ON ADVANCES IN THE THEORY OF
SEISMIC WAVE PROPAGATION
IN LAYERED VISCOELASTIC MEDIA
Roger D. Borcherdt
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
Outline
• History
LinearofSuperposition
 Brief
Advances in the principle
Theory of Viscoelastic Seismic Wave Propagation
(Boltzmann 1874) 1
 Discuss New Characteristics of Seismic Waves Implied by Theoretical Solutions for
Anelastic Media not Implied by Elasticity Theory
 Discuss Implications of these Advances for Seismology and Exploration Geophysics
Advances (1874 – 1960)
General Constitutive Law for Linear Viscoelastic Material Behavior
(Elastic and Anelastic)
 Linear
Superposition
principle (Boltzmann
1874) 1
• Linear
Superposition
principle
1874) 1 Integral transforms (Volterra1880 -1940, 2005) 2
 Theory(Boltzmann
of Linear Functionals,
 Rigorous Mathematical Theory
 Structures of the Theories of Viscoelasticity (Gross 1953)3
 Springs and Dashpot Representation of all linear Viscoelastic Behavior (Bland 1960) 4
 Fourth Order Tensor Relaxation and Creep Fncts. (Gurtin and Sternberg 19624 …

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.
1 Boltzmann
1874
2 Volterra
1880-1940, 2005
3 Gross
1953
4 Gurtin
and Sternberg 1962
5 Bland,
1960
Advances
Solutions 2& 3D Viscoelastic Wave Equations (Helmholtz Equations)
(1962-1973)
 Helmholtz Solutions
 Coordinate Variables – Incident Homogeneous Wave Single Boundary (1962 1a)
 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)
 Physical Characteristics: Anelastic P, SI and SII Waves (1971, 19732a; 19712b)
 Confirmation of Theory: Ultrasonic material testing (19703a)
1a Lockett,1962 ; 1b Buchen
1971
2a Borcherdt
1971, 1973 ; 2b Buchen 1971
3a Becker
and Richardson 1970
Advancements in
Fundamental Theoretical Solutions for Viscoelastic Media
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
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
1a
Borcherdt 1971, 1973;
2a Lockett
1b
Borcherdt 1971
1962; Cooper & Reiss 1966; Buchen 1971;
2c Borcherdt
1971, 1973, 1977, 1985;
3b Borcherdt,
2b Borcherdt
1988
1971, 1977, 1982
Advancements for
Multiple Layers, Source Problems, Ray Tracing, and Anisotropic Viscoelastic Media
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 …
 Source Problems 2
 Line Source near Welded Boundary 2a
 Numerical Simulation Line Source (memory variables) 2b
 Ray Tracing for Viscoelastic Media3
 Anisotropic Viscoelastic Media4
 Whole Space, Reflection-Refraction, Ray Tracing …
1a Borcherdt
2a Buchen
3 Buchen
2009;
1b Silva 1976; …
1971; 2b Carcione et al, 1987, 1988, 1993; …
1974; Krebes and Hron 1980; Cerveny 2001, 2003; Psencik et al, 1992; …
4 Carcione
1990, 1993; Cerveny & Psencik 2005, 2006, 2008, 2009, …
D0
(0)
vHs1 ; QHs1 1
(1)
vHsn ; QHsn 1
(n)
Dn
n
n
Reference
Hardback ISBN: 9780521898539
eBook
ISBN: 9780511577253
http://www.cambridge.org/catalogue/
General Mathematical Characterization of Viscoelastic Material Behavior
General Constitutive Law1
HLV Media

pij (t ) 
 rijkl (t   ) dekl ( )

Isotropic HLV Media

pkk 
 rK (t   ) dekk ( );
pij 


 rS (t   ) dekl ( )
for i  j

Material Parameters for HILV Media :
I - Complex Bulk ( K ) & Shear ( M ) moduli : K  i  RK 2  K R  iK I
  i  RS 2   R  i I
1
1
II - Phase Speed & Q-1 for Homogeneous S and P waves2 : vHS , vHP , QHS
, QHP
vHS
1 Boltzmann




2
2
 Κ R + 4 Μ R  2 1  QHP
ΚI + 4 Μ I
  R  2 1  QHS

1

1
3
3

 
; QHS  I ; vHP  
; QHP 

2

 1  1  Q 2
R
ρ
ΚR + 4 Μ R
   1  1  QHS
HP
3


1874; Gurtin and Sternberg 1962
2 Borcherdt
and Wennerberg 1985
Models for Viscoelastic Material Behavior1
1 Bland 1960
Equation of Motion –
General Vector Solutions for P, Elliptical S, and Linear S Waves
 

