A Poisson-like crust has and

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

The Solid Earth, Ch Surface waves, free oscillations and seismic tomography

Problem

A Poisson-like crust has

v_S = 3.4\,\mathrm{km\,s^{-1}}

and

v_P = 6.0\,\mathrm{km\,s^{-1}}

. A Rayleigh wave of period

T=15\,\mathrm{s}

propagates on its surface with speed

c_R = 0.9194\,v_S

. ((a)) The P and S displacement potentials beneath a Rayleigh wave decay with depth

z

e^{-kz\sqrt{1-c_R^2/v_P^2}}

and

e^{-kz\sqrt{1-c_R^2/v_S^2}}

respectively, where

k = 2\pi/\lambda

and

\lambda = c_R T

. Find the depth at which each potential has fallen to

1/e

of its surface value, and state which of the two decays more slowly with depth. ((b)) Elsewhere, in a region where a body-wave speed model increases linearly with depth,

v(z) = v_0 + gz

with

v_0=6.0\,\mathrm{km\,s^{-1}}

and

g=0.005\,\mathrm{s^{-1}}

(

z

\mathrm{km}

v

\mathrm{km\,s^{-1}}

). A ray with flat-Earth ray parameter

p=0.15\,\mathrm{s\,km^{-1}}

(defined by Snell's law

\sin\theta(z)/v(z)=p

for the angle

\theta

from the vertical) descends through this medium. Using

\mathrm{d}s = \mathrm{d}z/\cos\theta(z)

for the arc length along the ray, find the length of ray path lying between

z=40\,\mathrm{km}

and

z=80\,\mathrm{km}

: this length is the entry

G_{ij}

that this ray contributes to the row of the tomographic sensitivity matrix for the cell occupying that depth range.

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