Seismic source mechanism imaging

Ivan Lim Chen Ning & Paul SavaCenter for Wave Phenomena, Colorado School of Mines

ABSTRACT

Wavefield-based solutions are robust and facilitate straightforward seismic source pa-rameter estimation. The potential to access multicomponent stress and particle dis-placement data through new technology such as distributed acoustic sensing (DAS)allows us to formulate the source mechanism imaging problem using full anisotropicelastic wavefields. We propose a method that enables us to obtain the source locationand mechanism simultaneously, without the need for lengthy iterative inversion. Ourtechnique requires knowledge of the medium model parameters (also a requirement forany other wavefield-based method) for reverse-time wavefield extrapolation, and thesource wavelet (which can be reasonably well estimated) for the imaging condition.The source location and mechanism are inherently embedded in our imaging process.For high-resolution imaging, least-squares solutions diminish artifacts due to limitedreceiver coverage and illumination. We test our approach on multiple 3D numerical ex-periments using various combinations of source mechanisms, in order to demonstrateits capability to recover the source parameters.

Key words: moment tensor, seismic imaging

1 INTRODUCTION

Seismic source imaging is widely used in seismology, espe-cially for earthquake and microseismic monitoring and char-acterization. The source parameters such as location, excita-tion time, and focal mechanism are of great interest as theyprovide information about reservoirs subject to hydraulic frac-turing (Baig and Urbancic, 2010). The source information alsoallows us to quantify the extent (width and distance) of the in-duced fractures. Through the source mechanism, we can in-fer the fracture orientation that enables strategic decisions forborehole placement to improve reservoir production (Maxwellet al., 2010).

The source parameters can be obtained through wave-form tomography for simultaneous multi-source parameter in-version based on the amplitude and phase information of therecorded data. Waveform tomography utilizes adjoint wave-fields simulated by reverse time extrapolation (Tromp et al.,2005; Kim et al., 2011; Jarillo Michel and Tsvankin, 2014,2015, 2017). Alternatively, one can use reverse time imag-ing for source location (Artman et al., 2010; Li et al., 2017),or can adopt optimization-based source imaging for reducedtruncation artifacts and higher resolution (Fukahata et al.,2013; Bazargani and Snieder, 2015; Nakahara and Haney,2015). Chambers et al. (2014) investigate the use of diffrac-tion stacking to obtain the source mechanism through imag-

ing. Kawakatsu and Montagner (2008) as well as Gharti et al.(2011) outline the theory of reverse time imaging to providean approximate estimate of the seismic source mechanismby conventional amplitude-based inversions, i.e., using theGreens functions. Similarly, Montagner et al. (2012) demon-strate the use of reverse time wavefields to retrieve the sourcemechanism (moment tensor) from long period waves.

In this paper, we propose to use reverse-time wavefieldextrapolation to obtain the seismic source mechanism throughimaging without the need for lengthy iterative inversion. Ourmethod requires both stress and particle displacement data inorder to reconstruct accurately subsurface wavefields in het-erogeneous and anisotropic media. Although stress data arenot commonly acquired, we frame our method in the contextof new acquisition technologies such as distributed acousticsensing (DAS) that measures strain along optical fibers. Wecan obtain multicomponent DAS strain measurements usingnovel optical fiber configurations (Lim Chen Ning and Sava,2016, 2018, 2017). Assuming knowledge of the Earth’s mate-rial properties surrounding the optical fiber, we can computestress from strain via conventional constitutive relations (Akiand Richards, 2002). Without loss of generality, we can for-mulate our imaging approach with either stress or particle dis-placement data as special cases. The source image containsinformation about its source location and mechanism.

We demonstrate the proposed imaging process using

2 I. Lim Chen Ning & P. Sava

3D numerical simulations that mimic passive seismic experi-ments. We test our algorithm with multiple source mechanismconsisting of moment tensor or body force sources. The testedacquisition geometry includes both spherical receiver distribu-tions for full aperture analysis, as well as surface receivers toemulate practical microseismic observation arrays.

