Seismic waves and global seismology
Waves move energy (not matter!) in space-time
Spherical wave
W(r,t) = f(r-v*t)
Plane wave
W(x,t) = f(x-v*t)
Spherical wave propagating outward at velocity V (Sinusoidal source)
note the amplitude decay of the waves as it moves away from the source
Transverse (shear) wave propagating in 2-dimensions
Shear wave propagating in x-direction
Y
(0,0)
X
Spherical shear wave propagating in
radial-direction
Waves: wavelength, velocity, and amplitude
Note that both waves are propagating the disturbance in the medium (spring,rope) to the
right at a velocity (wave-speed). Also, the force that excites the spring-wave (a) is
perpendicular to the force that excites the rope-wave (b).
(a) SPRING: Note the compressed and dilatated regions of the spring which is where
elastic energy is present. The wavelength is define as the length of one cycle of the
waves compressed and dilitated regions.
(b) ROPE: Note the wiggle in the rope has peaks and trough that define the wavelength.
At a latter time the wave ‘wiggles’ (kinetic/potential energy) has move forward in the
direction of the wave’s propagation.
1-dimensional P and S wave propagation in a slinky
Note that the waves moves in time
Wave parameters: velocity, frequency, period, wavelength, amplitude
Waves may be graphed as a function of time
or distance. A single frequency wave will
appear as a sine wave in either case. From
the distance graph the wavelength may be
determined. From the time graph, the period
and frequency can be obtained. From both
together, the wave speed can be determined.
 : wave length (m)
T : wave period ( s )
v : velocity of wave (m / s )
f  1/ T (1/ s)
v   * f (m / s )
v   / T (m / s )
v  x / t ( m / s )
Longitudinal compressional P-wave
The deformation (a temporary elastic disturbance) propagates. Particle motion consists of
alternating compression and dilation (extension). Particle motion is parallel to the direction
of propagation (longitudinal). Material returns to its original shape after the wave passes.
The deformation (a temporary elastic disturbance) propagates. Particle motion consists of
alternating transverse motion. Particle motion is perpendicular to the direction of
propagation (transverse). The transverse particle motion shown here is vertical but can be
in any direction; however the Earth’s approximately horizontal layers tend to cause mostly
SV (in the vertical plane) or horizontal (SH) shear motions. Material returns to its original
shape after the wave passes.
The deformation propagates. Particle motion consists of alternating transverse
motions. Particle motion is horizontal and perpendicular to the direction of propagation
(transverse). To best view the horizontal particle motion, focus on the Y axis (red line) as
the wave propagates through it. Amplitude decreases with depth. Material returns to its
original shape after the wave passes.
The deformation (a temporary elastic disturbance) propagates. Particle
motion consists of elliptical motions (generally retrograde elliptical as shown
in the figure) in the vertical plane and parallel to the direction of
propagation. Amplitude decreases with depth. Material returns to its
original shape after the wave passes.
Review: Exciting a wave, longitudinal and transverse waves
(a) Pushing at the spring’s end with a
force over a distance (work) cause
the A-B coils to add elastic
(compressional) energy. This
energy then propagates to the
right.
(b) The end of the spring is pulled
back with a force over a distance
(work) to add elastic (dilitational)
energy.
Note relation between propagation direction
and wave vibration direction.
(a) Longitudinal P-wave
(b) Transverse S-wave
Concept: a wave pulse versus a sinusoidal wave
Two ‘kinds’ of waves:
•A harmonic sinusoidal wave
F(x,t) = A*sin( x– t) that oscillates up and
down across all space and time
(everywhere!). A is wave amplitude.
•A space-time localized wave pulse that is a
superposition of many different frequency
components moving in unison (phase).
Relation between wave raypaths and wavefronts
An explosion makes a force that
creates a wave pulse that propagates
outwards in time. The increasing
diameter circles are the wave-fronts
at successively increasing times.
Rays are a very useful concept that shows the
path that one parcel of the wavefront travels
along.
Draw the rays.
Each ray is always perpendicular to the
wavefront. The ray has an arrow to indicate the
direction of energy transport.
Measuring very small motions at the Earth’s surface in 3-dimensions
The ground motion (displacement, velocity,
acceleration) at the surface is measured by
seismometers.
Because space is three-dimensional, we will need to
record the ground motion in three perpendicular
directions (up-down, north-south, east-west).
A seismometer works by hanging a mass that is
from a frame attached firmly to the earth so that
the frame moves with the ground motion.
The trick to an inertial seismometer is that the
inertia of the mass suspended by the spring causes
the mass to ‘lag behind’ the motion of the frame.
This cause the mass to be displaced with respect to
the frame, and this displacement can be measured
by the ‘ruler’. In modern system the motion is
amplified a million times.
A modern seismometer: an inertial magnet inside a coil
As before, a mass (inertia) is hanging
from a spring attached the frame (case)
of the seismometer attached to the
earth.
As the inertial mass ‘lags’ behind the
motion of the case, there is differential
motion between the mass and the
electrical coil that can be measured via
electrical induction.
The trick is that the mass is magnetized
vertically (note poles). So as the magnet
(mass) moves with respect to the coil, a
current is induced in the coil circuit that
can be measured at the two wires. The
voltage associated with the current will
be proportional to the velocity of the
ground motion!
Spherical symmetry of the Earth
In a spherical geometry, the distance between two points on the earth’s surface is denoted
by the angle subtended from the center of the earth. This is called the epicentral angle (∆).
Therefore in (a), all the chords (A1 – B1, A2 - B2, etc ) have the SAME epicentral angle. And,
if the travel-times along each of these chords (seismic raypaths) is the same, then this
means that the earth velocity structure only varies with radius (spherically symmetric) and
does not vary with azimuth. (b) same as (a) except for larger epicentral angle.
Spherical symmetry: velocity only changes with radius!
V(r)
V(r)
V(r)
V(r)
V(r)
V(r)
V(r)
V(r)
The properties (e.g., seismic velocity) along the radial lines that start at the center of
the sphere are all are the same! This is spherical symmetry: also called radial
spherical symmetry.
Distance versus time travel-times for uniform and Velocity(r) earth
Around 1920, enough traveltime measurements from
earthquakes at different
epicentral angles (distances)
were measured. This is the
observed time curve below.
This showed that at greater
distances (epicentral angles)
the travel-times came in earlier
than for a uniform velocity
earth.
Ah hah! This means that the
earth’s velocity increases with
depth and is NOT uniform!
Plotting millions of P and S-wave travel-times reveals seismic phases
This is how the outer liquid iron and solid iron inner core were discovered.
Incident P-wave: P-Reflection, P-transmitted, S-reflection, S-transmitted
When a P-wave hits a velocity boundary, it makes three P-waves: a refracted, a
transmitted, and a reflected wave. But, it also forms a reflected and transmitted Swave too! Snell’s law still works, except when the outgoing ray is a S-wave, the S-wave
velocity must be used in Snell’s Law.
Refraction of a (P or S) ray due to velocity change: Snell’s Law
incident
reflected
transmitted
When a ray traverses a velocity
contrast (change), the ray MUST
refract. Why? Because otherwise
the wavefront would ‘tear’ apart
which the physics does not permit.
Also, in most all cases a reflected
wave is made.
All waves refract: e.g., seismic, light,
EM, water waves.
The refraction law: Snell’s Law. Note that all
angles between the ray and interface are
reckoned with respect to the surface normal.
sin(1 ) sin( 2 )

