The Elastic Wave Equation Equationsequations Ofof Motionmotion

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TheThe ElasticElastic WaveWave EquationEquation

• 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

• Reflection Seismology: an example from the Gulf of Mexico

Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion

2 ρ∂t ui = fi + ∂ jσ ij

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

σ ij = λθδ ij + 2µε ij

σ ij = λ∂ kukδ ij + µ(∂iu j + ∂ jui )

ρ∂ 2u = f + ∂ (λ∂ u δ + µ(∂ u + ∂ u )) t i i j k k ij i j j i ρ∂ 2u = f + λ∂ ∂ u + µ∂ ∂ u + µ∂ 2 u t i i i k k i j j j i

Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion –– hhomogeneousomogeneous mediamedia

ρ∂ 2u = f + λ∂ ∂ u + µ∂ ∂ u + µ∂ 2u t i i i k k i j j j i

We can now simplify this equation using the curl and div operators 2 2 2 2 ∇ •u = ∂iui ∇ = ∆ = ∂ x + ∂ y + ∂ z

and ∆u= ∇∇ •u-∇×∇×u

ρ∂ 2u= f+ (λ + 2µ)∇∇ •u- µ∇×∇×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 EquationsEquations ofof motionmotion –– PP waveswaves

ρ∂ 2u= (λ + 2µ)∇∇ •u- µ∇×∇×u t

Let us apply the div operator to this equation, we obtain ρ∂ 2θ = (λ + 2µ)∆θ t where Acoustic wave equation θ = ∇u= ∂iui P wave velocity or

2 1 λ + 2µ ∂ θ = ∆θ α = t α 2 ρ

Seismology and the Earth’s Deep Interior The elastic wave equation EquationsEquations ofof motionmotion –– sshearhear waveswaves

ρ∂ 2u= (λ + 2µ)∇∇ •u- µ∇×∇×u t

Let us apply the curl operator to this equation, we obtain ρ∂ 2∇×u = (λ + µ)∇×∇θ + µ∆(∇×u ) t i i

we now make use of ∇×∇θ = 0 and define

∇×ui = ϕi to obtain Shear wave Wave equation for velocity shear waves

2 1 µ ∂ ϕ = ∆ϕ β = t β 2 ρ

Seismology and the Earth’s Deep Interior The elastic wave equation ElastodynamicElastodynamic PotentialsPotentials

Any vector may be separated into scalar and vector potentials u = ∇Φ + ∇× Ψ

where P is the potential for Φ waves and Ψ the potential for shear waves ⇒ θ = ∆Φ ⇒ ϕ = ∇×u = ∇×∇× Ψ = −∆Ψ

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

∂ 2θ = α 2∆θ ∂ 2ϕ = β 2∆ϕ t t

Seismology and the Earth’s Deep Interior The elastic wave equation SeismicSeismic VelocitiesVelocities

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 SolutionsSolutions toto thethe wavewave equationequation -- ggeneraleneral

Let us consider a region without sources

∂ 2η = c2∆η t

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:

η(xi ,t) = G(a j x j ± ct)

Let us take a look at a 1-D example

Seismology and the Earth’s Deep Interior The elastic wave equation SolutionsSolutions toto thethe wavewave equationequation -- hharmonicarmonic

Let us consider a region without sources

∂ 2η = c2∆η t

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

ui (xi ,t) = Ai exp[ik(a j x j − ct)]

Seismology and the Earth’s Deep Interior The elastic wave equation SolutionsSolutions toto thethe wavewave equationequation -- hharmonicarmonic

… taking only the real part and considering only 1D we obtain u(x,t) = Acos[k(x − ct)]

2π 2π 2π k(x − ct) = kx − kct = x −ωt = x − t λ λ T

c wave speed kwavenumber λ wavelength Tperiod ω frequency A amplitude

Seismology and the Earth’s Deep Interior The elastic wave equation SphericalSpherical WavesWaves

2 2 Let us assume that η is a function ∂ η = c ∆η of the distance from the source t

2 1 r ∆η = ∂ 2η + ∂ η = ∂t η r r r c2 2

where we used the definition of the Laplace operator in spherical coordinates let us define η η = to obtain r

2 2 ∂ η = c ∆η with the known solution η = f (r −αt) t

Seismology and the Earth’s Deep Interior The elastic wave equation GeometricalGeometrical spreadingspreading

so a disturbance propagating away with spherical wavefronts decays like 1 1 η = f (r −αt) η ≈ 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 η ≈ r

Seismology and the Earth’s Deep Interior The elastic wave equation PlanePlane waveswaves

... what can we say about the direction of displacement, the polarization of seismic waves? u = ∇Φ + ∇× Ψ ⇒ u = P + S P = ∇Φ S = ∇× Ψ

... we now assume that the potentials have the well known form of plane harmonic waves Φ = Aexpi(k• x−ωt) Ψ = B expi(k• x−ωt)

P = Akexpi(k• x−ωt) S = k× B expi(k• x−ωt)

P waves are longitudinal as P is shear waves are transverse parallel to k because S is normal to the wave vector k

Seismology and the Earth’s Deep Interior The elastic wave equation HeterogeneitiesHeterogeneities

.. 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 BoundaryBoundary ConditionsConditions

... what happens when the material parameters change?