2 THEORY

We describe elastic-wave propagation using the second-orderpartial differential equations consisting of both stress tu (x, t)tensor and particle displacement uu (x, t) vector wavefields(Aki and Richards, 2002)

ρ∂2u

∂t2= ∇ · t + f , (1)

where x and t denote space and time, respectively. ρ (x) isthe density and f (x, t) is the external volume force. Under theassumption of linear elasticity the stress t (x, t) is related tostrain e (x, t) through the constitutive relation

t = c e + m , (2)

where m (x, t) is the seismic moment tensor source acting asstress perturbation. The relation between the strain e (x, t) ten-sor and particle displacement u (x, t) is

e =1

2{∇u + (∇u)ᵀ}

= Hu ,(3)

where the operator H captures the geometric relation betweenstrain and particle displacement. We express the seismic mo-ment tensor source as

m = sm M , (4)

where M (x) is a unitless matrix that determines the magni-tude of the moment tensor source. sm (t) denotes the momenttensor source wavelet with units of stress (force per unit area).Similarly, we express the body force vector as

f = sf F , (5)

where F (x) is a unitless vector that determines the magni-tude of the body force source. sf (t) represents the body forcesource wavelet with units of force density (force per unit vol-ume). We can interpret M (x) and F (x) as images that showthe source location and its associated mechanism. Followingthe notations in equations 4 and 5, we can define the momenttensor source term in matrix form as

m0

m1

m2

...

=

sm0

sm1

sm2

...

[

M]

(6)

and the corresponding body force source term asf0

f1

f2...

=

sf0

sf1

sf2

...

[

F]. (7)

The images M (x) and F (x) scale the respective sourcewavelets to generate the external sources m (x, t) and f (x, t)for wavefield extrapolation. We solve equations 1 and 2 usingthe finite-difference representation of the time-derivative lead-ing to the recursive relations for stress

t = cHu + M sm (8)

and particle displacement

u+ = 2u +∆t2

ρ

(∇ · t + F sf

)− u− . (9)

These recursive equations allow us to compute the subsequenttime wavefield u+ from wavefields at the current time u andat preceding time u−. In matrix form, we express the recurrentequations 8 and 9 as

I −cH I

− (∇ · ) I

− 2ρ

∆t2ρ

∆t20 . . .

ρ

∆t2− 2ρ

∆t2ρ

∆t2. . .

0ρ

∆t2− 2ρ

∆t2. . .

......

.... . .

t0t1t2...u0

u1

u2

...

=

m0

m1

m2

...f0f1f2...

(10)

where m and f are the input wavefields, i.e., the source, and tand u are propagating wavefields. We solve for t and u fromthe top, i.e., forward in time, using the moment tensor and thebody force sources. The forward operator takes a wavefieldd = [m, f ] and generates a wavefield w = [ t,u ] by propa-gating forward in time:

Lw = d . (11)

The adjoint operator takes the wavefield w = [ t,u ] and gen-erates the adjoint wavefield d∗ = [m∗, f∗ ] by propagatingbackward in time:

w = L†d∗ . (12)

Seismic source mechanism imaging 3

In matrix form, we can write equation 12 as

I − (∇ · )† I

−H† c I

− 2ρ

∆t2ρ

∆t20 . . .

ρ

∆t2− 2ρ

∆t2ρ

∆t2. . .

0ρ

∆t2− 2ρ

∆t2. . .

......

.... . .

m∗0m∗1m∗2

...f∗0f∗1f∗2...

=

t0t1t2...u0

u1

u2

...

.

(13)We solve for m∗ and f∗ from the bottom, i.e., backward intime, using the sources t and u. The time reversal iterativeequation for the adjoint strain is

m∗ = (∇ · )† f∗ + t (14)

and the adjoint particle displacement is

f∗− = 2f∗ +∆t2

ρ

(H† cm∗ + u

)− f∗+ , (15)

where m∗ denotes the adjoint strain and f∗ is the adjoint par-ticle displacement. Finally, we obtain the respective adjointsource mechanism image using the adjoint fields as zero lagcrosscorrelation with the moment tensor source wavelet

[M∗]

=[

sm0 sm

1 sm2 . . .

]m∗0

m∗1

m∗2...

(16)

and the body force source wavelet

[F∗]

=[

sf0 sf

1 sf2 . . .

]f∗0

f∗1

f∗2...

