What does a Distributed Acoustic Sensing Interrogator Measure?

A short review of how DAS interrogators derive different physical quantities from phase interference measurements.

Distributed Acoustic Sensing (DAS) interrogators use optical interferometry to measure vibrations in fibre-optic cables, but not all DAS data is the same – differences in interrogator construction mean that some systems may measure strain, strain-rate, or in the case of Terra15’s unique Treble interrogator, velocity. But what do these different measurements mean to a DAS user? Why is Terra15’s Treble interrogator different? And how can the different units for DAS data be interpreted? This article aims to answer these questions and help DAS users get the most out of their data.

How does a Typical DAS Interrogator Work?

In general terms, a DAS interrogator measures the movement of an optical fibre by transmitting many pulses of laser light into the fibre and measuring the phase of light reflected from tiny imperfections in the fibre by a process called Rayleigh backscatter. Any movement of the optical fibre changes the length of the optical path that the laser pulses travel, which results in phase shifts of the light backscattered to the interrogator. The phase shifts of reflected light are then compared to measure how much the optical fibre has stretched.

Figure 1: Typical DAS strain measurement, over gauge length LG. z is the length coordinate of the fibre.

Most DAS interrogators measure strain in the fibre, as shown in Figure 1. The optically measured interference phase Φ(z,t) is proportional to the strain, ε(z,t) = dL/LG, as below:

Where n is the refractive index of the fibre, LG is the gauge length, ξ is the photo-elastic constant, and λ is the laser wavelength in a vacuum.

The phase measurement is made by interfering light reflected from scattering sites separated by a short distance LG, called the gauge length of the system. As the fibre length expands or contracts due to acoustic signals, the phase measurement between the two scattering sites changes proportionally to the length change of the fibre section, dL. Strain in the fibre section is the ratio of the length change to the gauge length, dL/LG. By making this measurement every time a pulse is sent into the fibre, and at all points z along the fibre, the interrogator makes a continuous, distributed measurement of strain along the whole fibre, ε(z,t).

Why are Terra15 Interrogator Measurements Different?

For Terra15’s Treble and Treble+ interrogators, the phase is not measured across a gauge length, but between two pulse pairs backscattered from the same distance in the cable, where one pulse is delayed by a fixed delay T relative to the other (Figure 2). In other words, the Treble natively measures the rate of change of the entire fibre length from the interrogator to the backscatter point. This is a measurement of the fibre velocity, in (m/s), at the interrogated point. This is unique compared to other interrogator designs, which natively measure strain at the interrogated part of the fibre.

The measured optical interference phase for the Treble is proportional to the velocity of the fibre at position z according to the formula:

Where n is the refractive index of the fibre, T is the time between pulse pairs, ξ is the photo-elastic constant, and λ is the laser wavelength in a vacuum.

Figure 2: Treble velocity measurement

The difference in the native measurement means that raw data from the Treble needs to be interpreted and processed differently to other systems. This can involve either converting Treble data to strain-rate, processing the native velocity data, or converting to various other units.

Converting Treble Data to Strain-Rate

The Treble measurement of velocity along the fibre axis, v(z,t), is the spatial integral of strain-rate ∂ε(z,t)/∂t along the fibre axis.

Conversely, strain-rate is the spatial derivative of fibre velocity. This means that strain-rate can be calculated from Treble velocity data by performing a numerical derivative across a gauge length LG, as in Figure 3. Unlike most native strain interrogators, the gauge length for the Treble strain-rate conversion can be varied in post-processing. The resulting data has units of strain/s.

Figure 3: Treble velocity to strain-rate conversion

Effect of Pulse Length and Gauge Length

Selecting the correct pulse length and gauge length is critical to making high-quality velocity, strain-rate, or strain measurements with DAS. Because the velocity measurement of the Terra15 Treble is recorded without a gauge length, pulse length and gauge length can be selected independently. Both parameters have a significant impact on the sensitivity and spatial resolution of the Treble measurement.

Figure 4: Spatially narrow velocity signal broadened by pulse length.

A pulse length must be selected for all Treble measurements, either velocity or strain-rate. A longer pulse length means more optical power is sent into the fibre, reducing the noise floor of the measurement making the Treble more sensitive to quieter signals. On the other hand, a longer pulse length has the effect of broadening the position of signals smaller than the pulse length, as shown in Figure 4. Hence, the choice of pulse length is a compromise between better sensitivity and the ability to resolve signals at scales shorter than the pulse length. For this reason, the pulse length is often referred to as the spatial resolution of the measurement. Pulse length is typically set equal to half the shortest acoustic wavelength of the measured signal, since this choice maximises the signal to noise of the acoustic wavelengths of interest.

