High-resolution multicomponent distributed acoustic sensingIvan Lim Chen Ning & Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

The usage of Distributed Acoustic Sensing (DAS) ingeophysics is attractive due to its dense spatial sam-pling and low operation cost if the optical fiber is easilyaccessible. In the borehole environment, optical fibersfor DAS are often readily available as a part of othersensing tools, such as for temperature and pressure. Al-though the DAS system promises unlimited potential forreservoir monitoring and surface seismic acquisition, thesingle axial strain measurement of DAS along the fiberis inadequate to fully characterize the different wavemodes, thus making reservoir characterization difficult.We propose a configuration using five equally spaced he-lical optical fibers and a straight optical fiber to obtainsix different strain projections. This system allows usto reconstruct all components of the 3D strain tensorat any given location along the fiber. Using the con-dition number of the matrix product of the averagingand projection matrices, we can systematically searchfor the optimum design parameters for our configura-tion. Numerical examples demonstrate the effectivenessof our proposed method to successful reconstruction ofthe six component strain tensor from wavefields of arbi-trary complexity.

INTRODUCTION

Despite the recent technological advances in DAS, mul-ticomponent DAS remains a missing piece of the puzzleto capture the full character of the wavefield. In bore-hole application specifically, the usage of DAS focusesmainly on reservoir imaging (Mestayer et al., 2011; Ma-teeva et al., 2012, 2013; Wu et al., 2015; Zhan et al.,2015; Jiang et al., 2016) and velocity model updates (Wuet al., 2015; Li et al., 2015). Although, many examplesshow that DAS has the potential to provide low-costreservoir monitoring (Hornman et al., 2015; Dou et al.,2016; Chalenski et al., 2016), the conventional singlecomponent DAS measurements makes reservoir charac-terization challenging. Moreover, a typical DAS sys-tem that employs Coherent Optical Time-Domain Re-flectometry (COTDR) provides average strain measure-ment through analyzing the phase difference betweentwo points along the optical fiber separated by a dis-tance known as gauge length.

Since DAS acquires strain along the optical fiber, themeasurement is a projection of the surrounding straintensor as a function of the optical fiber position. Usingmultiple strain projections, it is possible to reconstructthe entire strain tensor; manipulating the geometry ofthe optical fiber allows us to obtain various strain pro-jections. Lim and Sava (2016) provide a basic workflowon how multicomponent DAS can be recovered usingdual optical fibers or a single chirping (variable pitch

angle) helical optical fiber. The underlying principleof the workflow is to group consecutive strain measure-ments along optical fiber(s) within a defined windowto perform the reconstruction. The drawback of thisworkflow is the assumption that the seismic wavelengthis significantly larger than the specified window for suc-cessfully strain tensor reconstruction. A larger seismicwavelength than the window gives us a slowly varyingstrain tensor which we assume to be invariant withinthe window.

To overcome the limitation of the workflow introducedby Lim and Sava (2016), we propose a configurationwith five equally spaced constant pitch angle helical op-tical fibers and a straight optical fiber. By doing so, weobtain six different strain projections at a given locationto avoid the need to group consecutive strain measure-ments along optical fiber(s) to get sufficient strain pro-jection for the reconstruction, thus overcoming the lim-itation of the workflow of Lim and Sava (2016). Six dif-ferent strain projections at a given location can be doneby using a configuration with five equally spaced con-stant pitch angle helical optical fibers, combined with astraight optical fiber. The configuration limits the engi-neering complexity required to build a multi-fiber cable,and also allows us to systematically analyze the effectof gauge length on strain tensor reconstruction. We re-construct the strain tensor using the information of thestrain projections G associated with the geometry of theoptical fiber, and the gauge length captured in an op-erator A defined by the DAS system. Since we capturethe effect of the gauge length, we remove the averag-ing effect in the reconstruction process, which allows usto use the reconstructed strain data as multicomponentgeophone point measurements.

We demonstrate a systematic way to choose the heli-cal optical fiber design parameters (diameter and pitchangle) and the gauge length characterizing our systemby analyzing the condition number of the Gram matrixLTL where L = AG. Using the chosen parameters, wereconstruct the full strain tensor through 3D syntheticexamples of arbitrarily complex seismic wavefields (Fig-ure 3d). This paper begins by reviewing the theoreticalaspects of the proposed configuration, followed by 3Dsynthetic examples. While discussing the results, wealso examine how the configuration parameters affectthe reconstruction, by analyzing the associated condi-tion number of the Gram matrix.

