A 27-point scheme for a 3D frequency-domain scalar wave equation based on an average-derivative method

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1 Geophysical Prospecting doi: 0./ A 7-point scheme for a 3D frequency-domain scalar wave equation based on an average-derivative method Jing-Bo Chen Key Laboratory of Petroleum Resources Research Institute of Geology and Geophysics Chinese Academy of Sciences P. O. Box 985 Beijing 0009 China Received August 0 revision accepted March 03 ABSTRACT Based on an average-derivative method and optimization techniques a 7-point scheme for a 3D frequency-domain scalar wave equation is developed. Compared to the rotated-coordinate approach the average-derivative optimal method is not only concise but also applies to equal and unequal directional sampling intervals. The resulting 7-point scheme uses a 7-point operator to approximate spatial derivatives and the mass acceleration term. The coefficients are determined by minimizing phase velocity dispersion errors and the resultant optimal coefficients depend on ratios of directional sampling intervals. Compared to the classical 7-point scheme the number of grid points per shortest wavelength is reduced from approximately 3 to approximately 4 by this 7-point optimal scheme for equal directional sampling intervals and unequal directional sampling intervals as well. Two numerical examples are presented to demonstrate the theoretical analysis. The average-derivative algorithm is also extended to a 3D frequency-domain viscous scalar wave equation. Key words: Scalar wave equation Average-derivative method. INTRODUCTION Recently full-waveform inversion FWI has been attracting increasing attention in the community of exploration geophysics. FWI is a full-wavefield-modelling-based data-fitting process to extract structural information of the subsurface from seismograms Tarantola 984; Gauthier Virieux and Tarantola 986; Pratt and Worthington 990; Pratt Shin and Hicks 998; Boonyasiriwat et al. 009; Virieux and Operto 009. Forward modelling constitutes an important part of FWI. Forward modelling can be classified into two categories: time-domain modelling and frequency-domain modelling. Compared to time-domain modeling Chen 009 frequency-domain modelling has its advantages: convenient manipulations of a single frequency multi-shot computation based on a direct solver and easy implementation of attenuation Jo Shin and Suh 996. Another advantage of frequencydomain modelling is also worth mentioning. No wavefield- chenjb@mail.iggcas.ac.cn storage issue occurs when constructing the gradient of FWI for frequency-domain modelling. This is not the case when constructing the gradient of FWI for time-domain modelling. This is because the forward source wavefield and backward receiver wavefield are computed in the opposite time direction Symes 007; Clapp 009. The main disadvantage of frequency-domain modelling is that it can be only performed in an implicit way by solving a set of linear equations. In comparison with time-domain modelling this disadvantage is particularly obvious when it comes to 3D computation. Therefore reducing the number of grid points per shortest wavelength is in great demand in particular when direct solution techniques are employed. Using a rotatedcoordinatesystemjoet al. 996 developed a 9-point operator to approximate the Laplacian and the mass acceleration terms in a D scalar wave equation. The coefficients are determined by obtaining the best normalized phase velocity dispersion curves. This 9-point scheme reduces the number of grid points per wavelength to approximately 4 and leads to significant reductions of computer memory and CPU time. 58 C 03 European Association of Geoscientists & Engineers