Figure 5: Measurement of a spatially localised strain, and dependence on gauge length (LG) for the conversion of Treble data from velocity to strain-rate. LG = Pulse Length is recommended. True signal in red; measured in blue.

For Treble measurements, the gauge length LG is selected for the conversion from velocity to strain-rate. Assuming that the pulse width is chosen such that it does not compromise the desired spatial frequency content of the measurement, the best signal-to-noise ratio is achieved by setting LG equal to the pulse length. Selecting LG > pulse length can improve the signal-to-noise ratio for long acoustic wavelength signals, while selecting LG < pulse length reduces signal levels and is not recommended.

Effect of Imbalanced Strain-Rate on Velocity

As previously described, the velocity measured by the Treble is the spatial integral of strain-rate along the fibre axis. The integration from strain-rate to velocity for a localized strain-rate signal in the fibre is illustrated in Figure 6. When the fibre is well-coupled to the surrounding medium, a disturbance which stretches one part of the fibre will be accompanied by an equal compression of the adjacent fibre. When this ‘balanced’ strain-rate is integrated to velocity, the adjacent positive and negative components will cancel out, and the resulting velocity measurement is a single peak at the location of the disturbance. This type of coupling and signal are commonly seen when fibre is cemented in place in a borehole or underground mine

Figure 6: Integration from strain-rate to velocity for a localised strain-rate signal measured in a fibre well-coupled to the surrounding medium.

In other cases, the strain-rate signal measured from a disturbance may be imbalanced – each stretch in the fibre may not be accompanied by an equal compression. Integrating the imbalanced strain-rate will cause a non-zero velocity to be measured at all locations along the fibre beyond the disturbance (Figure 7). Imbalanced strain-rate measurements often originate when the fibre is poorly coupled to the surrounding medium in one or more locations. At these locations the fibre may slip relative to the medium and measure an uneven stretch or compression, which is integrated to a non-zero velocity at more distant points along the fibre. This effect manifests as offsets in the raw velocity data originating at the locations of large strain-rate signals which propagate along the rest of the entire fibre length in streaks.

Figure 7: Illustration of an imbalanced strain-rate resulting in a velocity signal beyond the disturbance in the fibre.

Note that if it is desired to work with strain-rate data, a simple application of a gauge calculation removes the “streaking” effect in unbalanced raw velocity data. As a result, streaking does not occur in strain-rate data.

Using Local Velocity to Process Raw Velocity Data Containing Cable Slips

Since it is often useful to work directly with Treble data in its native units of velocity (m/s), a signal processing method of removing the streaking artifacts in raw velocity is provided. This allows for instance Treble data to be compared to particle velocity measurements made by other seismic instruments, such as geophones; and importantly for signal to noise ratio benefit. This signal processing method is implemented in a specific datatype which referred to as “local velocity”, which applies a spatial filter to remove the streaking effects in the data. An example of signal and streaking in raw velocity and signal only after processing to local velocity is shown in Figure 8.

Figure 8: Filtering raw Treble velocity data to local velocity

The local velocity filter is implemented by filtering the raw velocity in the spatial domain, and can be thought of as a form of DC shift removal. We use the approach of subtracting the moving average of the unfiltered velocity about the measurement location. This difference made clear by plotting the filter kernel as two separate components as shown in Figure 9.

Figure 9: The local velocity filter kernel consists of the difference between a delta and an exponentially weighted moving average. Position is normalised to the filter length scale (velocity_filter_length).

Due to the average removal, this filter only retains “local” signals, with a length scale proportional to the filter length. Mathematically, the filter has the following Fourier representation in k-space:

where k0 = 2π/velocity_filter_length.

The local velocity filter has the disadvantage of removing signal with long acoustic wavelengths, and care must be taken in the design of the appropriate length scale or cut-off wavenumber to ensure no spatial frequencies of interest are removed. It is zero-phase, and so does not influence the location of the signal.

Converting Treble Data to Other Units

For the purposes of further data interpretation, or comparison to other measurement systems, it is sometimes necessary to convert Treble measurements to other units. For example, to compare data with a pilot signal from a seismic source recorded by an accelerometer, DAS data should be converted to acceleration before trying to deconvolve (or cross-correlate) with the pilot signal. In addition to strain-rate and local velocity, the Treble native measurement of velocity can be used as a basis for obtaining several other commonly used units: strain, displacement or acceleration, as shown in Figure 9.

Figure 10: Relationship between native Treble velocity data and various other measurement units