1. EARTHQUAKE FOCAL MECHANISMS (FAULT PLANE SOLUTIONS)
FUNDAMENTALS of ENGINEERING SEISMOLOGY
EARTHQUAKE FOCAL MECHANISMS (FAULT PLANE SOLUTIONS)

2. - Focal mechanisms: body wave radiation pattern
- Focal mechanisms: stereographic fault plane representations
Credits:
-Some slides provided by Dino Bindi (INGV) and A. Kelly (USGS)
-Cox and Hart. Plate Tectonics – How it works.
-Stein and Wysession, An Introduction to seismology, earthquakes and
Earth structure

3. offsetting the features (time c). (Courtesy of S. Wesnousky)
Elastic rebound model of earthquakes assumes that between earthquakes, material
on the two sides of a fault undergoes relative motion. Because the fault is locked,
features across it that were linear at time (a), such as a fence, are slowly deformed
with time (b). Finally the strain becomes so great that the fault breaks in a earthquake,
offsetting the features (time c). (Courtesy of S. Wesnousky)
From S. Stein and M. Wysession

4. A minority of faults break the surface; how is the orientation of the fault plane and the direction of slip determined if the fault does not break to the surface?

5. which relative motion occurred during an earthquake.
To describe the geometry of a fault, we assume that the fault is a planar surface across
which relative motion occurred during an earthquake.
The fault is characterized by the normal vector n, while the slip in the fault plane is along the vector d. Several coordinate systems can be considered; one is aligned such
that the x1 axis is along the fault strike direction (intersection of the fault plane with the Earth surface); x3 axis points upward; x2 axis is perpendicular to the other two. The direction of x1 is selected such that the dip of the plane with respect to -x2 is less than 90 degrees. The direction of the motion is represented by the slip angle, l, measured counterclockwise in the fault plane from the x1 direction.
From S. Stein and M. Wysession

6. A very important figure describing sign conventions for focal mechanisms
From S. Stein and M. Wysession

7. Seismograms recorded at different distances and azimuth are used to study the geometry of faulting during an earthquake, known as focal mechanism. This operation uses the fact that the pattern of radiated seismic waves depends of the fault geometry. There are several methods to infer the focal mechanism, such as the analysis of the polarity of the first arrivals or the inversion of waveforms. We discuss only the former approach (e.g., see Introduction to Seismology by Stein and Wysession for details)
The first P-wave arrival varies between stations at different directions from an earthquake. The first motion is either compression (motion toward the station) or dilatation (motion away from station). The first motion defines 4 quadrants, divided by the fault and the auxiliary planes (nodal planes).
From S. Stein and M. Wysession

8. Focal Mechanism (Fault Plane Solution)

9. The seismic source is modelled as a point, but in reality it is rupture over a plane and relaxation over a volume… but, can use radiation pattern of first P-waves to determine the focal mechanism at the hypocenter

11. The Centroid Moment Tensor is the most complete description of the forces acting on the fault rupture.
If it is assumed that the fault rupture is PURE SHEAR (ie., that it involves no volume change) then the model of the forces acting on an equivalent point source is reduced to a much simpler system known as a DOUBLE COUPLE.

12. Radiation from a shear dislocation with slip S over area A in material with rigidity μ is identical to that from a double couple.

13. Faulting & Compressional Waves
P-wave radiation pattern

14. Earthquake on a vertical plane
Most earthquakes are double couple, in this situation the first motions are more complex. If we consider a strike slip earthquake on a vertical fault, then the ground motion can be divided into 4 quadrants - 2 with compression and 2 with dilation. The divisions between the compression and dilation are called the “nodal planes”.
Edited from Cox and Hart. Plate Tectonics – How it works.

15. Determination of nodal planes
If we had a network of seismic stations around such a strike slip earthquake then the some stations would have a compressional horizontal first motion on the and some would have a dilational motion. The distribution of the first motion can then be used to define the two nodal planes. One of these planes is the fault and the other is known as the “auxiliary plane”, unfortunately the first motions cannot tell us which is which, so we normally turn to the tectonic setting or knowledge of the fault from surface rupture or distribution of other earthquakes, to decide which plane is the fault plane.
Cox and Hart. Plate Tectonics – How it works.

16. N.B. The first motions from a slip on the actual fault plane and from a slip on the plane perpendicular to it (auxiliary plane) would be the same, so the first motions alone cannot resolve which plane is the actual fault plane. Additional information can often settle the question. Sometimes geologic or geodetic information indicates the fault. Often smaller aftershocks following the earthquake occur on, and thus delineate the fault plane.
In other cases, for earthquakes large enough, source effects can be used to determine the fault plane (e.g. directivity effects can destroy the symmetry of the radiation pattern)

17. Data on the surface, interpreted in 3D
The previous example was for a vertical fault. Now we consider a dipping fault, with a dip slip earthquake. In this case we will also see a pattern of dilation and compression first motions on the surface, which can be interpreted to find the nodal planes, but we have to think in 3-dimensions.
For the focal mechanism diagram we construct an imaginary sphere around the fault and each point on the surface of the sphere represents a unique direction from the focus. The ground motion in that direction is then recorded as either compression or dilation. To plot the focal mechanism diagram (beachball) we project half of the sphere onto paper using the stereographic projection.
Cox and Hart. Plate Tectonics – How it works.
Courtesy of A. Kelly

19. To determine the fault plane solution using the first motions, the observations at stations (Earth surface) have to be converted to observations over a sphere (of infinitesimal radius) around the source. The position on the sphere is determined by the take-off angle (computed from the slope of the travel-time curve).
A. Kelly, USGS
azimuth
S. Stein and M. Wysession
D. Boore

20. Determining the focal mechanism from P-wave first motions
Use azimuth and take off angle to determine where the P-wave intersects a sphere around the source
Plot those points of intersection on a stereographic projection, using a filled circle (or +) for compressions and an open circle for dilatations
Determine the two possible fault planes

21. P-wave first motions from 1989 Loma Prieta earthquake
P-wave first motions from 1989 Loma Prieta earthquake. Compressions are +. Lines divide stereonet into quadrants of compression and dilation. Strike=130, dip=70. Note: Because of symmetry, focal mechanism cannot distinguish fault plane from auxiliary plane. The two planes must be perpendicular to one another (how do you check this in the projection?).
Note some inconsistencies; this is to be expected when working with real data.