2
Equation of Motion :   





u  u

3 

Solutions of Helmholtz Equation 2G  k 2G  0 are solutions of Equation of Motion where
G  G0 exp   A  r  exp  iP  r  if and only if
P  P  A  A  Re  k 2  and P  A  P A cos( )   Im k 2  2
Helmholtz Equation implies :
1) Elastic Wave propagates if and only if A  0 (homogeneous) or A  P.
2) Anelastic Wave propagates if and only if A  0 (inhomogeneous) and A is not  P.
For u      and    0,
P wave : G   , G0  z0 , k  k P    

vHP
1  i Q 1 
1
HP
2
1  QHP

1 iQ
1


v

  
1 iQ
1 
v 
Elliptical S wave ( SI ) : G   , G0  z0 P  A P  A , k  k S    

1
HS
2
1  QHS
1
HS
2
1  QHS
HS
Linear S wave ( SII ) : G   , G0  z1 xˆ1  z2 xˆ2  z3 xˆ3 , k  k S
HS


Wave Speed – Homogeneous and Inhomogeneous S waves
General Viscoelastic Media :
vHS
Homogeneous Wave

Inhomogeneous wave
vS  vHS

2
1  1  QHS
1

2
1  QHS
sec 2 
Low - loss Viscoelastic media Q 1
 vHS
1:
R 

  
Homogeneous wave : vHS  
Inhomogeneous wave :
vS  vHS
2
1

2
2
1

Q
HS
 
  R
2
   1  1  QHS
2
1  QHS
sec 2 
 vHS
Absorption Coefficient – Homogeneous and Inhomogeneous S waves
General Viscoelastic Media :
Homogeneous Wave
AHS 
1
QHS

vHS 1  1  Q 2
HS
Inhomogeneous wave
AS  AHS

2
1  1  QHS

1
 AHS
2
2
cos

1  1  QHS sec 
1
Low - loss Viscoelastic media QHS
Homogeneous wave :
AHS 
1:
1
 QHS
vHS
2
1
Inhomogeneous wave for QHS  0 :
AS 
 1 

 AHS  AHS
2
2
1  1  QHS sec   cos  
2
Particle Motions of Viscoelastic Wave Fields
P waves :

uR  G0 k P exp[ AP  r ] 1P cos  P (t )   2 P sin  P (t )
where 1P  (k PR PP  k pI AP ) k P
2

and  2 P  n  (k pI PP  k pR AP ) k P
Elliptical S waves, G0  z0 PS  AS PS  AS = z0 nˆ :

uR  G0 k S exp   AS  r   1 1SI cos  SI  t    2 SI sin  SI  t 
where 1SI  n  ( k S P  k S A) k S
R
2
I

and  2 SI  n  (k SI P  k S A) k S
R
Linear S waves : G0  z1xˆ1  z3 xˆ3
uR  DSII exp   AS  r  cos t  PS  r  arg  DSII  xˆ2
2
2
Energy Densities and Energy Dissipation for Viscoelastic Wave Fields
1    2 FH
Mean energy flux:  I    I H  Y
(1   H )
where  = (vH , QH 1 ,  ), F =F(vH , QH 1 ,  ), and H =H(vH , QH 1 ,  ).
Mean kinetic energy density:  K    KH  Y
A

I
P

  K    KH 
H
Mean potential energy density: P   PH  Y (1   H )   P   PH 
Mean total energy density: E   EH  Y (
1    H
)   E   EH 
1  H
QHs 1
Mean rate energy dissipation:  D   DH  Y (1  
H )   D   DH 
QH 1
QHs 1
1  2 1 H
QH
Fractional energy loss for P and Elliptical S waves: QP , SI 1  [QP , SI 1 ]H
 QP , SI 1  [QP ,SI 1 ]H
1 H
2(1  H )
Fractional energy loss for Linear S waves: QSII 1  [QSII 1 ]H
 QSII 1  [QSII 1 ]H
1 H  1 H
Fractional energy loss for Elliptical S waves > Fractional energy loss for Linear S waves: QSI 1  QSII 1
Q-1 Ratios for Elliptical (SI) and Linear (SII) Anelastic S Waves
QSI1
 1  2H S

 1 HS

2 1  H S 
 1
1
 QHS  QSII  

 1 HS  1 HS
2
 1
QHS
tan 2 
 QHS where H S 

2
1  QHS

Waves Refracted at Anelastic Boundaries
in the Earth are Inhomogeneous
A
Q1-1  Q2 -1
P
Soil
A
P
Q1-1
1
Rock Q2
A
P
Tracing Inhomogeneous SII Wave in Layered Anelastic Media
(Phase and Amplitude)
Inhomogeneous Reflected & Refracted Anelastic Seismic Waves
If the incident SI wave is homogeneous and not normally incident then the Generalized Snell's Law implies:
1
1
1) if QHS
 QHP
, then the reflected P wave is inhomogeneous,
1
 1, then the transmitted P wave is inhomogeneous,
2) if QHS
 QHP
1
 1, then the transmitted S wave is inhomogeneous.
3) if QHS
 QHS
Incident General SII Wave
Specification of Incident SII Wave:
u1  D1 exp  Au1  r  exp i(t  Pu1  r)  xˆ2
where

