Introduction
Seismic waves: A primer
Computational Seismology 1
What are the governing equations for elastic wave propagation?
What are the most fundamental results in simple media?
How do we describe and input seismic sources (superposition principle)?
What are consequences of the reciprocity principle ?
What rheologies do we need (stress-strain relation)?
3-D heterogeneities and scattering
Green‘s functions, numerical solvers as linear systems
Goal: You know what to expect when running a wave simulation code!
Introduction Computational Geophysics and Data Analysis 2
Wave Equations
Introduction
The (anisotropic) elastic wave equation (strong form)
Computational Seismology 3
i ij
ij j
i
t u = ∂ + M + f
∂ 2 ( σ )
ρ
( k l l k )
kl
kl ijkl
ij
u u
c
∂ +
∂
=
= 2
ε 1
ε σ
This is the displacement – stress formulation where
Wave equation
Stress-strain relation
Strain-displacement relation
Introduction
The elastic wave equation – the cast
Computational Seismology 4
( k l l k )
kl
kl ijkl ij
i ij
ij j
i t
u u
c
f M
u
∂ +
∂
=
=
+ +
∂
=
∂
2 1
)
2 ( ε
ε σ
σ ρ
Mass density
Displacement vector Stress tensor (3x3) Moment tensor (3x3) Volumetric force
Tensor of elastic constants (3x3x3x3) Strain tensor (3x3)
) , (
) (
) , (
) , (
) , (
) , (
) (
t c
c
t f
f
t M
M
t t u
u
kl kl
ijkl ijkl
i i
ij ij
ij ij
i i
x x x
x x x x
ε ε
σ σ
ρ ρ
→
→
→
→
→
→
→
Introduction
3D to 1D
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Introduction
1D elastic wave equation
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Introduction
1D elastic wave equation
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f u
u = ∂ x ( µ ∂ x ) +
ρ
This is a scalar wave equation descriptive of transverse motions of a string
Introduction
The elastic wave equation
Computational Seismology 8
kl ijkl
ij
i ij
ij j
i t
c
f M
v
ε σ
σ ρ
=
+ +
∂
=
∂ ( )
This is the velocity – stress formulation, where
( )
i t i
i
k l l
k kl
u u
v
v v
∂
=
=
∂ +
∂
=
2
ε 1
( k l l k )
kl
kl ijkl ij
i ij
ij j
i t
u u
c
f M
u
∂ +
∂
=
=
+ +
∂
=
∂
2 1
)
2 ( ε
ε σ
σ
ρ
Introduction
3D acoustic wave equation
Computational Seismology 9
This is the constant density acoustic wave equation (sound
in a liquid or gas)
∂
∂
∂
→
∆
→
→
→
+
∆
=
2 2 2 2
) , (
) , (
) (
z y x
t x s
s
t x p
p
x c
c
s p
c p
P-velocity
Pressure
Sources .
Laplace Operator
This is equation is still tremendously important in
exploration seismics!
.
Introduction Computational Geophysics and Data Analysis 10
Rheologies
Introduction Computational Geophysics and Data Analysis 11
Stress and strain
To first order the Earth‘s crust deforms like an elastic body when the deformation (strain) is small.
In other words, if the force that causes the deformation is stopped the rock will go back to its original form.
The change in shape (i.e., the deformation) is
called strain, the forces that cause this strain
are called stresses.
Introduction
Stress-strain relation
Computational Geophysics and Data Analysis 12
The relation between stress and strain in general is described by the tensor of elastic constants c
ijklkl ijkl
ij c ε
σ =
From the symmetry of the stress and strain tensor and a thermodynamic condition if follows that the maximum number if independent constants of c
ijklis 21. In an isotropic body, where the properties do not depend on direction the relation reduces to
ij ij
ij λ δ µε
σ = Θ + 2
where l and m are the Lame parameters, q is the dilatation and d
ijis the Kronecker delta.
Generalised Hooke’s Law
Hooke’s Law
( xx yy zz ) ij
ij kk
ij ε δ ε ε ε δ
δ = = + +
Θ
Introduction
Other rheologies (not further explored in this course)
Computational Seismology 13
Viscoelasticity
• the loss of energy due to internal friction
• possibly frequency-dependent
• different for P and S waves (why?)
