Earthquake Source and Seismic Wave

Textbooks:
Shearer, P. M. (2019). Introduction to seismology. Cambridge University Press.
GEOPHYS 210: Earthquake Seismology by Eric Dunham
Segall, P. (2010). Earthquake and volcano deformation. Princeton University Press.

Earthquake faults

Earthquakes may be idealized as movement across a planar fault of arbitrary orientation

• strike: , the azimuth of the fault from north where it intersects a horizontal surface
  
• dip: , the angle from the horizontal
  
• rake: , the angle between the slip vector and the strike
  

Earthquake faults

Thrust faulting: reverse faulting on faults with dip angles less than 45
Overthrust faults: Nearly horizontal thrust faults
Strike-slip faulting: horizontal motion between the fault surfaces
Dip-slip faulting: vertical motion
Right-lateral strike–slip motion: standing on one side of a fault, sees the adjacent block move to the right
: left-lateral faulting
: right-lateral faulting
The San Andreas Fault: Right-lateral fault

Earthquake double couple

• An earthquake is usually modeled as slip on a fault, a discontinuity in displacement across an internal surface in the elastic media.
• Internal forces resulting from an explosion or stress release on a fault must act in opposing directions so as to conserve momentum.
• A force couple is a pair of opposing point forces separated by a small distance
• A double couple is a pair of complementary couples that produce no net torque

Moment tensor

We define the force couple as a pair of equal and opposite forces pointing in the direction and separated by a unit distance in the direction.

The magnitude of is the product of the force and the distance .

The condition that angular momentum be conserved requires that is symmetric (e.g., ).

Moment tensor

For example, right-lateral movement on a vertical fault oriented in the direction corresponds to the moment tensor representation

where is the scalar seismic moment:

where is the shear modulus, is the average fault displacement, and is the area of the fault.

The units for are Nm (or dynecm), the same as for force couples.

Global CMT catalog

Global Centroid Moment Tensor

Beach balls

Moment tensor

Because , there are two fault planes that correspond to a double-couple model.
In general, there are two fault planes that are consistent with distant seismic observations in the double-couple model.
The primary fault plane: The real fault plane
The auxiliary fault plane: The other plane

Eigenvectors

The moment tensor is a symmetric tensor, so it has three real eigenvalues and three orthogonal eigenvectors.

Tension axis
Pressure axis

Non-double couple sources

Isotropic part of :

where

Decomposing into isotropic and deviatoric parts:
where , free from isotropic sources by may contain non-double couple sources

Non-double couple sources

Diagonalize by rotating to coordinates of principal axes:

where .

Because , .
For a pure double couple source, and .

Non-double couple sources

We can decompose into a best-fitting double couple and a non-double couple part :

The complete decomposition of the original is:

Non-double couple sources

Example of the decomposition of a moment tensor into isotropic, best-fitting double couple, and compensated linear vector dipole terms:

The decomposition of into and is unique because we have defined as the best-fitting double couple, i.e., minimizing the CLVD part.

Non-double couple sources

A measure of the misfit between and a pure double-couple source is provided by the ratio to the remaining eigenvalue with the largest magnitude to the largest eigenvalue of

: pure double couple
: close to pure CLVD

Physically, non-double-couple components can arise from simultaneous faulting on faults of different orientations or on a curved fault surface.

Green's function

Review:

The momentum equation:

The stress-strain relation:

The strain:

Green's function

We consider a unit impulse source at point at time .

The unit force function is a useful concept because more realistic sources can be described as a sum of these force vectors.

For every , there is a unique that describes the Earth’s response, which could be computed if we knew the Earth’s structure to sufficient accuracy.

Green's function

We define the notation:

where is the Green's function, is the displacement, and is the force.

Assuming that can be computed, the displacement resulting from any body force distribution can be computed as the sum or superposition of the solutions for the individual point sources.

Green's function + Moment tensor

The moment tensor provides a general representation of the internally generated forces that can act at a point in an elastic medium. Although it is an idealization, it has proven to be a good approximation for modeling the distant seismic response for sources that are small compared to the observed seismic wavelengths.

So the displacement resulting from a force couple at is given by:

Radiation patterns

Solving for the Green's function is rather complicated. Here we consider the simple case of a spherical wavefront from an isotropic source.
Review: The solution for the P-wave potential:

where is the P-wave velocity, is the distance from the point source, and is the source-time function.

