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1.3.2. Sensitivity and Dynamic Range
ОглавлениеDAS 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 the coefficient depends on the phase‐detection approach. For a classical phase‐locked homodyne, only half of the photons reach the photodetector when the interferometer operates in quadrature, and so the noise is . 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 . Direct photodetection at λ = 1550nm is not sufficiently sensitive, so DAS usually uses an erbium doped fiber amplifier (EDFA) to boost the signal, which introduces additional noise. This noise can be simply represented by a noise figure NF ≈ 3, which can be reached with appropriate optical filtering as explained in Kirkendall & Dandridge (2004). In this case, the shot noise limit is then given by:
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:
The maximum strain comparison for the first and second order tracking algorithm ε1 and ε2 (Equations 1.50–1.51) is presented in Figure 1.30 for FS = 50kHz and L0 = 10m. The second order algorithm can deliver measurements of fiber strain up to fiber breakage point (~10%) at frequencies of around 10 Hz.
Figure 1.29 Displacement noise comparison of DAS (with and without engineered fiber) with seismometer and geophone. 30 m DAS data are for synthetic gauge length.
Figure 1.30 Maximum strain comparison of first and second order algorithms for DAS.
We can now estimate the maximum DAS dynamic range D as:
(1.52)
Using the real noise level εmin = 0.03nanostrain from Miller et al. (2016), we can estimate D = 99dB for a maximum value ε1 = 2.9μstrain. This estimation gives the practical upper limit for seismic DAS at 100 Hz using Rayleigh scattering. Generally speaking, the second order tracking algorithm has limited applicability for a conventional DAS because flicker noise pulses can reach π and destroy measurements in accordance with Equation 1.49. Nevertheless, 120dB was achieved in Parker et al. (2014) when the fiber elongation zone was significantly smaller than the gauge length and pulsewidth, such that the flicker noise was suppressed. However, when a continuous seismic signal expands the reflectivity zone, then the reflection can disappear, and the signal has ambiguity. Fortunately, in engineered fibers, the scatter center zones are well defined, and so the reflectivity change is negligible. As a result, we can optimistically estimate a maximum D = 167dB for engineered fiber using εmin = 1picostrain and maximum ε2 = 220μstrain—see Figure 1.30.
The dynamic range of DAS with engineering fiber was tested during a dry alluvium geology series of chemical explosions, including 50,000 kg TNT‐equivalent at 300‐m depth‐of‐burial (Abbott et al., 2019). “Two orders of magnitude more data relative to traditional geophones/accelerometers” was successfully recorded.
Summarizing, we can conclude that theoretical estimations demonstrate that the performance of DAS with engineered fiber can potentially exceed that of conventional geophones and seismometers. In general, given that the overall sensitivity of a DAS system is a function of the coupling, cable, fiber, electronics, and digital signal processing, field data is most convincing, and, in the next section, we will discuss some examples of high definition seismic and microseismic data that demonstrate the benefits of the engineered fiber DAS solution as compared to conventional DAS and geophones.
