Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation. (English) Zbl 07923418
Summary: Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson’s constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.
MSC:
| 74D05 | Linear constitutive equations for materials with memory |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 41A05 | Interpolation in approximation theory |
| 35R11 | Fractional partial differential equations |
| 33F05 | Numerical approximation and evaluation of special functions |
Keywords:
viscoelasticity; fractional derivative; sum-of-exponentials approximation; generalized Gaussian quadrature; fast algorithm; rheological modelReferences:
| [1] | Liu, H.; Anderson, D. L.; Kanamori, H., Velocity dispersion due to anelasticity; implications for seismology and mantle composition, Geophys. J. Int., 47, 41-58, 1976 |
| [2] | Kjartansson, E., Constant Q-wave propagation and attenuation, J. Geophys. Res., 84, B9, 4737-4748, 1979 |
| [3] | Carcione, J. M.; Cavallini, F.; Mainardi, F.; Hanyga, A., Time-domain seismic modeling of constant-Q wave propagation using fractional derivatives, Pure Appl. Geophys., 159, 1714-1736, 2002 |
| [4] | Ursin, B.; Toverud, T., Comparison of seismic dispersion and attenuation models, Stud. Geophys. Geod., 46, 293-320, 2002 |
| [5] | Emmerich, H.; Korn, D., Incorporation of attenuation into time-domain computations of seismic wave fields, Geophysics, 52, 9, 1252-1264, 1987 |
| [6] | Yang, P.; Brossier, R.; Métivier, L.; Virieux, J., A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium, Geophys. J. Int., 207, 1, 129-149, 2017 |
| [7] | Tromp, J., Seismic wavefield imaging of Earth’s interior across scales, Nat. Rev. Earth Env., 1, 1, 40-53, 2020 |
| [8] | Mirzanejad, M.; Tran, K. T.; Wang, Y., Three-dimensional Gauss-Newton constant-Q viscoelastic full-waveform inversion of near-surface seismic wavefields, Geophys. J. Int., 231, 3, 1767-1785, 2022 |
| [9] | Wang, Y.; Harris, J. M.; Bai, M.; Saad, O. M.; Yang, L.; Chen, Y., An explicit stabilization scheme for Q-compensated reverse time migration, Geophysics, 87, 3, F25-F40, 2022 |
| [10] | Marques, S. P.C.; Creus, C. J., Computational Viscoelasticity, 2012, Springer Science & Business Media |
| [11] | Blanch, J. O.; Robertsson, J. O.A.; Symes, W. W., Modeling of a constant Q; methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique, Geophysics, 60, 1, 176-184, 1995 |
| [12] | Day, S. M.; Minster, J. B., Numerical simulation of attenuated wavefields using a Padé approximant method, Geophys. J. Int., 78, 1, 105-118, 1984 · Zbl 0587.73150 |
| [13] | Park, S. W.; Schapery, R. A., Methods of interconversion between linear viscoelastic material functions. Part I - a numerical method based on Prony series, Int. J. Solids Struct., 36, 11, 1653-1675, 1999 · Zbl 0942.74012 |
| [14] | Moczo, P.; Kristek, J., On the rheological models used for time-domain methods of seismic wave propagation, Geophys. Res. Lett., 32, Article L01306 pp., 2005 |
| [15] | Zhu, T.; Carcione, J. M.; Harris, J. M., Approximating constant-Q seismic propagation in the time domain, Geophys. Prospect., 61, 5, 931-940, 2013 |
| [16] | Cao, D.; Yin, X., Equivalence relations of generalized rheological models for viscoelastic seismic-wave modeling, Bull. Seismol. Soc. Am., 104, 1, 260-268, 2014 |
| [17] | Blanc, É.; Komatitsch, D.; Chaljub, E.; Lombard, B.; Xie, Z., Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation, Geophys. J. Int., 205, 1, 427-439, 2016 |
| [18] | Groby, J.; Tsogka, C., A time domain method for modeling viscoacoustic wave propagation, J. Comput. Acoust., 14, 2, 201-236, 2006 · Zbl 1198.74033 |
| [19] | Wang, N.; Xing, G.; Zhu, T.; Zhou, H.; Shi, Y., Propagating seismic waves in VTI attenuating media using fractional viscoelastic wave equation, J. Geophys. Res., Solid Earth, 127, 4, Article e2021JB023280 pp., 2022 |
| [20] | Robertsson, J. O.A.; Blanch, J. O.; Symes, W. W., Viscoelastic finite-difference modeling, Geophysics, 59, 9, 1444-1456, 1994 |
| [21] | Wang, E.; Liu, Y.; Ji, Y.; Chen, T.; Liu, T., Q full-waveform inversion based on the viscoacoustic equation, Appl. Geophys., 16, 1, 77-91, 2019 |
| [22] | Carcione, J. M.; Kosloff, D.; Kosloff, R., Wave propagation simulation in a linear viscoelastic medium, Geophys. J., 95, 597-611, 1988 · Zbl 0659.73031 |
| [23] | Hanyga, A.; Seredyńska, M., Power-law attenuation in acoustic and isotropic anelastic media, Geophys. J. Int., 155, 3, 830-838, 2003 |
| [24] | Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010, World Scientific: World Scientific Singapore · Zbl 1210.26004 |