22. Seismic “Beach Balls”
Project the pattern of initial motions intersecting an imaginary sphere around the source (the “focal sphere”) onto a flat surface.
We use the radiation patterns of P-waves to construct a graphical representation of earthquake faulting geometry (two planes intersecting one another at right angles)
The symbols are called “Focal Mechanisms” or “Beach Balls”, and they contain information on the fault orientation and the direction of slip.

23. Focal Mechanism
When mapping the focal sphere to a circle (beachball) two things happen:
Lines (vectors) become points
Planes become curved lines

24. Stereographic projection
A method of projecting half a sphere onto a circle.
e.g. planes cutting vertically through the sphere plot as straight lines
To visualize the stereographic projection imagine looking straight down onto half a sphere, although this is not exactly the projection it is close enough to help understand what the data plotted using this projection are showing. The easiest data to plot on a stereographic projection are vertical planes. Imagine looking straight down onto a bowl cut by vertical planes, the line where the plane meets the bowl would appear to be a straight line cutting across the bowl. This will be demonstrated with a model in the session. Likewise a plane dipping at any other angle will appear to be a curved line running across the circle, with the amount of curvature varying with the dip of the plane (again this will be demonstrated with a model in the session).
Courtesy of A. Kelly
Images from

25. This is a lower hemisphere projection
Focal mechanisms: Shown by stereographic projection. Graphic shows stereographic projection of a fault plane (2D projection of plane onto focal sphere)
This is a lower hemisphere projection

26. -Azimuths are along the great circle -Dip angles are along the equator
-NS-striking Planes with different dip
are meridians
From S. Stein and M. Wysession

27. the meridian corresponding to 60°. Finally, rotate back the stereonet.
If the plane is striking in other azimuths, they can be displayed by rotating the
stereonet
E.g : strike 45° , dip 60°
First rotate the steronet such that the strike 45° is vertical. Then, select
the meridian corresponding to 60°. Finally, rotate back the stereonet.
From S. Stein and M. Wysession

28. Courtesy of A. Kelly

29. To plot a point, the procedure is the same.
E.g. station with azimuth 40° and take-off 60°
Rotate the stereonet of 40° clockwise (the equator is along azimuth 40°). Select the dip=30° (remember, take-off are measured with respect to vertical, dip with respect to horizontal  dip=90-take-off). Then, turn back the stereonet
From S. Stein and M. Wysession

30. Focal Mechanisms
Beachballs always have two curved lines separating the quadrants. That means that beachballs show two planes.
But there is only one fault plane.
The other plane is called the auxiliary plane.
The planes are orthogonal.
Seismologists cannot tell which is which from seismograms alone, so we always show both of the possible solutions.

31. Azimuth of these fault planes all strike north, with dips as shown
Azimuth of these fault planes all strike north, with dips as shown. Rotate stereonet for other strike azimuths. 
Projection of the normal to the plane; the auxiliary plane must go through this point. Does this define a unique auxiliary plane?

32. Same N-S fault, different slip direction
These diagrams show how the focal mechanism solution changes with different slip direction on the same north-south fault.
Same N-S fault, different slip direction
Stein and Wysession, An Introduction to seismology, earthquakes and Earth structure

33. + -
Normal dip-slip
fault
Reverse dip-slip

34. Fault types and “Beach Ball” plots
So returning to our early examples we can see the characteristic patterns of strike slip, normal and thrust faults.
USGS

35. Stein and Wysession

36. The Principal Mechanisms
which is the fault plane?

37. Focal Mechanisms
It is often possible to make an educated guess as to which of the two possible planes is the actual fault plane:
Normal earthquakes – work with gravity so are often steep
Thrust events – work against gravity so are often shallow

38. Focal Mechanisms
For strike slip focal mechanisms we can often determine the fault plane by its orientation with respect to the fault.
Ridge
Plate A
Plate B
(right or left lateral ??)
Ridge

39. Summary of focal mechanism concepts
Direction of first motions (up, down) can be used to divide focal sphere into quadrants of compression and dilation (stereographic projection).
The planes separating these quadrants are the fault plane and its auxiliary plane (at 90 degrees to fault plane). The planes are orthogonal.
Thus first motions can be used to determine fault orientation (provided we can find some information to distinguish fault plane from auxiliary plane).

40. Summary
The focal mechanism of an earthquake is a graphical description of the type of faulting
Although the fault type can be determined from seismograms, the particular fault plane cannot be determined.
There are always 2 possibilties:
The real plane, known as the fault plane
The fake plane, known as the auxiliary plane

41. Summary
The fault plane can be distinguished from the auxiliary plane by considering:
The steepness of the fault
Reverse = shallow
Normal = steep
Nearby morphological features
The strike of the transform segment for instance
The distribution of aftershocks
Aftershocks tend to line up along the fault plane (but not always of course)
Directivity effects on the waveforms

42. Faulting and Plate Tectonics

43. Stein and Wysession

44. Stein and Wysession

45. End