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1 Computation of qp- and qs-wave Rays raveltimes Slowness Vector and Polarization in General Anisotropic Media Junwei Huang 1 * Juan M. Reyes-Montes 1 and R. Paul Young 2 1 Applied Seismology Consultants Ltd 5 Claremont Buildings Claremont Bank Shrewsbury SY1 1RJ UK 2 University of oronto 170 College Street oronto Ontario M5S 3E3 Canada *junwei@appliedseismology.com Summary Based on Lax-Friedrichs scheme we developed a fast sweeping method of calculating the direct traveltime field for both quasi-p- and quasi-s-wave in general anisotropic media. he ray vector and the polarization can be obtained from the slowness vector at any given points in the model and the seismic ray path can be traced following the opposite direction of ray vectors. We illustrate the efficiency and accuracy of this algorithm using 2-D and 3-D numerical examples. Introduction A variety of mechanisms can cause seismic anisotropy such as alignments of mineral grains preferential orientation of fractures sequences of fine lithological layers and non-hydrostatic stress. In particular strong layering in shale deposits causes a non-negligible deviation from homogeneous isotropic velocity models when processing seismic data from this type of lithology. Accounting anisotropy in both active and passive seismic data processing has thus become a routine and critical step. he high frequency asymptotics for wave propagation in elastic solids reduces the isotropic seismic wave equation to a type of static Hamilton-Jacobi equation the so-called eikonal equation simply expressed as (1) where the operator takes the gradient of the traveltime field. calculates its norm and c is the propagation speed. In a general anisotropic medium however there is no simple explicit expression for the traveltime field and approximations such as weak anisotropy (homsen 1986) and first-order perturbation (Jech and Pšenčik 1989) must be incorporated to simplify the problem. In addition although it is convex in terms of slowness for qp-wave the anisotropic eikonal equation for one of the qs-waves is non-convex and conventional methods (e.g. sai et al. 2003; Sethian and Vladimirsky 2003; Mensch and Farra 1999) can only be used to solve the traveltime for qp-wave. Kao et al. (2004) presented the first fast numerical method based on discretizing the Hamilton-Jacobi equation directly without assuming convexity of the Hamiltonian and thus can be used to solve both qp- and qs-wave traveltime. In addition unlike seismic ray tracing Lax-Friedrichs scheme based fast sweeping method only needs the eigenvalues (Musgrave 1970) to update the traveltime field iteratively. In the following sections after showing that the analytical expressions of eikonal equations from Christoffel matrix are available only for isotropic and transversely isotropic medium we provide the workflow of solving the general anisotropic eikonal equation using Lax-Friedrichs sweeping scheme. We demonstrate the accuracy and convergence rate of our method on a 2-D isotropic medium and compare the three vectors: ray slowness and polarization in a 3-D homogeneous triclinic medium. Analytical Eikonal Equations he general anisotropic medium is fully characterized by 21 independent elastic parameters. Following the Voigt recipe the four-order density normalized elastic tensor can be expressed by a 6x6 symmetric matrix: A11 A12 A13 A14 A15 A16 A12 A22 A23 A24 A25 A 26. (2) A13 A23 A33 A34 A35 A36 A A14 A24 A34 A44 A45 A46 A15 A25 A35 A45 A55 A 56 A16 A26 A36 A46 A56 A66 he element of the Christoffel matrix Γ 3x3 are given by where p 1 p 2 p 3 are the three component of the slowness vector p i.e. the direction of the phase velocity and normal

2 Computation of Rays raveltimes Slowness and Polarization in General Anisotropic Medium to the wavefront. he matrix has three eigenvalues G m (xp) and eigenvectors g m (xp) with m=1 for qp-wave and m=23 for two qs-waves. If the slowness vector is known at a given point x in the model the ray vector r = [r 1 r 2 r 3 ] generally not in parallel with the slowness vector can be calculated as (Červenỳ 2001). (3) Using the element values of isotropic and transversely isotropic medium A33 A33 2A44 A33 2A A33 2A44 A33 A33 2A (4) A33 2A44 A33 2A44 A A A A A44 A11 A11 2A66 A A11 2A66 A11 A (5) A13 A13 A A A A A66 in Christoffel matrix Γ 3x3 we obtain the following eigenvalues and eigenvectors respectively: P P S S 0 G A p p p g p p p S G A p p p g p 0 p (6a) G A p p p g p p S L N GqP M N gqp p1 p3 p2 p A A L N Gq S1 M N gq S1 p1 p3 p2 p A A G A p A p p g p p qs qs 2 where L ( A33 A44 ) p3 ( A11 A44 ) p1 ( A11 A44 ) p2 M ( A A ) p ( A A ) p ( A A ) p N A A A A p A A A A p A 2A A A p p A 2A A A p (6b) A11 4A13 A33 A44 A11 A33 2A13 A44 p p 3. A11 4A13 A33 A44 A11 A33 2A13 A44 p2 In the isotropic case with letting the square root of the eigenvalue in (6a) equal to one we obtain eq. (1) with c=v p for P-wave and c=v s for S-wave. he ray direction is identical to the slowness direction and the polarization of P-wave is orthogonal to the polarization of S-wave. It is easy to verify that using the analytical expressions in (6b) the angle between the ray vector slowness