modeling of frequency-domain elastic-wave equation with an...

Modeling of frequency-domain elastic-wave equationwith an average-derivative optimal method

Jing-Bo Chen1 and Jian Cao1

ABSTRACT

Based on an average-derivative method, we developed anew nine-point numerical scheme for a frequency-domainelastic-wave equation. Compared with the classic nine-pointscheme, this scheme reduces the required number of gridpoints per wavelength for equal and unequal directional spac-ings. The reduction in the number of grid points increases asthe Poisson’s ratio becomes larger. In particular, as the Pois-son’s ratio reaches 0.5, the numerical S-wave phase velocityof this new scheme becomes zero, whereas the classicalscheme produces spurious numerical S-wave phase velocity.Numerical examples demonstrate that this new scheme pro-duces more accurate results than the classical scheme at ap-proximately the same computational cost.

INTRODUCTION

Frequency-domain numerical modeling of seismic-wave equationswas pioneered by Lysmer and Drake (1972) to describe the wavepropagation in the earth. In analyzing the accuracy of finite-differenceand finite-element modeling of the scalar- and elastic-wave equa-tions, Marfurt (1984) further examines frequency-domain numericalmodeling. One of Marfurt’s important research results is that fre-quency-domain finite-element solutions using a weighted averageof consistent and lumped masses lead to accurate results. Pratt andWorthington (1990) present in detail the implementation of theclassical five-point scheme for frequency-domain 2D acoustic-waveequation and apply it to multisource crosshole tomography. Based onthe spatial approximations in Kelly et al. (1976), Pratt (1990) devel-ops a classic nine-point scheme for frequency-domain 2D elastic-wave equation. Because of its small bandwidth in the resultingimpedance matrix, this scheme is very efficient. To reduce numerical

dispersion, however, this classical scheme requires a large number ofgrid points per wavelength.Based on a rotated coordinates system and averaging mass accel-

eration terms, Jo et al. (1996) develop an optimal nine-point schemefor 2D scalar-wave equation. The optimization coefficients areobtained by minimizing the phase velocity errors. This scheme re-duces the number of grid points per wavelength from 13 to 4 incomparison with the classical five-point scheme. For frequency-domain numerical modeling, Jo et al.’s (1996) method is an impor-tant optimization approach. This approach is then generalized to avariable-density case (Hustedt et al., 2004) and a 3D case (Opertoet al., 2007). Schemes involving more grid points have also beendeveloped (Shin and Sohn, 1998; Min et al., 2000; Gu et al., 2013).These schemes have higher accuracy but lower efficiency.For frequency-domain elastic-wave modeling, Štekl and Pratt

(1998) propose an optimal nine-point scheme for frequency-domain2D elastic-wave equation. The construction of the optimal schemeincorporates Jo et al.’s (1996) method and staggered-grid technique.This optimal nine-point scheme is particularly desirable because itnot only reduces the number of grid points per wavelength but alsoremains approximately the same in terms of computational cost.However, this optimal scheme suffers from two limitations: (1) It re-quires equal directional grid intervals and (2) when generalized to the3D case, it becomes very complicated and requires a lot of transfor-mations (Operto et al., 2007).To address these two limitations, we will develop an average-

derivative optimal method for the frequency-domain elastic-waveequation. This method is a generalization of the correspondingapproach for the acoustic-wave equation (Chen, 2012, 2014) andresults in an average-derivative optimal nine-point scheme forfrequency-domain 2D elastic-wave equation. This new average-derivative optimal nine-point scheme not only preserves the advan-tages of the optimal scheme proposed by Štekl and Pratt (1998) butalso accommodates arbitrary directional grid intervals. In addition,this average-derivative optimal nine-point scheme can be easily

Manuscript received by the Editor 21 January 2016; revised manuscript received 15 June 2016; published online 23 September 2016.1Institute of Geology and Geophysics, Key Laboratory of Petroleum Resources Research, Chinese Academy of Sciences, Beijing, China and University of

Chinese Academy of Sciences, Beijing, China. E-mail: [email protected]; [email protected].© 2016 Society of Exploration Geophysicists. All rights reserved.

T339

GEOPHYSICS, VOL. 81, NO. 6 (NOVEMBER-DECEMBER 2016); P. T339–T356, 20 FIGS., 3 TABLES.10.1190/GEO2016-0041.1

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generalized to a 27-point scheme for the frequency-domain 3D elas-tic-wave equation.In the next section, we will present the average-derivative

numerical method for the frequency-domain elastic-wave equation.This is followed by a numerical dispersion analysis. Then, wepresent numerical examples to demonstrate the theoretical analysis.

AN AVERAGE-DERIVATIVE METHOD

We consider the 2D elastic-wave equation in the frequency do-main:

∂∂x

�η∂u∂x

�þ ∂

∂z

�μ∂u∂z

�þ ∂

∂x

�λ∂w∂z

�

þ ∂∂z

�μ∂w∂x

�þ ρω2u ¼ 0; (1)

∂∂x

�μ∂w∂x

�þ ∂

∂z

�η∂w∂z

�þ ∂

∂x

�μ∂u∂z

�

þ ∂∂z

�λ∂u∂x

�þ ρω2w ¼ 0; (2)

where u and w represent the horizontal and vertical displacementcomponents, respectively, ρ is the density, ω is the circular frequency,λ and μ are the Lamé parameters, and η ¼ λþ 2μ.We will now discretize equations 1 and 2. Set um;n ≈ uðmΔx;

nΔzÞ and wm;n ≈ wðmΔx; nΔzÞ, where Δx and Δz are the grid in-tervals in the x- and z-directions, respectively. First, we deal withequation 1 and it includes five terms. For the first two terms, wediscretize them with an average-derivative method (Chen, 2012):

∂∂x

�η∂u∂x

�≈

1

Δx2

×hηmþ1

2;numþ1;n − ðηmþ1

2;nþ ηm−1

2;nÞum;nþ ηm−1

2;num−1;n

i;

(3)

∂∂z

�μ∂u∂z

�≈

1

Δz2

×hμm;nþ1

2~um;nþ1 − ðμm;nþ1

2þ μm;n−1

2Þ ~um;n þ μm;n−1

2~um;n−1

i;

(4)

where

ηmþ12;n¼1

2ðηm;nþηmþ1;nÞ; ηm−1

2;n¼1

2ðηm−1;nþηm;nÞ;

ηm;nþ12¼1

2ðηm;nþηm;nþ1Þ; ηm;n−1

2¼1

2ðηm;n−1þηm;nÞ;

μmþ12;n¼

1

2ðμm;nþμmþ1;nÞ; μm−1

2;n¼

1

2ðμm−1;nþμm;nÞ;

μm;nþ12¼1

2ðμm;nþμm;nþ1Þ; μm;n−1

2¼1

2ðμm;n−1þμm;nÞ; (5)

and

umþj;n¼1−γ12

umþj;nþ1þγ1umþj;nþ1−γ12

umþj;n−1; j¼−1;0;1;

~um;nþj¼1−γ22

umþ1;nþjþγ2um;nþjþ1−γ22

um−1;nþj; j¼−1;0;1:

(6)

Here, γ1 and γ2 are the weighted coefficients.For the third and fourth terms in equation 1, we use the centered

differences:

