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
The San Andreas Fault: Right-lateral fault
We define the force couple
The magnitude of
The condition that angular momentum be conserved requires that is symmetric (e.g.,
For example, right-lateral movement on a vertical fault oriented in the
where
where
The units for
Because
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
The moment tensor
Tension axis
Pressure axis
Isotropic part of
where
Decomposing
where
Diagonalize
where
Because
For a pure double couple source,
We can decompose
The complete decomposition of the original
Example of the decomposition of a moment tensor into isotropic, best-fitting double couple, and compensated linear vector dipole terms:
The decomposition of
A measure of the misfit between
Physically, non-double-couple components can arise from simultaneous faulting on faults of different orientations or on a curved fault surface.
Review:
The momentum equation:
The stress-strain relation:
The strain:
We consider a unit impulse source
The unit force function is a useful concept because more realistic sources can be described as a sum of these force vectors.
For every
We define the notation:
where
Assuming that
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
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
The displacement field = the gradient of the displacement potential:
Define
So the displacement field is:
The first term decays as
The second term decays as
The first term decays as
The second term decays as
These waves cause no permanent displacement, and their displacements are given by the first time derivative of the source-time function.
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
The far-field S displacements:
where
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.
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.
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.
The magnitude of the equivalent body forces is
The scalar seismic moment of the earthquake; units of dyn-cm, or N-m
The Haskell source model
The rise time:
For a long, narrow fault, we assume that the rupture propagates along the fault of length
The total duration
The apparent rupture duration time
Rupturing directly toward us:
Rupturing directly away from us:
The apparent rupture duration for a seismic phase with local horizontal phase velocity
where
The changes in
Since
The rupture length
The true rupture duration is
The average rupture velocity is
High-frequency (2–4 Hz) envelopes from teleseismic P-wave observations of the 2004 Sumatra earthquake.
We have
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.
Earthquakes may be idealized as movement across a planar fault of arbitrary orientation
The Haskell source model
How to quantify the size of an earthquake?
The original magnitude scale is based on the maximum amplitude recorded on a standard Wood-Anderson torsion seismograph.
An approximate empirical formula has been derived for
The local magnitude can be calculated by
where
where A is the ground displacement in microns, T is the dominant period of the measured waves,
For Rayleigh waves on vertical instruments:
Since the strongest Rayleigh wave arrivals are generally at a period of 20 s, this expression is often written as
A boxcar pulse in the time domain produces a sinc function in the frequency domain:
The far-field amplitude spectrum for the Haskell fault model may be expressed as
where
where
We can approximate
The saturation of the and scales for large events helped motivate development of the moment magnitude
where is the moment measured in N-m.
The local strength of ground shaking as determined by damage to structures and the perceptions of people who experienced the earthquake.
Moment
Stress drop
large D
Moment
Stress drop
Area under displacement pulse
Radiation patterns, e.g.:
Assuming dimensions are scaled proportionally, displacement D will increase by b
Are different sized events self-similar?
Self-similarity predicts smaller events are “shifted” versions of larger events.
Following
From model fit obtained:
Is the stress drop close to the absolute stress level (weak faults) or just a small portion (strong faults)?
For simple slip-weakening friction
where
The radiation efficiency should not be confused with the seismic efficiency
The seismic efficiency is more difficult to estimate than the radiation efficiency because it depends upon the poorly constrained absolute stress level on the fault.
P-wave | S-wave |
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## Earthquake recurrence model  ( Shimazaki and Nakata, 1980)