U.S. Geological Survey

Menlo Park, CA

borcherdt@usgs.gov

Workshop Active and Passive Seismics in Laterally Inhomogeneous MediaLoučeň Castle, Czech Republic

June 8-12, 2015

Roger D. Borcherdt

ON ADVANCES IN THE THEORY OF ON ADVANCES IN THE THEORY OF SEISMIC WAVE PROPAGATION SEISMIC WAVE PROPAGATION

IN LAYERED VISCOELASTIC MEDIAIN LAYERED VISCOELASTIC MEDIA

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

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 19623 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 19624 …

Structures of the Theories of Viscoelasticity (Gross 1953)3

1 Boltzmann 1874

Linear Superposition principle (Boltzmann 1874) 1

Helmholtz Solutions Coordinate Variables – Incident Homogeneous Wave Single Boundary (1962 1a)

AdvancesSolutions 2& 3D Viscoelastic Wave Equations (Helmholtz Equations)

(1962-1973)

1a Lockett,1962 ; 1b Buchen 1971

Confirmation of Theory: Ultrasonic material testing (19703a)

3a Becker and Richardson 1970

Physical Characteristics: Anelastic P, SI and SII Waves (1971, 19732a; 19712b)

General Vector Solutions: Generalized Snell’s Law (app. velocity and attenuation along boundary constant) 19712a

Incident General (Inhomogeneous or Homogeneous) P, SI, and SII Waves (19712a

Two Types Anelastic S Waves: Elliptical SI and Linear SII Waves (1971, 19732a)

2a Borcherdt 1971, 1973 ; 2b Buchen 1971

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 SEllip

tical S Inho

m. 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 In

hom

. P

Inhom. P

2c Borcherdt 1971, 1973, 1977, 1985; 3b Borcherdt, 1988

Inhomog. Linear S

(0)

(1)

(n)

1 1

1; Hs Hsv Q

1; n nHs Hsv Q

n

n

0D

nD

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 Media4

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 Media3

ReferenceReference

http://www.cambridge.org/catalogue/

Hardback ISBN: 9780521898539eBook ISBN: 9780511577253

General Mathematical Characterization of Viscoelastic Material BehaviorGeneral Mathematical Characterization of Viscoelastic Material Behavior

1

( ) ( ) ( )ij ijkl klp t r t de

Media HLV

General Constitutive Law

2 1 1 , , ,HS HP HS HPv v Q Q -1II - Phase Speed & Q for Homogeneous S and P waves :

( ) ( ); ( ) ( ) forkk K kk ij S klp r t de p r t de i j

Isotropic HLV Media

2

2K R I

S R I

K i R K iK

i R i

I - Complex Bulk ( ) & Shear ( ) moduli :

Material Parameters for HILV Media : K M

1 Boltzmann 1874; Gurtin and Sternberg 1962

2 21 1

2 2

4 42 1 2 13 3; ; ;41 1 1 1 3

HS HPR R I IR IHS HS HP HP

R R RHS HP

Q QΚ + Μ Κ + Μv Q v Q

ρ Κ + ΜQ Q

2 Borcherdt and Wennerberg 1985

Models for Viscoelastic Material BehaviorModels for Viscoelastic Material Behavior1

1 Bland 1960

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

2 2

2 2

0

Re cos( ) Im 2

0

exp exp r

P P A A k P A P A k

G k G

G G A r iP

and

Solutions of Helmholtz Equation are solutions of Equation of Motion where

if and only if

2 3

u u

Equation of Motion :

0A A 2) Anelastic Wave propagates if and only if (inhomogeneous) and is not P.

0A A P

Helmholtz Equation implies :

1) Elastic Wave propagates if and only if (homogeneous) or .

1 20 0

1 20 0

0 1 1 2

( )

( )

and 0,

, , 1 1 1

, , 1 1 1

ˆ ˆ,

P HP HPHP

S HS HSHS

SI

SII

u

G G z k k iQ Qv

G G z P A P A k k iQ Qv

G G z x z

For

P wave :

Elliptical S wave :

Linear S wave :

1 22 3 3ˆ , 1 1 1S HS HS

HS

x z x k k iQ Qv

Wave Speed – Homogeneous and Inhomogeneous S wavesWave Speed – Homogeneous and Inhomogeneous S waves

1 1

RHS

Q

v

Homogeneous wave :

Low - loss Viscoelastic media :

2 2

2

1 1 secS HS HS

HS

v v vQ

Inhomogeneous wave :

