Download Seismo 1: Body waves

Seismo 1: Body waves Barbara Romanowicz Ins%tut de Physique du Globe de Paris, Univ. of California, Berkeley Les Houches, 6 Octobre 2014 Seismology
•  Seismology is the study of the generation
and propagation of elastic waves through
the Earth
•  Wave generation: source seismology
–  Earthquakes
–  Tremor/micro-seismicity
–  Explosions/mine collapses
–  Ocean-solid earth interactions (Earth’s hum..)
•  Wave propagation: structural seismology
–  Oil/gas/mineral exploration
–  Crust, Mantle, Core structure
u(t) = S(t) ! P(t) ! I(t)
Typically : Z (verFcal) , N, E (2 horizontals oriented orthogonally) VonReubeur Paschwitz 1889 “Reading the report on this earthquake in NATURE (June 13, p. 162), I was struck by its coincidence in Fme with a very singular perturbaFon registered by two delicate horizontal pendulums at the Observatories of Potsdam and Wilhelmshaven.” VonRebeur Paschwitz, E., Nature, 40, July 25, 1889 Global Seismology
•  Seismology is the most powerful technique to sample and
constrain the present day physical structure of the Earth’s
interior.
–  Seismic waves are generated by natural earthquakes and bounce
around the earth’s interior
–  For large enough earthquakes (M~5) they can be observed as far as
the antipode of the earthquake
•  Seismology was essential in defining the 1D structure of the Earth
Global seismology
•  Powerful imaging techniques (seismic tomography)
have been developed to infer physical properties
inside the earth’s mantle and core from observations
of records of natural earthquakes (“seismograms”)–
similarly to techniques used in medicine (CAT-scan,
IRM..ultrasound..)
Surface wave
P
S
SS
Body waves
Energy travels through the Interior of the earth
Rays bend and reflect due to variations of physical
properties of the Earth’s interior
"-> Snell’s law
  Two types of body waves: P waves and S waves
  P waves faster than S waves
  S waves do not propagate through liquids
Surface wave
P
S
SS
Surface waves
  Energy travels along the surface
of the earth and decays with depth
  Dispersive: speed varies with
frequency
 Arrive after the main body waves
  Contain most of the long period
energy radiated by an earthquake
  Large amplitudes dominate
seismograms
Amplitude Distance (x) Period = Frequency -­‐1 T = 1/f Wavelength = Wavenumber-­‐1 λ= 1/k Speed (c): c = λf = f/k 10 From Stein and Wysession, 2003
What are elastic waves?
•  In an elastic continuum, perturbing the position
of one particle will result in a restoring force
that will accelerate that particle back and past
the equilibrium position
–  In an elastic medium, stress and strain
are linearly related (for small strains, and
assuming perfect elasticity)
–  In 1D, this is the familiar Hooke’s Law:
F = -­‐ k x The constant k is the spring stiffness
•  Perturbation in position of one particle
will accelerate its neighbors
Constitutive relationship
•  In 1D:
– 
F=-kx
•  In 3D, x becomes the strain tensor, and F becomes
the stress tensor:
k=3 l=3
! ij = cijkl " kl # $$ cijkl " kl
k=1 l=1
10/7/14 13 Tractions and the stress tensor
•  The stress tensor describes
forces acting across infinitesimal
planes within a continuum (i.e.
tractions)
•  Its units are: N/m2 a.k.a. Pa
•  Can vary with position, describes
both compression / estension and
shearing
•  Stress tensor is symmetric, in 3D
has 6 independent components
Pressure: direcFon depends on surface 10/7/14 Gravity: direcFon always down 15 Internal deformation
•  How do we describe (small) internal deformations?