∂∂x

�λ∂w∂z

�≈

1

4ΔxΔz½λmþ1;nðwmþ1;nþ1 − wmþ1;n−1Þ

− λm−1;nðwm−1;nþ1 − wm−1;n−1Þ�; (7)

∂∂z

�μ∂w∂x

�≈

1

4ΔxΔz½μm;nþ1ðwmþ1;nþ1 − wm−1;nþ1Þ

− μm;n−1ðwmþ1;n−1 − wm−1;n−1Þ�: (8)

For the fifth term in equation 1, we use the weighted average:

ρω2u≈ρm;nω2½cum;nþdðumþ1;nþum−1;nþum;nþ1þum;n−1Þ

þeðumþ1;nþ1þum−1;nþ1þumþ1;n−1þum−1;n−1Þ�; (9)

where c and d are the weighted coefficients ande ¼ ð1 − c − 4dÞ∕4.Substituting equations 3–9 into equation 1, and performing the

same discretizations for equation 2, we obtain an average-derivativenine-point scheme for equations 1 and 2:

1

Δx2hηmþ1

2;numþ1;n−ðηmþ1

2;nþηm−1

2;nÞum;nþηm−1

2;num−1;n

i

þ 1

Δz2hμm;nþ1

2~um;nþ1−ðμm;nþ1

2þμm;n−1

2Þ ~um;nþμm;n−1

2~um;n−1

i

þ 1

4ΔxΔz½λmþ1;nðwmþ1;nþ1−wmþ1;n−1Þ

−λm−1;nðwm−1;nþ1−wm−1;n−1Þ�

þ 1

4ΔxΔz½μm;nþ1ðwmþ1;nþ1−wm−1;nþ1Þ

−μm;n−1ðwmþ1;n−1−wm−1;n−1Þ�þρm;nω

2½cum;nþdðumþ1;nþum−1;nþum;nþ1þum;n−1Þþeðumþ1;nþ1þum−1;nþ1þumþ1;n−1þum−1;n−1Þ�¼0; (10)

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1

Δx2hμmþ1

2;nwmþ1;n−

�μmþ1

2;nþμm−1

2;n

�wm;nþμm−1

2;nwm−1;n

i

þ 1

Δz2hηm;nþ1

2~wm;nþ1−

�ηm;nþ1

2þηm;n−1

2

�~wm;nþηm;n−1

2~wm;n−1

i

þ 1

4ΔxΔz½μmþ1;nðumþ1;nþ1−umþ1;n−1Þ

−μm−1;nðum−1;nþ1−um−1;n−1Þ�

þ 1

4ΔxΔz½λm;nþ1ðumþ1;nþ1−um−1;nþ1Þ

−λm;n−1ðumþ1;n−1−um−1;n−1Þ�þρm;nω

2½cwm;nþdðwmþ1;nþwm−1;nþwm;nþ1þwm;n−1Þþeðwmþ1;nþ1þwm−1;nþ1þwmþ1;n−1þwm−1;n−1Þ�¼0: (11)

Figure 1 shows the stencil of the average-derivative nine-pointscheme 10 and 11. If we take γ1 ¼ γ2 ¼ c ¼ 1 and d ¼ 0, the aver-age-derivative nine-point scheme 10 and 11 reduces to the classicalnine-point scheme (Pratt, 1990):

1

Δx2hηmþ1

2;numþ1;n−

�ηmþ1

2;nþηm−1

2;n

�um;nþηm−1

2;num−1;n

i

þ 1

Δz2hμm;nþ1

2um;nþ1−

�μm;nþ1

2þμm;n−1

2

�um;nþμm;n−1

2um;n−1

i

þ 1

4ΔxΔz½λmþ1;nðwmþ1;nþ1−wmþ1;n−1Þ

−λm−1;nðwm−1;nþ1−wm−1;n−1Þ�

þ 1

4ΔxΔz½μm;nþ1ðwmþ1;nþ1−wm−1;nþ1Þ

−μm;n−1ðwmþ1;n−1−wm−1;n−1Þ�þρm;nω

2um;n¼0; (12)

1

Δx2hμmþ1

2;nwmþ1;n−

�μmþ1

2;nþμm−1

2;n

�wm;nþμm−1

2;nwm−1;n

i

þ 1

Δz2hηm;nþ1

2wm;nþ1−

�ηm;nþ1

2þηm;n−1

2

�wm;nþηm;n−1

2wm;n−1

i

þ 1

4ΔxΔz½μmþ1;nðumþ1;nþ1−umþ1;n−1Þ

−μm−1;nðum−1;nþ1−um−1;n−1Þ�

þ 1

4ΔxΔz½λm;nþ1ðumþ1;nþ1−um−1;nþ1Þ

−λm;n−1ðumþ1;n−1−um−1;n−1Þ�þρm;nω

2wm;n¼0: (13)

The average-derivative nine-point scheme 10 and 11 can be easilygeneralized to an average-derivative 27-point scheme for the 3Dfrequency-domain elastic-wave equation (Appendix A).

NUMERICAL DISPERSION ANALYSIS

We will now perform numerical dispersion analysis for the aver-age-derivative nine-point scheme 10 and 11. Set

�uv

�¼

�u0v0

�eiðkxxþkzzÞ; (14)

where u0 and v0 are the constants that are not equal to zero simul-taneously and kx and kz are the horizontal and vertical wavenum-bers, respectively.Substituting equation 14 into the scheme 10 and 11 and setting the

determinant of the resulting matrix to be zero, we obtain theP- and S-wave dispersion relations:

ω2 ¼ 1

2EΔx2

�ðα2 þ β2ÞðExx þ EzzÞ

þ ðα2 − β2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx − EzzÞ2 þ 4E2

xz

q �; (15)

ω2 ¼ 1

2EΔx2

�ðα2 þ β2ÞðExx þ EzzÞ

− ðα2 − β2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx − EzzÞ2 þ 4E2

xz

q �; (16)

where α and β are the P- and S-wave velocities, respectively, and

E ¼ cþ 2dðcosðkxΔxÞ þ cosðkzΔzÞÞþ 4e cosðkxΔxÞ cosðkzΔzÞ; (17)

Exx ¼ ½ð1 − γ1Þ cosðkzΔzÞ þ γ1�½2 − 2 cosðkxΔxÞ�; (18)

Ezz ¼ r2½ð1 − γ2Þ cosðkxΔxÞ þ γ2�½2 − 2 cosðkzΔzÞ�; (19)

Exz ¼ r sinðkxΔxÞ sinðkzΔzÞ; (20)

where r ¼ Δx∕Δz. Here without loss of generality, we supposethat Δx⩾Δz.Let αph and βph denote the compressional and shear phase veloc-

ities, respectively. They are defined as

αph ¼ω

kp; βph ¼

ω

ks; (21)

where kp and ks are the compressional and shear wavenumbers,respectively.From equations 15–21, we can obtain the normalized compres-

sional and shear phase velocities:

Frequency-domain elastic modeling T341

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αphα

¼ Gs

2πR

1

2E

�ð1þR2ÞðExxþEzzÞ

þð1−R2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx−EzzÞ2þ4E2

xz

q �12

; (22)