THEORY

DAS measures axial strain along the optical fiber, andtherefore, it captures different projections of the sur-rounding strain tensor as a function of the location andgeometry of the optical fiber. To describe the projec-

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Figure 1: Fiber geometry with (a) five equally spaced vectors of 20◦ pitch angle and a straight vector in the middle.(b) Right pentagonal pyramid using the respective vectors in (a) and sharing the same origin. (c) The singular valuesof the corresponding Gram matrix.

tions, we use the intrinsic coordinate system of a curveas described by Pisano (1988) to express the local coor-dinate system of the optical fiber with respect to a globalcoordinate system. The tangent vector along the opticalfiber is sufficient to relate the surrounding strain tensorto the axial strain measurement. The axial strain andthe surrounding strain tensor can be related through thestrain tensor coordinate transformation relationship as(Young and Budynas, 2002)

ε̄ = RεRT , (1)

where the ε̄ and R denote the transformed strain tensorand the transformation (also known as rotation) matrixrespectively. We rearrange equation 3 as

b = Gm , (2)

where b and m are the vectorized transformed and orig-inal strain tensor respectively. The matrix G is the ex-pansion of equation 3 using the transformation matrixR and contains all the geometric information about theoptical fiber. Using equation 2, the single optical fiberDAS measurement can be expressed as

d = WAGm , (3)

where W is an operator that defines the channel spac-ing (usually equal to the gauge length) which refers tothe distance between consecutive average strain mea-surements within a gauge length, A is an operator de-scribing strain averaging within a gauge-length, and Grepresents the projection operator which depends on thegeometry of the system. At every single point along theoptical fiber, we obtain the projected axial strain. Anexample of such operation is given as

b[bi]

=G[

Gxx,i Gyy,i Gzz,i Gxy,i Gxz,i Gyz,i

] mεxx,iεyy,iεzz,iεxy,iεxz,iεyz,i

,

(4)where the index i denotes points along the optical fiber.This process is repeated for every point along the opti-cal fiber to produce the projected strain vector b. We

represent the combination operators of WAG as a lin-ear operator L, which allows us to reconstruct the straintensor m in a least-squares sense as

m = (LTL)−1LTd . (5)

To achieve accurate reconstruction using equation 5, theGram matrix LTL has to be invertible which is inferredfrom the condition number associated with its singularvalues. Therefore, it is advantageous to use the condi-tion number as an indicator of strain tensor reconstruc-tion capabilities. This also provides us with an oppor-tunity to access various optical fiber system designs.

Lim and Sava (2016) perform the strain tensor recon-struction by grouping consecutive strain measurementsalong the optical fiber within a defined window. Thegrouping of consecutive measurements allows us to ob-tain a full rank Gram matrix for the strain tensor recon-struction. Their method requires as little as one opticalfiber (a chirping helical optical fiber) to perform the re-construction. However, this approach has the drawbackthat it assumes a seismic wavelength significantly largerthan the reconstruction window. Our proposed configu-ration uses five equally spaced helical optical fibers withconstant pitch angle. A constant pitch angle is less man-ufacturing challenging and allows us to obtain measure-ments at the same portion in space along all the heli-cal optical fibers. We represent individual optical fibersin our proposed configuration with tangent vectors toconceptually visualize the associated measurements asshown in Figure 1a using a pitch angle of 20◦. Thelowest condition number is around 20◦, for which weobtain sufficient horizontal and vertical strain projec-tions to reconstruct all the strain components. The fiveequally spaced optical fibers provide enough azimuthalcoverage for accurate strain reconstruction. Figure 1bshows that by using the same origin for all the vectors,we obtain a right pentagonal pyramid. Using the projec-tion matrix G of the individual vectors, we evaluate thesingular values of GTG as shown in Figure 1c, whichindicates that our configuration is full rank, as all thesingular values are nonzero indicating that this configu-ration can reconstruct the entire strain tensor.

We analyze the effects of the gauge length A on our

Hi-res multicomponent DAS

g (m)0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

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Figure 2: Condition number as a function of gaugelength g using five helical optical fiber with diameterof 1 in and 20◦ pitch angle, and a straight optical fiber.The stars mark the condition number for gauge lengthsof 0.1 and 1.0 m.

configuration by evaluating the condition number of theGram matrix. Figure 2 illustrates the effect of gaugelength between 0.05 and 1.10 m on the condition num-ber of the Gram matrix for a cable with a diameter of1 in and a pitch angle of 20◦. We observe that there aremany local minima (low condition number) throughoutFigure 2 and the corresponding gauge lengths at thetroughs of the graph are optimal for strain tensor re-construction. For a gauge length of 1.0 m which corre-sponds to a trough, the condition number is in the orderof 104 which implies that the reconstruction is likely tobe significantly affected by noise. The oscillating char-acteristic of the condition number in Figure 2 shows thatsystematic reduction of the gauge length ensures a lowcondition number for high reconstruction accuracy.

