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The energy of wavefield is gradually attenuated in all real materials, which is a fundamental feature and more obvious in the media containing liquid and gas. Because the viscosity effect is not considered in the classical wave theory, the actual wavefield is different from the simulated scenario based on the assumption of complete elasticity so that the application of wavefield does not meet the expectations in engineering technology, such as geophysical exploration. In the rock physics field, the well-known constant-Q theory gives a linear description of attenuation and Q is regarded as independent of the frequency. The quality factor Q is a parameter for calculating the phase difference between stress and strain of the media, which, as an index of wavefield attenuation behavior, is inversely proportional to the viscosity. Based on the constant-Q theory, a wave equation can be directly obtained by the Fourier transform of the dispersion relation, in which there is a fractional time differential operator. Therefore, it is difficult to perform the numerical simulation due to memory for all historical wavefields. In this paper, the dispersion relation is approximated by polynomial fitting and Taylor expansion method to eliminate the fractional power of frequency which is uncomfortably treated in the time domain. And then a complex-valued wave equation is derived to characterize the propagation law of wavefield in earth media. Besides the superiority of numerical simulation, the other advantage of this wave equation is that the dispersion and dissipation effects are decoupled. Next, a feasible numerical simulation strategy is proposed. The temporal derivative is solved by the finite-difference approach, moreover, the fractional spatial derivative is calculated in the spatial frequency domain by using the pseudo-spectral method. In the process of numerical simulation, only two-time slices, instead of the full-time wavefields, need to be saved, so the demand for data memory significantly slows down compared with solving the operator of the fractional time differential. Following that, the numerical examples prove that the novel wave equation is capable and efficient for the homogeneous model. The research work contributes to the understanding of complex wavefield phenomena and provides a basis for treating the seismology problems.
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Keywords:
- viscoelastic media /
- wave equation /
- numerical simulation /
- pseudo-spectral method
[1] Yang P, Brossier R, Metivier L 2018 SIAM J. Sci. Comput. 40 B1101
Google Scholar
[2] Chen H, Zhou H, Yao Y 2020 Geophysics 85 S169
Google Scholar
[3] Keating S, Innanen K 2020 Geophysics 85 R397
Google Scholar
[4] Zhang, W, Shi Y 2019 Geophysics 84 S95
Google Scholar
[5] Liu H P, Anderson D L, Kanamori H 1976 Geophys. J. Int. 47 41
Google Scholar
[6] Emmerich H, Korn M 1987 Geophysics 52 1252
Google Scholar
[7] Zhu T, Carcione J M, Harris J M 2013 Geophys. Prospect. 61 931
Google Scholar
[8] Kjartansson E 1979 Geophys. Prospect. 84 4737
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Google Scholar
[11] Lu J F, Hanyga A 2004 Geophys. J. Int. 159 688
Google Scholar
[12] Podlubny I 1999 Fractional Differential Equations (California: Academic Press) pp270−217
[13] Yang J, Zhu H 2018 Geophys. J. Int. 215 1064
Google Scholar
[14] Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282
Google Scholar
[15] Dorodnitsyn V, Kozlov R 2010 J. Eng. Math. 66 253
Google Scholar
[16] 方刚, 张斌 2013 62 154502
Google Scholar
Fang G, Zhang B 2013 Acta Phys. Sin. 62 154502
Google Scholar
[17] Li H X, Tao C H, Liu C, Huang G N, Yao Z A 2020 Chin. Phys. B 29 064301
Google Scholar
[18] 周聪, 王庆良 2015 64 239101
Google Scholar
Zhou C, Wang Q 2015 Acta Phys. Sin. 64 239101
Google Scholar
[19] Zhang Z J, Wang G J, Harris J M 1999 Phys. Earth Planet. Inter. 114 25
Google Scholar
[20] 董良国, 马在田, 曹景忠 2000 地球 43 856
Google Scholar
Dong L G, Ma Z T, Cao J Z 2000 Chin. J. Geophys 43 856
Google Scholar
[21] 孟路稳, 程广利, 张明敏, 尚建华 2017 海军工程大学学报 29 57
Meng L W, Cheng G L, Zhang M M, Shang J H 2017 J. Naval Univ. Eng. 29 57
[22] 杜启振, 刘莲莲, 孙晶波 2007 56 6143
Google Scholar
Du Q, Liu L, Sun J 2007 Acta Phys. Sin. 56 6143
Google Scholar
[23] 唐春安 1997 岩石力学与工程学报 4 75
Tang C A 1997 Chin. J. Rock Mech. Eng. 4 75
[24] Carcione J M 2014 Wave Fields in Real Media (Amsterdam: Elsevier Science) p75
[25] Chen W, Holm S 2004 J. Acoust. Soc. Am. 115 1424
Google Scholar
[26] Carcione J M 2010 Geophysics 75 A53
Google Scholar
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图 1 拟合曲线 (Q = 30)
Figure 1. Fitting curve (Q = 30).