2 A 7-point scheme for a 3D frequency-domain scalar wave equation 59 Hustedt Operto and Virieux 004 and Operto Virieux and Sourbier 007 generalized the rotated-coordinate method to a variable density case and a 3D case respectively. A disadvantage of the rotated-coordinate method is that equal directional sampling intervals are required. To overcome the disadvantage of the rotated optimal 9-point scheme Chen 0 developed a new 9-point finite-difference scheme for a D scalar wave equation based on an average-derivative approach Chen 00; Chen 008. This new scheme imposes no restriction of equal directional sampling intervals and reduces the number of grid points per shortest wavelength to approximately 4 for both equal and unequal directional sampling intervals. Furthermore due to its flexibility and simplicity the average-derivative method can be easily extended to the viscous scalar wave equation and 3D wave equation. In Chen 0 the average-derivative method was discussed mainly in terms of a D frequency-domain scalar wave equation and the 3D case was only briefly examined. In fact the average-derivative method for a 3D wave equation is more attractive because it is not only concise but also applies to both equal and unequal directional sampling intervals. On the other hand the construction of 3D finite-difference schemes based on the rotated-system approach is not only very complicated but also restricted to equal directional sampling intervals because a lot of coordinate-transformations based on equal grid directional intervals are needed. The present paper is a further development of Chen 0 and I will explore the averagederivative method for a 3D frequency-domain scalar wave equation in detail including coefficients optimization dispersion analysis generalization and numerical examples. A 7-POINT SCHEME FOR A 3D WAVE EQUATION The 3D frequency-domain scalar wave equation reads: P x + P y + P z + ω P = 0 v where P is the pressure wavefield ω is the circular frequency and vx y z is the velocity. A classical 7-point scheme for equation is see Fig. a: P m+ l n P m l n + P m l n x + P m l+ n P m l n + P m l n y + P m l n+ P m l n + P m l n + ω P z vmln m l n = 0 where P m l n Pm x l y n z v m l n vm x l y n z and x y and z are directional sampling intervals in the x- y- andz-directions respectively. As can be seen later within the phase velocity error of % the classical 7-point scheme requires approximately 3 grid points per shortest wavelength. In order to reduce the numerical dispersion of scheme very fine grids are required. This leads to a huge amount of computer storage and CPU time. Therefore reducing the number of grid points required per shortest wavelength is needed. To this aim Chen 0 developed an average-derivative approach. Using a weighted-average technique the averagederivative method provides a family of approximations to derivatives from which the optimization approximation can be determined to greatly improve dispersion accuracy. This method is particularly advantageous when it comes to the 3D frequency-domain wave equation because no complicated coordinate transforms are needed. An average-derivative optimal 7-point scheme for equation can be obtained as follows see Fig. b: P m+ l n P m l n + P m l n x + P m l n+ P m l n + P m l n z + ˆP m l+ n ˆP m l n + ˆP m l n y + ω cp vmln m l n + da+ eb + fc = 0 3 where P r l n = α P r l+ n + P r l n+ + P r l n + P r l n + α P r l+ n+ + P r l n+ + P r l+ n + P r l n + 4α 4α P r l n r = m + m m ˆP m s n = β P m+ s n + P m s n+ + P m s n + P m s n + β P m+ s n+ + P m+ s n + P m s n+ + P m s n + 4β 4β P m s n s = l + l l P m l q = γ P m+ l q + P m l+ q + P m l q + P m l q + γ P m+ l+ q + P m+ l q + P m l+ q + P m l q + 4γ 4γ P m l q q = n + n n A = P m l+ n + P m l n+ + P m l n + P m l n + P m+ l n + P m l n B = P m+ l+ n + P m+ l n+ + P m+ l n + P m+ l n + P m l+ n + P m l n+ + P m l n + P m l n +P m l+ n+ + P m l n+ + P m l+ n + P m l n C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

