alt="StartRoot 2 slash upper N EndRoot"/>. For both heterodyne and/or homodyne phase detection, the photons number halves again (Kazovsky, 1989), as sine and cosine signal components should be measured independently, and so the noise rises 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:
(1.47)
The comparison in Figure 1.28 demonstrates that DAS sensitivity is compatible with geophones. The noise spectrum data for Sercel SG5‐SG10 was adapted from Fougerat et al. (2018), and the seismometer Streckeisen STS‐2 data from Ringler & Hutt (2010) and Wielandt & Widmer‐Schnidrig (2002).
The sensitivity of DAS can even be improved at low frequencies by extending the gauge length from L0 = 10m to L0 = 30m, but at the cost of increased noise at frequencies of more than 70 Hz. Also, 30 m data for DAS with engineered fiber is presented with synthetic gauge length optimization (Equation 1.44). It is worth mentioning that this optimization can be effectively applied to DAS with engineered fiber only, as it has no significant pink noise and can be effectively spatially averaged. As is clear from Figure 1.29, the performance of DAS with engineered fiber can reach seismometers, and it is deep below Peterson’s low noise model level (Peterson, 1993). So, the engineered fiber antenna is an equivalent of a set of multiple seismic stations and can be used for passive seismic applications. Moreover, DAS with engineered fiber has unique sensing capability below 1 Hz, where gravitational wave detectors have limited environmental isolation (Matichard et al., 2015); DAS can be potentially used for such applications.
We now turn our attention to the increase in dynamic range achievable using DAS with an engineered fiber. The acoustic algorithm transforms DAS intensity signals into a phase shift proportional to the fiber elongation value. The algorithm is based on an ambiguous function such as ATAN(x), which give a valid result only inside a limited region (Itoh, 1982). As was analyzed in Section 1.1 (titled ‘Distributed Acoustic Sensor (DAS) Principles and Measurements’), a set of different algorithms can be used, depending on the order of phase tracking. For the first and second order, we have:
For limits (Equations 1.48–1.49), it is clear that the maximum recoverable strain ε1,2 will depend on the algorithm order 1 or 2, and can also be increased using a higher sampling frequency Fs. For a harmonic signal cos(2πFt), we can normalize strain results as: