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  • Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method

    Jing-Bo Chen1 and Jian Cao1

    ABSTRACT

    Based on an average-derivative method, we developed a new nine-point numerical scheme for a frequency-domain elastic-wave equation. Compared with the classic nine-point scheme, this scheme reduces the required number of grid points per wavelength for equal and unequal directional spac- ings. The reduction in the number of grid points increases as the Poisson’s ratio becomes larger. In particular, as the Pois- son’s ratio reaches 0.5, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classical scheme 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 equations was pioneered by Lysmer and Drake (1972) to describe the wave propagation in the earth. In analyzing the accuracy of finite-difference and finite-element modeling of the scalar- and elastic-wave equa- tions, Marfurt (1984) further examines frequency-domain numerical modeling. One of Marfurt’s important research results is that fre- quency-domain finite-element solutions using a weighted average of consistent and lumped masses lead to accurate results. Pratt and Worthington (1990) present in detail the implementation of the classical five-point scheme for frequency-domain 2D acoustic-wave equation and apply it to multisource crosshole tomography. Based on the 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 resulting impedance matrix, this scheme is very efficient. To reduce numerical

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

    eration terms, Jo et al. (1996) develop an optimal nine-point scheme for 2D scalar-wave equation. The optimization coefficients are obtained by minimizing the phase velocity errors. This scheme re- duces the number of grid points per wavelength from 13 to 4 in comparison 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 a variable-density case (Hustedt et al., 2004) and a 3D case (Operto et al., 2007). Schemes involving more grid points have also been developed (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-domain 2D elastic-wave equation. The construction of the optimal scheme incorporates Jo et al.’s (1996) method and staggered-grid technique. This optimal nine-point scheme is particularly desirable because it not only reduces the number of grid points per wavelength but also remains 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 the 3D 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-wave equation. This method is a generalization of the corresponding approach for the acoustic-wave equation (Chen, 2012, 2014) and results in an average-derivative optimal nine-point scheme for frequency-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) but also 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: chenjb@mail.iggcas.ac.cn; chris40474047@163.com. © 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, we present 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 displacement components, 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 with equation 1 and it includes five terms. For the first two terms, we discretize them with an average-derivative method (Chen, 2012):

    ∂ ∂x

    � η ∂u ∂x

    � ≈

    1

    Δx2

    × h ηmþ1

    2 ;nūmþ1;n − ðηmþ1

    2 ;nþ ηm−1

    2 ;nÞūm;nþ ηm−1

    2 ;nūm−1;n

    i ;

    (3)

    ∂ ∂z

    � μ ∂u ∂z

    � ≈

    1

    Δz2

    × h μm;nþ1

    2 ~um;nþ1 − ðμm;nþ1

    2 þ μm;n−1

    2 Þ ~um;n þ μm;n−1

    2 ~um;n−1

    i ;

    (4)

    where

    ηmþ12;n¼ 1

    2 ðηm;nþηmþ1;nÞ; ηm−12;n¼

    1

    2 ðηm−1;nþηm;nÞ;

    ηm;nþ1 2 ¼1 2 ðηm;nþηm;nþ1Þ; ηm;n−1

    2 ¼1 2 ðηm;n−1þηm;nÞ;

    μmþ1 2 ;n¼

    1

    2 ðμm;nþμmþ1;nÞ; μm−1

    2 ;n¼

    1

    2 ðμm−1;nþμm;nÞ;

    μm;nþ1 2 ¼1 2 ðμm;nþμm;nþ1Þ; μm;n−1

    2 ¼1 2 ðμm;n−1þμm;nÞ; (5)

    and

    ūmþj;n¼ 1−γ1 2

    umþj;nþ1þγ1umþj;nþ 1−γ1 2

    umþj;n−1; j¼−1;0;1;

    ~um;nþj¼ 1−γ2 2

    umþ1;nþjþγ2um;nþjþ 1−γ2 2

    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

    4ΔxΔz ½λmþ1;nðwmþ1;nþ1 − wmþ1;n−1Þ

    − λm−1;nðwm−1;nþ1 − wm−1;n−1Þ�; (7)

    ∂ ∂z

    � μ ∂w ∂x

    � ≈

    1

    4ΔxΔz ½μm;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 and e ¼ ð1 − c − 4dÞ∕4. Substituting equations 3–9 into equation 1, and performing the

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

    1

    Δx2 h ηmþ1

    2 ;nūmþ1;n−ðηmþ1

    2 ;nþηm−1

    2 ;nÞūm;nþηm−1

    2 ;nūm−1;n

    i

    þ 1 Δz2

    h μm;nþ1

    2 ~um;nþ1−ðμm;nþ1

    2 þμm;n−1

    2 Þ ~um;nþμm;n−1

    2 ~um;n−1

    i

    þ 1 4ΔxΔz

    ½λmþ1;nðwmþ1;nþ1−wmþ1;n−1Þ −λm−1;nðwm−1;nþ1−wm−1;n−1Þ�

    þ 1 4ΔxΔz

    ½μm;nþ1ðwmþ1;nþ1−wm−1;nþ1Þ −μm;n−1ðwmþ1;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; (10)

    T340 Chen and Cao

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