NUMERICAL EXAMPLE

Using synthetic examples of a complex wavefield, we il-lustrate the reconstruction of the 3D strain tensor fromaxial strain measurements along the proposed opticalfiber geometry using different gauge lengths. We sim-ulate complex wavefields through triplications with a ve-locity model containing a low-velocity Gaussian anomaly,as shown in Figure 3c, using elastic finite-difference mod-eling. We use smaller than usual gauge lengths such as0.1 m which are possible using specially designed opti-cal fibers, as indicated by Farhadiroushan et al. (2016).However, we also perform the analysis using a gaugelength of 1.0 m to show the effect of a more conven-tional fiber system.

Our experiment setup with a source indicated by a dotand receivers indicated by a straight line of coordinates(xb, yb) is shown in Figure 3a. Figure 3b, shown in astrain tensor matrix layout, represents our target straintensor reconstruction observed along the receiver loca-tion at (xb, yb). The horizontal and vertical axes of theindividual panels represent the reconstructed measure-ments along the optical fiber and time respectively. Weperform the reconstruction by adding random noise with30% of the maximum amplitude of the data and in the

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Figure 3: (a) Schematic representation of a DAS ex-periment depicting the source (dot) and receiver (line)locations. (b) The ideal strain tensor that we wouldlike to reconstruct from DAS measurements. (c) TheP-wave velocity model containing a low velocity Gaus-sian anomaly designed to produce wavefield triplica-tions. The S-wave velocity is half of the P-wave velocity.(d) A snapshot of the vertical displacement wavefield.

data frequency band. Using a gauge length of 0.1 m, wecan reconstruct the strain tensor as shown in Figure 4a.The difference plot in Figure 4b shows no signal leak-age and only contain primarily random noise. Figure 4cshows the reconstruction results by increasing the gaugelength to 1.0 m. We observe stronger arrivals althoughthe results are noisy. The difference plot illustrated inFigure 4d contains noise primarily.

DISCUSSION

We demonstrate full strain tensor reconstruction witha high level of accuracy using six optical fibers (fiveequally spaced helical optical fiber with a pitch angleof 20◦ and a straight optical fiber). In our results, weshow that by using a small but achievable gauge lengthwith low condition number such as 0.1 m provides arobust strain tensor reconstruction due to the low con-dition number of the Gram matrix. A larger diameter ofthe helical configuration allows for a larger gauge lengthwhich improves the reconstruction results in noisier en-vironments. A relaxed diameter dimension is more ap-plicable for surface seismic acquisition. Using our pro-posed configuration, we can analyze the design parame-ters for the helical optical fibers systematically, as shown

Hi-res multicomponent DAS

(a) (b)

(c) (d)

Figure 4: Strain tensor reconstructed from data containing random noise with 30% of the maximum data amplitudeand band-limited to the data band with five equally spaced helical optical fibers and a straight optical fiber usinga gauge length of (a) 0.1 and (c) 1.0 m. Panels (b) and (d) are the difference between the ideal strain tensor inFigure 3b and the respective reconstructed tensor in (a) and (c).

in Figure 2. The design goal is to obtain parametersthat have the lowest possible condition number of theGram matrix consist of the averaging A and projectionmatrix G, while also satisfying engineering constraintsfor optical fiber construction. Numerous configurationof equivalent robustness and quality are possible.

CONCLUSION

We demonstrate that high resolution multicomponentdistributed acoustic sensing data is achievable by us-ing strain projections along several optical fibers to re-construct all components of the 3D strain tensor. Fiveequally spaced helical optical fibers, together with astraight optical fiber can be used for reconstruction with-out the need to group consecutive strain measurementsalong the optical fiber, within a defined window as shownby Lim and Sava (2016). We thus overcome the re-

quirement that the seismic wavelength is significantlylarger than the window and achieve high spatial resolu-tion. This method opens the possibility for an acquisi-tion of shorter seismic wavelengths, which aids imagingand reservoir characterization applications. Numericalexamples show that our method can reconstruct the full3D strain tensor for wavefields of arbitrary complexity,and in the presence of substantial noise in the band ofthe seismic data.

ACKNOWLEDGMENTS

We would like to thank the sponsors of the Center forWave Phenomena, whose support made this researchpossible and Martin Karrenbach for fruitful discussions.The reproducible numeric examples in this paper use theMadagascar open-source software package (Fomel et al.,2013) freely available from http://www.ahay.org.

Hi-res multicomponent DAS

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