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Seismology and the Earth’s Deep Interior 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 LawReflection coefficientsFree surface
Reflection Seismology: an example from the Gulf of Mexico
Seismology and the Earth’s Deep Interior The elastic wave equation
Equations of motion
What are the solutions to this equation? At first we lookat infinite homogeneous isotropic media, then:
ijjiit fu 2
ijijij 2
)( ijjiijkkij uuu
)(2ijjiijkkjii uuufu
t
ijjikkiii uuufujt
22
Seismology and the Earth’s Deep Interior The elastic wave equation
Equations of motion – homogeneous media
We can now simplify this equation using the curl and div operators
2222zyx
ijjikkiii uuufujt
22
u-ufu )2(2
t
iiuu
and u-uu
… 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
u-uu )2(2
t
Let us apply the div operator to this equation, we obtain
)2(2
t
where
iuiu or
22
t 2
P wave velocity
Acoustic wave equation
Seismology and the Earth’s Deep Interior The elastic wave equation
Equations of motion – shear waves
u-uu )2(2
t
Let us apply the curl operator to this equation, we obtain
)()(2ii uu
t
we now make use of and define0
to obtain
22
t
iiu Shear wavevelocityWave equation for
shear waves
Seismology and the Earth’s Deep Interior The elastic wave equation
Elastodynamic Potentials
Any vector may be separated into scalar and vector potentials
u
Shear waves have no change in volume
22
t
P-waves have no rotation
where F is the potential for P waves and Y the potential for shear waves
u
22
t
Seismology and the Earth’s Deep Interior The elastic wave equation
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
Seismology and the Earth’s Deep Interior The elastic wave equation
Solutions to the wave equation - general
Let us consider a region without sources
Where n could be either dilatation or the vector potential and c is either P- or shear-wave velocity. The general solution to this equation is:
22 ct
Let us take a look at a 1-D example
)(),( ctxaGtx jji
Seismology and the Earth’s Deep Interior The elastic wave equation
Solutions to the wave equation - harmonic
Let us consider a region without sources
22 ct
The most appropriate choice for G is of course the use of harmonic functions:
)](exp[),( ctxaikAtxu jjiii
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[),( ctxkAtxu
tT
xtxkctkxctxk
222
)(
c wave speed
k wavenumber
l wavelength
T period
w frequency
A amplitude
Seismology and the Earth’s Deep Interior The elastic wave equation
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 coordinateslet us define
to obtain
22 ct
trr cr 22
2 12r
r
22 ct
with the known solution )( trf
Seismology and the Earth’s Deep Interior The elastic wave equation
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)
rtrf
r
1)(
1
r
1
Seismology and the Earth’s Deep Interior The elastic wave equation
Plane waves
... what can we say about the direction of displacement, thepolarization of seismic waves?
u SPu
P S
... we now assume that the potentials have the well known form of plane harmonic waves
)(exp tiA xk )(exp tiB xk
)(exp tiA xkkP )(exp tiB xkkS
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
Wavenumber, slowness, phase velocity
Seismology and the Earth’s Deep Interior The elastic wave equation
Reflection at an interface
Seismology and the Earth’s Deep Interior The elastic wave equation
Boundary Conditions
... what happens when the material parameters change?
r1 v1
r2 v2
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 The elastic wave equation
Reflection and Transmission – Snell’s Law
What happens at a (flat) material discontinuity?
Medium 1: v1
Medium 2: v2
i1
i2
2
1
2
1
sin
sin
v
v
i
i
But how much is reflected, how much transmitted?
Seismology and the Earth’s Deep Interior The elastic wave equation
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
1122
1122
A
R
1122
112
A
T
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
uin 2-D this yields
yxzz
zxxzy
zyxx
u
u
u
an incoming P wave has the form
)(exp0 txaiA jj
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
where
)'tan('exp
)tan(exp)tan(exp
31
31310
tcfxxikB
ctexxikActexxikA
cek
ec
cos
cos
fk
fc
cos'
cos'
P
Pr
SVr
e f
what we know is that
0
0
zz
xz
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
22
2
0
22
22
0
)tan1(tantan4
)tan1(tan4
)tan1(tantan4
)tan1(tantan4
ffe
fe
A
BR
ffe
ffe
A
AR
PS
PP
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 ...
P PSV
ij
The reflected amplitudes can be described by the scattering matrix S
dudu
dudu
SSSP
PSPPS
Seismology and the Earth’s Deep Interior The elastic wave equation
Example
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
There is no conversion only SH waves are reflected or transmitted
dudu
dudu
SSSS
SSSSS
SH
Seismology and the Earth’s Deep Interior The elastic wave equation
Example
Seismology and the Earth’s Deep Interior The elastic wave equation
Case 3: Solid-solid interface
To account for all possible reflections and transmissions we need 16 coefficients, described by a 4x4 scattering matrix.
P PrSVr
PtSVt
Seismology and the Earth’s Deep Interior The elastic wave equation
Case 4: Solid-Fluid interface
At a solid-fluid interface there is no conversion to SV in the lower medium.
P PrSVr
Pt
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
Refractions – waveform effects
Seismology and the Earth’s Deep Interior The elastic wave equation
Reservoir example
Seismology and the Earth’s Deep Interior The elastic wave equation
Example: Gulf of Mexico
Seismology and the Earth’s Deep Interior The elastic wave equation
Summary
Reflection and transmission coefficients can be derived using plane waves and boundary conditions
Cases include: solid-solid solid-fluid fluid-solid solid-free surface fluid-free surface
The angular dependence of R-T coefficients is complex. Their behaviour allows constraining the change of physical properties.