(COMPUTER) Assume that the Rayleigh wave phase velocity at periods between
50 and 500 s can be approximated by the polynomial representation
c(T) = 4.020 − 1.839 × 10−3 T + 3.071 × 10−5 T 2 − 3.549 × 10−8 T 3,
where c is the phase velocity in km/s and T is the period in seconds.
(a) Plot both the group and phase velocity dispersion curves as a function of
period for T = 50 to 500 s, using (8.42) to generate the U(T) curve (this can
be done either analytically or through numerical differencing on the computer
after converting c(T) points to ω(k) points).
(b) Use this relationship to write a computer program to compute Rayleigh wave
synthetic seismograms at source–receiver ranges of 0, 30, 90, and 150 degrees. Define your seismograms to be 1.5 hours long with a digitization interval of 5 s. Construct the synthetics as a sum of cosine functions at frequencies
from 0.002 to 0.02 Hz with a frequency spacing of 0.0002 Hz. Apply a phase
shift to each frequency component to account for the propagation distance.
Make a plot of the synthetic seismogram at each range.
(c) Pick two adjacent peaks in your synthetic Rayleigh wave at 150◦ and measure
their time separation. Using this as an approximation for the period T , show
that the arrival time is in reasonable agreement with the group velocity curve
plotted in part (a).
(d) Note: You may notice a “wrap-around’’ phase at 5000 s at zero distance. Note
that 5000 = 1/df where df = 0.0002 Hz. To get rid of this phase, use a smaller
value for df (which will push it back to later times) or simply window it out
of your plot. Your synthetics will not be accurate past 180◦ unless a π/2 phase
advance is added to correct for the effect of the focusing at the antipode (see
Aki and Richards, 2002, p. 351). An additional π/2 phase advance occurs for
each additional epicentral or antipodal passage.
Hint: Here is the key part of a FORTRAN program to solve this problem:
In this case, we apply a Hanning taper (sin2 function) to smoothly reduce
the amplitudes close to the frequency limits. This minimizes ringing and
other artifacts in the final synthetic seismogram that are caused by the finite
bandwidth. If you are curious why your synthetics are missing the fast waves
at very long periods, try removing the Hanning taper. You will now see the
early-arriving, long-period energy, but your synthetics will suffer ringing
from the abrupt frequency limits in the calculation.