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 can be measured by adding a few consecutive downsampled signals. From a physical point of view, it means that the combination of multiple gauge lengths L0 can be used to form a single long gauge length. The SNR for the resultant gauge length j L0 will decrease proportionally to
The ultimate spectral response of DAS with standard (Equation 1.30) and engineered (Equation 1.43) fiber compared to that from a geophone array is shown in Figure 1.28. The pulsewidth of the DAS is the same as distance between scatter centers in engineered fiber τ = LS = 5m, and the gauge length is the same as the distance between geophones LG = L0 = 10m. In summary, downsampling of the DAS signal with engineered fiber can improve the spectral response as compared to standard fiber with the same gauge length. However, DAS with standard fiber can provide a wide spectral response without aliasing, as is shown in Figure 1.28.
1.3.2. Sensitivity and Dynamic Range
DAS sensitivity can be calculated for a fundamental limit—the shot noise generated by the number of photons detected. Let us estimate the photon number N per second based on input peak power P0 = 1 W, which is near to the maximum optical connector power damage threshold (De Rosa, 2002). The backscattered intensity can be found from the typical scattering coefficient for SM fiber RBS = 82dB for a 1 ns pulse (Ellis, 2007). For an optical pulsewidth τ = 50ns, the energy quant for λ = 1550nm is hυ = 1.28 · 10−19 J. We consider a relatively short fiber length, L = 2000m, to neglect nonlinear effects (Martins et al., 2013) and suppose that light is collected over an integration length LP = 5m:
Figure 1.28 Ultimate SNR spectral response of DAS with standard and engineered fiber and geophone antenna. Pulse width of DAS is the same as distance between scatter centers along engineered fiber—5 m, and gauge length of DAS is the same as distance between geophones—10 m.
The shot or Poisson noise limit for phase measurement Φmin is proportional to