βphβ

¼Gs

2π

1

2E

��1þ 1

R2

�ðExxþEzzÞ

þ�1−

1

R2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx−EzzÞ2þ4E2

xz

q �12

; (23)

where R ¼ β∕α ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0.5 − σ∕1 − σ

p, σ is the

Poisson’s ratio, andGs ¼ 2π∕ksΔx is the numberof grid points per shear wavelength. For the nor-malized P-wave phase velocity (equation 22), theexpressions kxΔx and kzΔz should be replaced by

kxΔx ¼ 2πR sin θ

Gs; kzΔz ¼

2πR cos θ

Gsr; (24)

and for the normalized S-wave phase velocity (equation 23), the ex-pressions kxΔx and kzΔz should be replaced by

kxΔx ¼ 2π sin θ

Gs; kzΔz ¼

2π cos θ

Gsr; (25)

where θ is the propagation angle.The coefficients γ1, γ2, c, and d are determined by minimizing the

phase error:

Eðγ1;γ2;c;d;r;σÞ¼Z Z �

1−αphð~k;θ;γ1;γ2;;c;d;r;σÞ

α

�2

þ�1−

βphð~k;θ;γ1;γ2;;c;d;r;σÞβ

�2d~kdθ; (26)

where ~k ¼ 1∕Gs.

•

••

• •

•

•

•

•w

m,num,n

um−1,n u

m+1,nw

m+1,nwm−1,n

um−1,n+1

um−1,n−1

wm−1,n+1

wm−1,n−1

um,n+1

um,n−1

wm,n+1

wm,n−1

um+1,n+1

um+1,n−1

wm+1,n+1

wm+1,n−1

γ1

(1−γ1)/2

(1−γ1)/2

γ2(1−γ

2)/2 (1−γ

2)/2

x

z

Figure 1. The stencil of the average-derivative nine-point scheme.

Table 1. The optimization coefficients for different σ withr � Δx∕Δz � 1.

γ1 γ2 c d

σ ¼ 0.1 0.6814 0.6814 0.6279 0.0923

σ ¼ 0.15 0.6702 0.6702 0.6249 0.0931

σ ¼ 0.2 0.6577 0.6577 0.6249 0.0938

σ ¼ 0.25 0.6439 0.6439 0.6254 0.0936

σ ¼ 0.3 0.6277 0.6277 0.6265 0.0934

σ ¼ 0.35 0.6079 0.6079 0.6283 0.0929

σ ¼ 0.4 0.5828 0.5828 0.6308 0.0923

σ ¼ 0.45 0.5488 0.5488 0.6337 0.0916

σ ¼ 0.5 0.5 0.5 0.5096 0.1226

Table 2. The optimization coefficients for different σ withr � Δx∕Δz � 1.25.

γ1 γ2 c d

σ ¼ 0.1 0.7334 0.6528 0.6284 0.0929

σ ¼ 0.15 0.7179 0.6448 0.6282 0.0930

σ ¼ 0.2 0.7009 0.6356 0.6285 0.0930

σ ¼ 0.25 0.6875 0.6211 0.6285 0.0929

σ ¼ 0.3 0.6646 0.6087 0.6303 0.0924

σ ¼ 0.35 0.6535 0.6001 0.7183 0.0491

σ ¼ 0.4 0.6053 0.5710 0.6344 0.0914

σ ¼ 0.45 0.5619 0.5419 0.6373 0.0907

σ ¼ 0.5 0.5 0.5 0.5145 0.1214

Table 3. The optimization coefficients for different σ withr � Δx∕Δz � 2.

γ1 γ2 c d

σ ¼ 0.1 0.9987 0.6118 0.6297 0.0926

σ ¼ 0.15 0.9649 0.6062 0.6296 0.0926

σ ¼ 0.2 0.9294 0.5994 0.6297 0.0926

σ ¼ 0.25 0.8854 0.5923 0.6302 0.0924

σ ¼ 0.3 0.8414 0.5830 0.6502 0.0826

σ ¼ 0.35 0.7818 0.5699 0.6321 0.0919

σ ¼ 0.4 0.7160 0.5546 0.6550 0.0810

σ ¼ 0.45 0.6288 0.5331 0.7050 0.0566

σ ¼ 0.5 0.5 0.5 0.5306 0.1173

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The integration in equation 26 is usually used to simplify theoptimization process (Jo et al., 1996). The ranges of ~k and θ aretaken as ½0; 0.25� and ½0; π∕2�, respectively. We use a constrainednonlinear optimization program fmincon in MATLAB to deter-mine the optimization coefficients. This program consists of three

kinds of algorithms: trust region reflective, active set, and interiorpoint. The program fmincon is widely used in constrained non-linear optimizations. The optimization coefficients for different σwith r ¼ Δx∕Δz ¼ 1, r ¼ Δx∕Δz ¼ 1.25, and r ¼ Δx∕Δz ¼ 2

are listed in Tables 1, 2, and 3, respectively.

0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

αβ

βα

ph/

Classical nine-point scheme

0° 15° 30° 45°

0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

Average-derivative nine-point scheme

0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

ph/

αβ

βα

ph/

ph/


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

Averag-derivative nine-point scheme

0° 15° 30° 45°

0° 15° 30° 45°0° 15° 30° 45°

Figure 2. Normalized compressional and shearphase velocity of the classical nine-point schemeand the average-derivative optimal nine-pointscheme for different propagation angles θ. Here,σ ¼ 0.25 and r ¼ Δx∕Δz ¼ 1.

0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

ph/


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

ββ

αα

ph/

αα

ph/

ββ

ph/


0° 15° 30° 45°0° 15° 30° 45°

0° 15° 30° 45° 0° 15° 30° 45°



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Now, we make a numerical dispersion analysis and we first con-sider the case in which r ¼ Δx∕Δz ¼ 1. Figure 2 shows normalizedcompressional and shear phase velocity of the classical nine-pointscheme and the average-derivative optimal nine-point scheme fordifferent propagation angles θ and for σ ¼ 0.25. Compared with

the classical nine-point scheme, the average-derivative optimalnine-point scheme reduces the phase velocity errors, particularlyfor the shear phase velocity. This difference in errors reduction be-tween compressional and shear phase velocity probably results fromthe optimization method. Within the phase errors of 2%, the aver-

0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

ph/


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs

ββ

αα

ph/

αα

ph/

ββ

ph/


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0° 15° 30° 45° 0° 15° 30° 45°

0° 15° 30° 45°0° 15° 30° 45°


0 0.05 0.1 0.15 0.2 0.250.7

0.8

0.9

1

1.1

1.2

1/Gp


0 0.05 0.1 0.15 0.2 0.250.7

0.8

0.9

1

1.1

1.2

1/Gp

αα

βα

ph/

αα

ph/


0 0.05 0.1 0.15 0.2 0.25−0.5

0

0.5

1/Gp

ph/ α

β ph/


0 0.05 0.1 0.15 0.2 0.25−0.5

0

0.5

1/Gp


0° 15° 30° 45° 0° 15° 30° 45°

0° 15° 30° 45° 0° 15° 30° 45°


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age-derivative optimal nine-point scheme requires four grid pointsper shear wavelength, whereas for the classical nine-point scheme,it requires 10 grid points per shear wavelength. Figure 3 showsnormalized compressional and shear phase velocity of the classicalnine-point scheme and the average-derivative optimal nine-pointscheme for different propagation angles θ and for σ ¼ 0.4. We

can see that as σ increases, the shear phase velocity errors becomelarger for the classical nine-point scheme. In this case, within thephase errors of 2%, the average-derivative optimal nine-point schemestill requires four grid points per shear wavelength, whereas for theclassical nine-point scheme, it requires 20 grid points per shear wave-length.

