principles and applications of microearthquake networks

6.2. Fault- Plctrze Solutiori 141
that for locating earthquakes: later phases are difficult to identify because
of limited dynamic range in recording and the common use of only
vertical-component seismometers. As discussed earlier, deriving a faultplane
solution amounts to finding the two orthogonal nodal planes which
separate the first motions of P-waves into compressional and dilatational
quadrants on the focal sphere. The actual procedure consists of three
steps:
(1) First arrival times of P-waves and their corresponding directions of
motion for an earthquake are read from vertical-component seismograms.
Normally one uses the symbol U or C or + for up motions, and either D
or - for down motions.
(2a) Information for tracing the observed first motions of P-waves
back to the focal sphere is available from the earthquake location procedure
using Geiger's method. The position of a given seismic station on the
surface of the focal sphere is determined by two angles CY and p. a is the
azimuthal angle (measured clockwise from the north) from the earthquake
epicenter to the given station. /3 is the take-off angle (with respect to the
downward vertical) of the seismic ray from the earthquake hypocenter to
the given station. The former is computed from the coordinates of the
hypocenter and of the given station. The latter is determined in the course
of computing the travel time derivatives as described in Section 4.4. Accordingly,
if first motions of P-waves (y) are observed at a set of rn
stations, we will have a data set (ak. Pk, yk), k = 1, 2, . . . , rn, describing
the polarities of first motions on the focal sphere; Yk will be either a C for
compression or a D for dilatation.
(2b)
Since it is not convenient to plot data on a spherical surface, we
need some means of projecting a three-dimensional sphere onto a piece of
paper. Various projection techniques have been employed to plot the
P-wave first motion data. The most commonly used are stereographic or
equal-area projection. The stereographic projection has been used extensively
in structural geology to describe and analyze faults and other
geological features. Readers are referred to some elementary texts (e.g.,
Billings, 1954, pp. 482-488; Ragan, 1973, pp. 91-102) for a discussion of
this technique. The equal-area projection is very similar to the stereographic
projection and is preferred for fault-plane solutions because area on
the focal sphere is preserved in this projection (see Fig. 27 for comparison).
A point on the focal sphere may be specified by (R, a, p), where R is
the radius, a is the azimuthal angle, and p is the take-off angle. In equalarea
projection, these parameters (R, a, p) are transformed to plane polar
coordinates (r, 0) by the following formulas (Maling, 1973, pp. 141-144):