1. U.S. Geological Survey Menlo Park, CA borcherdt@usgs.gov Workshop Active and Passive Seismics in Laterally Inhomogeneous Media Loučeň Castle, Czech Republic June 8-12, 2015 Roger D. Borcherdt ON ADVANCES IN THE THEORY OF SEISMIC WAVE PROPAGATION IN LAYERED VISCOELASTIC MEDIA

2. Outline Linear Superposition principle (Boltzmann 1874) 1  Brief History of Advances in the Theory of Viscoelastic Seismic Wave Propagation  Discuss Implications of these Advances for Seismology and Exploration Geophysics  Discuss New Characteristics of Seismic Waves Implied by Theoretical Solutions for Anelastic Media not Implied by Elasticity Theory

3. Advances (1874 – 1960) General Constitutive Law for Linear Viscoelastic Material Behavior (Elastic and Anelastic) Linear Superposition principle (Boltzmann 1874) 1  1953 --“The Theory of Viscoelasticity is approaching completion. Further progress is likely to made in applications rather than fundamental principles.” Gross, B. 1953, Mathematical Structures of the Theories of Viscoelasticity, Hermann et Cie, Paris.  1960 -- “Application of the general theory of viscoelasticity to other than one-dimensional wave propagation is incomplete.” Hunter, S. C. 1960. Viscoelastic Waves, Progress in Solid Mechanics, I, p 1-57. 2 Volterra 1880-1940, 2005  Theory of Linear Functionals, Integral transforms (Volterra1880 -1940, 2005) 2  Rigorous Mathematical Theory 4 Gurtin and Sternberg 1962 3 Gross 1953  Springs and Dashpot Representation of all linear Viscoelastic Behavior (Bland 1960) 4 5 Bland, 1960  Fourth Order Tensor Relaxation and Creep Fncts. (Gurtin and Sternberg 1962 4 …  Structures of the Theories of Viscoelasticity (Gross 1953) 3 1 Boltzmann 1874  Linear Superposition principle (Boltzmann 1874) 1

4.  Helmholtz Solutions  Coordinate Variables – Incident Homogeneous Wave Single Boundary ( 1962 1a ) Advances Solutions 2& 3D Viscoelastic Wave Equations (Helmholtz Equations) (1962-1973) 1a Lockett,1962 ; 1b Buchen 1971  Confirmation of Theory: Ultrasonic material testing (1970 3a ) 3a Becker and Richardson 1970  Physical Characteristics: Anelastic P, SI and SII Waves (1971, 1973 2a ; 1971 2b )  General Vector Solutions:  Generalized Snell’s Law (app. velocity and attenuation along boundary constant) 1971 2a  Incident General (Inhomogeneous or Homogeneous) P, SI, and SII Waves (1971 2a  Two Types Anelastic S Waves: Elliptical SI and Linear SII Waves (1971, 1973 2a ) 2a Borcherdt 1971, 1973 ; 2b Buchen 1971

5. Advancements in Fundamental Theoretical Solutions for Viscoelastic Media 1a Borcherdt 1971, 1973; 1b Borcherdt 1971  Half-space  Incident Inhomogeneous P, Linear S (SII), and Elliptical S (SI) (1971, 1988) 1a  Rayleigh-type Surface Waves (1971, 1973) 1a  Reflection-Refraction Coefficients for Volumetric Strain (1988) 1b Elliptical S Inhom. P 2a Lockett 1962; Cooper & Reiss 1966; Buchen 1971; 2b Borcherdt 1971, 1977, 1982  Single Welded Boundary  Incident Homogenous P, SV, and SH (1962, 1966, 1971) 2a  Incident Inhomogeneous P, Linear SII, and Elliptical SI (1971, 1977, 1982) 2b  Physical (numerical) characteristics in low-loss media (1971, 1985) 2c  Volumetric strain Body and Surface Waves (1988) 2d Elliptical S Inhom. P Elliptical S Inhom. P 2c Borcherdt 1971, 1973, 1977, 1985; 3b Borcherdt, 1988

6. Inhomog. Linear S (0) (1) (n)  Stack of Welded Boundaries (Multiple Layers)  Incident Inhomogeneous P, SII, and SI Waves (Thompson Haskell Formulation; 2009) 1a  Love Type Surface Waves –  Variational perturbation approximation (1976) 1b  General Solution Model Independent (2009) 1a … 1a Borcherdt 2009; 1b Silva 1976; … Advancements for Multiple Layers, Source Problems, Ray Tracing, and Anisotropic Viscoelastic Media  Source Problems 2  Line Source near Welded Boundary 2a  Numerical Simulation Line Source (memory variables) 2b 2a Buchen 1971; 2b Carcione et al, 1987, 1988, 1993; …  Anisotropic Viscoelastic Media 4  Whole Space, Reflection-Refraction, Ray Tracing … 4 Carcione 1990, 1993; Cerveny & Psencik 2005, 2006, 2008, 2009, … 3 Buchen 1974; Krebes and Hron 1980; Cerveny 2001, 2003; Psencik et al, 1992; …  Ray Tracing for Viscoelastic Media 3

7. Reference http://www.cambridge.org/catalogue/ Hardback ISBN: 9780521898539 eBook ISBN: 9780511577253

8. General Mathematical Characterization of Viscoelastic Material Behavior 1 Boltzmann 1874; Gurtin and Sternberg 1962 2 Borcherdt and Wennerberg 1985