0 0.05 0.1 0.15 0.2 0.250.7

0.8

0.9

1

1.1

1.2

1/Gp

αα

αβ

ph/

αα

ph/


0 0.05 0.1 0.15 0.2 0.250.7

0.8

0.9

1

1.1

1.2

1/Gp


0 0.05 0.1 0.15 0.2 0.25−0.5

0

0.5

1/Gp

ph/ α

β ph/


0 0.05 0.1 0.15 0.2 0.25−0.5

0

0.5

1/Gp


0° 15° 30° 45°0° 15° 30° 45°

0° 15° 30° 45°0° 15° 30° 45°


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


0 0.05 0.1 0.15 0.2 0.250.9

0.95

1

1.05

1.1

1/Gs


αα

ph/

ββ ph

/

αα

ph/

ββ ph

/

0° 15° 30° 45°0° 15° 30° 45°

0° 15° 30° 45°0° 15° 30° 45°



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In Figure 4, we show the case in which σ ¼ 0.5. In this case,the theoretical shear phase velocity should be zero. For the classicalnine-point scheme, it produces spurious shear waves and cannotbe used in this case as a result. For the average-derivative optimalnine-point scheme, the S-wave phase velocity is zero. Within thephase errors of 5%, the average-derivative optimal nine-pointscheme requires four grid points per compressional wavelength. Fig-ures 5, 6, and 7 show the situation in which r ¼ Δx∕Δz ¼ 2. Be-cause of the relatively smaller Δz, the corresponding phase errorsare all smaller.In our research, we are only concerned with phase velocity

errors. For completeness, however, we have derived the normal-ized compressional group velocity αgr∕α and shear group velocityβgr∕β:

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u

AnalyticalClassic nine-point scheme Average-derivative scheme

−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u

Classic nine-point scheme Average-derivative scheme

−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

a)

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u

−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u

−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

b)







Figure 9. Frequency-domain seismograms (ω ¼30π) computed with the analytical formula (greenline), the classic nine-point scheme (blue line), andthe average-derivative scheme (red line). (a) σ ¼0.25 and (b)σ ¼ 0.4. Here, Δx∕Δz ¼ 1.

∇∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇∇∇∇∇∇∇

Perfectly matched layers

*

receivers

source

Figure 8. Schematic of the homogeneous model.

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αgrα

¼ 1

4Eωα

�ð1þ R2ÞðExx;k þ Ezz;kÞ þ ð1 − R2Þ

×ðExx − EzzÞðExx;k − Ezz;kÞ þ 4ExzExz;kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðExx − EzzÞ2 þ 4E2xz

p�

−Ek

4E2ωα

�ð1þ R2ÞðExx þ EzzÞ

þ ð1 − R2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx − EzzÞ2 þ 4E2

xz

q �; (27)

βgrβ

¼ 1

4Eωβ

��1þ 1

R2

�ðExx;k þ Ezz;kÞ þ

�1 −

1

R2

�

×ðExx − EzzÞðExx;k − Ezz;kÞ þ 4ExzExz;kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðExx − EzzÞ2 þ 4E2xz

p�

−Ek

4E2ωβ

��1þ 1

R2

�ðExx þ EzzÞ þ

�1 −

1

R2

�

×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx − EzzÞ2 þ 4E2

xz

q �; (28)

where

−500 0 500

−1

−0.5

0

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1

Offset (m)

Nor

mal

ized

am

plitu

de

u


−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u


−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

a)

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u

−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u

−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

b)







Figure 10. Frequency-domain seismograms (ω ¼60π) computed with the analytical formula (greenline), the classic nine-point scheme (blue line), andthe average-derivative scheme (red line). (a) σ ¼0.25 and (b)σ ¼ 0.4. Here, Δx∕Δz ¼ 1.


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ωα¼

1

2E

�ð1þR2ÞðExxþEzzÞþð1−R2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx−EzzÞ2þ4E2

xz

q �12

;

ωβ¼

1

2E

��1þ 1

R2

�ðExxþEzzÞþ

�1−

1

R2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðExx−EzzÞ2þ4E2

xz

q �12

;

Ek¼−2d�sinðkxΔxÞsinθþ

sinðkzΔzÞcosθr

�−4esinðkxΔxÞcosðkzΔzÞ

×sinθ−4ercosðkxΔxÞsinðkzΔzÞcosθ;

Exx;k¼−1−γ1r

sinðkzΔzÞcosθ½2−2cosðkxΔxÞ�þ2½ð1−γ1ÞcosðkzΔzÞþγ1�sinðkxΔxÞsinθ;Ezz;k¼−r2ð1−γ2ÞsinðkxΔxÞsinθ½2−2cosðkzΔzÞ�þ2r½ð1−γ2ÞcosðkxΔxÞþγ2�sinðkzΔzÞcosθ;Exz;k¼rcosðkxΔxÞsinðkzΔzÞsinθþsinðkxΔxÞcosðkzΔzÞcosθ: (29)

Compared with the phase velocity, the group velocity has largererrors. The average-derivative optimal nine-point scheme still hassmaller errors in comparison with the classical nine-point scheme.Because we mainly deal with the phase velocity, we have not shownthe results here.

NUMERICAL EXAMPLES

In this section, numerical examples are presented to make com-parisons between the average-derivative optimal nine-point schemeand the classic nine-point scheme.

A homogeneous model

We first consider a homogeneous velocity model for which an ana-lytical solution is available. The P-wave velocity is 3500 m∕s. We

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u


−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u


−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

a)

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u

−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u

−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

b)







Figure 11. Frequency-domain seismograms(ω ¼ 30π) computed with the analytical formula(green line), the classic nine-point scheme (blueline), and the average-derivative scheme (red line).(a) σ ¼ 0.25 and (b) 0.4. Here, Δx∕Δz ¼ 2.

