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Today’s lecture (Part 1)
- Fault geometry
- First motions
- Stereographic fault plane representation
1 Fault geometry
There are three main types of faults:
- Normal faults
- Reverse faults
- Stike-slip faults
We can think of these three types of faults as forming a set of basis functions. All
faults can be described as a combination of these 3 basis faults.
The fault geometry is described in terms of the orientation of the fault plane and
the direction of slip along the plane. The geometry of this model is shown in figure
The dip angle δ is the angle between the fault plane and the horizontal.
The slip angle, λ is the angle between the slip-vector and the horizontal.
The stike angle φ is used to orientate this system relative to the geographic one. It
is defined as the angle in the plane of the earth’s surface measured clockwise from
north to the x 1 axis.
We can use the slip angle, λ to specify the type of motion on the fault.
λ = 0 implies left-lateral (sinstral) fault motion
λ = 180 implies right-lateral (dextral) fault motion
λ = 270 implies normal faulting (extension)
λ = 90 implies reverse faulting
Most earthquakes consist of a combination of these motions and have a slip angle
between these values.
Seismologists refer to the direction of slip in an earthquake and the orientation
on the fault on which it occurs as the ‘focal mechanism’. They typically display the
focal mechanisms on maps as a ‘beach-ball’ symbol. We will talk more about this
Figure 1: Basic types of faulting. Strike-slip motion can be right- or left-lateral. Dip
slip-faulting can occur as either reverse or normal faulting (1)
Right-lateral strike slip
Dip slip (reverse) Dip slip (normal)
Left-lateral strike slip
Figure by MIT OpenCourseWare.
in inverting seismic observations for fault models. Additional geologic or geodetic
information is needed to identify which is the actual fault plane.
3 Stereographic projections
The fault geometry can be found from the distribution of data on a sphere around the
focus. We can trace rays from the earthquake onto a hemisphere using the eikonal
equation. We can then use a stereographic projection to transform the hemisphere
to a plane. The graphic construction that allows us to do this is called a stereonet
(figure 3)
Consider how planes will appear on this net.
-A vertically dipping, N-S striking plane will plot as a straight line.
-A N-S striking plane with a different type will appear as a curve going from top to
bottom.
-A horizontal plane will appear as a perimeter.
-Different type of fault will appear differently on a stereonet (see figure 5). For
example, a four-quadrant ‘checkerboard’ indicates pure strike-slip motion.
5 The Harvard-CMT catalogue
Seismologists collect data in real-time when there is an earthquake and produce a
representation of the focal-mechanism of an earthquake as a ‘beach-ball’ diagram.
These are listed as entries in the Harvard-CMT catalogue. Each entry includes the
following data:
-the location of the earthquake
-the time at which it occurred
-the depth of the earthquake
-the half-duration of the earthquake
Note that there are four different measurements of the magnitude of the earth
quake listed in the Harvard-CMT catalogue. Note also that 2 sets of values for the
strike, slip and dip directions of the fault plane are listed. This is because we are
unable to distinguish between the fault plane and the auxiliary plane when these
entries are produced.
200701010105A NORTHERN MID-ATLANTIC RI
Date: 2007/ 1/ 1 Centroid Time: 1: 5:16.1 GMT Lat= 32.75 Lon= -39. Depth= 12.0 Half duration= 0. Centroid time minus hypocenter time: 2. Moment Tensor: Expo=23 -2.790 0.458 2.330 -0.701 -1.890 1. Mw = 5.0 mb = 4.8 Ms = 0.0 Scalar Moment = 3.49e+ Fault plane: strike=210 dip=28 slip=- Fault plane: strike=23 dip=63 slip=-
Figure by MIT OpenCourseWare.
12.510 Lecture Notes 4.30.2008 (Part 2)
MOMENT TENSOR
To know the source properties from the observed seismic displacements, the solution of the equation of
motion can be separated as below.
(1)
Where is the displacement, is the force vector. The Green’s function gives the displacement at
point
u (^) i f (^) j Gij
x that results from a unit force function applied at point. Internal forces, , must act in opposing
directions, - , at a distance so as to conserve momentum (force couple). For angular momentum
conservation, there also exists a complementary couple that balances the forces (double couple). There are
nine different force couples as shown in Figure 1.
x o f
f d
Figure 1. The nine different force couples for the components of the moment tensor. [Adapted from Shearer 1999]
12.510 Lecture Notes 4.30.2008 (Part 2)
be time dependent, so that M o ( t )= μ D ( t ) s ( t ). The right-hand side time dependent terms become the
source time function, x(t), thus the seismic moment function is given by
(7)
We can diagonalize the moment matrix (6) to find principal axes. In this case, the principal axes are at 45° to
the original x 1 and x 2 axes.
(8)
The principal axes become tension and pressure axis. The above matrix represents that x 1 ′ coordinate is the
tension axis, T, and x ′ 2 is the pressure axis, P. (Figure 2)
Figure 2. The double-coupled forces and their rotation along the principal axes. [Adapted from Shearer 1999]
*adapted from 5.4.2005 Lecture Notes
