Focal mechanism solutions of earthquakes
The seismic waves generated by earthquakes, when recorded at seismograph stations around the world, can be used to determine the nature of the faulting associated with the earthquake, to infer the orientation of the fault plane and to gain information on the state of stress of the lithosphere. The result of such an analysis is referred to as a focal mechanism solution or fault plane solution. The technique represents a very powerful method of analyzing movements of the lithosphere, in particular those associated with plate tectonics. Information is available on a global scale as most earthquakes with a magnitude in excess of 5.5 can provide solutions, and it is not necessary to have recorders in the immediate vicinity of the earthquake, so that data are provided from regions that may be inaccessible for direct study.
According to the elastic rebound theory, the strain energy released by an earthquake is transmitted by the seismic waves that radiate from the focus. Consider the fault plane shown in Fig. 2.4 and the plane orthogonal to it, the auxiliary plane. The first seismic waves to arrive at recorders around the earthquake are P waves, which cause compression/dilation of the rocks through which they travel. The shaded quadrants, defined by the fault and auxiliary planes, are compressed by movement along the fault and so the first motion of the P wave arriving in these quadrants corresponds to a compression. Conversely, the unshaded quadrants are stretched or dilated by the fault movement. The first motion of the P waves in these quadrants is thus dilational. The region around the earthquake is therefore divided into four quadrants on the basis of the P wave first motions,
Compression 
Dilation  
_______M 
1________  
Focus' 
'Compression  
Dilation  
Fault plane S Auxiliary plane Fault plane S Figure 2.4 Quadrantal distribution of compressional and dilational P wave first motions about an earthquake. defined by the fault plane and the auxiliary plane. No P waves propagate along these planes as movement of the fault imparts only shearing motions in their directions; they are consequently known as nodal planes. Simplistically, then, a focal mechanism solution could be obtained by recording an earthquake at a number of seismographs distributed around its epicenter, determining the nature of the first motions of the P waves, and then selecting the two orthogonal planes which best divide compressional from dilational first arrivals, that is, the nodal planes. In practice, however, the technique is complicated by the spheroidal shape of the Earth and the progressive increase of seismic velocity with depth that causes the seismic waves to follow curved travel paths between the focus and recorders. Consider Fig. 2.5. The dotted line represents the continuation of the fault plane, and its intersection with the Earth's surface would represent the line separating compressional and dilational first motions if the waves generated by the earthquake followed straightline paths. The actual travel paths, however, are curved and the surface intersection of the dashed line, corresponding to the path that would have been followed by a wave leaving the focus in the direction of the fault plane, represents the actual nodal plane. It is clear then, that simple mapping of compressional and dilational first motions on the Earth's surface cannot readily provide the focal mechanism solution. However, the complications can be overcome Fault plane Focal sphere Compression JNodal plane Dilation Compression Fault plane Focal sphereFigure 2.5 Distribution of compressional and dilational first arrivals from an earthquake on the surface of a spherical Earth in which seismic velocity increases with depth. by considering the directions in which the seismic waves left the focal region, as it is apparent that compressions and dilations are restricted to certain angular ranges. A focal mechanism solution is obtained firstly by determining the location of the focus by the method outlined in Section 2.1.4. Then, for each station recording the earthquake, a model for the velocity structure of the Earth is used to compute the travel path of the seismic wave from the focus to the station, and hence to calculate the direction in which the wave left the focal region. These directions are then plotted, using an appropriate symbol for compressional or dilational first motion, on an equal area projection of the lower half of the focal sphere, that is, an imaginary sphere of small but arbitrary radius centered on the focus (Fig. 2.5). An equal area net, which facilitates such a plot, is illustrated in Fig. 2.6. The scale around the circumference of such a net refers to the azimuth, or horizontal component of direction, while dips are plotted on the radial scale from 0° at the perimeter to 90° at the center. Planes through the focus are represented on such plots by great circles with a curvature appropriate to their dip; hence a diameter represents a vertical plane. Let us assume that, for a particular earthquake, the fault motion is strikeslip along a near vertical fault plane. This plane and the auxiliary plane plot as orthogonal great circles on the projection of the focal sphere, as shown on Fig. 2.7. The lineation defined by the intersection of these planes is almost vertical, so it is apparent that the direction of movement along the fault is orthogonal to this intersection, that is, near horizontal. The two shaded and two unshaded regions of the projection defined by the nodal planes now correspond to the directions in which compressional and dilational Figure 2.6 Lambert equal area net. first motions, respectively, left the focal region. A focal mechanism solution is thus obtained by plotting all the observational data on the projection of the focal sphere and then fitting a pair of orthogonal planes which best divide the area of the projection into zones of compressional and dilational first motions. The more stations recording the earthquake, the more closely defined will be the nodal planes. 
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