vector and the polarization for qp-wave is nonzero and form an orthogonal coordinate. We will quantify the angle difference in the numerical example section. Obtaining analytical expressions for other less symmetrical anisotropy (e.g. orthorhombic) becomes significantly challenging. he Lax-Friedrichs sweeping scheme only requires the value of and thus we can numerically solve the general anisotropic eikonal equation (7) with a point source at x 0 and. Lax-Friedrichs Sweeping Algorithm Kao et al. (2004) gave a workflow of solving general static Hamilton-Jacobi equations in 2-D medium. Here we provide a workflow specifically tailored for eq. (7) in 3-D. 1. Initialization. We assign zero value to (x 0 ) at the source grid point and a large positive value to other grid points where the value should be larger than the possible largest arrival time. 2. Alternating Sweeping. At each iteration we perform eight alternating sweeps covering all the inner grid points whereas the grid points at the edge will be calculated in step 3. Each sweep is independent and can be easily parallelized on a shared memory machine. 3. Enforcing computational boundary conditions. After each sweep we enforce computational boundary condition. In our case for a point source inside the model there must be only outflow energy at the computational boundary. 4. Convergence check. Given the convergence criterion ε > 0 we check if is satisfied where n is the iteration number. Otherwise we return to step 2. At step 2 the following formulae are used to update the traveltime at grid point (ijk): i j k i 1 j k i 1 j k i j1 k i j1 k i j k 1 i j k 1 1 Gm x (8a) x y z i 1 j k i 1 j k i j1 k i j1 k i j k 1 i j k 1 x 2x y 2y z 2z n 1 n min (8b) i j k i j k i j k where 1 x y z and σ x σ y σ z are artificial viscosities satisfying x y z. (8c) According to our experience σ x = σ y = σ z = 5.0 is adequate for qp-wave and σ x = σ y = σ z = 3.0 for qs-wave. o enforce the outflow computational boundary condition in 3-D we first update the grid points on boundary planes

3 Computation of Rays raveltimes Slowness and Polarization in General Anisotropic Medium using the inner grid points and then the grid points on the edges using grid points inside the planes. For example assuming N x N y N z are the grid points along X Y and Z axes i=0 1 N x -1 j=0 1 N y -1 and k=0 1 N z -1 on the bottom XY boundary plane (k=0) we impose the following conditions: min max 2 (9a) i j0 i j1 i j2 i j2 i j0 min max 2 0 j0 1 j0 2 j0 2 j0 0 j0 min max 2 N x 1 j0 N x 2 j0 N x 3 j0 N x 2 j0 N x 1 j0 min max 2 i00 i10 i20 i20 i00 i N y 10 i N y 20 i N y 30 i N y 30 i N y 10 min max 2 (9b) (9c) Eq. (9a) ensures the energy at points on XY plane is from inside of the model and eq. (9b) and (9c) ensure the energy at the edge points is from inside of the XY plane. Equations (9) are trivially modified for computational boundary conditions on other planes. he details can be found in Kao et al. (2004). o evaluate its numerical efficiency and stability we apply the algorithm to a large 3-D model with strong triclinic anisotropy. o increase the anisotropy the elastic stiffness tensor is modified from a weak triclinic medium (Mah and Schmitt 2003) (10) A where only the upper part of the symmetric matrix is shown and the plus/minus signs mark the modification of the original weak triclinic elastic tensor. he density is assumed to be 1000 kg/m 3 for simplicity. Numerical Examples We first apply the algorithm to eq. (1) with c=1. A single point source is located at the center (x 0 y 0 ) = ( ) of the 2-D modeling region 1.0 x 1.0. he exact solution can be used to evaluate the convergence order and accuracy of the method. able 1 shows the apparent convergence order and L error as the mesh size approaches zero. able 1: he 2-D isotropic eikonal equation case: the error decreases with mesh size linearly (first-order). he convergence order p is calculated from and. he mesh size in b is selected in all other cases as a trade-off between computation time and accuracy. Mesh size L -error (e) Convergence Order (p) 1/ / / / / / able 2: he numerical efficiency and stability of solving the traveltime field for both qp- and qs-wave in a strong triclinic medium. Wave σ x=σ y=σ z CPU time (sec) Iteration No. qp qs-fast qs-slow Figure 1: he isosurfaces of the traveltime field for both qp- and qs-wave in a 3-D anisotropic medium. Significant elongation of the wavefront and shear wave splitting are consistent with the strong elastic stiffness assigned to the elastic tensor (see eq. 10). Note that the isosurface values for qp- and qs-waves are different for display purpose. able 3: he numerical efficiency and stability of solving the traveltime field for both qp- and qs-wave in a weak triclinic medium. Wave σ x=σ y=σ z CPU time (sec) Iteration No. qp qs-fast qs-slow able 2 presents the CPU time and number of iterations for solving the traveltime on 8 million grid points (200 x 200 x 200) for qp qs-fast and qs-slow waves on a single core. he traveltime will converge after several iterations and the isosurface of the traveltime field are shown in Figure 1. We found that increasing artificial viscosity values would increase the number of iterations and a trade-off should be found between a small viscosity value and the stability of the algorithm. In this study we use transversely isotropic