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take a constant density ρ ¼ 2000 kg∕m3. For the Poisson’s ratio, wetake σ ¼ 0.25 and 0.4, respectively. The corresponding S-wavevelocities are 2021 and 1429 m∕s, respectively. The numbers ofhorizontal and vertical grid points are both 101. To make a compari-son with the analytical solution in the full space, perfectly matchedlayer (PML) boundary conditions are used on the four sides of thecomputational domain (Appendix B). Figure 8 shows the schematicof the homogeneous model.We compute the frequency-domain numerical solutions and com-

pare them with the analytical solution (Min et al., 2000). First, wetake the circular frequency ω to be 30π. According to the criterionof four grid points per shear wavelength, the sampling intervals forσ ¼ 0.25 and 0.4 are 32 and 22 m, respectively. A single vertical

force is applied at the center of the model with a Ricker wavelettime history, and a receiver array is placed at the 17th layer inthe z-direction. Second, the circular frequency ω is taken to be60π. According to the criterion of four grid points per shear wave-length, the sampling intervals for σ ¼ 0.25 and 0.4 are 16 and 11 m,respectively. A receiver array is placed at the 34th layer in the z-direction.Figures 9 and 10 show frequency-domain seismograms (u and w)

computed with the analytic formula, the classical nine-point scheme,and the average-derivative optimal nine-point scheme for ω ¼ 30πand 60π, respectively, where r ¼ Δx∕Δz ¼ 1. The relative errorsof the two nine-point schemes in comparison with the analyticalsolution are also displayed. Figures 11 and 12 show the correspond-

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

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ized

am

plitu

de

u


−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u


−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

a)

−500 0 500

−1

−0.5

0

0.5

1

Offset (m)

Nor

mal

ized

am

plitu

de

u

−500 0 500

0

20

40

60

80

100

Offset (m)

Rel

ativ

e er

ror

(%)

u

−500 0 500−1

−0.5

0

0.5

1

Nor

mal

ized

am

plitu

de

w

−500 0 500

0

20

40

60

80

100

Rel

ativ

e er

ror

(%)

w

b)







Figure 12. Frequency-domain seismograms(ω ¼ 60π) computed with the analytical formula(green line), the classic nine-point scheme (blueline), and the average-derivative scheme (red line).(a) σ ¼ 0.25 and (b) 0.4. Here, Δx∕Δz ¼ 2.


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ing results for the case in which r ¼ Δx∕Δz ¼ 2. From these fig-ures, we can see that the results of the average-derivative optimalnine-point scheme are in relatively good agreement with the ana-lytical results. The results of the classic nine-point scheme exhibitlarge errors due to numerical dispersion. It should be noted that thecomputational cost of the classic nine-point scheme and the aver-age-derivative optimal nine-point scheme is approximately thesame.

Lamb’s problem

To address Lamb’s problem, a proper numerical treatment of thefree-surface boundary condition is essential. In our example, we use

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time (s)

Nor

mal

ized

am

plitu

de

u


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time (s)

Nor

mal

ized

am

plitu

de

w


a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time (s)

Nor

mal

ized

am

plitu

de

u


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time (s)

Nor

mal

ized

am

plitu

de

w


b)

Figure 14. Time-domain seismograms of Lamb’sproblem computed with the analytical solution(green line), the classic nine-point scheme (blueline), and the average-derivative optimal nine-pointscheme (red line) at the free-surface with an offsetof 660 m: Poisson’s ratio (a) σ ¼ 0.25 and(b) σ ¼ 0.4.

PML layers

source*

∇receiver

Figure 13. Schematic of the half-space model.

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an improved free-surface expression presented in Cao et al. (2016).This improved free-surface expression is based on the parameter-modified expression for the free surface (Mittet, 2002; Xu et al.,2007). The analytical solution is calculated with the Cagniard-deHoop method (Aki and Richards, 1980).A finite-size model of 1200 × 240 m is considered in our com-

putation. We take two Poisson’s ratios σ ¼ 0.25 and 0.4. The cor-responding grid intervals are Δx ¼ 6 m, Δz ¼ 6 m and Δx ¼ 4 m,Δz ¼ 4 m, respectively. To simulate wave propagation in the half-space, PML boundary conditions are used on the other three sides ofthe model (Figure 13). A single vertical force is applied at the depthof 48 m. Figure 14 shows the time-domain seismograms (u and w)computed with the analytic solution, the classical nine-pointscheme, and the average-derivative optimal nine-point scheme atthe free-surface with an offset of 660 m. In comparison with theclassical nine-point scheme, the average-derivative optimal nine-point scheme agrees with the analytical solution better. The remain-ing errors are probably due to the inaccuracies in numerical treat-ment of the free-surface boundary condition.

Overthrust model

We now consider part of the overthrust model with a Poisson’sratio of 0.25 and a constant density of 2300 kg∕m3 (Figure 15). Onthe top side, we use a centered finite-difference free-surface boun-dary condition (Alterman and Karal, 1968). For the remaining threesides, PML boundary conditions are applied. The numbers of hori-zontal and vertical grid points are nx ¼ 275 and nz ¼ 75, respec-tively. We take Δx ¼ 10 m and Δz ¼ 5 m. A single vertical forcewith a Ricker wavelet time history is applied at (x ¼ 1370 m andz ¼ 50 m). Two receiver lines are set at the depth of 30 and 180 m.The circular frequency ω is taken to be 60π.For this model, an analytic solution is not available. To make

comparisons, we use a method of finer grids. Figure 16 shows fre-quency-domain seismograms (u and w) at receiver line 1 computedwith the classic nine-point scheme with Δx ¼ 10 m and Δz ¼ 5 m,the classic nine-point scheme with Δx ¼ 5 m and Δz ¼ 2.5 m, andthe average-derivative optimal nine-point scheme with Δx ¼ 10 m

and Δz ¼ 5 m. Figure 17 shows the corresponding results atreceiver line 2. We can see that the results of the average-derivative

optimal nine-point scheme on a coarse grid are closer to those of theclassic nine-point scheme on a fine grid. This demonstrates that theaverage-derivative optimal nine-point scheme is more accurate thanthe classical nine-point scheme when these two schemes are com-puted on the grid with the same grid intervals.

Figure 15. Part of the overthrust model (P-wave velocity). We takeσ ¼ 0.25 and a constant density ρ ¼ 2000 kg∕m3.

−1000 −500 0 500 1000

−1

−0.5

0

1

u

Offset (m)

Nor

mal

ized

am

plitu

de classic nine-point scheme with Δx = 10 m and Δz = 5 m classic nine-point scheme with Δx = 5 m and Δz = 2.5 m average-derivative scheme with Δx = 10 m and Δz = 5 m

−1000 −500 0 500 1000−1

−0.5

0

0.5

0.5

1

w

receiver line 1

Nor

mal

ized

am

plitu


Figure 16. Frequency-domain seismograms (upper plot: u andlower plot: w) at receiver line 1 computed with the classic nine-pointschemewithΔx ¼ 10 m andΔz ¼ 5 m (blue line), the classic nine-point scheme with Δx ¼ 5 m and Δz ¼ 2.5 m (green line), and theaverage-derivative scheme with Δx ¼ 10 m and Δz ¼ 5 m (redline).

−1000 −500 0 500 1000

−1

−0.5

0

0.5

1

u

Offset (m)

Nor

mal

ized

am

plitu

de

−1000 −500 0 500 1000−1

−0.5

0

0.5

1

w

receiver line 2

Nor

mal

ized

am

plitu


classic nine-point scheme with Δx = 10 m and Δz = 5 m classic nine-point scheme with Δx = 5 m and Δz = 2.5 m average-derivative scheme with Δx = 10 m and Δz = 5 m

Figure 17. Frequency-domain seismograms (upper plot: u andlower plot: w) at receiver line 2 computed with the classic nine-pointschemewithΔx ¼ 10 m andΔz ¼ 5 m (blue line), the classic nine-point scheme with Δx ¼ 5 m and Δz ¼ 2.5 m (green line), and theaverage-derivative scheme with Δx ¼ 10 m and Δz ¼ 5 m (redline).


