Seismic Rays - Quantitative Plate Tectonics - page 261

q 1

q 1

p 2 c i

T.p/ D 2 X
i

1
pm i

s i h i

p 2 c iC1

D I

q s i p 2 I s i >p (9.57)

i D 0;1;:::;n 1

(9.60)

where h i is the thickness of the i -th layer. When
the velocity model has continuous gradients, the
approximate solutions ( 9.56 )and( 9.57 )arein-
adequate, because of the large number of quasi-
homogeneous layers that would be needed to
obtain reliable results. In this instance, a better
approach is to assume a velocity model such
that c D c ( z ) is a broken line and evaluate the
expressions ( 9.53 )and( 9.55 ) for each segment.
For example, if c i ( i D 0,2, :::, n )are n C 1rep-
resentative control points at depths z i along an
experimental velocity profile, we can assume that
the velocity varies linearly between any pair of
successive control points, so that:

Similarly,

z iC1 s iC1

Z Z

s 2 . z /d z
p s 2 . z / p 2 D

ds
p s 2

1
m i

T i .p/ D

p 2

z i s i

m i ln s C p s 2

p 2 LJ LJ LJ

s iC1

1

D

s i

c i 2 p 2 i LJ LJ LJ

m i ln h m i . z z i /Cc i C q

z iC1

1 1 1

Πm i . z z i /

D

C

z i

ln 1

q 1 p 2 c iC1

v iC1 1 C

1
m i

D

ln v i

1 C

q 1 p 2 c i

(9.61)

c i C 1 c i
z iC1 z i

c. z / D . z z i / C c i m i . z z i / C c i I

The final arrival time T ( p ) and range X ( p )
are then obtained summing these contributions
over the index i and multiplying the result by
two. Let us consider now the generalization of
the previous solutions to the spherical Earth. We
know that in this instance the horizontal distance
X is substituted by the angular distance ,so
that dX D Rd , R being the Earth's radius. In
this case, the ray parameter coincides with the
slowness at the turning point only when c is a
continuous and mostly increasing function of the
depth. To generalize ( 9.49 ), we start as before
from ( 9.43 ). However, in this case Snell's law
in the form ( 9.40 ) must be substituted by ( 9.42 ).
Therefore,

for z i z z iC1 I i D 0;1; ;n 1 (9.58)

In this case, the variation of slowness with
depth will be given by:

ds
dc dc. z / D

1
c 2 . z / dc. z /

ds. z / D

m i
c 2 . z / d z D m i s 2 . z /d z I

D

for z i < z < z iC1 I i D 0;1; ;n 1
(9.59)

If we substitute the functions ( 9.58 ) into the
integrals ( 9.53 )and( 9.55 ) and change the inte-
gration variable to ds , we have that each segment
gives a contribution X i to the total range X that
can be expressed as follows:

dT

d D R dT dX D R p R D p

(9.62)

This is the generalization of ( 9.49 ) to a spher-
ical Earth. Solutions for the arrival time T ( p )and
the range X ( p ) can be found easily using the same
approach discussed previously for the flat Earth
approximation. We obtain:

z iC1

Z

d z
p s 2 . z / p 2

X i .p/ D p

z i

LJ LJ LJ LJ LJ

p s 2

s iC1

Z

s iC1

p 2
m i ps

p
m i

ds
s 2 p s 2

D

p 2 D

Z

R

s i

s i

LJ LJ LJ LJ LJ LJ LJ

q 1

1
r p r 2 s 2 .r/ p 2 dr (9.63)

z iC1

.p/ D 2p

c i 2 p 2 Œm i . z z i / C

D

pm i

r min

z i