2

2 2

1 1

1 1 sec

HS

S HS HS

HS

Qv v v

Q

Inhomogeneous wave

2

2

2 1

1 1

HSRHS

HS

Qv

Q

Homogeneous Wave

General Viscoelastic Media :

Absorption Coefficient – Homogeneous and Inhomogeneous S wavesAbsorption Coefficient – Homogeneous and Inhomogeneous S waves

1

1

1

2

HS

HSHS

HS

Q

QA

v

Homogeneous wave :

Low - loss Viscoelastic media :

1

2 2

0 :

2 1

cos1 1 sec

HS

S HS HS

HS

Q

A A AQ

Inhomogeneous wave for

2

2 2

1 1 1

cos1 1 sec

HS

S HS HS

HS

QA A A

Q

Inhomogeneous wave

1

21 1

HSHS

HS HS

QA

v Q

Homogeneous Wave

General Viscoelastic Media :

Particle Motions of Viscoelastic Wave FieldsParticle Motions of Viscoelastic Wave Fields

0 1 2

2 2

1 2

exp[ ] cos ( ) sin ( )

where ( ) and ( )R I I R

R P P P P P P

PP P P p P P p P p P P

u G k A r t t

k P k A k n k P k A k

P waves :

������������������������������������������������������������������������������������

2 2

1 2

0 0 0

0 1 2

where ( ) and ( )

ˆ

exp 1 cos sin

R I RSI S S S SI SI S S

S S S S

R S S SI SI SI SI

n k P k A k n k P k A k

G z P A P A z n

u G k A r t t

=Elliptical S waves, :

������������������������������������������������������������������������������������

0 1 1 3 3

2

ˆ ˆ

ˆexp cos argR SII S S SII

G z x z x

u D A r t P r D x

Linear S waves :

Energy Densities and Energy Dissipation for Viscoelastic Wave Fields

1

11 1 1 1

, , , ,

1 1 1 1

1 2

Fractional energy loss for P and Elliptical S waves: [ ] [ ]1

2(1 )Fractional energy loss for Linear S waves: [ ] [ ]

1 1

Fract

Hs

HP SI P SI H P SI P SI H

SII SII H SII SII H

QH

QQ Q Q Q

HH

Q Q Q QH H

1 1ional energy loss for Elliptical S waves > Fractional energy loss for Linear S waves: SI SIIQ Q

A��������������

P��������������

��������������I

1 1 1

1 2Mean energy flux:

(1 )

where = ( , , ), =F( , , ), and =H( , , ).

HH

H H H H H H

FHY

v Q F v Q H v Q

����������������������������I I

Mean kinetic energy density:

Mean potential energy density: (1 )

1Mean total energy density: ( )

1

Mean rate energy dissipation:

H HH

H H H

H HH

Y

Y

HY

K K K K

P P P P

E E E E

D1

1(1 )Hs

H HH

QY H

Q

D D D

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

22

2

1 1 1 1 where tan1

2 11 2

1 1 1HS

S

HS

SSSI HS SII HS

S S S

QH

Q

HHQ Q Q Q

H H H

PA

Refracted

Inhomogeneous P Wave

Incident P Wave

PA

PA

RefractedInhomogeneous S Wave

Waves Refracted at Anelastic BoundariesWaves Refracted at Anelastic Boundaries in the Earth are Inhomogeneous in the Earth are Inhomogeneous

-11 Q

12 Q

Soil

RockP

A

Incident P Wave

-1 -11 2 Q Q

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

Inhomogeneous Reflected & Refracted Anelastic Seismic Waves Inhomogeneous Reflected & Refracted Anelastic Seismic Waves

1 1

1 1

If the incident SI wave is homogeneous and not normally incident then the Generalized Snell's Law implies:

1) if , then the reflected P wave is inhomogeneous,

2) if , the

HS HP

HS HP

Q Q

Q Q

1 1

n the transmitted P wave is inhomogeneous,

3) if , then the transmitted S wave is inhomogeneous.HS HSQ Q

Incident General SII Wave

Specification of Incident SII Wave:

