Seismic Rays - Quantitative Plate Tectonics - page 257

Fig. 9.5 In a spherical
homogeneous Earth,
seismic rays are straight
lines but the incidence
angle changes with depth.
In this instance, the
quantity r sin™( r )is
invariant

only upon the distance r from the Earth's center,
so that s D s ( r )and™ D ™( r ). In this instance, the
vector invariant assumes the form:

p D r s.r/ d d— (9.41)

Also in this case the seismic rays lie on vertical
planes, and it is easy to prove that Snell's law
assumes the form:

Fig. 9.6 A downward propagating plane wave from a
seismic source at the Earth's surface. The material is
homogeneous, so that the seismic rays are straight lines
and the incidence angle ™ is constant

p D s.r/r sin ™.r/ (9.42)

In fact, in this instance sin™ is not anymore
invariant when the slowness is constant (Fig. 9.5 ).
Conversely, the example of Fig. 9.5 shows that in
the spherical context the quantity r sin™ will be
conserved for s ( r ) D const . It is important to note
that both the flat approximation ( 9.40 )andthe
spherical conservation law ( 9.42 ) do not take into
account of lateral inhomogeneities , so that they
are only applicable when the seismic velocity
depends from a unique parameter ( z or r ).

T
s D x sin ™

— D (9.43)

Therefore, by Snell's law:

T
x D s sin ™ D p

(9.44)

This relation implies that a measurement of
arrival times at different seismic stations allows to
determine the seismic parameter p . It also implies
that p is the apparent slowness of the wavefront
in the horizontal direction. Therefore, p is some-
times referred to as the horizontal slowness of the
seismic ray. Let us assume now that the wave
strikes a horizontal discontinuity in the elastic
parameters, so that we can separate the material
into an upper layer with velocity c 1 and a lower
layer with higher velocity c 2 > c 1 (Fig. 9.7 ).
For s 2 < s 1 , Snell's law predicts an increase

9.4 1-D Velocity Models

Let us consider now a plane wave that propagates
through a homogeneous material with constant
slowness s (Fig. 9.6 ). The wavefronts at time T
and T C T are separated by a distance —
along a seismic ray. At the Earth's surface, the
wavefront arrivals have a different separation x ,
which depends from both — and the slowness s :