The displacement field = the gradient of the displacement potential:

Radiation patterns

Define as the delay time:

So the displacement field is:

The first term decays as , and is called the near-field term, which represents the permanent static displacement due to the source
The second term decays as , and is called the far-field term, which represents the dynamic response (the transient seismic waves radiated by the source that cause no permanent displacement)

Radiation patterns

The first term decays as , and is called the near-field term, which represents the permanent static displacement due to the source

The second term decays as , and is called the far-field term, which represents the dynamic response - the transient seismic waves radiated by the source.
These waves cause no permanent displacement, and their displacements are given by the first time derivative of the source-time function.

Radiation patterns

Most seismic observations are made at sufficient distance from faults that only the far-field terms are important.
Consider the far-field P-wave displacement for a double-couple source, assuming the fault is in the plane with motion in the direction:

The far-field radiation pattern for P-waves

• The fault plane and the auxiliary fault plane form nodal lines of zero motion.
• The outward pointing vectors in the compressional quadrant.
• The inward pointing vectors in the dilatational quadrant.
• The tension (T axis) is in the middle of the compressional quadrant;
• The pressure (P axis) is in the middle of the dilatational quadrant.

The far-field radiation pattern for S-waves

The far-field S displacements:

where is the S-wave velocity.

There are no nodal planes, but there are nodal points.
S-wave polarities generally point toward the T axis and away from the P axis.

Beach balls

Plotting beach balls

Projection of the focal sphere onto an equal-area lower-hemisphere map.
The numbers around the outside circle show the fault strike in degrees.
The circles show fault dip angles with 0° dip (a horizontal fault) to 90° dip (a vertical fault).
The curved line ABC shows the intersection of the fault with the focal sphere.

Review: Basic types of faulting

Basic types of faulting

Body force

First-motion polarity

Radiation pattern

Radiation pattern

Earthquake Focal Mechanism

Earthquake Focal Mechanism

Earthquake Focal Mechanism

Earthquake Focal Mechanism

Earthquake Focal Mechanism

Earthquake Focal Mechanism

P/T axes v.s. Compressional/Dilatational quadrants

The Pressure/Tension (P/T) axes are defined inside the beach ball;
The Compressional/Dilatational quadrants are defined outside the beach ball;

The tension (T axis) is in the middle of the compressional quadrant;
The pressure (P axis) is in the middle of the dilatational quadrant.

Moment tensor and Beach ball

Moment tensor and Radiation pattern

Moment tensor decomposition

Magnitude

The magnitude of the equivalent body forces is
The scalar seismic moment of the earthquake; units of dyn-cm, or N-m

Time-dependent seismic moment

The Haskell source model

The rise time: describes the slip duration at any point on a fault.

Rupture propagation

For a long, narrow fault, we assume that the rupture propagates along the fault of length from left to right at a rupture velocity
The total duration of the rupture is

Far field the apparent rupture time

The apparent rupture duration time for P-waves:
Rupturing directly toward us:
Rupturing directly away from us:

Doppler effect

General apparent rupture duration

The apparent rupture duration for a seismic phase with local horizontal phase velocity at the observing station as

where is the station azimuth relative to the rupture direction.

The changes in as a function of receiver location are termed directivity effects.

Rupture Length

Since and
The rupture length is

The true rupture duration is

The average rupture velocity is

Example: 2004 Sumatra Earthquake Directivity

High-frequency (2–4 Hz) envelopes from teleseismic P-wave observations of the 2004 Sumatra earthquake.

Example: 2004 Sumatra Earthquake Directivity

We have s, ; and assume , so:

• km
• s
• km/s

The Haskell fault model

Rupture velocity

The rupture velocity is generally observed to be somewhat less than the shear-wave velocity for most earthquakes.
Anomalously fast ruptures sometimes exceed the local S-wave velocity and are termed supershear ruptures.

Earthquake rupture simulation

Eric Dunham

Source spectra

A boxcar pulse in the time domain produces a sinc function in the frequency domain

Source spectra

The far-field amplitude spectrum for the Haskell fault model may be expressed as

where is a scaling term that includes geometrical spreading, etc

where .

Source spectra

We can approximate for and for

The amplitude spectrum for the Haskell fault model.

Earthquake rupture (notebook)

Seismic wave propagation (notebook)

P-wave	S-wave

Earthquake recurrence model

( Shimazaki and Nakata, 1980)

where