A thick continental crust becomes stiffer

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Rays and travel times: from arrivals to a layered planet

Problem

A thick continental crust becomes stiffer, and so faster, with depth. Rays traced through it for five values of the ray parameter

p

give the following ranges

X(p)

, the horizontal distance from source to the point where each ray re-emerges. Data: for

p=0.1500, 0.1557, 0.1732, 0.2039, 0.2500\,\mathrm{s\,km^{-1}}

the ranges are

X=266.7, 251.3, 208.2, 142.0, 0\,\mathrm{km}

respectively. The last value,

X=0

, is the ray that turns essentially at the surface, so

p_0=0.2500\,\mathrm{s\,km^{-1}}

is the surface slowness. Find the turning depth of the ray with

p_1 = 0.1500\,\mathrm{s\,km^{-1}}

from this table alone, using the flat-layer Herglotz-Wiechert result

z_t(p_1) = \frac{1}{\pi}\int_{p_1}^{p_0}\frac{X(p)\,\mathrm{d}p}{\sqrt{p^2-p_1^2}}.

((a)) The integrand is singular at

p=p_1

. Using the substitution

p=p_1\cosh w

of Worked example 5.2, compute

w=\cosh^{-1}(p/p_1)

for each row of the table and confirm that the five values of

w

are evenly spaced (to the precision the table gives); state the resulting step size

h

and the upper limit

w_{\max}

. ((b)) Apply Simpson's rule with that step to

\displaystyle\int_0^{w_{\max}}X(p_1\cosh w)\,\mathrm{d}w

, and hence find

z_t(p_1)

. ((c)) State, in one sentence, the condition on the speed profile between the surface and the turning depth under which this recovered value can be trusted.

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