v1
v2
For a wave reflected from a flat
interface, the angle of incidence
EQUALS the angle or reflection.
incidence   reflection
Snell’s Law derivation
An explosion makes a wavefield at (S). The waves travel outwards as represented by
the wavefront at different times. Where the wave hits the velocity interface, the
waves refracts (and reflects).
What determine the angle of the refraction into the rock-2 layer ?
T0
λ1
Ѳ
v1
v2
1
Derivation of
Snell’s Law
T1
Ѳ
Ѳ
A
Ѳ
Ѳ
1
1
2
2
B
Ѳ
2
sin(1 ) v1 1
 
sin( 2 ) v2 2
λ2
T0

sin(1 )  1
AB

sin( 2 )  2
AB
T1

AB  1
AB 
sin(1 )
2
sin( 2 )
sin(1 )
1

sin( 2 )
2
v1  f * 1
sin(1 ) sin( 2 )

v1
v2
v2  f * 2
The ratios of the velocities and wavelengths and sin(angles) are all equal!
Otherwise, the wavefield would ‘tear’ apart.
Tracing rays using Snell’s Law in multiple layer medium
Tracing a raypath through multiple layers is
simple. It is just the process of using Snell’s
Law sequentially each successive interface.
Note that the angle (i1) at the top and the
bottom (i1’) of a layer is the same.
If the lower layer’s velocity increases
( v2 > v1 ), the ray refracts AWAY from the
interface normal. In the converse, the ray
refracts TOWARDS the normal.
sin(1 )
v1

sin( 2 ) v2
If the deeper layers have a monotonic
increase in velocity, the ray will continue to
flatten out with depth. Eventually, the ray
will reach its turning depth where it goes
exactly horizontal and will start to go up!
Using Snell’s Law for a spherical geometry (not Cartesian)
r1 sin(1 ) r2 sin( 2 )

