(COMPUTER) Construct a P-wave travel time curve for Earth using the PREM
model (seeAppendixA). Your program should first read in the depth and P velocity
for the different layers in the model. Next, apply the Earth-flattening transformation
(4.45)–(4.46) to convert these depths and velocities to their flat-Earth equivalent
values. Then, use the subroutine LAYERXT (provided in Fortran in Appendix D
and in the supplemental web material as a Matlab script) to trace rays through this
model and produce a P-wave T(X) curve, using 201 values of the ray parameter p
equally spaced between 0.0017 and 0.1128 s/km. Your program can be structured
as described in Exercise 4.8; however, you should convert the X values returned
by LAYERXT from kilometers to degrees along the Earth’s surface. Now produce
plots of:
(a) T(X) with X = 0 to 180◦, T = 0 to 25 minutes, and no reduction velocity. If
you connect the individual T(X) points with a line, be careful to avoid filling
in the shadow zone between P and PKP. Compare your result with Figure
4.20.
(b) T(X) with X = 10 to 35◦, T = 50 to 100 s, and a reduction velocity of
0.1 degree/s. This should produce an enlarged view of the triplications asso-
ciated with the upper mantle discontinuities at 400 and 670 km depth in the
PREM model. On the plot, label the travel time branch that represents: (1)
rays that turn above 400 km, (2) rays that turn at 400 km, (3) rays that turn
between 400 and 670 km, (4) rays that turn at 670 km, and (5) rays that turn
below 670 km.
Note: The flat-Earth transformation blows up at the center of the Earth and your
program may produce strange results at small r values; thus do not attempt to
transform the (r = 0, z = 6,371) level in PREM. As a kluge, simply change
the final depth in the model to 6360 km. This means that you will not be able to
include the vertical ray that goes straight through the center of the inner core; this
is why you are asked to use p = 0.0017 as a minimum ray parameter. Warning:
Your computed P travel times are only approximate, owing to the relatively coarse
sampling of PREM in Appendix 1. The true PREM model does not contain linear
velocity gradients between depth points, as the LAYERXT subroutine assumes.