1 11 1 2ˆexp r exp ( r)u uu D A i t P x

1 1 1 11 3 1 3ˆ ˆ ˆ ˆsin cosRu R u u uP k x d x P x x

1 1 1 1 1 11 3 1 3ˆ ˆ ˆ ˆsin cosIu I u u u u uA k x d x A x x

1

1

2 2

2

1 1 sec

1 1

HS uu

HS HS

QP

v Q

1

1

2 2

2

1 1 s

1,

ec

1

HS uu

HS HS

QA

v Q

1 1

1 1 1

2 2 2 2

2 2

1 1 sec 1 1 secsin sin

1 1 1 1

HS u HS uu u u

HS HS HS

Q Qk i

v Q Q

1

elastic case:

sin uHS

kv

where

2 2. . Sd p v k k 1

21

1 1

HSS

HS HS

Qk i

v Q

and

Generalized Snell’s Law

1 1 2 2 1 1

1 2 1

1 2 1

sin sin sin

or in terms of velocity

sin sin sin

R u u u u u u

u u uR

u u u

k P P P

k

v v v

1 1 1 2 2 2 1 1 1sin sin sinI u u u u u u u u uk A A A

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.

Generalized Snell’s Law

1 1 2 2 1 1

1 2 1

1 2 1

sin sin sin

or in terms of velocity

sin sin sin

R u u u u u u

u u uR

u u u

k P P P

k

v v v

1 1 1 2 2 2 1 1 1sin sin sinI u u u u u u u u uk A A A

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.

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

1

2 21 1 2

2 2 sin S R HSHS HS u

RS HS

k vQ Q and

k v

• 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.

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

Response of Multilayered Viscoelastic MediaResponse of Multilayered Viscoelastic Mediato Incident Inhomogeneous Waves to Incident Inhomogeneous Waves

Response of Viscoelastic LayerResponse of Viscoelastic LayerIncident Homogeneous and Inhomogeneous SII WavesIncident Homogeneous and Inhomogeneous SII Waves

Elliptical S W

ave

Water

Stainless Steel

P W

ave

source receiver

Anelastic Reflection CoefficientsAnelastic Reflection CoefficientsNondestructive Testing for Metal ImpuritiesNondestructive Testing for Metal Impurities ( (Becker and Richardson, 1970) Becker and Richardson, 1970)

(Empirical Confirmation of Theory )(Empirical Confirmation of Theory )

Sea Floor Mapping of Q (age?)Sea Floor Mapping of Q (age?)

Viscoelastic Rayleigh-Type Surface WaveViscoelastic Rayleigh-Type Surface Wave

Propagation and Attenuation VectorsFor Component P and S solutions Tilt of Particle Motion Orbit

Viscoelastic Rayleigh-Type Surface WaveViscoelastic Rayleigh-Type Surface WaveTilt and Amplitude versus DepthTilt and Amplitude versus Depth

Love-Type Surface Waves Love-Type Surface Waves Multilayered Viscoelastic MediaMultilayered Viscoelastic Media

Viscoelastic Period Equation – Love-Type Surface WavesViscoelastic Period Equation – Love-Type Surface Waves

1 2

2

2 11 1

1 1

122 2 2

21 2 22 2

1

0,

1 arctan 1 1

HS HS

HS

HS HSHS HS

Q Q

vd n

v vv v

= Special Case: Elastic Period Equation with

L L L

L

v v v

v

2 2 1

11 21

1 1 1

1

1 1

-122

2 -1-1 22 1-1

12

-1

11

1 1

11

1arctan

11

HS HS HS

HSHS HSHS

HS HS HS

HS

HS HS

i Qvi

v i QQ

ia v

dii

aQ

iv

Viscoelastic Period Equation:

L L

L

LL

a v

a

vv

1 1 2

1 1 1 1 2 2

1 1

1 1 1 1 1

22-1 -1

22-1 -1

11 1

1 11 1

HS HS HS

HS HS HS HS HS HS

HS HS

HS HS HS HS HS

Q Qvi n

v v v

Q Qi i

a v v

L L

L L L

v v

a v v

2where 1 1,2 , and .HS j HS jQ j

L L IR

, v a = -kk

Solution Viscoelastic Period Equation : L Lv ,a

Solution Curves -- Fundamental ModeSolution Curves -- Fundamental ModeAbsorption Coefficient and Phase Speed DispersionAbsorption Coefficient and Phase Speed Dispersion ,L Lv a

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

1 1 ( , , , )HS HP HS HPv v Q Q Angle of Incidence (Travel Path) and Media Properties

Body Wave Characteristics depend on:

Accurate Models of Linear Material Behavior for Seismology require Inhomogeneous Waves

Thank You

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 energy2) Energy Balance equations, Energy flux at boundaries due to interaction 3) Amplitude reflection-refraction phase and amplitude coefficients.