v1
v2
Note that the angle at the top and the
bottom of a spherical shell are NOT
the same! This is because the ‘layers’
are curved.
But, if one is just calculating the angles
on either side of an interface, then the
two radius values (r1 and r2 ) are the
same and the Cartesian form of Snell’s
Law is operative (i.e., the radius scaling
cancels out in the spherical Snell’s Law.
Radial Earth velocity models, Ray paths, and travel-time curves
s1
s2
(a) Representing the earth’s velocity structure as many
shells of constant velocity.
(b) Tracing ray-paths through the constant velocity shells.
Each ray has a different ‘take-off’ angle.
(c) Travel-time curve. The changes is the slope (slowness
units s/m) of the curve are directly related to the
changes in velocity with depth. The farther the distance
a ray travels, the deeper the ray dives before turning
around to come back to the surface.
The compressional (bulk) and shear (rigidity) modulus: P and S wave velocity


P
: Pisthe pressure applied to
dv
v
 
the sphere, vis volume, dvis volumechange.
restoring elastic force (stress)
V
mass of parcel
Vp 
F
: F is applied shear force,
A * d
Ais area, d shear angle .
  43 

Vs 


Average radial velocity structure of Earth
The Vp and Vs velocity profile of the Earth. The crust, mantle lithosphere, upper mantle,
transition zone, lower mantle, outer core, inner core are the primary divisions of the
planet’s velocity profile. This was not know until the 1950’s.
Why is the shear velocity of the outer core zero? What is velocity contrast at the coremantle boundary?
Radius and volume of earth main chemical/phase subdivisions:
inner and outer core, mantle, crust
Radius of Earth
6371 km
Depth* to base of the crust (average)
35 km
Depth* to base of lithosphere (average)
100 km
Depth* to base of upper mantle
670 km
Depth* to core-mantle boundary
2885 km
Depth* to outer core-inner core boundary
5155 km
Phase names: P, PP, PS, PPP, PKP(P’), PcP, PcS and Shadow zones
The different raypaths with both P and S-wave ‘legs’ have been named. A ‘P’ denotes a Pwave leg and an ‘S’ denotes a S-wave leg. PcP denotes a reflection off the core-mantle
boundary. PP and PPP are free-surface multiple reflections. ‘K’ is used for a core traversing
wave such as PKP.
Because there is a very large velocity decrease across the core-mantle boundary, Snell’s Law
predict the waves will refract ‘towards the normal’. This refraction creates a ‘shadow zone’
for both the P- and S-waves at epicentral distances >97°.
More seismic phases
and raypaths
Travel times of P-waves for ∆ = 0-180⁰
Note that the rays that take off at the source dive progressively deeper into the mantle
before reaching their turning depths after which the ray comes back up.
For the core-traversing P-wave, there are 2-3 P-wave arrivals due to the strong refraction
effects of the low velocity outer core and the high velocity inner core.
Most all the
seismic phases
for planet earth
North-ridge, California earthquake recording by global seismic network stations
The four effects that make a seismogram: earthquake source,
propagation, site response, and instrument
Earthquake in Nevada (Feb. 2008) recorded by vertical component of many seismometers.
Can you identify the P-wave, the Love and Rayleigh waves and measure their velocities ?
Low velocity zone: a region just under the lithosphere often thought to
be the weak asthenosphere where plate shear accululates
Lithosphere
A combination of increasing temperature and pressure affect the rock modulus to
make a LVZ. The LVZ generally extends to NO deeper than about 200 km depth.
Shear velocity North America from surface wave measurement inversion
Tomography: make whole earth 3-d images of velocity structure
from millions of P and S-wave travel-time measurements!
Note high velocity (blue) slab
image under north Turkey
going into lower mantle
Blue is seismically fast and red is seismiclly slow material. Note the slab image in
lower mantle under SE coast of North America in both the P and S-wave images.
Mean vertical average crustal shear velocity
Shear velocity (m/s)
Note oceans have mean velocity of 3.3 km/s and
continents a mean velocity of 3.4-3.9 km sec.
Passive margin/delta sediments at3.0 km/s.
At right is error map which is always important to
understand. If errors are big, results will be no good.
Mean Moho depth from surface wave measurement ‘inversions’
Note 8-12 km beneath oceans and 28-80 km beneath continents.
What causes this crustal thickness difference between oceans and continents?
Crustal thickness from surface wave seismic
measurement inversion
note oceans at 10 km and continents at >35
km