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Salt model

We now consider a salt model that contains various Poisson’sratios including σ ¼ 0.5. Figure 18 shows the P-wave velocity,S-wave velocity, density, and Poisson’s ratio of the model (Houseet al., 2000). The parameters of this salt model are nx ¼ nz ¼ 251,Δx ¼ 5 m, and Δz ¼ 4 m. The optimization coefficients forthis directional grid interval are listed in Table 2. For a Poisson’sratio that is not covered in Table 2, we use linear interpolationto obtain the corresponding optimization coefficients. An explosiveRicker source with peak frequency of 15 Hz is applied at(x ¼ 625 m and z ¼ 120 m), and a receiver array is set at the depthof 40 m.Figure 19 shows monochromatic wavefields of 15 and 30 Hz

computed with the classic nine-point scheme and the average-derivative optimal nine-point scheme. Due to the spurious S-wavesgenerated by the classical nine-point scheme, the wavefieldscomputed with the classical nine-point scheme exhibit badly con-taminated results in the water, particularly for the wavefield of15 Hz. These spurious S-waves are a kind of artifacts producedby numerical discretizations. The reason why we call themspurious S-waves is because these artifacts stem from the S-wavenumerical dispersion relation of the classical nine-point schemewhen the Poisson’s ratio is 0.5. On the other hand, the wavefieldscomputed with the average-derivative optimal nine-point schemeshow clean results in the water. This is because the numericalS-wave phase velocity of the average-derivative optimal nine-pointscheme becomes zero in the water. We show the time-domain seis-mograms computed with the classic nine-point scheme and theaverage-derivative optimal nine-point scheme in Figure 20. Thesetime-domain seismograms are obtained by temporal Fourier trans-form of frequency-domain wavefields. In comparison with the seis-mogram computed with the average-derivative optimal nine-pointscheme, the seismogram computed with the classical nine-pointscheme is dominated by low-velocity spurious S-waves and thusbecome unacceptable.

Figure 18. The salt model: (a) P-wave velocity,(b) S-wave velocity, (c) density, and (d) Poisson’sratio.

Figure 19. (a) Monochromatic wavefields of the salt model com-puted with the classic nine-point scheme and (b) the average-deriva-tive optimal nine-point scheme.

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CONCLUSIONS

We have developed an average-derivative optimal nine-pointscheme for frequency domain 2D elastic-wave equation. Comparedwith the classical nine-point scheme, this new nine-point schemereduces the number of grid points per shear wavelength from 10or 20 to 4 for σ ¼ 0.25 or 0.4. For σ ¼ 0.5, the S-wave phase veloc-ity of the new nine-point scheme is zero, whereas the classical nine-point scheme produces spurious S-waves. These conclusions applyto arbitrary directional grid intervals. Numerical examples demon-strate that the new nine-point scheme produces more accurate mod-eling results in comparison with the classical nine-point scheme atapproximately the same computational cost. This new nine-pointscheme is also generalized to an average-derivative 27-point schemefor frequency-domain 3D elastic-wave equation.

ACKNOWLEDGMENTS

We would like to thank the anonymous reviewers for valuablesuggestions. This work is supported by National Natural ScienceFoundation of China under grant nos. 41474104 and 41274139.

APPENDIX A

3D CASE

We consider the 3D elastic-wave equation in the frequency do-main

∂∂x

�η∂u∂x

�þ ∂∂y

�μ∂u∂y

�þ ∂∂z

�μ∂u∂z

�þ ∂∂x

�λ∂v∂y

�

þ ∂∂y

�μ∂v∂x

�þ ∂∂x

�λ∂w∂z

�þ ∂∂z

�μ∂w∂x

�þρω2u¼0; (A-1)

∂∂x

�μ∂v∂x

�þ ∂∂y

�η∂v∂y

�þ ∂∂z

�μ∂v∂z

�þ ∂∂y

�λ∂w∂z

�

þ ∂∂z

�μ∂w∂y

�þ ∂∂y

�λ∂u∂x

�þ ∂∂x

�μ∂u∂y

�þρω2v¼0; (A-2)

∂∂x

�μ∂w∂x

�þ ∂∂y

�μ∂w∂y

�þ ∂∂z

�η∂w∂z

�þ ∂∂z

�λ∂u∂x

�

þ ∂∂x

�μ∂u∂z

�þ ∂∂z

�λ∂v∂y

�þ ∂∂y

�μ∂v∂z

�þρω2w¼0; (A-3)

where u, v, and w are the particle displacements in the x-, y-, andz-directions, respectively.Using the same approach as the 2D case, an average-derivative

27-point scheme for equations A-1–A-3 can be obtained asfollows:

1

Δx2hηmþ1

2;l;numþ1;l;n−

�ηmþ1

2;l;nþηm−1

2;n

�um;nþηm−1

2;l;num−1;l;n

i

þ 1

Δy2hμm;l;nþ1

2um;lþ1;n−

�μm;lþ1

2;nþμm;l−1

2;n

�um;l;nþμm;l−1

2;num;l−1;n

i

þ 1

Δz2hμm;l;nþ1

2~um;l;nþ1−

�μm;l;nþ1

2þμm;l;n−1

2

�~um;l;nþμm;l;n−1

2~um;l;n−1

i

þ 1

4ΔxΔy½λmþ1;l;nðwmþ1;lþ1;n−wmþ1;l−1;nÞ−λm−1;l;nðwm−1;lþ1;n−wm−1;l−1;nÞ�

þ 1

4ΔxΔy½μm;lþ1;nðwmþ1;lþ1;n−wm−1;lþ1;nÞ−μm;l−1;nðwmþ1;l−1;n−wm−1;l−1;nÞ�

þ 1

4ΔxΔz½λmþ1;l;nðwmþ1;l;nþ1−wmþ1;l;n−1Þ−λm−1;l;nðwm−1;l;nþ1−wm−1;l;n−1Þ�

þ 1

4ΔxΔz½μm;nþ1ðwmþ1;l;nþ1−wm−1;l;nþ1Þ−μm;l;n−1ðwmþ1;l;n−1−wm−1;l;n−1Þ�

þρm;l;nω2ðcum;l;nþdAþeBþfCÞ¼0; (A-4)

Figure 20. (a) Time-domain seismograms computed with the clas-sic nine-point scheme and (b) the average-derivative optimal nine-point scheme.