. (17)

We represent the forward cascading operations in equations 6,7 and 10 through forward operator

Gn = G†w (18)

and the equivalent adjoint cascading operations in equa-tions 13, 16 and 17 by the adjoint operator

Gn = G†w . (19)

The forward operator G generates data w comprising of stresst and particle displacement u from source mechanism image nof moment tensor M and body force F sources. The adjoint op-erator G† forms an image representing the source mechanismand is analogous to the gradient for the centroid-moment ten-sor inversion method (Kim et al., 2011). Since we formulateboth the forward (equation 18) and adjoint (equation 19) oper-ations, we can perform least-squares imaging to obtain high-

resolution source mechanism images as

n =(G†G

)−1

G†w . (20)

We can interpret G†G as a point spread function (PSF) thatblurs (via convolution) the image n. Least-squares imagingdeconvolves the PSF to generate high-resolution images.

3 NUMERICAL EXAMPLES

To analyze the theoretical performance of our proposedmethod, we perform a passive seismic numerical simulationwith full aperture geometry. This ideal coverage enables us todiscount the possibility of truncation artifacts through inade-quate wavefield sampling to impair our analysis. We also showanalysis of our technique with surface seismic acquisition todemonstrate the feasibility of our technique in a realistic sce-nario. The following examples cover reverse-time migrationsand least-squares solutions. We present the source mechanismimages using the tensor-vector matrix layout

Mxx Mxy Mxz

Fx Myy Myz

Fy Fz Mzz

,

which combines the stress tensor (in red) in its natural matrixrepresentation together with the particle displacement vectorwavefield in a combined 3× 3 plot.

We set up our perfect aperture experiment using a con-stant elastic model with the source (in red) and spherical re-ceiver (in black) geometry in Figure 1(a), emulating a pas-sive seismic example. Note that the receivers are nearly evenlydistributed on the sphere using a Fibonacci lattice (Gonzalez,2010). Table 1 represents the six different combinations ofsource mechanisms for analysis using the same acquisition ge-ometry. Table 1 shows a summary of the figures and the cor-responding source mechanisms. The image consists of threedifferent slices of the 3D cube along the principal axes cen-tered at the source location.

The first experiment in Table 1 resembles a double-couple earthquake source in the xz-plane. Figure 2 showsnine panels that populate the moment tensor matrix(M xx,M yy,M zz,M yz,M xz,M xy) and the body force vec-tor (F x, F y, F z). We observe components of source imagethat are symmetric in some axes, i.e., the image panel of F x

along the vertical axis or the image panel of F z along the hor-izontal axis. However, of particular interest is the image panelof M xz that is symmetric along both major axes which co-incide with our exact source mechanism at M xz = −1 asthe only nonzero component. The negative blue dot in panel

4 I. Lim Chen Ning & P. Sava

Experiment M xx M yy M zz M yz Mxz Mxy Fx F y F z

1 0 0 0 0 0 -1 0 0 0

2 1 1 1 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 1

Table 1. Summary of all the 3D source mechanism experiment.

M xz shows the source location. Table 2 shows a summary ofthe image symmetry with the corresponding elements of thesource mechanism. We obtain higher resolution source mech-anism images through the least-squares formulation in equa-tion 20. Comparing Figure 2 to Figure 3, the least-squares im-ages are sharper and provide a more compact representationof the source image due to the removal of the PSF blurringeffect. For the remaining examples, we focus on the sourceimages from least-squares.

We demonstrate the ability of our method to identifysource mechanism with multiple nonzero components such asan explosion. The source images in Figure 4 that display sym-metry are on the main diagonal of the moment tensor imagesthat indicates an explosive source. The source images consistof stacked spheres along the components associated with theprincipal axis, i.e., M xx, M yy , and M zz along x-, y-, and z-axis respectively. Besides that, we perform least-squares imag-ing on body force source such as a seismic vibrator source. InFigure 5, we use a vertical body force as our source mecha-nism. The only panel that show symmetry is F z which showsthe capability of our method to identify both moment tensorand body force sources. This is important for scenarios wherethe recorded data is insufficient to describe the moment tensorsource, we can describe them through its body force equiva-lent.

