The Elastic Wave Equation The Elastic Wave Equation

The Elastic Wave Equation The Elastic Wave Equation

• Elastic waves in infinite homogeneous isotropic media

Numerical simulations for simple sources

• Plane wave propagation in infinite media

Frequency, wavenumber, wavelength

• Conditions at material discontinuities

Snell’s Law

Reflection coefficients Free surface

Equations of motion Equations of motion

What are the solutions to this equation? At first we look at infinite homogeneous isotropic media, then:

ij j i

i

t

u f 

 

  

ij ij

ij

 

   2

) (

ij k k

ij

   u     u   u



 ^{( )} 

2

i j j

i ij

k k j

i

i

f u u u

t

u        

   



i j

j i k

k i i

i

f u u u

u

2

        

   



Equations of motion – homogeneous media Equations of motion – homogeneous media

We can now simplify this equation using the curl and div operators

and

… this holds in any coordinate system …

This equation can be further simplified, separating the wavefield into curl free and div free parts

2 2

2 2

z y

x

   











i j

j i k

k i i

i

f u u u

u

t

2

2

        

   



u -

u f

u         

   



( 2 )

t

i i

u u  





u -

u

u       



Equations of motion – P waves Equations of motion – P waves

Let us apply the div operator to this equation, we obtain

where

or

P wave velocity Acoustic wave

equation

u -

u

u        

   



( 2 )

t









 

 (  2 ) 

t

u

u  





 





  



t





   ^{ 2}

Equations of motion – shear waves Equations of motion – shear waves

Let us apply the curl operator to this equation, we obtain

we now make use of and define

to obtain Shear wave velocity Wave equation for

shear waves

u -

u

u        

   



( 2 )

t

) (

)

2

(

i

i

u

t

  u         

    



 0





 





  





  

i

u

 





Elastodynamic Potentials Elastodynamic Potentials

Any vector may be separated into scalar and vector potentials

Shear waves have no change in volume P-waves have no rotation

where F is the potential for P waves and Y the potential for shear waves

Y





 F

 u 

F



 





  



t

Y



 Y

















  u





  



Seismic Velocities Seismic Velocities

Material and Source P-wave velocity (m/s) shear wave velocity (m/s)

Water 1500 0

Loose sand 1800 500

Clay 1100-2500

Sandstone 1400-4300

Anhydrite, Gulf Coast 4100

Conglomerate 2400

Limestone 6030 3030

Granite 5640 2870

Granodiorite 4780 3100

Diorite 5780 3060

Basalt 6400 3200

Dunite 8000 4370

Gabbro 6450 3420

Solutions to the wave equation - general Solutions to the wave equation - general

Let us consider a region without sources

Where h could be either dilatation or the vector potential and c is either P- or shear-wave velocity. The general solution to this equation is:

Let us take a look at a 1-D example

h h  



c

t

) (

) ,

( x

t  G a

x

 ct

h

Solutions to the wave equation - harmonic Solutions to the wave equation - harmonic

Let us consider a region without sources

The most appropriate choice for G is of course the use of harmonic functions:

h h  



c

t

)]

( exp[

) ,

( x t A ik a x ct

u





Solutions to the wave equation - harmonic Solutions to the wave equation - harmonic

… taking only the real part and considering only 1D we obtain

c wave speed

k wavenumber

 wavelength

T period

w frequency

A amplitude

)]

( cos[

) ,

( x t A k x ct

u  

T t x

t x

kct kx

ct x

k 

 w 



 2 2

) 2

(       

Spherical Waves Spherical Waves

Let us assume that h is a function of the distance from the source

where we used the definition of the

Laplace operator in spherical coordinates let us define

to obtain r

with the known solution

h h  



c

t

h h

h

h

c

r

2

 2   1 







r h  h h

h  



c

t

h  f ( r   t )

Geometrical spreading Geometrical spreading

so a disturbance propagating away with spherical wavefronts decays like

... this is the geometrical spreading for spherical waves, the amplitude decays proportional to 1/r.

r

If we had looked at cylindrical waves the result would have been that the waves decay as (e.g. surface waves)

t r r

r f

) 1 1 (





  h

h

r

 1

h

Plane waves Plane waves

... what can we say about the direction of displacement, the polarization of seismic waves?

... we now assume that the potentials have the well known form of plane harmonic waves

shear waves are transverse because S is normal to the wave

vector k P waves are longitudinal as P is

parallel to k

Y





 F



u   u  P  S

F



P  S    Y

) (

exp i t

A   w



F k x Y  B exp i ( k  x  w t ) )

(

exp i t

A   w

 k k x

P S  k  B exp i ( k  x  w t )

Heterogeneities Heterogeneities

.. What happens if we have heterogeneities ?