3 60 J.-B. Chen a mln b m ln m l+n mln ml+n m+ln m+l+n ml+n m l n m l+n ml n ml+n m+l n m+l+n m ln mln m+ln m ln mln m+ln ml n mln+ m l n ml n m l+n+ m ln+ mln+ ml+n+ m+l n m+ln+ m+l+n+ y x m l n+ ml n+ m+l n+ y x z z Figure Schematic of the classical 7-point scheme a and the average-derivative optimal 7-point scheme b. C = P m+ l+ n+ + P m+ l n+ + P m+ l+ n + P m+ l n + P m l+ n+ + P m l n+ + P m l+ n + P m l n where α α β β γ γ c d and e are coefficients and f = c 6d e 8. In equation 3 the approximations of the derivatives are weighted averages of three kinds of approximations and therefore I call equation 3 the average-derivative optimal 7-point scheme. The motivation of the average-derivative method is to provide a family of approximations to the derivatives from which the optimization approximation can be chosen to reduce dispersion errors Chen 0. Compared to the 3D scheme based on the rotated-coordinate method Operto et al. 007 the average-derivative 7-point scheme 3 is not only concise but also applies to the situation where the directional sampling intervals x y and z are different. In addition the average-derivative optimal 7-point scheme 3 also includes the classical 7-point scheme as a special case because when α = 0 α = 0 β = 0 β = 0 γ = 0 γ = 0 c = d = 0ande = 0 scheme 3 becomes scheme. OPTIMIZATION AND DISPERSION ANALYSIS In this section I perform optimization of coefficients of the average-derivative optimal 7-point scheme 3 and make corresponding dispersion analysis. Substituting Px y zω = P 0 e ik x x+k y y+k z z into equation 3 and assuming a constant v one obtains the discrete dispersion relation: ω v = 4sin kx x where E x + 4r ky sin y x c + dã+ 4e B + 8 f C E x = α cosk y y + cosk z z E y + 4r kz sin z + 4α cosk y ycosk z z + 4α 4α E y = β cosk x x + cosk z z + 4β cosk x xcosk z z + 4β 4β E z = γ cosk x x + cosk y y r = x y + 4γ cosk x xcosk y y + 4γ 4γ r = x z à = cosk x x + cosk y y + cosk z z B = cosk x xcosk y y + cosk x xcosk z z + cosk y ycosk z z C = cosk x xcosk y ycosk z z. In equation 4 I suppose that x = max{ x y z}. For other cases a similar analysis can be performed. If y = max{ x y z} r and r should be defined by r = y x E z 4 C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

4 A 7-point scheme for a 3D frequency-domain scalar wave equation 6 Table Optimization coefficients for α α β β γ γ c d and e. α α β β γ γ c d e r = r = r = r = r = r = 3 r = r = r = r = r = r = and r = y respectively. If z = max{ x y z} r z and r should be defined by r = z and r x = z respectively. y From equation 4 the normalized phase velocity can be derived as follows: V ph { v = sin π sin θ cos φ G E x +r sin π sin θ sin φ r G π G {c+dã+4e B+8 f C} } E y +r sin π cos θ r G E z where V ph is the phase velocity k x = ksin θ cos φ k y = ksin θ sin φ k z = kcos θ G = π θ is the propagation angle φ is the azimuth angle k x and π sin θ sin φ E x = α cos r G π sin θ sin φ + 4α cos r G + 4α 4α π sin θ cos φ E y = β cos G π sin θ cos φ + 4β cos G + 4β 4β π sin θ cos φ E z = γ cos G π sin θ cos φ + 4γ cos G + cos cos + cos cos + cos cos π cos θ r G π cos θ r G π cos θ r G π cos θ r G π sin θ sin φ r G π sin θ sin φ r G + 4γ 4γ π sin θ cos φ π sin θ sin φ Ã = cos + cos G r G 5 π cos θ + cos r G π sin θ cos φ π sin θ sin φ B = cos cos G r G π sin θ cos φ π cos θ + cos cos G r G π sin θ sin φ π cos θ + cos cos r G r G π sin θ cos φ π sin θ sin φ C = cos cos G r G π cos θ cos. r G In equation 5 if y = max{ x y z} G should be defined by G = π.if z = max{ x y z} G should be defined by G = π. k z k y The coefficients α α β β γ γ c d and e are determined by minimizing the phase error: [ PE = V phθφ k; α α β β γ γ c d e v d kdθdφ 6 where k = G. The ranges of k θ and φ are taken as [0 0.5] [0 π ]and [0 π ] respectively. The range of k depends on the number of grid points involved in the scheme. The more grid points are involved the larger the range of k can be taken. For scheme 3 the range [0 0.5] is a reasonable choice and a larger range can lead to a degradation of dispersion accuracy. A constrained non-linear optimization program fmincon in Matlab is used to determine the optimization coefficients. The optimization coefficients for different r = x y and ] C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

