Elastic Wave Equations in Seismology
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
Equations of motion
What are the solutions to this equation? At first we look
at infinite homogeneous isotropic media, then:
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
Equations of motion – P waves
Let us apply the
div
operator to this equation, we obtain
where
or
P wave velocity
Acoustic wave
equation
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
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
is the potential for P
waves and
the potential for shear waves
Seismology and the Earth’s Deep Interior
Seismic Velocities
Seismology and the Earth’s Deep Interior
Solutions to the wave equation - general
Let us consider a region without sources
Where
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
Seismology and the Earth’s Deep Interior
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:
Seismology and the Earth’s Deep Interior
Solutions to the wave equation - harmonic
… taking only the real part and considering only 1D we obtain
Seismology and the Earth’s Deep Interior
Spherical Waves
Let us assume that
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
Seismology and the Earth’s Deep Interior
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)
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
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?
Seismology and the Earth’s Deep Interior
Boundary Conditions
... what happens when the material parameters change?
1
v
1
2
v
2
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.
Seismology and the Earth’s Deep Interior
Reflection and Transmission – Snell’s Law
What happens at a (flat) material discontinuity?
Medium 1: v
1
Medium 2: v
2
i
1
i
2
But how much is reflected, how much transmitted?
Seismology and the Earth’s Deep Interior
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.
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
Reflection and Transmission – Ansatz
... here a
i
are the components of the vector normal to the
wavefront : a
i
=(cos e, 0, -sin e), where e is the angle between
surface and ray direction, so that for the
free surface
where
P
P
r
SV
r
e
f
what we know is that
Seismology and the Earth’s Deep Interior
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.
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
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
SH example
Seismology and the Earth’s Deep Interior
SH relation
Seismology and the Earth’s Deep Interior
Polarity effects
Seismology and the Earth’s Deep Interior
Example for crust SH case
Seismology and the Earth’s Deep Interior
Seismology and the Earth’s Deep Interior
Case 3: Solid-solid interface
To account for all possible reflections and transmissions
we need 16 coefficients, described by a 4x4 scattering
matrix.
P
P
r
SV
r
P
t
SV
t
Seismology and the Earth’s Deep Interior
Case 4: Solid-Fluid interface
At a solid-fluid interface there is no conversion to SV in
the lower medium.
P
P
r
SV
r
P
t
Seismology and the Earth’s Deep Interior
Reflection coefficients - example
Seismology and the Earth’s Deep Interior
Reflection coefficients - example
Summary
Seismology and the Earth’s Deep Interior
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
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AVO amplitude versus offset
AVA amplitude versus angle
Explore the fundamentals of elastic wave equations for seismology, including topics such as plane wave propagation, reflection coefficients, and wavefield simplifications using curl and div operators. Learn about P-waves, shear waves, and elastodynamic potentials in the context of infinite homogeneous isotropic media. Gain insights into wave propagation in the Earth's deep interior.
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Presentation Transcript
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 Seismology and the Earth s Deep Interior The elastic wave equation
Equations of motion 2 = + u f t i i j ij What are the solutions to this equation? At first we look at infinite homogeneous isotropic media, then: = + 2 ij ij ij = + + ( ) u ( u u ij k k ij i j j i ) ) i = + + + 2 ( u f u u u i i j k k ij i j j t = + + + 2 2 u f u u u i i i k k i j j i t j Seismology and the Earth s Deep Interior The elastic wave equation
Equations of motion homogeneous media = + + + 2 2 u f u u u i i i k k i j j i t j We can now simplify this equation using the curl and div operators = i u = = = + + 2 2 x 2 y 2 z iu u - u u and = + ( + 2 ) - u 2 u f u t this holds in any coordinate system This equation can be further simplified, separating the wavefield into curl free and div free parts Seismology and the Earth s Deep Interior The elastic wave equation
Equations of motion P waves = ( + 2 ) - u 2 u u t Let us apply the div operator to this equation, we obtain t = ( + 2 2 ) where u Acoustic wave equation = = iu P wave velocity i or + 2 = 2 2 = t Seismology and the Earth s Deep Interior The elastic wave equation
Equations of motion shear waves = ( + 2 ) - u 2 u u t Let us apply the curl operator to this equation, we obtain ) ( i u t + = = = Wave equation for shear waves + 2 ( ) u i 0 we now make use of and define iu Shear wave velocity to obtain i = 2 2 = t Seismology and the Earth s Deep Interior The elastic wave equation
Elastodynamic Potentials Any vector may be separated into scalar and vector potentials + = u where is the potential for P waves and the potential for shear waves = = = = u P-waves have no rotation Shear waves have no change in volume t = = 2 2 2 2 t Seismology and the Earth s Deep Interior The elastic wave equation
Seismic Velocities Material and Source Water P-wave velocity (m/s) 1500 shear wave velocity (m/s) 0 Loose sand 1800 500 Clay Sandstone 1100-2500 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 Seismology and the Earth s Deep Interior The elastic wave equation
