Sources generating eleastic waves generate different amplitudes as a function of takeoff angle and azimuth from the source. The convention in seismology is to envision a sphere surrounding the source and the to plot the signal amplitude on the sphere as the point where the ray from the source to the receiver intersects the sphere. Such a plot can be performed for any set of measurements, but typically one plots the P-wave, SV-wave, SH-wave amplitude or the S-wave polarization at this point.  Rather than plotting this on an actual sphere, the upper or lower half of this focal sphere is projected onto a piece of paper tangent to the pole.

The convention of the takeoff angle is that a value of 0 represents a ray going directly downward from the source into the earth, while a value of 180 represents a ray propagating upward. The azimuth is measured with respect to local north, with values of 0, 90, 180, and 270 representing north, east, south and west.

The lower hemisphere is defined by the combination of all azimuths and takeoff angles from 0 to 90 degrees, while the upper hemisphere has takeoff angles of 90 to 180 degrees.

The purpose of this section is to exercise the synthetics seismogram programs to ensure that the conventions used for the Green's functions and the focal mechanism are correct. We will accomplish this by making wholespace synthetics (because of speed) using the programs mkmod96, hprep96, hwhole96, hpulse96, fmech96 and f96tosac .  We then use gsac to convert the cylindrical coordinate traces to spherical radial, longitudinal and latitudinal and to create the plot.  We also use fmplot to plot the theoretical amplitudes on the focal spheres, the program CAL to annotate the plot, plotnps to convert from CALPLOT graphics to Encapsulated PostScript. We use the ImageMagick tool convert, available on LINUX,Cygwin and OSX, to convert the Encapsulated PostScript to a Portable Network Graphics (PNG) file. In addition we use the command gawk (locally called awk) as a calculator.

Trace Rotation

The computer programs in seismology package creates Z - vertical, R - radial and T - transverse component synthetics for a cylindrical coordinate system, with the convention that Z is positive up, R is positive away from the source, and T is positive in a direction of increasing azimuth (using the convention above).

On a sphere we want the rotation to create the far-field P, SV and SH waveforms, with P positive in a radial direction, SV positive as a point on the ray moves in the direction of increasing takeoff angle, and SH positive in the direction of increasing azimuth.  The necessary transformation is

      R = Uradial = -UZ*COS(IO) + UR*SIN(IO)
 T = Utheta = UZ*SIN(IO) + UR*COS(IO)
P = Uphi = UT

where IO is the takeoff angle in radians ( angle in radians = 3.1415926 * angle in degrees / 180)


The scripts used are as follow
Lower hemisphere script
Upper hemisphere script


Consider a double-couple focal mechanism with strike = 0, dip = 45 and rake = 45 degrees. We will consider takeoff angles 0f 10, 30, 50 and 70 degrees for the lower hemisphere run and 110, 130, 150 and 170 degrees for the upper hemisphere computations.

Each of the lines below present a figure with several components. There are three columns of traces, arranged to highlight the far-field P (R), SV (T) and SH (P) components of motion.  Each plot is annotated as IO_AZ_CMP, where IO is the takeoff angle in degrees AZ is the azimuth in degrees and CMP is the trace component, e.g., R, T or P.  For each component the traces are plotted downward in order of increasing azimuth. All traces of each component are plotted using the same amplitude scale to be able to compare amplitudes. All traces represent ground displacement.

To the right the P, SV and SH amplitudes are plotted on the focal sphere. This plot shows an amplitude value of each of 30 azimuths and 10 takeoff angles, which are 10 degrees apart.  The + and - symbols indicate whether the amplitudes are positive or negative, and the size of the symbol indicates relative amplitudes. The black curves in these figures are nodal curves indicating the zero amplitude contour.

In the far-field, on can prove that the direction of vector S -motion on the focal sphere is in the local direction of maximum increase of P-wave amplitude on the sphere (gradient on the spehrical surface).

When looking at the synthetics you will note some spikes as well as some linear trends. The spikes are the far-field body-wave arrivals which are superimposed on the near-field (linear ) trends. Focus on the P-arrival on the Radial component, and the S arrivals on the SV and SH components.