| [25] | Schmidt, A.; Gaul, L., On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems, Mech. Res. Commun., 33, 1, 99-107, 2006 · Zbl 1192.74153 |
| [26] | Carcione, J. M., Theory and modeling of constant-Q P- and S-waves using fractional time derivatives, Geophysics, 74, 1, 1787-1795, 2009 |
| [27] | Zhu, T., Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation, Geophysics, 82, 4, WA1-WA10, 2017 |
| [28] | Diethelm, K., An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives, Numer. Algorithms, 47, 4, 361-390, 2008 · Zbl 1144.65017 |
| [29] | Huang, Y.; Li, Q.; Li, R.; Zeng, F.; Guo, L., A unified fast memory-saving time-stepping method for fractional operators and its applications, Numer. Math., Theory Methods Appl., 15, 3, 679-714, 2022 · Zbl 1524.65349 |
| [30] | Sun, F.; Gao, J.; Liu, N., The approximate constant Q and linearized reflection coefficients based on the generalized fractional wave equation, J. Acoust. Soc. Am., 145, 1, 243-253, 2019 |
| [31] | Liu, S.; Yang, D.; Dong, X.; Liu, Q.; Zheng, Y., Numerical simulation of the wavefield in a viscous fluid-saturated two-phase VTI medium based on the constant-Q viscoelastic constitutive relation with a fractional temporal derivative, Chin. J. Geophys., 61, 6, 2446-2458, 2018 |
| [32] | Christensen, R., Theory of Viscoelasticity: An Introduction, 2012, Elsevier |
| [33] | Yuan, L.; Agrawal, O. P., A numerical scheme for dynamic systems containing fractional derivatives, J. Vib. Acoust., 124, 2, 321-324, 2002 |
| [34] | Blanc, E.; Chiavassa, G.; Lombard, B., Wave simulation in 2D heterogenous transversely isotropic porous media with fractional attenuation: a Cartesian grid approach, J. Comput. Phys., 275, 118-142, 2014 · Zbl 1349.76807 |
| [35] | Xiong, Y.; Guo, X., A short-memory operator splitting scheme for constant-Q viscoelastic wave equation, J. Comput. Phys., 449, Article 110796 pp., 2022 · Zbl 07524792 |
| [36] | Jiang, S., Fast evaluation of the nonreflecting boundary conditions for the Schrödinger equation, ProQuest LLC, 2001, New York University: New York University Ann Arbor, MI, Thesis (Ph.D.) |
| [37] | Jiang, S.; Greengard, L., Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension, Comput. Math. Appl., 47, 6-7, 955-966, 2004 · Zbl 1058.35012 |
| [38] | Beylkin, G.; Monzón, L., Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal., 28, 2, 131-149, 2010 · Zbl 1189.65035 |
| [39] | Jiang, S.; Zhang, J.; Zhang, Q.; Zhang, Z., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21, 3, 650-678, 2017 · Zbl 1488.65247 |
| [40] | Li, J-R., A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31, 6, 4696-4714, 2010 · Zbl 1208.26014 |
| [41] | Bremer, J.; Gimbutas, Z.; Rokhlin, V., A nonlinear optimization procedure for generalized Gaussian quadratures, SIAM J. Sci. Comput., 32, 4, 1761-1788, 2010 · Zbl 1215.65045 |
| [42] | Ma, J.; Rokhlin, V.; Wandzura, S., Generalized Gaussian quadrature rules for systems of arbitrary functions, SIAM J. Numer. Anal., 33, 3, 971-996, 1996 · Zbl 0858.65015 |
| [43] | Yarvin, N.; Rokhlin, V., Generalized Gaussian quadratures and singular value decompositions of integral operators, SIAM J. Sci. Comput., 20, 2, 699-718, 1998 · Zbl 0932.65020 |
| [44] | Hanyga, A., Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. A, 458, 933-957, 2002 · Zbl 1153.35347 |
| [45] | Lu, J. F.; Hanyga, A., Wave field simulation for heterogenous porous media with singular memory drag force, J. Comput. Phys., 208, 2, 651-674, 2005 · Zbl 1329.76338 |
| [46] | Fowler, M.; Zaki, T. A.; Meneveau, C., A multi-time-scale wall model for large-eddy simulations and applications to non-equilibrium channel flows, J. Fluid Mech., 974, A51, 2023 · Zbl 07767356 |
| [47] | Chapman, C. H.; Hobro, J. W.D.; Robertsson, J. O.A., Correcting an acoustic wavefield for elastic effects, Geophys. J. Int., 197, 1196-1214, 2014 |
| [48] | Gimbutas, Z.; Marshall, N. F.; Rokhlin, V., A fast simple algorithm for computing the potential of charges on a line, Appl. Comput. Harmon. Anal., 49, 3, 815-830, 2020 · Zbl 1446.31009 |
| [49] | Serkh, K., On the Solution of Elliptic Partial Differential Equations on Regions with Corners. ProQuest LLC, 2016, Yale University: Yale University Ann Arbor, MI, Thesis (Ph.D.) · Zbl 1349.35049 |
| [50] | Kambo, N. S., Error of the Newton-Cotes and Gauss-Legendre quadrature formulas, Math. Comput., 24, 110, 261-269, 1970 · Zbl 0264.65018 |
| [51] | Luchko, Y., Algorithms for evaluation of the Wright function for the real arguments’ values, Fract. Calc. Appl. Anal., 11, 1, 57-75, 2008 · Zbl 1145.33005 |
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