5 6 J.-B. Chen point scheme φ= 5 o point scheme φ= 5 o /G point scheme φ=5 o 5 o /G point scheme φ=5 o 5 o /G point scheme φ=3 5 o /G point scheme φ=3 5 o /G /G Figure Normalized phase velocity curves of the classical 7-point scheme and the average-derivative optimal 7-point scheme for fixed azimuth angle φ and different propagation angles θ when r = andr =. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

6 A 7-point scheme for a 3D frequency-domain scalar wave equation point scheme φ= 5 o point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G /G Figure 3 Continuation of Fig.. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

7 64 J.-B. Chen point scheme θ= 5 o point scheme θ= 5 o /G point scheme θ=5 o 5 o /G point scheme θ=5 o 5 o /G point scheme θ=3 5 o /G point scheme θ=3 5 o /G /G Figure 4 Normalized phase velocity curves of the classical 7-point scheme and the average-derivative optimal 7-point scheme for fixed propagation angle θ and different azimuth angles φ when r = andr =. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

8 A 7-point scheme for a 3D frequency-domain scalar wave equation point scheme θ= 5 o point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G /G Figure 5 Continuation of Fig. 4. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

9 66 J.-B. Chen In order to obtain an overall estimation of the phase velocity errors varying with φ and θ Fig. 6 shows the following relative phase velocity error: V RE = ph φθg v v 80o φ 8 θ and G = 4. 7 Figure 6 The relative phase velocity error for the classical 7-point scheme a and the average-derivative optimal 7-point scheme b when r = andr =. r = x when x = max{ x y z} are listed in Table. z One can see that the coefficients α α β β γ γ and e vary with r and r and the changes in coefficients c and d are small. I now perform numerical dispersion analysis. First I consider the case where r = andr = which corresponds to the equal directional intervals x = y = z. Figures and 3 show normalized phase velocity curves of the classical 7- point scheme and the average-derivative optimal 7-point scheme 3 for fixed azimuth angle φ and different propagation angles θ. Figures 4 and 5 show normalized phase velocity curves of the classical 7-point scheme and the averagederivative optimal 7-point scheme 3 for fixed propagation angle θ and different azimuth angles φ. From thesefigures one can conclude that within the phase velocity error of % the classical 7-point scheme requires approximately 3 grid points per shortest wavelength while the average-derivative optimal 7-point scheme 3 requires approximately 4 points. For the classical 7-point scheme the maximum relative error is 9.97% while for the average-derivative optimal 7- point scheme 3 the maximum relative error is 0.3%. Second I consider the case where r = and r = 3 which corresponds to the unequal directional intervals x = y = 3 z. Figures 7 and 8 show normalized phase velocity curves of the classical 7-point scheme and the averagederivative optimal 7-point scheme 3 for fixed azimuth angle φ and different propagation angles θ. Figures 9 and 0 show normalized phase velocity curves of the classical 7- point scheme and the average-derivative optimal 7-point scheme 3 for fixed propagation angle θ and different azimuth angles φ. From these figures one can conclude that within the phase velocity error of % the classical 7-point scheme requires approximately 3 grid points per shortest wavelength while the average-derivative optimal 7-point scheme 3 requires approximately 4 points. Figure shows the relative phase velocity error equation 7. For the classical 7-point scheme the maximum relative error is 9.97% while for the average-derivative optimal 7-point scheme 3 the maximum relative error is 0.5%. For other cases on r and r a similar analysis can be made. The common conclusion is that within the phase velocity error of % and for equal and unequal directional sampling intervals the classical 7-point scheme requires approximately 3 grid points per shortest wavelength while the average-derivative optimal 7-point scheme 3 requires approximately 4 grid points. GENERALIZATION OF SCHEME3 Due to its flexibility and simplicity scheme 3 can be easily extended to the 3D viscous scalar wave equation. The 3D viscous scalar wave equation reads: x ρ P + x y ρ P + y z ρ P + ω P = 0 8 z κ C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