Depending on the kind of

reflection part or all of the signal is reflected or transmitted.

• What happens at a free surface?

• Can a P wave be converted in an S wave or vice versa?

• How big are the amplitudes of the

reflected waves?

Boundary Conditions Boundary Conditions

... what happens when the material parameters change?



v



v

welded interface

At a material interface we require continuity of displacement and traction A special case is the free surface condition, where the surface

tractions are zero.

Reflection and Transmission – Snell’s Law Reflection and Transmission – Snell’s Law

What happens at a (flat) material discontinuity?

Medium 1: v

Medium 2: v

i

i

But how much is reflected, how much transmitted?

2 1 2

1

sin sin

v v i

i 

Reflection and Transmission coefficients Reflection and Transmission coefficients

Medium 1: r1,v1

Medium 2: r2,v2

T

Let’s take the most simple example: P-waves with normal incidence on a material interface

A R

At oblique angles conversions from S-P, P-S have to be considered.

1 1 2

2

1 1 2

2



















  A R

1 1 2

2

1

2 1













 

A

T

Reflection and Transmission – Ansatz Reflection and Transmission – Ansatz

How can we calculate the amount of energy that is transmitted or reflected at a material discontinuity?

We know that in homogeneous media the displacement can be described by the corresponding potentials

in 2-D this yields

an incoming P wave has the form

Y





 F

 u 

y x z

z

z x x

z y

z y x

x

u u u

Y



 F





Y



 Y





Y



 F





) (

exp i a x t

A w 





F

Reflection and Transmission – Ansatz Reflection and Transmission – Ansatz

... here a

are the components of the vector normal to the wavefront : a

=(cos e, 0, -sin e), where e is the angle between surface and ray direction, so that for the free surface

where

P

P

SV

e f

what we know is that

) ' tan

( ' exp

) tan

( exp )

tan (

exp

3 1

3 1

3 1

0

t c f

x x

ik B

ct e

x x

ik A

ct e

x x

ik A





 Y











 F

e c k

c e

w

 w









cos cos

f k

c f

cos '

' cos

 w







0 0





zz xz





Reflection and Transmission – Coefficients Reflection and Transmission – Coefficients

... putting the equations for the potentials (displacements) into these equations leads to a relation between incident and

reflected (transmitted) amplitudes

These are the reflection coefficients for a plane P wave incident on a free surface, and reflected P and SV waves.

2 2

2 0

2 2

2 2

0

) tan

1 ( tan

tan 4

) tan

1 ( tan

4

) tan

1 ( tan

tan 4

) tan

1 ( tan

tan 4

f f

e

f e

A R B

f f

e

f f

e A

R A

PS PP







 









 



Case 1: Reflections at a free surface Case 1: Reflections at a free surface

A P wave is incident at the free surface ...

P P

SV i j

The reflected amplitudes can be described by the scattering matrix S

 

 



 

d u d

u

d u d

u

S S S

P

P S P

S P

Case 2: SH waves Case 2: SH waves

For layered media SH waves are completely decoupled from P and SV waves

There is no conversion only SH waves are reflected or transmitted

SH

 

 



  ^{u d u d} S S S

S

S S S

S S

SH example

SH example

SH relation SH relation

) cos(

) cos(

) cos(

2

) cos(

) cos(

) cos(

) cos(

2 2

2

´ 1

1 1

1 1

1

2 2

2 1

1 1

2 2

2 1

1 1











































 



 

d d

u d

S S

S

S

Polarity effects

Polarity effects

Example for crust SH case

Example for crust SH case

Case 3: Solid-solid interface Case 3: Solid-solid interface

To account for all possible reflections and transmissions we need 16 coefficients, described by a 4x4 scattering matrix.

P SV

P

P

SV

Case 4: Solid-Fluid interface Case 4: Solid-Fluid interface

At a solid-fluid interface there is no conversion to SV in the lower medium.

P SV

P

P

Reflection coefficients - example

Reflection coefficients - example

Reflection coefficients - example

Reflection coefficients - example

Summary Summary

 In homogeneous full space P and S waves are solutions to the elastic wave equation

 P waves are compressional (curl-free) and S waves are transversal (div-free)

 Material discontinuities (stress continuity) leads to transmission and reflection coefficients and conversions for each wave type

 Information from reflected waves can be used to inver the change of material properties

 AVO amplitude versus offset

 AVA amplitude versus angle