We repeat surface receivers (Figure 1(b)) to demonstrateour method with using surface monitoring methods to test therobustness against wavefield truncation artifacts. The limitedaperture which translate to inadequate sampling of the wave-field may lead to inaccurate identification of the source mech-anism from the artifacts. However, we demonstrate the robust-ness of our method through least-squares to minimize artifacts.For comparison with the full aperture scenario, we use thesame double-couple earthquake source example in Figure 2but acquired using surface receivers. The least-squares solu-tion in Figure 6 shows that only panel M xz describes the sym-metry for source mechanism identification, although a reduc-tion in compactness of the source image due to the limitedaperture of a single-sided illumination from the top.

4 DISCUSSION

We demonstrate the ability to locate and identify the seismicsource mechanism accurately. Since our formulation is derivedfrom the elastic anisotropic wave equation, our method canbe applied to media of arbitrary anisotropy and heterogene-ity given the media parameters are known (also a requirement

Experiment M xx M yy M zz M yz Mxz Mxy Fx F y F z

symmetry no no no no yes no no no no

values 0 0 0 0 1 0 0 0 0

Table 2. xz-plane double-couple source mechanism and image sym-metry comparison.

for any other wavefield-based method). The source mechanismimaging method provides a fast and robust way to obtain thesource parameters without the need for lengthy iterative in-version. However, to reduce aperture and source wavelet re-lated artifacts, we are required to perform least-squares imag-ing that would increase the computational cost. Our imagingmethod and conventional inversion-based methods are relatedin that the source images form the gradient for waveform in-versions. The imaging condition in equation 16 is compara-ble to the moment tensor gradient defined from the Frechetderivatives by Kim et al. (2011) in their centroid-moment ten-sor inversion method. Besides the moment tensor image, ourmethod also provides the body force image which is essen-tial when the observed data are not capable to not fully de-scribe the moment tensor source. A natural extension of ourapproach is to use all of the source images (M and F) to per-form waveform-based inversion to obtain medium parameters.The least-squares imaging provides high-resolution source im-ages, and could accelerate waveform inversions.

5 CONCLUSIONS

We demonstrate a wavefield-based imaging method that canrobustly and accurately locate seismic sources and identify itsmechanism. Our technique requires both stress and particledisplacement data, but can be formulated using either stressor particle displacement as special cases. This methodologyleverages strain and displacement acquisition using novel mul-ticomponent DAS acquisition. Our source images naturallyidentify the source mechanism as they are individual imagesfor the respective moment tensor and body force components.The ability to obtain the source location and mechanism with-out the need for lengthy iterative processes could potentiallyallow practitioners to make timely decisions about the reser-voirs for immediate business impact. Our method using theleast-squares solution facilitates fast and accurate waveforminversion for source and medium parameters. The method-ology presented in this paper applies to media of arbitraryanisotropy and heterogeneity.

6 ACKNOWLEDGMENTS

We would like to thank the sponsors of the Center for WavePhenomena, whose support made this research possible andOscar Jarillo Michel for fruitful discussions. The reproducible

Seismic source mechanism imaging 5

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Figure 1. 3D microseismic experiment setup with receivers providing (a) ideal spherical coverage and (b) surface seismic acquisition. The red dotrepresents the source location.

numeric examples in this paper use the Madagascar open-source software package (Fomel et al., 2013) freely availablefrom http://www.ahay.org.

REFERENCES

Aki, K., and P. G. Richards, 2002, Quantitative seismology.Artman, B., I. Podladtchikov, and B. Witten, 2010, Source

location using time-reverse imaging: Geophysical Prospect-ing, 58, 861–873.

Baig, A., and T. Urbancic, 2010, Microseismic moment ten-sors: A path to understanding frac growth: The LeadingEdge, 29, 320–324.

Bazargani, F., and R. Snieder, 2015, Optimal source imag-ing in elastic media: Geophysical Journal International, 204,1134–1147.

Chambers, K., B. D. Dando, G. A. Jones, R. Velasco, andS. A. Wilson, 2014, Moment tensor migration imaging:Geophysical Prospecting, 62, 879–896.