ρ1 v1 welded interface

ρ v 2 2 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 ReflectionReflection andand TransmissionTransmission –– SSnell’snell’s LawLaw

What happens at a (flat) material discontinuity?

sin i1 v1 Medium 1: v1 i1 = sin i2 v2

Medium 2: v2

But how much is reflected, how much transmitted?

Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission coefficientscoefficients

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

R A R ρ α − ρ α = 2 2 1 1 Medium 1: r1,v1 A ρ2α2 + ρ1α1

Medium 2: r2,v2 T 2ρ α = 1 1 T A ρ2α2 + ρ1α1

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

Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– AAnsatznsatz

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 = ∇Φ + ∇× Ψ in 2-D this yields

ux = ∂ xΦ − ∂ y Ψz

u y = ∂ z Ψx − ∂ xΨz

uz = ∂ zΦ + ∂ xΨy an incoming P wave has the form ω Φ = A expi (a x −αt) 0 α j j Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– AAnsatznsatz

... 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

Φ = A0 expik(x1 − x3 tan e − ct) + Aexpik(x1 + x3 tan e − ct)

Ψ = B expik'(x1 + x3 tan f − c't)

α β e f where c = c'= cose cos f Pr ω ω ω k = cose = k'= cos f α c β P SVr

what we know is that σ xz = 0

σ zz = 0

Seismology and the Earth’s Deep Interior The elastic wave equation ReflectionReflection andand TransmissionTransmission –– CCoefficientsoefficients

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

A 4 tan e tan f − (1− tan 2 f )2 RPP = = 2 2 A0 4 tan e tan f + (1− tan f ) A 4 tan e − (1− tan 2 f ) RPS = = 2 2 A0 4 tan e tan f + (1− tan f )

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 CaseCase 1:1: ReflectionsReflections atat aa freefree surfacesurface

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

⎛ Pu Pd Su Pd ⎞ S = ⎜ ⎟ ⎝ Pu Sd Su Sd ⎠

Seismology and the Earth’s Deep Interior The elastic wave equation CaseCase 2:2: SHSH waveswaves

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

There is no conversion only SH waves are reflected or transmitted

⎛ Su Sd Su Sd ⎞ S = ⎜ ⎟ ⎝ Su Sd Su Sd ⎠

Seismology and the Earth’s Deep Interior The elastic wave equation CaseCase 3:3: Solid-solidSolid-solid interfaceinterface

SVr P P r

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 CaseCase 4:4: Solid-FluidSolid-Fluid interfaceinterface

SVr P P r

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 ReflectionReflection coefficientscoefficients -- eexamplexample

Seismology and the Earth’s Deep Interior The elastic wave equation RefractionsRefractions –– wwaveformaveform effectseffects

Seismology and the Earth’s Deep Interior The elastic wave equation ScatteringScattering andand AttenuationAttenuation

Propagating seismic waves loose energy due to

• geometrical spreading

e.g. the energy of spherical wavefront emanating from a point source is distributed over a spherical surface of ever increasing size

• intrinsic attenuation

elastic wave propagation consists of a permanent exchange between potential (displacement) and kinetic (velocity) energy. This process is not completely reversible. There is energy loss due to shear heating at grain boundaries, mineral dislocations etc.

• scattering attenuation

whenever there are material changes the energy of a wavefield is scattered in different phases. Depending on the material properties this will lead to amplitude decay and dispersive effects.

Seismology and the Earth’s Deep Interior The elastic wave equation IntrinsicIntrinsic attenuationattenuation

How can we describe intrinsic attenuation? Let us try a spring model:

The equation of motion for a damped harmonic oscillator is

γ mx +γx + kx = 0 ε = γ k mϖ 0 x + x + x = 0 1/ 2 m m ⎛ k ⎞ 2 ϖ 0 = ⎜ ⎟ x+ εϖ 0 x +ω0 x = 0 ⎝ m ⎠

where ε is the friction coefficient.

Seismology and the Earth’s Deep Interior The elastic wave equation QQ

The solution to this system is

−εϖ 0t 2 x(t) = A0e sin(ϖ 0t 1−ε )

so we have a time-dependent amplitude of

ϖ t − 0 − εϖ 0 t 2 Q A (t ) = A 0 e = A 0 e

and defining 1 A π ε = δ = ln 1 Q = 2Q A2 δ

Q is the energy loss per cycle. Intrinsic attenuation in the Earth is in general described by Q.

Seismology and the Earth’s Deep Interior The elastic wave equation DispersionDispersion effectseffects

What happens if we have frequency independent Q, i.e. each frequency looses the same amount of energy per cycle? −( fπQv)x A(x) = A0e

high frequencies – more oscillations – more attenuation low frequencies – less oscillations – less attenuation

Consequences:

- high frequencies decay very rapidly - pulse broadening

In the Earth we observe that Qp is large than QS. This is due to the fact that intrinsic attenuation is predominantly caused by shear lattice effects at grain boundaries.

Seismology and the Earth’s Deep Interior The elastic wave equation QQ inin thethe EarthEarth

Rock Type Qp QS

Shale 30 10

Sandstone 58 31

Granite 250 70-250

Peridotite 650 280 Midmantle 360 200 Lowermantle 1200 520 Outer Core 8000 0

Seismology and the Earth’s Deep Interior The elastic wave equation ScatteringScattering

Seismology and the Earth’s Deep Interior The elastic wave equation