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1

Δx2

�μmþ1

2;l;nvmþ1;l;n−

�μmþ1

2;l;nþμm−1

2;n

�vm;nþηm−1

2;l;num−1;l;n

�

þ 1

Δy2

�ηm;l;nþ1

2vm;lþ1;n−

�ηm;lþ1

2;nþηm;l−1

2;n

�vm;l;nþμm;l−1

2;nvm;l−1;n

�

þ 1

Δz2

�μm;l;nþ1

2~vm;l;nþ1−

�μm;l;nþ1

2þμm;l;n−1

2

�~vm;l;nþμm;l;n−1

2~vm;l;n−1

�

þ 1

4ΔyΔz½λm;lþ1;nðwm;lþ1;nþ1−wm;lþ1;n−1Þ−λm;l−1;nðwm;l−1;nþ1−wm;l−1;n−1Þ�

þ 1

4ΔzΔy½μm;l;nþ1ðwm;lþ1;nþ1−wm;l−1;nþ1Þ−μm;l;n−1ðwm;lþ1;n−1−wm;l−1;n−1Þ�

þ 1

4ΔyΔx½λm;lþ1;nðumþ1;lþ1;n−um−1;lþ1;nÞ−λm;l−1;nðumþ1;l−1;n−um−1;l−1;nÞ�

þ 1

4ΔxΔy½μmþ1;l;nðumþ1;lþ1;n−umþ1;l−1;nÞ−μm−1;l;nðum−1;lþ1;n−um−1;l−1;nÞ�

þρm;l;nω2ðcvm;l;nþdAþeBþfCÞ¼0; (A-5)

1

Δx2hμmþ1

2;l;nwmþ1;l;n−

�μmþ1

2;l;nþμm−1

2;n

�wm;nþμm−1

2;l;nwm−1;l;n

i

þ 1

Δy2hμm;l;nþ1

2wm;lþ1;n−

�μm;lþ1

2;nþμm;l−1

2;n

�wm;l;nþμm;l−1

2;nwm;l−1;n

i

þ 1

Δz2hηm;l;nþ1

2~wm;l;nþ1−

�ηm;l;nþ1

2þηm;l;n−1

2

�~wm;l;nþηm;l;n−1

2~wm;l;n−1

i

þ 1

4ΔzΔx½λmþ1;l;nðumþ1;l;nþ1−um−1;l;nþ1Þ−λm−1;l;nðumþ1;l;n−1−um−1;l;n−1Þ�

þ 1

4ΔxΔz½μmþ1;l;nðumþ1;l;nþ1−umþ1;l;n−1Þ−μm−1;l;nðum−1;l;nþ1−um−1;l;n−1Þ�

þ 1

4ΔzΔy½λm;l;nþ1ðvm;lþ1;nþ1−vm;l−1;nþ1Þ−λm;l;n−1ðvm;lþ1;n−1−vm;l−1;n−1Þ�

þ 1

4ΔyΔz½μm;lþ1;nðvm;lþ1;nþ1−vm;lþ1;n−1Þ−μm;l−1;nðvm;l−1;nþ1−vm;l−1;n−1Þ�

þρm;l;nω2ðcwm;l;nþdAþeBþfCÞ¼0; (A-6)

where

pmþj;l;n¼α1ðpmþj;lþ1;nþpmþj;l;nþ1þpmþj;l−1;nþpmþj;l;n−1Þþα2ðpmþj;lþ1;nþ1þpmþj;l−1;nþ1þpmþj;lþ1;n−1þpmþj;l−1;n−1Þþð1−4α1−4α2Þpmþj;l;n; j¼−1;0;1;

pm;lþj;n¼β1ðpmþ1;lþj;nþpm;lþj;nþ1þpm−1;lþj;nþpm;lþj;n−1Þþβ2ðpmþ1;lþj;nþ1þpmþ1;lþj;n−1þpm−1;lþj;nþ1þpm−1;lþj;n−1Þþð1−4β1−4β2Þpm;lþj;n; j¼−1;0;1;

~pm;l;nþj¼γ1ðpmþ1;l;nþjþpm;lþ1;nþjþpm−1;l;nþjþpm;l−1;nþjÞþγ2ðpmþ1;lþ1;nþjþpmþ1;l−1;nþjþpm−1;lþ1;nþjþpm−1;l−1;nþjÞþð1−4γ1−4γ2Þpm;l;nþj; j¼−1;0;1;

A¼ðpm;lþ1;nþpm;l;nþ1þpm;l−1;nþpm;l;n−1þpmþ1;l;nþpm−1;l;nÞ;B¼ðpmþ1;lþ1;nþpmþ1;l;nþ1þpmþ1;l−1;nþpmþ1;l;n−1þpm−1;lþ1;nþpm−1;l;nþ1

þpm−1;l−1;nþpm−1;l;n−1þpm;lþ1;nþ1þpm;l−1;nþ1þpm;lþ1;n−1þpm;l−1;n−1Þ;C¼ðpmþ1;lþ1;nþ1þpmþ1;l−1;nþ1þpmþ1;lþ1;n−1þpmþ1;l−1;n−1

þpm−1;lþ1;nþ1þpm−1;l−1;nþ1þpm−1;lþ1;n−1þpm−1;l−1;n−1Þ; (A-7)

where p represents u, v, or w, and α1, α2, β1, β2, γ1, γ2, c, d, and eare coefficients. Here, f ¼ ð1 − c − 6d − 12eÞ∕8.

APPENDIX B

IMPLEMENTATION OF PML BOUNDARYCONDITIONS

The 2D frequency-domain elastic-wave equations with PMLboundary conditions are

1

ξx

∂∂x

�η

ξx

∂u∂x

�þ 1

ξz

∂∂z

�μ

ξz

∂u∂z

�þ 1

ξx

∂∂x

�λ

ξz

∂w∂z

�

þ 1

ξz

∂∂z

�μ

ξx

∂w∂x

�þ ρω2u ¼ 0; (B-1)

1

ξx

∂∂x

�μ

ξx

∂w∂x

�þ 1

ξz

∂∂z

�η

ξz

∂w∂z

�þ 1

ξx

∂∂x

�μ

ξz

∂u∂z

�

þ 1

ξz

∂∂z

�λ

ξx

∂u∂x

�þ ρω2w ¼ 0; (B-2)

where

ξxðxÞ ¼ 1 −icxω

cos

�π

2

xLx

�; (B-3)

ξzðzÞ ¼ 1 −iczω

cos

�π

2

zLz

�; (B-4)

where Lx and Lz denote the width of the PML layer in x-, and z-di-rections, respectively. The coordinates x and z are local coordinateswhose origins are located at the outer edges of the PML layers. Thescalars cx and cz are determined by trial and error (Operto etal., 2007).The corresponding average-derivative optimal nine-point scheme

becomes

1

ξxmΔx2

�ηmþ1

2;n

ξxmþ12

umþ1;n−�ηmþ1

2;n

ξxmþ12

þηm−1

2;n

ξxm−12

�um;nþ

ηm−12;n

ξxm−12

um−1;n

�

þ 1

ξznΔz2

�μm;nþ1

2

ξznþ12

~um;nþ1−�μm;nþ1

2

ξznþ12

þμm;n−1

2

ξzn−12

�~um;nþ

μm;n−12

ξzn−12

~um;n−1

�

þ 1

4ξxmΔxΔz

�λmþ1;n

ξznðwmþ1;nþ1−wmþ1;n−1Þ−

λm−1;n

ξznðwm−1;nþ1−wm−1;n−1Þ

�

þ 1

4ξznΔxΔz

�μm;nþ1

ξxmðwmþ1;nþ1−wm−1;nþ1Þ−

μm;n−1

ξxmðwmþ1;n−1−wm−1;n−1Þ

�

þρm;nω2½cum;nþdðumþ1;nþum−1;nþum;nþ1þum;n−1Þ

þeðumþ1;nþ1þum−1;nþ1þumþ1;n−1þum−1;n−1Þ�¼0; (B-5)