Pu1  k R xˆ1  d  R xˆ3  Pu1 sin u1  xˆ1  cos u1  xˆ3


Au1  k I xˆ1  d  I xˆ3  Au1 sin u1   u1  xˆ1  cos u1   u1  xˆ3
and
d   p.v. kS2  k 2
kS 
 
vHS 

1 i
Pu1 

vHS
1
2
1  QHS
sec  u1
2
2
1  1  QHS




1
QHS
2
1  1  QHS

2
2
  1  1  QHS sec  u1
k
sin u1  i
2
vHS 
1  1  QHS

1 
Au1 

2
1  QHS
sec  u1
2
2
1  1  QHS

vHS

sin u1   u1  


2
1  1  QHS
sec2  u1
1
2
1  QHS
,
elastic case:
k

sin  u1
v HS
Generalized Snell’s Law
Real part of k implies:
k R  Pu1 sin u1  Pu2 sin u2  Pu1 sin u1
or in terms of velocity
kR


sin u1
vu1

sin u2
vu2

sin u1
vu1
Imaginary part of k implies:
k I  Au1 sin u1   u1   Au2 sin u2   u2   A u1 sin u1   u1 
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
Real part of k implies:
k R  Pu1 sin u1  Pu2 sin u2  Pu1 sin u1
or in terms of velocity
kR


sin u1
vu1

sin u2
vu2

sin u1
vu1
Imaginary Part of k implies:
k I  Au1 sin u1   u1   Au2 sin u2   u2   A u1 sin u1   u1 
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
• 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.
• 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
QHS
 1
 QHS
2
kS 2   R vHS
and sin u1  2 
 2
 R vHS
kS
2
Near-Surface Reflection & Refraction Coefficients
Inhomogeneous Linear S Wave Incident on a Soil Boundary
Response of Multilayered Viscoelastic Media
to Incident Inhomogeneous Waves
Response of Viscoelastic Layer
Incident Homogeneous and Inhomogeneous SII Waves
Anelastic Reflection Coefficients
Nondestructive Testing for Metal Impurities (Becker and Richardson, 1970)
(Empirical Confirmation of Theory )
source
Water
Stainless Steel
receiver
Sea Floor Mapping of Q (age?)
Viscoelastic Rayleigh-Type Surface Wave
Propagation and Attenuation Vectors
For Component P and S solutions
Tilt of Particle Motion Orbit
Viscoelastic Rayleigh-Type Surface Wave
Tilt and Amplitude versus Depth
Love-Type Surface Waves
Multilayered Viscoelastic Media
Viscoelastic Period Equation – Love-Type Surface Waves
Viscoelastic Period Equation:

 1  i a L

aHS1

d1

  i 
vL
 v

L
 
 vHS1



-1 

 i  v 2 1  
1

i
Q
HS
HS
HS
2
2
1

  2

2
-1
-1
2


2 
 1 vHS1 1   HS1 1  i QHS2 
-1



v L QHS1
  
2
 
2
-1
-1
vHS1 1   HS1   






QHS2
 
 
a L v L QHS1
v L vHS1





1

i

1

i
arctan

  
2
aHS1 vHS1 1   HS1   vHS1 vHS2 
1   HS2  
-1
  

  
QHS




1
1  i
  
 
2
2

-1
-1
1   HS1    








Q
Q

  
a vL
v
HS1
 
1  i L
   L 1  i HS1  


 
aHS1 vHS1 1   HS1   vHS1 
1   HS1  








2
where  HS j  1  QHS
j
 j  1, 2  ,
vL 

kR
, and a L = -k I .
Solution Viscoelastic Period Equation :  vL ,aL 
1
1
Special Case : Elastic Period Equation with QHS

Q
HS
1
2
1
 v2
 v2
 
 HS

d1   2L  1   arctan  2 2 2
 1 vHS
vL
 vHS
 
1
1

 

2

v
L
 1
2

vHS
2

= 0,


v L2
 1   n 
2


vHS1
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Solution Curves -- Fundamental Mode
Absorption Coefficient and Phase Speed Dispersion
 vL , aL 
Summary
 General Viscoelasticity Characterizes Linear Material Behavior (Elastic & Anelastic)
 Solutions of Fundamental Seismic Problems for General Linear (Viscoelastic) Media
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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
Anelastic Seismic Waves are Inhomogeneous
 Wave Speed, Damping, Particle Motions, Energy Flux … vary with Inhomogeneity
 Body Wave Characteristics depend on:
1
1
Angle of Incidence (Travel Path) and Media Properties (vHS , vHP , QHS
, QHP
)
 Accurate Models of Linear Material Behavior for Seismology require
Inhomogeneous Waves
 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
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 energy
2) Energy Balance equations, Energy flux at boundaries due to interaction
3) Amplitude reflection-refraction phase and amplitude coefficients.