图 2 数值模拟模型
Figure 2. Numerical simulation model.
图 3 600 ms波场快照
Figure 3. Wavefield snapshot at 600 ms.
图 4 不同时刻的1维波场
Figure 4. One dimensional wavefield at different times.
图 5 波场记录
Figure 5. Wavefield record.
图 6 记录频谱
Figure 6. Record spectrum.
图 7 组合波场
Figure 7. Combined wavefield.
图 8 部分波场
Figure 8. Part wavefield.
图 9 复杂模型波场模拟 (a) 速度模型; (b) 0.3 s波场; (c) 0.4 s波场; (d) 0.5 s波场
Figure 9. Wavefield simulation for the complex model: (a) Velocity; (b) wavefield at 0.3 s; (c) wavefield at 0.4 s; (a) wavefield at 0.5 s.
表 1 不同Q值拟合系数
Table 1. Fitting coefficients of different Q values.
Q 5 10 20 30 60 100 5000 a1 0.8144 0.9081 0.9545 0.9698 0.9850 0.991 0.9998 a2 58.452 30.728 15.662 10.499 5.276 3.1716 0.0636 a3 –2078.2 –1133.1 –587.58 –396.08 –200.14 –120.57 –2.4255 
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[1] Yang P, Brossier R, Metivier L 2018 SIAM J. Sci. Comput. 40 B1101
Google Scholar
[2] Chen H, Zhou H, Yao Y 2020 Geophysics 85 S169
Google Scholar
[3] Keating S, Innanen K 2020 Geophysics 85 R397
Google Scholar
[4] Zhang, W, Shi Y 2019 Geophysics 84 S95
Google Scholar
[5] Liu H P, Anderson D L, Kanamori H 1976 Geophys. J. Int. 47 41
Google Scholar
[6] Emmerich H, Korn M 1987 Geophysics 52 1252
Google Scholar
[7] Zhu T, Carcione J M, Harris J M 2013 Geophys. Prospect. 61 931
Google Scholar
[8] Kjartansson E 1979 Geophys. Prospect. 84 4737
[9] Carcione J M 2008 Geophysics 74 T1
[10] Carcione J M, Cavallini F, Mainardi F, Hanyga A 2002 Pure Appl. Geophys. 159 1719
Google Scholar
[11] Lu J F, Hanyga A 2004 Geophys. J. Int. 159 688
Google Scholar
[12] Podlubny I 1999 Fractional Differential Equations (California: Academic Press) pp270−217
[13] Yang J, Zhu H 2018 Geophys. J. Int. 215 1064
Google Scholar
[14] Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282
Google Scholar
[15] Dorodnitsyn V, Kozlov R 2010 J. Eng. Math. 66 253
Google Scholar
[16] 方刚, 张斌 2013 62 154502
Google Scholar
Fang G, Zhang B 2013 Acta Phys. Sin. 62 154502
Google Scholar
[17] Li H X, Tao C H, Liu C, Huang G N, Yao Z A 2020 Chin. Phys. B 29 064301
Google Scholar
[18] 周聪, 王庆良 2015 64 239101
Google Scholar
Zhou C, Wang Q 2015 Acta Phys. Sin. 64 239101
Google Scholar
[19] Zhang Z J, Wang G J, Harris J M 1999 Phys. Earth Planet. Inter. 114 25
Google Scholar
[20] 董良国, 马在田, 曹景忠 2000 地球 43 856
Google Scholar
Dong L G, Ma Z T, Cao J Z 2000 Chin. J. Geophys 43 856
Google Scholar
[21] 孟路稳, 程广利, 张明敏, 尚建华 2017 海军工程大学学报 29 57
Meng L W, Cheng G L, Zhang M M, Shang J H 2017 J. Naval Univ. Eng. 29 57
[22] 杜启振, 刘莲莲, 孙晶波 2007 56 6143
Google Scholar
Du Q, Liu L, Sun J 2007 Acta Phys. Sin. 56 6143
Google Scholar
[23] 唐春安 1997 岩石力学与工程学报 4 75
Tang C A 1997 Chin. J. Rock Mech. Eng. 4 75
[24] Carcione J M 2014 Wave Fields in Real Media (Amsterdam: Elsevier Science) p75
[25] Chen W, Holm S 2004 J. Acoust. Soc. Am. 115 1424
Google Scholar
[26] Carcione J M 2010 Geophysics 75 A53
Google Scholar
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