9. Models for Viscoelastic Material Behavior Models for Viscoelastic Material Behavior 1 1 Bland 1960

10. Equation of Motion – General Vector Solutions for P, Elliptical S, and Linear S Waves

11. Wave Speed – Homogeneous and Inhomogeneous S waves

12. Absorption Coefficient – Homogeneous and Inhomogeneous S waves

13. Particle Motions of Viscoelastic Wave Fields

14. Energy Densities and Energy Dissipation for Viscoelastic Wave Fields

15. Q -1 Ratios for Elliptical (SI) and Linear (SII) Anelastic S Waves

16. P A Refracted Inhomogeneous P Wave Incident P Wave P A P A Refracted Inhomogeneous S Wave Waves Refracted at Anelastic Boundaries in the Earth are Inhomogeneous Soil Rock P A Incident P Wave

17. Tracing Inhomogeneous SII Wave in Layered Anelastic Media (Phase and Amplitude)

18. Inhomogeneous Reflected & Refracted Anelastic Seismic Waves

19. Incident General SII Wave Specification of Incident SII Wave: where and

20. Generalized Snell’s Law Real part of k implies: Imaginary part of k implies: Theorem. Generalized Snell’s Law – For the problem of a general SII wave incident on a welded viscoelastic boundary in a plane perpendicular to the boundary, (1) the reciprocal of the apparent phase velocity along the boundary of the general reflected and refracted waves is equal to that of the given general incident wave, and (2) the apparent attenuation along the boundary of the general reflected and refracted waves is equal to that of the given general incident wave.

21. Generalized Snell’s Law Real part of k implies: Imaginary Part of k implies: Theorem 5.4.15. Generalized Snell’s Law – For the problem of a general SII wave incident on a welded viscoelastic boundary in a plane perpendicular to the boundary, (1) the reciprocal of the apparent phase velocity along the boundary of the general reflected and refracted waves is equal to that of the given general incident wave, and (2) the apparent attenuation along the boundary of the general reflected and refracted waves is equal to that of the given general incident wave.

22. Conditions for Homogeneity of the Reflected and Transmitted Waves Transmitted SII wave : Theorem 5.4.21. For the problem of a general SII wave incident on a welded viscoelastic boundary, if the incident SII wave is homogeneous and not normally incident, then the transmitted SII wave is homogeneous if and only if Reflected SII Wave: Theorem 5.4.20. For the problem of a general SII wave incident on a welded viscoelastic boundary, the reflected SII wave is homogeneous if and only if the incident SII wave is homogeneous.

23. Near-Surface Reflection & Refraction Coefficients Inhomogeneous Linear S Wave Incident on a Soil Boundary

24. Response of Multilayered Viscoelastic Media to Incident Inhomogeneous Waves

25. Response of Viscoelastic Layer Incident Homogeneous and Inhomogeneous SII Waves

26. Elliptical S Wave Water Stainless Steel P Wave sourcereceiver Anelastic Reflection Coefficients Nondestructive Testing for Metal Impurities ( Becker and Richardson, 1970) (Empirical Confirmation of Theory )

27. Sea Floor Mapping of Q (age?)

28. Viscoelastic Rayleigh-Type Surface Wave Propagation and Attenuation Vectors For Component P and S solutions Tilt of Particle Motion Orbit

29. Viscoelastic Rayleigh-Type Surface Wave Tilt and Amplitude versus Depth

30. Love-Type Surface Waves Multilayered Viscoelastic Media

31. Viscoelastic Period Equation – Love-Type Surface Waves

32. Solution Curves -- Fundamental Mode Absorption Coefficient and Phase Speed Dispersion

33.  Whole Space (P, SI, SII waves)  Reflection-Refraction, Multiple Layers,  Rayleigh-Type, Love-Type Surface Waves  Some Source Problems, Numerical Simulations, …  Anisotropic Media, Weakly Attenuating Media Summary  Anelastic Seismic Waves are Inhomogeneous  Wave Speed, Damping, Particle Motions, Energy Flux … vary with Inhomogeneity  Future Advances Likely to be:  Solution of Viscoelastic Source Problems (Harmonic and Transient)  Synthetic & Inversion Algorithms based on Inhomogeneous Wave Fields  Applications in Seismology and Exploration Geophysics  General Viscoelasticity Characterizes Linear Material Behavior (Elastic & Anelastic)  Solutions of Fundamental Seismic Problems for General Linear (Viscoelastic) Media  Body Wave Characteristics depend on:  Accurate Models of Linear Material Behavior for Seismology require Inhomogeneous Waves

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35. Correspondence Principle Concept: Solutions to certain steady-state problems in viscoelasticty can inferred from the solutions to corresponding problems in elastic media upon replacement of of real material parameters by complex material parameters. Bland (1960, p65) states: The correspondence principle can be used to obtain solutions to problems in viscoelasticity only if : 1) a solution for the corresponding problem in elastic media exists, 2) no operation in obtaining the elastic solution would have a corresponding operation in viscoelastic media involving separating the complex modulus into real and imaginary parts, 3) the boundary conditions for the two problems are identical. Examples where the Correspondence Principal does not work: 1) Dissipation and storage of energy 2) Energy Balance equations, Energy flux at boundaries due to interaction 3) Amplitude reflection-refraction phase and amplitude coefficients.