• described by Q
• Not easy to implement numerically for time-domain methods Porosity
• Effects of pore space (empty, filled, partially filled) on stress-strain
• Frequency-dependent effects
• Additional wave types (slow P wave)
• Highly relevant for reservoir wave propagation Plasticity
• permanent deformation due to changes in the material as a function of deformation or stress
• resulting from (micro-) damage to the rock mass
• often caused by damage on a crystallographic scale
• important close to the earthquake source
• not well constrained by observations
Introduction Computational Geophysics and Data Analysis 14
Seismic Waves
Introduction Computational Geophysics and Data Analysis 15
Consequences of the equations of motion
What are the solutions to this equation? At first we look at infinite homogeneous isotropic media, then:
ij j i
i f
u σ
ρ = + ∂
ρ µ λ + 2
p
=
v ρ
= µ v
sapproximately S-waves
s
p
v
v = 3
P-waves
Introduction
Boundary conditions: external and internal interfaces
Computational Seismology 16
´j
=
ijn
j= 0
t σ
Traction is zero n
perpendicular to free surface (needs special attention with most numerical methods)
At internal interfaces we speak of welded contact Normal tractions are continuous (they are usually not directly implemented, except fluid-solid)
n σ n
σ
1ˆ =
2ˆ
Introduction
Seismic wave types Surface waves waves
Computational Geophysics and Data Analysis 17
Love waves – transversely
polarized – superposition of SH waves in layered media Non-existing in half space
Always dispersive in layered media
Rayleigh waves – polarized in the plane through source and receiver –
superposition of P and SV waves
Non-dispersive in half space
Dispersive in layered media
Introduction Computational Geophysics and Data Analysis 18
Surface wave dispersion
Introduction Computational Geophysics and Data Analysis 19
Data Example
theoretical experimental
Introduction
Real vs. numerial dispersion
Computational Seismology 20
Introduction
Surface waves summary
Computational Seismology 21
Elastic surface waves (Love and Rayleigh) in nature generally show
dispersive behavior (later we will see that there is also dispersive behaviour due to numerical effects!)
Surface waves are a consequence of the free-surface boundary condition . We thus might expect that – when using numerical approximations there
might be differences concerning the accurate implementation of this boundary condition.
The accurate simulation of surface waves plays a dominant role in global and regional (continental scale) seismology and is usually not so
important in exploration geophysics.
Introduction Computational Seismology 22
Reflection, Transmission
Introduction
Reflection and transmission at boundaries oblique incidence - conversion
Computational Geophysics and Data Analysis 23
P S
rP
rP
tS
tP waves can be converted to S waves and vice versa. This creates a quite complex behavior of wave amplitudes and wave forms at interfaces. This behavior can be used to constrain the properties of the material interface.
incoming P-wave
reflections
transmissions Material 1
Material 2
Interface
Introduction Computational Geophysics and Data Analysis 24
Analytical solutions
Introduction 25
„delta“-generating function
Spatial (or temporal) source function
bc stands for boxcar
Introduction
Analytical solutions for acoustic wave equation (Green‘s function)
Computational Seismology 26
Introduction
Analytical solutions
Computational Seismology 27
Introduction Computational Geophysics and Data Analysis 28
Seismic sources
Introduction Computational Geophysics and Data Analysis 29
Radiation from a point double-couple source
Geometry we use to express the seismic wavefield radiated by point double-couple source with area A and slip Du
Here the fault plane is the x
1x
2- plane and the slip is in x
1-direction.
Which stress components are affected?
Introduction Computational Geophysics and Data Analysis 30
Radiation from a point source (M
zx=M
xz=M
0)… one of the most important results of
seismology!
… Let’s have a closer look …
u ground displacement as a function of space and r timedensity
r distance from source Vs shear velocity
Vp P-velocity N near field IP/S intermediate field FP/S far field
M0 seismic moment
Introduction Computational Geophysics and Data Analysis 31
Radiation from a point source (M
zx=M
xz=M
0)Near field term contains the static
deformation
Intermediate terms
Far field terms:
the main ingredient for source
inversion, ray
theory, etc.
Introduction
Elastic waves 2D
Computational Seismology 32
Explosion Double couple
Introduction Computational Geophysics and Data Analysis 33
Beachballs and moment tensor
explosion - implosion
vertical strike slip fault
vertical dip slip fault
45° dip thrust fault
compensated linear vector dipoles
Introduction
Translation, Divergence, Rotation, and all that (M4, 3km away)
Computational Seismology 34
Introduction Computational Geophysics and Data Analysis 35
Source mechanisms
Basic fault types and their appearance in the focal
mechanisms. Dark
regions indicate
compressional P-
wave motion.
Introduction
Radiation patterns of a double couple point sources
Computational Geophysics and Data Analysis 36
Far field P – blue
Far field S - red
Introduction Computational Geophysics and Data Analysis 37
Seismic moment M
0A t
u M 0 = µ ∆ ( )
M
0seismic moment m rigidity
< ∆ u(t)> average slip
A fault area Note that the far-field
displacement is proportional
to the moment rate!
Introduction Computational Geophysics and Data Analysis 38
Source time function
Introduction 39
Displacement, Velocity, Acceleration
Introduction
The superposition principle
Computational Seismology 40
Introduction
Discrete representation of finite sources
Computational Seismology 41
Introduction
Superposition principle
Computational Seismology 42
Introduction
The Earth (or a numerical solver) as a linear system
Computational Seismology 43
Introduction Computational Seismology 44
⊗
⊗
=
=
Green‘s function Source time function Seismogram
anal yt ic al num er ic al
Numerical
seismogram using peak source time function
Introduction
Source-receiver reciprocity
Computational Seismology 45
Introduction Computational Seismology 46
Introduction
In other words
Computational Seismology 47
Seismogram through
random model
A B
A -> B
B -> A
Introduction
Time reversal – reverse acoustics
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forward
reverse
Introduction
Practical example – Valhall active experiment
Computational Seismology 49
Introduction
Full waveform inversion – Inverse Problems
Computational Seismology 50
Sirgue et al., 2010
Introduction
Summary
To understand seismic wave propagation the following concepts need to be understood:
Computational Geophysics and Data Analysis 51