10 A 7-point scheme for a 3D frequency-domain scalar wave equation point scheme φ= 5 o point scheme φ= 5 o /G point scheme φ=5 o 5 o /G point scheme φ=5 o 5 o /G point scheme φ=3 5 o /G point scheme φ=3 5 o /G /G Figure 7 Normalized phase velocity curves of the classical 7-point scheme and the average-derivative optimal 7-point scheme for fixed azimuth angle φ and different propagation angles θ when r = andr = 3. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

11 68 J.-B. Chen point scheme φ= 5 o point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G point scheme φ= 5 o /G /G Figure 8 Continuation of Fig. 7. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

12 A 7-point scheme for a 3D frequency-domain scalar wave equation point scheme θ= 5 o point scheme θ= 5 o /G point scheme θ=5 o 5 o /G point scheme θ=5 o 5 o /G point scheme θ=3 5 o /G point scheme θ=3 5 o /G /G Figure 9 Normalized phase velocity curves of the classical 7-point scheme and the average-derivative optimal 7-point scheme for fixed propagation angle θ and different azimuth angles φ when r = andr = 3. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

13 70 J.-B. Chen point scheme θ= 5 o point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G point scheme θ= 5 o /G /G Figure 0 Continuation of Fig. 9. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

14 A 7-point scheme for a 3D frequency-domain scalar wave equation 7 * Δ Figure Schematic of the homogeneous model. The symbol represents the source and the receiver. Figure The relative phase velocity error for the classical 7-point scheme a and the average-derivative optimal 7-point scheme b when r = andr = 3. where ρx y z is the density and κx y z is the complex bulk modulus that accounts for attenuation in one of the following two ways: κx y z = ρx y zv x y z i 9 Q κx y z = ρx y z vx y z + πvx y zq ln ω r ω sgnω + i 0 vx y zq where vx y z is the real velocity Qis the attenuation factor i is the unit of imaginary numbers sgn is the sign function and ω r is a reference frequency Operto et al An average-derivative optimal 7-point scheme for equation 8 is x [ ρ m+ l n + ρ m l n P m+ l n P m l n ] ρ m+ l n + ρ m l n P m l n [ + y [ + z ρ m l+ n + ρ m l n ρ m l n+ + ρ m l n ˆP m l+ n ˆP m l n ] ρ m l+ n + P m l n+ + ρ m l n+ P m l n ] ρ m l n ρ m l n ˆP m l n P m l n + ω cpm κm l n + da+ eb + fc = 0 l n where ρ m+ l n = ρ m l n +ρ m+ l n ρ m l n = ρ m l n +ρ m l n ρ m l+ n = ρ m l n +ρ m l+ n ρ m l n = ρ m l n +ρ m l n ρ m l n+ = ρ m l n +ρ m l n+ ρ m l n = ρ m l n +ρ m l n. NUMERICAL EXAMPLES In this section I present two numerical examples to verify the theoretical analysis on the classical 7-point scheme and the average-derivative optimal 7-point scheme 3. First I consider a homogeneous velocity model with a velocity of 3000 m/s Fig.. In this case an analytical solution is available to make comparisons with numerical solutions. Horizontal and vertical samplings are nx = 5 ny = 5 and nz = 4 respectively. A Ricker wavelet with a peak frequency of 5 Hz is placed at the centre of the model as a source and a C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

15 7 J.-B. Chen a.5 b.5 Amplitude Amplitude Time s Time s c.5 d.5 Analytical 7 point 7 point Amplitude Amplitude Time s Time s Figure 3 Seismograms computed with the analytical method a the classical 7-point scheme b the average-derivative optimal 7-point scheme c and the superimposed results d. Here r = andr =. a.5 b.5 Amplitude Amplitude Time s Time s c.5 d.5 Analytical 7 point 7 point Amplitude Amplitude Time s Time s Figure 4 Seismograms computed with the analytical method a the classical 7-point scheme b the average-derivative optimal 7-point scheme c and the superimposed results d. Here r = andr = 3. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

