other, either at the launch path of the IU or by a time‐delay loop of fiber in the receiver path of the IU. This time delay corresponds to what is called the “gauge length,” the corresponding length of fiber it takes to create (half of) this time delay. Independent of the design, the effect of this time delay is to allow the system to compare the phase of the backscattered light at all points of the sensing fiber, each separated by gauge length. Figure 2.2 shows an example where the gauge is a physical loop of fiber creating a time delay of the backscattered light at the receiver path of the IU.
Figure 2.2 shows that the IU outputs are two signals—I and Q. The I signal is the interference of the direct optical path and the long optical path through the gauge, and the Q signal is created from the quadrature of the I signal. The phase of the vector spanned by the I and Q signals provides the differential phase measurement related to the strain in the fiber. Figure 2.2, right, shows how the phase is computed from the I and Q signals using:
The arctangent function returns a value within the phase range of –π/2 to +π/2. To obtain a continuous function of phase as a function of time, the computed phase values must be unwrapped by adding or subtracting integer multiples of π to the time series (Tribolet, 1977). The relative strain ε(z, t) at each point z on the fiber at time t can be computed from this phase difference by (Equation 10, SEAFOM, 2018):
where λ is the wavelength of the laser light in vacuum used by the DAS interrogator, n is the refractive index of the glass, γ is the photoelastic scaling factor (γ ≈ 0.78) for axial strain on silica fiber (Giallorenzi et al., 1982), and G is the gauge length. Strain rate can be computed from Equation 2.2 by taking the time derivative.
2.3.2. Synchronizing Source Information and Time Stamps
Data acquisition can generally be separated into either passive or active systems. For active systems, such as VSP and surface seismic acquisition, the firing of seismic sources is under control of the operator. In passive systems, data acquisition starts, and the desired signals (e.g., earthquakes or ambient noise) are not under our control. Regardless of the application, a reference time signal is necessary; thus, it is important for a global positioning system (GPS) time to be incorporated into the acquired data sets. For active sources, additional information is necessary, such as the origin time of the source, source signature, etc. Some systems directly record this information into the DAS data set in real time, while other systems record this information separately and combine it with DAS data later by comparing the GPS timestamps from DAS and source recording systems.
2.4. ACQUISITION PARAMETER SELECTION
2.4.1. Gauge Length
For DAS data acquisition, gauge length is among the most important choices to make. Gauge length imparts a wavenumber spectrum response that acts like an array of closely spaced single sensors. A longer gauge length decreases the usable frequency spectrum of the seismic signal; however, a long gauge length provides a better SNR because of the cancellation of random noise over the length of the gauge. Actually, the SNR, with respect to ambient (white) noise, increases as the square root of the gauge length.
Gauge length imparts notches (i.e., zeros) in the wavenumber domain according to sinc(kG), where k is the wavenumber and G is the gauge length; notch locations are at k = n/G, where n is the (integer) order of the notch. Because k = 2πf/c, where f is frequency and c is the apparent wave velocity, the effect of these notches depends on both frequency and apparent velocity of each wave in the seismic data.
The solid black lines in Figure 2.3 (top) show the locations of the first notch as a function of gauge length (horizontal axis) and frequency (vertical axis). Each line is associated with a specific apparent velocity that is labeled next to it, and the corresponding −20 dB point for each apparent velocity is represented by the dotted line. The area between the notch and the −20 dB point is grayed out. Looking at a 15 m gauge length, for a 500 m/s wave, the first spectral notch is located at 33 Hz, and the −20 dB point is located at 30.7 Hz. This means that, for all practical purposes, if a 15 m gauge is used, we can only effectively record waves traveling 500 m/s below 30.7 Hz; likewise, looking at a wave traveling 1000 m/s, we can only effectively record frequencies below 61.4 Hz. For waves traveling at the speed of water (1500 m/s), we are limited to frequencies under 92 Hz; however, if a 40 m gauge is used, the high frequency limits for waves traveling 500, 1000, 1500, 2000, and 2500 m/s are 11.5, 23.0, 34.5, 46.0, and 57.5 Hz, respectively.
Figure 2.3 (Top) Location of first spectral notch for a range of gauge lengths; solid black line represents the spectral notch associated with the labeled velocity, and the dotted line represents the corresponding −20 dB point. (Bottom) Relative signal‐to‐noise improvement using different gauge lengths compared to a 5 m gauge.
Two trends are easily observed: (1) larger gauge lengths can be used if the apparent velocities of interest are large, and (2) spectral notches are not as limiting on the upper frequencies for small gauges. It might appear that the conclusion should be to always use a small gauge length; but, as previously mentioned, the SNR improves as the gauge length increases (Figure 2.3, bottom). The vertical axis shows the SNR improvement as compared to a 5 m gauge, in dB; thus, the improvement using a 15 m gauge is 4.8 dB, while the 40 m gauge provides 9 dB improvement. Therefore, the goal is to use as large a gauge length as possible without damaging the desired frequency content of the seismic waves being recorded.
For earthquake‐monitoring applications where the frequency content is typically much less than 2 Hz, the notches will not affect the recorded spectrum, and thus a long gauge is appropriate. For a Gulf of Mexico deepwater VSP, it could be that seismic energy reflecting from the reservoir has a maximum frequency content of 20 Hz because of the earth’s attenuation. Figure 2.3 (top) shows that any gauge length from 5 to 50 m can capture all seismic waves faster than 1000 m/s; thus, it might be worth the added SNR benefits to use a long gauge. However, if, instead, the application is shallow and unconsolidated formations with extremely slow P‐ and S‐waves, and a high‐frequency source is used, then an extremely short gauge should be used at the expense of SNR.
2.4.2. Sampling Rate
The IU sends a pulse of light into the fiber with a sampling rate of 10 kHz or higher. The time between these pulses are set so that