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1

ξxmΔx2

�μmþ1

2;n

ξxmþ12

wmþ1;n−�μmþ1

2;n

ξxmþ12

þμm−1

2;n

ξxm−12

�wm;nþ

μm−12;n

ξxm−12

wm−1;n

�

þ 1

ξznΔz2

�ηm;nþ1

2

ξznþ12

~wm;nþ1−�ηm;nþ1

2

ξznþ12

þηm;n−1

2

ξzn−12

�~wm;nþ

ηm;n−12

ξzn−12

~wm;n−1

�

þ 1

4ξxmΔxΔz

�μmþ1;n

ξznðumþ1;nþ1−umþ1;n−1Þ−

μm−1;n

ξznðum−1;nþ1−um−1;n−1Þ

�

þ 1

4ξznΔxΔz

�λm;nþ1

ξxmðumþ1;nþ1−um−1;nþ1Þ−

λm;n−1

ξxmðumþ1;n−1−um−1;n−1Þ

�

þρm;nω2½cwm;nþdðwmþ1;nþum−1;nþwm;nþ1þwm;n−1Þ

þeðwmþ1;nþ1þwm−1;nþ1þwmþ1;n−1þwm−1;n−1Þ�¼0: (B-6)

We can rewrite schemes B-5 and B-6 as

c1um−1;n−1þc2um;n−1þc3umþ1;n−1þc4um−1;nþc5um;n

þc6umþ1;nþc7um−1;nþ1þc8um;nþ1þc9umþ1;nþ1

þd1wm−1;n−1þd2wmþ1;n−1þd3wm−1;nþ1þd4wmþ1;nþ1¼0;

(B-7)

~d1wm−1;n−1þ ~d2wm;n−1þ ~d3wmþ1;n−1þ ~d4wm−1;nþ ~d5wm;n

þ ~d6wmþ1;nþ ~d7wm−1;nþ1þ ~d8wm;nþ1þ ~d9wmþ1;nþ1

þ ~c1um−1;n−1þ ~c2umþ1;n−1þ ~c3um−1;nþ1þ ~c4umþ1;nþ1þ¼0;

(B-8)

where

c1¼1

ξxmΔx2

ηm−12;n

ξxm−12

1−γ12

þ 1

ξznΔz2

μm;n−12

ξzn−12

1−γ22

þρm;nω2e;

c2¼−1

ξxmΔx2

�ηmþ1

2;n

ξxmþ12

þηm−1

2;n

ξxm−12

�1−γ12

þ 1

ξznΔz2

μm;n−12

ξzn−12

γ2þρm;nω2d;

c3¼1

ξxmΔx2

ηmþ12;n

ξxmþ12

1−γ12

þ 1

ξznΔz2

μm;n−12

ξzn−12

1−γ22

þρm;nω2e;

c4¼1

ξxmΔx2

ηm−12;n

ξxm−12

γ1−1

ξznΔz2

�μm;nþ1

2

ξznþ12

þμm;n−1

2

ξzn−12

�1−γ22

þρm;nω2d;

c5¼−1

ξxmΔx2

�ηmþ1

2;n

ξxmþ12

þηm−1

2;n

ξxm−12

�γ1−

1

ξznΔz2

�μm;nþ1

2

ξznþ12

þμm;n−1

2

ξzn−12

�γ2þρm;nω

2c;

c6¼1

ξxmΔx2

ηmþ12;n

ξxmþ12

γ1−1

ξznΔz2

�μm;nþ1

2

ξznþ12

þμm;n−1

2

ξzn−12

�1−γ22

þρm;nω2d;

c7¼1

ξxmΔx2

ηm−12;n

ξxm−12

1−γ12

þ 1

ξznΔz2

μm;nþ12

ξznþ12

1−γ22

þρm;nω2e;

c8¼−1

ξxmΔx2

�ηmþ1

2;n

ξxmþ12

þηm−1

2;n

ξxm−12

�1−γ12

þ 1

ξznΔz2

μm;nþ12

ξznþ12

γ2þρm;nω2d;

c9¼1

ξxmΔx2

ηmþ12;n

ξxmþ12

1−γ12

þ 1

ξznΔz2

μm;nþ12

ξznþ12

1−γ22

þρm;nω2e;

d1¼1

4ξxmΔxΔzλm−1;n

ξznþ 1

4ξznΔxΔzμm;n−1

ξxm;

d2¼−1

4ξxmΔxΔzλmþ1;n

ξzn−

1

4ξznΔxΔzμm;n−1

ξxm;

d3¼−1

4ξxmΔxΔzλm−1;n

ξzn−

1

4ξznΔxΔzμm;nþ1

ξxm; d4¼

1

4ξxmΔxΔzλmþ1;n

ξznþ 1

4ξznΔxΔzμm;nþ1

ξxm;

~d1¼1

ξxmΔx2

μm−12;n

ξxm−12

1−γ12

þ 1

ξznΔz2

ηm;n−12

ξzn−12

1−γ22

þρm;nω2e;

~d2¼−1

ξxmΔx2

�μmþ1

2;n

ξxmþ12

þμm−1

2;n

ξxm−12

�1−γ12

þ 1

ξznΔz2

ηm;n−12

ξzn−12

γ2þρm;nω2d;

~d3¼1

ξxmΔx2

μmþ12;n

ξxmþ12

1−γ12

þ 1

ξznΔz2

ηm;n−12

ξzn−12

1−γ22

þρm;nω2e;

~d4¼1

ξxmΔx2

μm−12;n

ξxm−12

γ1−1

ξznΔz2

�ηm;nþ1

2

ξznþ12

þηm;n−1

2

ξzn−12

�1−γ22

þρm;nω2d;

~d5¼−1

ξxmΔx2

�μmþ1

2;n

ξxmþ12

þμm−1

2;n

ξxm−12

�γ1−

1

ξznΔz2

�ηm;nþ1

2

ξznþ12

þηm;n−1

2

ξzn−12

�γ2þρm;nω

2c;

~d6¼1

ξxmΔx2

μmþ12;n

ξxmþ12

γ1−1

ξznΔz2

�ηm;nþ1

2

ξznþ12

þμm;n−1

2

ξzn−12

�1−γ22

þρm;nω2d;

~d7¼1

ξxmΔx2

μm−12;n

ξxm−12

1−γ12

þ 1

ξznΔz2

ηm;nþ12

ξznþ12

1−γ22

þρm;nω2e;

~d8¼−1

ξxmΔx2

�μmþ1

2;n

ξxmþ12

þμm−1

2;n

ξxm−12

�1−γ12

þ 1

ξznΔz2

ηm;nþ12

ξznþ12

γ2þρm;nω2d;

~d9¼1

ξxmΔx2

μmþ12;n

ξxmþ12

1−γ12

þ 1

ξznΔz2

ηm;nþ12

ξznþ12

1−γ22

þρm;nω2e;

~c1¼1

4ξxmΔxΔzμm−1;n

ξznþ 1

4ξznΔxΔzλm;n−1

ξxm; ~c2¼−

1

4ξxmΔxΔzμmþ1;n

ξzn−

1

4ξznΔxΔzλm;n−1

ξxm;

~c3¼−1

4ξxmΔxΔzμm−1;n

ξzn−

1

4ξznΔxΔzλm;nþ1

ξxm; ~c4¼

1

4ξxmΔxΔzμmþ1;n

ξznþ 1

4ξznΔxΔzλm;nþ1

ξxm:

(B-9)

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modeling of frequency-domain elastic-wave equation with an...

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