Seismic waves and global seismology

Waves move energy (not matter!) in space-time

Spherical waveW(r,t) = f(r-v*t)

Plane waveW(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

X

Y

(0,0)

Shear wave propagating in x-direction 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 slinkyNote 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.

: ( )

: ( )

: ( / )

1/ (1/ )

* ( / )

/ ( / )

/ ( / )

wave length m

T wave period s

v velocity of wave m s

f T 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.

Draw the rays.

Rays are a very useful concept that shows the path that one parcel of the wavefront travels along.

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!

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.

V(r)V(r)

V(r)V(r)

V(r)V(r)

V(r)V(r)

Distance versus time travel-times for uniform and Velocity(r) earth

Around 1920, enough travel-time 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 S-wave 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

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.

1 2

1 2

sin( ) sin( )

v v

For a wave reflected from a flat interface, the angle of incidence EQUALS the angle or reflection.

incidence reflection

reflectedincident

transmitted

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 ?

v1v2

T0

T1

Ѳ1Ѳ1

Ѳ2 Ѳ2Ѳ2

Ѳ1A B

T0 T1

λ1

λ2

1 11

1

2 22

2

sin( ) sin( )

sin( ) sin( )

ABAB

ABAB

1 2 1 2

1 2 1 2

1 1 2 2

sin( ) sin( ) sin( ) sin( )

* *

v v

v f v f

1 1 1

2 2 2

sin( )

sin( )

v

v

The ratios of the velocities and wavelengths and sin(angles) are all equal!Otherwise, the wavefield would ‘tear’ apart.

Derivation of Snell’s Law

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.

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!

1 1

2 2

sin( )

sin( )

v

v

Using Snell’s Law for a spherical geometry (not Cartesian)

1 1 2 2

1 2

sin( ) sin( )r r

v v

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.

43

p sV V

The compressional (bulk) and shear (rigidity) modulus: P and S wave velocity

( )restoring elastic force stressV

mass of parcel

.

:

, ,

P Pisthe pressureapplied todvv

thesphere visvolume dvisvolumechange

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FF is applied shear force

A d

Ais area d shear angle

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 core-mantle boundary?

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

Radius and volume of earth main chemical/phase subdivisions: inner and outer core, mantle, crust

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 P-wave 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