be recovered from Equation 1.34 using A0 from Equation 1.14 as:
where A0 = 115nm. Equation 1.38 can also be represented in convolution as:
The main parameter for spatial resolution is still the gauge length L0, and the sampling distance can be chosen to have two points per gauge length LS = L0/2. We are considering here the physical spatial sampling, which is defined by the optical configuration, keeping in mind that the photocurrent sampling can have a higher rate. The difference from conventional fiber is an absence of averaging, as the detected signal is deterministic for engineered fiber, and excessive noise from non‐averaged components will hence disappear. Also, the generated optical field can be significantly larger than with conventional Rayleigh backscattering, so the shot noise limitation can be reduced significantly.
The velocity field can be recovered by spatial integration starting from a motionless point as:
(1.40)
So Equation 1.39 can be transformed to:
Formally, the engineered fiber DAS signal expression (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