10/7/14 CIDER 2014 -­‐ Seismology #1 -­‐ Ved Lekic 16 Strain and Rotation tensors
•  No net rotation only if ∂ui/∂xj – ∂uj/∂xi = 0 •  Decompose deformation into Strain and Rotation matrices:
Strain tensor is symmetric
+ Rotation tensor is anti-symmetric
•  Strain is a dimensionless quantity (ΔL/L ) •  Strain due to passage of seismic waves are typically <10-5
10/7/14 1
!ij = (ui, j + u j,i )
2
17 Elasticity: Linear stress-strain
relationship
! ij = cijkl" kl
Hooke s law
Strains <10-4
•  cijkl is the elastic tensor
–  Reduced from 81 to 21 independent elements
(symmetries)
•  Isotropic material: properties are the same in
all directions:
–  there are only 2 independent elements, λ and µ
(Lamé Parameters)
! ij = "#ij$ kk + 2µ$ij
Isotropic medium
•  In an isotropic medium, wave speed depends on
NEITHER the direction of propagation NOR
polarization direction
•  Relationship between stress and strain is simple:
! ij = "#ij$ kk + 2µ$ij
µ=
10/7/14 ! xy
2" xy
•  Shear modulus: measure of the resistance
of material to shear.
•  Bulk modulus: measure of incompressibility
of a material. Given by hydrostatic stress
divided by fractional volume change.
•  Poisson’s ratio: lateral contraction of a
cylinder divided by longitudinal extension.
19 Isotropic medium
Described by distribution of (ρ, λ, µ) within the medium
or: (ρ, κ, µ)
= “elastic parameters”
or: (ρ ,α ,β)
P velocity:
S velocity:
λ + 2µ
α=
ρ
µ
β=
ρ
Bulk sound velocity:
! <"
κ
Vφ =
ρ
4 2
Vφ = α − β
3
2
2
Momentum  Seismic Wave Equation
2
" ui
! 2 = " j # ij
"t
•  Homogeneous momentum equation governs
seismic waves outside of source regions
•  Summation convention: repeated index ! ij = "# ij$k uk + µ($i u j + $ j ui )
+ manipulaFon and a vector idenFty: •  Constitutive relation
for isotropic
medium, where we
used 2εjk = (∂uk/∂xj + ∂uk/∂xj) •  Seismic Wave Equation for an isotropic medium
10/7/14 21 Seismic Wave Equation
Gradient: Vector
describing the
direction and
magnitude of change
in a quantity
10/7/14 Divergence: Scalar
describing volume
change
Curl: Vector describing
infinitesimal rotation
(think paddle wheel)
22 Homogeneous medium
•  In a homogeneous medium, or in the high-frequency approximation, the
gradients of λ and μ can be neglected:
•  Taking the divergence of the equation (remember
):
Compressional wave (Pwave) propagating at
speed α •  Taking the curl of the equation (remember
and
):
"
Shear wave (S-wave)
involving no volume change
propagating at speed β 10/7/14 23 Polarisation of P and S waves
S waves are shear waves No volume change, shear and rotaFon P waves and S waves have different velociFes P waves are compressional waves cause volume change – compression And rarefacFon No rotaFon Similar to sound waves in air From Stein and Wysession, 2003 Terminology
θ
Great c
ircle
Terminology
Terminology
Shallow earthquake From Stein and Wysession, 2003 Plane waves
•  SoluFons to the P and S equaFons are waves: u(x,t)= f(x ± vt), v = velocity •  Plane waves: s is the slowness vector •  ParFcle displacement at a frequency ω: (Figure from Stein and Wysession, 2003) k is the wavenumber vector 10/7/14 30 Wavefronts and raypaths
Ray theory
•  Simple and fast:
–  Used extensively in earthquake location, focal
mechanisms, inversion for structure in crust and
mantle
•  Shortcomings
–  High frequency approximation: fails at long periods
–  Does not predict non geometrical effects i.e.
diffracted waves, head waves
–  Limitations in predicting effects of heterogeneity
on waveforms
Horizontal & vertical slownesses
•  Slowness: u = 1 / c
•  Horizontal slowness: p = ΔT/ΔX
•  Vertical slowness:
10/7/14 •  We can measure how quickly
a wave sweeps across a small,
dense array of seismometers
figures from Shearer, 2009 33 Snell’s Law – the ray parameter
For the wavefield to
remain continuous
along interface
between regions with
different velocities,
waves will change
direction in addition to
their speed.
10/7/14 34 Snell’s Law – the ray parameter
For wavefield to remain
continuous along
interface between
regions with different
velocities, waves will
change direction in
addition to their speed.