16 A 7-point scheme for a 3D frequency-domain scalar wave equation 73 receiver is set halfway horizontally between the source and the boundary. The maximum frequency used in the computation is 70 Hz. I first consider the case where r = and r =. According to the criterion of 4 grid points per smallest wavelength horizontal sampling interval is determined by dx = 3000/70/4 m m and dy = dx and dz = dx. For the analytical solution the following formula is used: [ { Px y z t = F r exp iω r } ] F f t v Figure 5 Schematic of the salt dome model. where F and F are forward and inverse Fourier transformations with respect to time respectively f t is the Ricker wavelet and r = x x 0 + y y 0 + z z 0 where x 0 y 0 z 0 is the source position. Figure 6 Sections of a 70 Hz monochromatic wavefield. Cross-line section for y = 39 m a in-line section for x = 39 m b depth section for z = 59 m c and depth section for z = 60 m d. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

17 74 J.-B. Chen Figure 3 shows the results computed with the analytical formula the classical 7-point scheme and the average-derivative optimal scheme 3. The simulation result with the average-derivative optimal 7-point scheme 3 is in good agreement with the analytical result while the result with the classical 7-point scheme exhibits errors due to numerical dispersion. Figure 4 shows the results for the case where r = andr = 3. In this case dx = 3000/70/4 m m and dy = dx/ anddz = dx/3. Again the simulation result with the average-derivative optimal 7-point scheme 3 is better than that of the classical 7-point scheme in terms of agreement with the analytical result. Second I consider a heterogeneous velocity model and examine the performance of the average-derivative optimal 7-point scheme 3 in a heterogeneous model. Figure 5 shows a salt dome velocity model. The velocity of the salt dome is 4000 m/s and the velocity of the overburden is 3000 m/s. The sampling numbers the Ricker wavelet and the maximum frequency used in this example are the same as those used in the homogeneous velocity model. In this example I consider another case where r = andr =. The source is placed at x = 75 m y = 75 m z = 77 m. PML Perfectly Matched Layer boundary conditions are used see Appendix in which L x = L y = L z = 0 and c x = c y = c z = 80. Figure 6 shows the sections of the 70 Hz monochromatic wavefield computed with the average-derivative optimal 7-point scheme 3. Because of the symmetry of the velocity model the cross-line and in-line sections are basically the same. The two depth sections correspond to the positions inside and above the salt dome respectively. No visible dispersion and boundary reflections are observed. CONCLUSIONS I have presented an average-derivative optimal 7-point scheme for the 3D frequency-domain wave equation in detail. The optimization coefficients are obtained by minimizing the phase velocity errors and they vary with the ratios of directional sampling intervals. Based on dispersion analysis a conclusion is drawn that within the phase velocity error of % and for equal and unequal directional sampling intervals the classical 7-point scheme requires approximately 3 grid points per shortest wavelength and the average-derivative optimal 7-point scheme requires approximately 4 grid points. Two numerical examples demonstrate the theoretical analysis. Natural generalization to the 3D viscous case and incorporation with PML boundary conditions show the great flexibility and broad applicability of the 3D average-derivative optimal method. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China under grant nos and REFERENCES Boonyasiriwat C. P. Valasek P. Routh W. Cao G. T. Schuster and B. Macy 009 An efficient multiscale method for time-domain waveform tomography: Geophysics 746 WCC59-WCC68. Chen J.-B. 00 New schemes for the nonlinear Schrödinger equation: Applied Mathematics and Computation Chen J.-B. 008 Variational integrators and the finite element method: Applied Mathematics and Computation Chen J.-B. 009 Lax-Wendroff and Nyström methods for seismic modelling: Geophysical Prospecting Chen J.-B. 0 An average-derivative optimal scheme for frequency-domain scalar wave equation: Geophysics 77 6 T0- T0. Clapp R. G. 009 Reverse time migration with random boundaries: 79th Annual International Meeting SEG Expanded Abstracts Clayton R. and B. Engqusit 977 Absorbing boundary conditions for scalar and elastic wave equations: Bulletin of the Seismological Society of America Gauthier O. J. Virieux and A. Tarantola 986 Two-dimensional nonlinear inversion of seismic waveforms: Numerical results: Geophysics HustedtB. S. Operto and J. Virieux 004 Mix-grid and staggeredgrid finite-difference methods for frquency-domain acoustic wave modelling: Geophysical Journal International Jo C.-H. C. Shin and J. H. Suh 996 An optimal 9-point finitedifference frequency-space -D scalar wave extrapolator: Geophysics Operto S. J. Virieux P. Amestoy J.-Y. L Excellent L. Giraud and H. B. H. Ali 007 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study: Geophysics 75 SM95-SM. Operto S. J. Virieux and F. Sourbier 007 Documentation of FWTD program version 4.8: Frequency-domain full-waveform modeling/inversion of wide-aperture seismic data for imaging D scalar media: Technical report N o 007-SEISCOPE project. Pratt R. G. C. Shin and G. J. Hicks 998 Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International Pratt R. G. and M.-H. Worthington 990 Inverse theory applied to multi-source cross-hole tomography Part I: acoustic wave-equation method: Geophysical Prospecting Symes W. M. 007 Reverse-time migration with optimal checkpointing: Geophysics 75 SM3-SM. C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