Solutions to the wave equation - general Let us consider a region without sources t = 2 2 c Where could be either dilatation or the vector potential and c is either P- or shear-wave velocity. The general solution to this equation is: = ( , ) ( ) x t G a x ct i j j Let us take a look at a 1-D example Seismology and the Earth s Deep Interior The elastic wave equation
Solutions to the wave equation - harmonic Let us consider a region without sources t = 2 2 c The most appropriate choice for G is of course the use of harmonic functions: = ( , ) exp[ ( )] u x t A ik a x ct i i i j j Seismology and the Earth s Deep Interior The elastic wave equation
Solutions to the wave equation - harmonic taking only the real part and considering only 1D we obtain = ( , ) cos[ ( )] u x t A k x ct 2 2 2 = = = ( ) k x ct kx kct x t x t T c k wave speed wavenumber wavelength T period frequency A amplitude Seismology and the Earth s Deep Interior The elastic wave equation
Spherical Waves Let us assume that is a function of the distance from the source t = 2 2 c 2 1 r = r + r = 2 2 t 2 r c where we used the definition of the Laplace operator in spherical coordinates let us define = r to obtain = t = 2 2 ( ) f r t c with the known solution Seismology and the Earth s Deep Interior The elastic wave equation
Geometrical spreading so a disturbance propagating away with spherical wavefronts decays like 1 1 = ( ) f r t r r r ... this is the geometrical spreading for spherical waves, the amplitude decays proportional to 1/r. If we had looked at cylindrical waves the result would have been that the waves decay as (e.g. surface waves) 1 r Seismology and the Earth s Deep Interior The elastic wave equation
Plane waves ... what can we say about the direction of displacement, the polarization of seismic waves? + = u = P S = + u P S = ... we now assume that the potentials have the well known form of plane harmonic waves ) ( exp t i A = x k = k x B exp ( ) i t = = k k x P exp ( ) k B k x S A i t exp ( ) i t shear waves are transverse because S is normal to the wave vector k P waves are longitudinal as P is parallel to k Seismology and the Earth s Deep Interior The elastic wave equation
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? Seismology and the Earth s Deep Interior The elastic wave equation
Boundary Conditions ... what happens when the material parameters change? 1v1 welded interface 2v2 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. Seismology and the Earth s Deep Interior The elastic wave equation
Reflection and Transmission Snells Law What happens at a (flat) material discontinuity? sin i v = 1 1 Medium 1: v1 i1 sin i v 2 2 i2 Medium 2: v2 But how much is reflected, how much transmitted? Seismology and the Earth s Deep Interior The elastic wave equation
Reflection and Transmission coefficients Let s take the most simple example: P-waves with normal incidence on a material interface R A 1 R = 2 2 1 + 1 Medium 1: r1,v1 A 2 2 1 Medium 2: r2,v2 1 + 2 T = 1 1 A T 2 2 1 At oblique angles conversions from S-P, P-S have to be considered. Seismology and the Earth s Deep Interior The elastic wave equation
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 + = u in 2-D this yields u = = x x y z u y z x x z = + u z z x y an incoming P wave has the form = exp ( ) A i a x t 0 j j Seismology and the Earth s Deep Interior The elastic wave equation
Reflection and Transmission Ansatz ... here ai are the components of the vector normal to the wavefront : ai=(cos e, 0, -sin e), where e is the angle between surface and ray direction, so that for the free surface = + + exp ( tan ) exp ( tan ) A ik x x e ct A ik x x e ct 0 1 3 1 3 = + exp ( ' tan ' ) B ik x x f c t 1 3 f e cos = = where c ' c Pr e cos f = = cos k e = ' cos k f c SVr P = 0 what we know is that xz = 0 zz Seismology and the Earth s Deep Interior The elastic wave equation
Reflection and Transmission Coefficients ... putting the equations for the potentials (displacements) into these equations leads to a relation between incident and reflected (transmitted) amplitudes 2 2 4 tan tan 1 ( tan ) A e f f = = R PP + 2 2 4 tan tan 1 ( tan ) A e f f 0 2 4 tan e 1 ( + tan ) B e f = = R PS 2 2 4 tan tan 1 ( tan ) A f f 0 These are the reflection coefficients for a plane P wave incident on a free surface, and reflected P and SV waves. Seismology and the Earth s Deep Interior The elastic wave equation
Case 1: Reflections at a free surface A P wave is incident at the free surface ... i j P P SV The reflected amplitudes can be described by the scattering matrix S = P P P S P u d u d S S S S u d u d Seismology and the Earth s Deep Interior The elastic wave equation
Case 2: SH waves For layered media SH waves are completely decoupled from P and SV waves SH There is no conversion only SH waves are reflected or transmitted S S S S u d u d = S S S S S u d u d Seismology and the Earth s Deep Interior The elastic wave equation
SH example Seismology and the Earth s Deep Interior The elastic wave equation
SH relation cos( ) cos( ) = 1 1 1 2 2 2 S S d u + cos( ) cos( ) 1 1 1 2 2 2 2 cos( + ) = 1 1 1 S S d d cos( ) cos( ) 1 1 1 2 2 2 Seismology and the Earth s Deep Interior The elastic wave equation
Polarity effects Seismology and the Earth s Deep Interior The elastic wave equation
Example for crust SH case Seismology and the Earth s Deep Interior The elastic wave equation
Case 3: Solid-solid interface SVr Pr P SVt Pt To account for all possible reflections and transmissions we need 16 coefficients, described by a 4x4 scattering matrix. Seismology and the Earth s Deep Interior The elastic wave equation
Case 4: Solid-Fluid interface SVr Pr P Pt At a solid-fluid interface there is no conversion to SV in the lower medium. Seismology and the Earth s Deep Interior The elastic wave equation
Reflection coefficients - example Seismology and the Earth s Deep Interior The elastic wave equation
Reflection coefficients - example Seismology and the Earth s Deep Interior The elastic wave equation
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 Seismology and the Earth s Deep Interior The elastic wave equation