Fomel, S., P. Sava, I. Vlad, Y. Liu, and V. Bashkardin, 2013,Madagascar: open-source software project for multidimen-sional data analysis and reproducible computational experi-ments: Journal of Open Research Software, 1.

Fukahata, Y., Y. Yagi, and L. Rivera, 2013, Theoretical re-lationship between back-projection imaging and classicallinear inverse solutions: Geophysical Journal International,196, 552–559.

Gharti, H. N., V. Oye, D. Kuhn, P. Zha, et al., 2011, Simulta-neous microearthquake location and moment-tensor estima-

tion using time-reversal imaging: 81st Annual InternationalMeeting, SEG, Expanded Abstracts, 1632–1637.

Gonzalez, A., 2010, Measurement of areas on a sphere us-ing Fibonacci and latitude–longitude lattices: MathematicalGeosciences, 42, 49.

Jarillo Michel, O., and I. Tsvankin, 2014, Gradient calcula-tion for waveform inversion of microseismic data in VTImedia: Journal of Seismic Exploration, 23, 201–207.

——–, 2015, Estimation of microseismic source parametersby 2D anisotropic waveform inversion: Journal of SeismicExploration, 24, no. 5, 379–400.

——–, 2017, Waveform inversion for microseismic velocityanalysis and event location in VTI media: Geophysics, 82,WA95–WA103.

Kawakatsu, H., and J.-P. Montagner, 2008, Time-reversalseismic-source imaging and moment-tensor inversion: Geo-physical Journal International, 175, 686–688.

Kim, Y., Q. Liu, and J. Tromp, 2011, Adjoint centroid-moment tensor inversions: Geophysical Journal Interna-tional, 186, 264–278.

Li, L., I. Abakumov, H. Chen, X. Wang, D. Gajewski, et al.,2017, Comparison and generalization of correlation-basedseismic source imaging methods: 87th Annual InternationalMeeting, SEG, Expanded Abstracts, 2913–2917.

Lim Chen Ning, I., and P. Sava, 2016, Multicomponent dis-tributed acoustic sensing: 86th Annual International Meet-ing, SEG, Expanded Abstracts, 5597–5602.

——–, 2017, High-resolution multicomponent distributedacoustic sensing: 87th Annual International Meeting, SEG,Expanded Abstracts, 941–946.

——–, 2018, Multicomponent distributed acoustic sensing:

6 I. Lim Chen Ning & P. Sava

Figure 2. Source mechanism reverse time migration using the experiment setup in Figure 1(a) with seismic moment tensor M =

{0,0,0,0,0,−1}.

Concept and theory: Geophysics, 83, P1–P8.Maxwell, S. C., J. Rutledge, R. Jones, and M. Fehler, 2010,

Petroleum reservoir characterization using downhole micro-seismic monitoring: Geophysics, 75, 75A129–75A137.

Montagner, J.-P., C. Larmat, Y. Capdeville, M. Fink, H.Phung, B. Romanowicz, E. Clevede, and H. Kawakatsu,2012, Time-reversal method and cross-correlation tech-niques by normal mode theory: a three-point problem: Geo-physical Journal International, 191, 637–652.

Nakahara, H., and M. M. Haney, 2015, Point spread functionsfor earthquake source imaging: An interpretation based on

seismic interferometry: Geophysical Journal International,202, 54–61.

Tromp, J., C. Tape, and Q. Liu, 2005, Seismic tomography,adjoint methods, time reversal and banana-doughnut ker-nels: Geophysical Journal International, 160, 195–216.

Seismic source mechanism imaging 7

Figure 3. Source mechanism Least-squares reverse time migration corresponding to the images in Figure 2 with seismic moment tensor M =

{0,0,0,0,0,−1}.

8 I. Lim Chen Ning & P. Sava

Figure 4. Source mechanism Least-squares reverse time migration for seismic moment tensor M = {1,1,1,0,0,0}.

Seismic source mechanism imaging 9

Figure 5. Source mechanism Least-squares reverse time migration for body force vector F = {0,0,1}.

10 I. Lim Chen Ning & P. Sava

Figure 6. Source mechanism Least-squares reverse time migration using the surface seismic acquisition geometry in Figure 1(b) for seismicmoment tensor M = {0,0,0,0,0,−1}.