18 A 7-point scheme for a 3D frequency-domain scalar wave equation 75 Tarantola A. 984 Inversion of seismic reflection data in the acoustic approximation: Geophysics Virieux J. and S. Operto 009 An overview of full-waveform inversion in exploration geophysics: Geophysics 746 WCC-WCC6. APPENDIX COEFFICIENTS WITH PERFECTLY MATCHED LAYER BOUNDARY CONDITIONS FOR THE AVERAGE-DERIVATIVE 7-POINT SCHEME The 3D frequency-domain scalar wave equation with PML boundary conditions reads: P + P + P x ξ x x ξ y y ξ y y ξ z z ξ z z ξ x + ω v P = 0 where ξ x x = ic x ω cos π ξ y z = ic y π ω cos ξ z z = ic z π ω cos x L x y L y z L z A A A3 A4 where L x L y andl z denote the width of the PML layer in the x- y- andz-directions respectively. The coordinates x y and z are local coordinates whose origins are located at the outer edges of the PML layers. The scalars c x c y and c z depend on L x L y andl z respectively and are determined by trial and error Operto et al For a function f t in time its Fourier Fω is defined as: Fω = f te iωt dt. A5 where + y ξ zn z ξ zn+ + ξ zn ˆP m l+ n + ˆP + ξ m l n yl ˆP m l n P m l n+ + P ξ zn+ ξ m l n zn P m l n + ω cpm vmln l n + da+ eb + fc = 0 A7 + = ξ x m + + = ξ x m + + = ξ y l + + = ξ y l + ξ zn+ = ξ z n + ξ zn+ ξ zn = ξ z n + ξ zn. Using the ordering in Fig. A scheme A7 can be rewritten as: c P m l n + c P m l n + c 3 P m+ l n + c 4 P m l n + c 5 P m l n + c 6 P m+ l n + c 7 P m l+ n + c 8 P m l+ n + c 9 P m+ l+ n + c 0 P m l n + c P m l n + c P m+ l n + c 3 P m l n + c 4 P m l n + c 5 P m+ l n + + c 6 P m l+ n + c 7 P m l+ n + c 8 P m+ l+ n + c 9 P m l n+ + c 0 P m l n+ + c P m+ l n+ If one uses another definition: Fω = f te iωt dt A6 + c P m l n+ + c 3 P m l n+ + c 4 P m+ l n+ A8 + c 5 P m l+ n+ + c 6 P m l+ n+ + c 7 P m+ l+ n+ = 0 then ω in equations A A4 should be replaced by ω. The average-derivative optimal 7-point scheme for equation A becomes: x + + P m+ l n P m l n + + P m l n where c = α + β + γ + f ω x ξ yl y ξ zn ξ zn z v mln c = + + ξ xm + γ + e ω ξ zn ξ zn z v mln α x + β y C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