4 Computation of Rays raveltimes Slowness and Polarization in General Anisotropic Medium reference medium to derive the first-order analytical expression of anisotropic eikonal equation similar to Mensch and Farra (1999) but for both qp- and qs-waves. We can then estimate a feasible artificial viscosity value and choose the smaller one between the user defined (5 for qp-wave and 3 for qs-wave) and the estimated value. Next we solve the traveltime and derive slowness vector ray vector and polarization for a more realistic anisotropic medium i.e. the original weak triclinic medium shown in eq. (10) without modification. he CPU time is much less than the strong triclinic case (see able 3). he angle difference between the slowness ray and polarization is generally nonzero (Figure 2) and better quantified in Figure 3 for qp-wave. In general homogeneous-anisotropic media the angle difference between the slowness and ray and the slowness and polarization can be up to 20 o whereas the angle difference between the ray and the polarization is much smaller i.e. ~5 o. Figure 3: he angle difference as a function of θ and Φ on the isosurface shown in Figure 2b. In this triclinic case the deviation of the ray vector from the polarization is smaller than that of slowness vector. Figure 4: An example showing the irreversibility of the slowness path in a weak triclinic medium. As the source (S1) and the receiver (R1) switches positions the slowness path following the gradient of the traveltime field is irreversible. o trace the seismic ray path the ray vector must be calculated at each step from the slowness vector and as expected the ray path is reversible. Conclusions Figure 2: (a) he isosurfaces of qp qs-fast and qs-slow wavefronts in a weak triclinic medium. (bcd) show the angle difference between the slowness vector (blue arrow) ray vector (black arrow) and the polarization (red arrow) for grid point on an isosurface. In general the angle difference is nonzero. In applications where sources significantly outnumber receivers (e.g. Huang et al. 2013) receivers are usually regarded as virtual sources and the seismic reciprocity is applied to obtain the traveltime and ray path from an arbitrary grid to the receivers as if the source is located at the grid. In anisotropic medium care must be taken when tracing the ray path from the traveltime field since the ray direction must be calculated from the slowness at each step using eq. (3). Figure 4 gives an example of two paths following slowness vector and ray vectors. As the source and receiver position switches ray path is reversible whereas the slowness path is not as it is the phase velocity direction instead of the group velocity (energy) direction. In this paper we present a Lax-Friedrichs scheme-based fast sweeping method to calculate the direct traveltime field for both qp- and qs-waves in general anisotropic media. We give the workflow of updating traveltime and enforcing computational boundary conditions for 3-D cases. he convergence test on a 2-D example verifies its accuracy and the linear convergence rate (first order). he numerical efficiency and stability are demonstrated using a strong triclinic model. We quantify the difference between three vectors: slowness ray and polarization in a weak triclinic model and show that the ray path is reversible with smaller angle deviation from the polarization than the slowness. Acknowledgement We thank the members of staff at Applied Seismology Consultants Ltd for their support in this work. he authors are grateful to Dr. Leon homsen for the communication regarding anisotropy.

5 Computation of Rays raveltimes Slowness and Polarization in General Anisotropic Medium REFERENCE Červenỳ V Seismic ray theory Cambridge University Press 725 pages. Huang J. J. Reyes-Mountes and R.P. Young 2013 Automated microseismic event location using finite difference traveltime calculation and enhanced waveform stacking 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London UK. Jech J. and Pšenčik I First-order approximation method for anisotropic media Geophys. J. Int Kao Y.C. S. Osher and J. Qian 2004 Lax- Friedrichs sweeping scheme for static Hamilton-Jacobi equations Journal of Computational Physics Mensch. and Farra V Computation of qpwave rays traveltimes and slowness vectors in orthorhombic media. Geophysical Journal International 138: doi: /j x x Mah M. and D.R. Schmitt 2003 Determination of the complete elastic stiffnesses from ultrasonic phase velocity measurements Journal of Geophysical Research Vol 108 No. B doi: /2001jb Musgrave M.J.P Crystal Acoustics: Introduction to the Study of Elastic Waves and Vibrations in Crystals Hen-Day San Francisco 288 pages. Sethian J. and A. Vladimirsky 2003 Ordered Upwind Methods for Static Hamilton--Jacobi Equations: heory and Algorithms SIAM Journal on Numerical Analysis 41: homsen L Weak elastic anisotropy: Geophysics sai Y.-H.R. L.-. Cheng S. Osher H.-K. Zhao 2003 Fast sweeping algorithms for a class of Hamilton Jacobi equations SIAM J. Numer. Anal. 41 (2)