiber spectral response for special sampling equal to gauge length (black) and half of gauge length (gray).
DAS with engineered fiber combines the benefits of a distributed sensor, giving full coverage, with the high sensitivity of point sensors such as geophones. The scatter centers are precisely engineered along the length of the fiber and not distributed randomly as for standard fiber (see Figure 1.28). This allows the backscattered signal to be downsampled precisely and optimum spectral response to be obtained.
The DAS signal with engineered fiber, as expressed in Equation 1.39, can be considered as a staircase function with differential velocity sampling LS: when sampled over each staircase distance LS, the expression in the square brackets will be eliminated from Equation 1.39, and, therefore, the corresponding sinc function in Equation 1.43 will also be eliminated. As a result, the DAS signal with engineered fiber will be defined by (v(z) − v(z − L0)), or comb filters in the spectral domain:
Equation 1.43 also includes a gain that can be obtained from synthetic gauge length optimization. With this approach, low spectral frequencies