19 76 J.-B. Chen c 3 = α + β + γ + f ω + x ξ yl y ξ zn ξ zn z v mln c 4 = α x + γ + e ω ξ zn ξ zn z v mln c 5 = + + ξ xm + + ξ yl α x β y β + 4γ 4γ + d ω y ξ zn ξ zn z v mln c 6 = + α x + γ + e ω ξ zn ξ zn z v mln + + ξ yl + + ξ yl β y c 7 = α + β + γ + f ω x ξ yl + y ξ zn ξ zn z v mln c 8 = + + ξ xm + γ + e ω ξ zn ξ zn z v mln α x + + β y c 9 = α + β + γ + f ω + x ξ yl + y ξ zn ξ zn z v mln c 0 = α + β x ξ yl y ξ zn c = ξ zn + ξ zn+ ξ zn + + ξ xm + ξ zn+ ξ zn γ + e ω z vmln c = α + β + x ξ yl y ξ zn + ξ zn+ ξ zn c 3 = 4α 4α x ξ yl ξ zn c 4 = + ξ zn+ ξ zn + + ξ xm α + 4β 4β x y γ + d ω z vmln γ + e ω z vmln + + ξ yl γ + d ω z vmln 4α 4α x β y ξ zn + + ξ yl + ξ zn+ ξ zn c 5 = 4α 4α + x ξ yl ξ zn + ξ zn+ ξ zn 4β 4β y 4γ 4γ z c 6 = α + β x ξ yl + y ξ zn c 7 = ξ zn + ξ zn+ ξ zn + + ξ xm + ξ zn+ ξ zn + c ω vmln + + ξ yl γ + d ω z vmln γ + e ω z vmln c 8 = α + β + x ξ yl + y ξ zn + ξ zn+ ξ zn β y α + 4β 4β x + y γ + d ω z vmln γ + e ω z vmln c 9 = α + β + γ + f ω x ξ yl y ξ zn ξ zn+ z v mln c 0 = + + ξ xm + γ + e ω ξ zn ξ zn+ z v mln α x + β y c = α + β + γ + f ω + x ξ yl y ξ zn ξ zn+ z v mln c = α x c 3 = + γ + e ω ξ zn ξ zn+ z v mln + + ξ xm + + ξ yl α x β y β + 4γ 4γ + d ω y ξ zn ξ zn+ z v mln c 4 = + α x + γ + e ω ξ zn ξ zn+ z v mln + + ξ yl + + ξ yl β y C 03 European Association of Geoscientists & Engineers Geophysical Prospecting

20 A 7-point scheme for a 3D frequency-domain scalar wave equation 77 c 7 c 8 c 9 m l+n ml+n c 4 m ln c 5 mln c 6 m+ln c m l n c ml n c 3 m+l n c 6 c 7 c 8 m l+n ml+n c 3 m ln c c 4 mln 5 m+ln c 0 c c m l n ml n m+l+n m+l+n m+l n c 5 c m l+n+ c 6 ml+n+ 7 c m ln+ c c 3 mln+ 4 m+ln+ c 9 m l n+ c 0 ml n+ c m+l n+ y m+l+n+ x Figure A Schematic of the average-derivative optimal 7-point scheme and its coefficients with PML boundary conditions. z c 5 = α + β + γ + f ω x ξ yl + y ξ zn ξ zn+ z v mln c 6 = + + ξ xm α x + + β y + γ + e ω ξ zn ξ zn+ z v mln c 7 = α + β + γ + f ω. + x ξ yl + y ξ zn ξ zn+ z v mln C 03 European Association of Geoscientists & Engineers Geophysical Prospecting