•  Snell’s Law:
sin !1 sin ! 2
=
=p
v1
v2
•  Ray parameter (horizontal slowness)
remains the same along the ray path
figure from garnero.asu.edu 10/7/14 35 Layered / shell medium
By specifying a ray
parameter and a
starting location, we
fix the ray-path
through a layered or
spherically-symmetric
Earth
The deepest depth
the ray can reach is
one where:
sin 90 1
= =p
v1
v1
10/7/14 figure from garnero.asu.edu 36 In general, some of the energy is transmitted, some reflected, and, in
the P-SV case, some converted
SH case
P-SV case
Angles of reflection/transmission depend only on velocities
Amplitudes depend on impedance (ρ v)
Travel-time curves
figures from Shearer, 2009 • 
T(X) = τ(p) + pX Ray parameter
(horizontal slowness) is
the slope of the traveltime curve
•  Different rays observed
at different distances
•  Shallowing of slope dT/dX
with distance  slowness
10/7/14 decreases with depth
• 
38 Ray paths in spherical geometry,
when velocity increases with depth
Seismic wavefield
P waves S waves Seismic waves take a
variety of paths and
sample all parts of the
Earth
They provide most
complete and direct
information on presentday state of the deep
interior
10/7/14 40 Wavefronts and raypaths
10/7/14 41 Triplications
figures from Shearer, 2009 •  Depth ranges in which velocity increases
rapidly with depth can produce triplications:
multiple rays (with different horizontal
slownesses) arrive at the same time
•  Transition zone discontinuities produce
triplications
•  At epicentral distances > 30° rays bottom in
the lower mantle and no triplications are
10/7/14 (typically) observed
42 Steep gradients, such as upper-mantle discontinuities,
create triplications
Shadow zones
Rays cannot bottom
in a depth range
where velocity
decreases with
depth*
Low-velocity zones
produce shadow
zones
Prominent shadow
zone in PKP waves
due to low velocities
in the outer core
10/7/14 figures from Shearer, 2009 44 Low velocity layers create shadow zones
Naming body waves
P or S
means
K
means
I or J
P or S transmission thru mantle
P transmission thru outer core
means
P or S transmission thru inner core
i = reflection off inner core
c = reflection off CMB
Basics: paths Basics: paths Example: ParFcle moFon plots for SKS+SKKS and Sdiff From Stein and Wysession, 2003 Basics: paths Basics: paths Basics: paths Basics: paths Basics: paths Basics: paths PredicFons from IASP91 model Shearer, 1996 •  Densité moyenne de la terre ρ = 5.515 kg/m3 •  Densité des roches en surface ~ 3 kg/m3 –  => densité augmente avec la profondeur –  => La terre ne peut être enFèrement composée de roches –  Emil Wiechert (1898) suggère que la terre est comme un météorite géant avec un noyau de fer-­‐nickel qui, plus lourd que les roches s’est enfoncé au centre de la terre par un processus similaire à la séparaFon du fer de sa gangue dans une enclume. D’où l’idée d’une terre” à deux couche”s avec un noyau de fer/nickel entouré d’une couche de roche. The1910
earth
La terre circa
as seen ~1910
Proof of the existence of a core:
Oldham (1906)
Richard Oldham 1858-1936
First director of the Seismological Service of India
?
?
S
P
Radius of the core estimated by Oldham: ~2600km
Oldham, 1906
Liquid state of the core
•  Suggested for a long time from the study of earth’s
solid tides, which are significantly larger than what
would be expected if the earth was rigid throughout
•  Confirmed by the absence of observations of S waves
beyond the distance of penetration into the core
•  Jeffreys (1926) combined tide data and S wave data
to construct the first proof of lack of rigidity in the
core
Sir Harold Jeffreys
1891-1989
Mean rigidity of the earth (from solid
tides) is lower than that of the mantle
calculated from S wave travel times)
“The rigidity of the Earth’s central core”, Geophys. J. (1926)
P wave shadow zone
Shadow zones due to the presence of
the liquid core
Discovery of the inner core:
1936
Observes “P-like” phases in the
shadow of the core
Calls the new seismic phase P’
Writes a paper whose title is the
shortest ever: “ P’ ”
Inge Lehmann 1888-1993
->Existence of a region at the
center of the core with different
elastic properties than the external
part of the core.