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
where visibility, V = 0.5, includes all other system imperfections such as polarization mismatch. Equation 1.45 represents the white noise level for 1 second time integration of the DAS signal. For engineered fiber, the number of photons can be up to 100 times larger than for conventional Rayleigh backscattering, so the noise will be 10 times smaller.
Another advantage of DAS with engineered fiber is a wider dynamic range that is defined as the ratio of the maximum detectable signal to the noise level. The typical geophone bandwidth is ΔF = 100Hz, so the minimum strain level εmin detectable for DAS for gauge length L0 = 10m within the same detection bandwidth is:
where A0 = 115nm is the elongation corresponding to one radian phase shift (Equation 1.14).
Experimental measurements with conventional fiber DAS found a value three times higher, at 0.03nanostrain (Miller et al., 2016). In this case, there was some extra flicker noise, as discussed earlier (see Figure 1.11). Here, a spiky noise structure corresponds to algorithm discontinuities that amplify photodetector noise, with a spectrum after DAS signal time integration, which is ∝F−1. The typical low frequency limit when excessive noise starts to dominate over shot noise is between 10 and 100 Hz, depending on the fiber conditions.
For engineered fiber (Farhadiroushan et al., 2021), reflectivity can be engineered to be hundreds of times higher than the normal Rayleigh level, without any significant problems with crosstalk, such that R = 100 · RBS · τ = − 45dB. As a result, sensitivity is ten times higher, at around 1picostrain, which corresponds to a 100x (20 dB) improvement in acoustic signal sensitivity.
It is important to compare the shot noise level of DAS with the noise level of high‐sensitivity geophones and seismometers. The DAS white noise value should be added to flicker noise with coefficient μ and corrected for spatial filtering (Equation 1.46) as:
