Seismic Rays - Quantitative Plate Tectonics - page 255

where the slowness s has been expressed as a
function of T through r . Finally, comparing ( 9.28 )
with ( 9.29 ) gives the following expression for the
geometrical spreading at T :

J.T;¥;§/ D sD.T;¥;§/

(9.30)

Let us consider now the integral of the vector
field r T over the surface of dV . It is given by:
I

r T dS D s.T C dT/dS.T C dT/

Fig. 9.3 The volume element formed by two eikonal
surfaces and a tube

dV

s.T/dS.T/ D Œs.T C dT/J.T C dT/

s.T/J.T/d¥d§

decreases progressively to zero at the point of
convergence. Let us consider now the volume
element dV formed by a ray tube and two closely
spaced wavefronts at T D T 0 and T D T 0 C dT ,
asshowninFig. 9.3 .
To evalu ate dV , we observe that the triple
product a b c of three vectors represents the
volume of the parallelepiped formed by the three
vectors. Therefore:

Therefore, using the definition of divergence
(see Appendix 1 ) and( 9.29 ), we have:

I
dV r

1
dV

2 T

r Drr T D lim

dV

T dS

! 0

Πs . T

dT / J . T dT / s.T/J.T/ d ¥ d §

C C

D lim

dV

D

1
s.T/ J.T/d¥d§dT

! 0

s.T/Œs.T C dT/J .T C dT/ s.T/J.T/

LJ LJ LJ LJ d r

d¥ d¥ d§ d§ LJ LJ LJ LJ

D lim

dV

d d— d— d r

J.T/dT ! 0

dV D

s.T/
J.T/

d
dT .s.T/J.T//

D

LJ LJ LJ LJ d r

d¥

LJ LJ LJ LJ

(9.31)

d d— dr

d

D d—d¥d§

Substituting this expression into the transport
equation ( 9.23 ) and dividing by s gives:

LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ LJ

@x
@—

@y
@—

@ z
@—

(9.28)

@x
@¥

@y
@¥

@ z
@¥

2s.T/ dA

J.T/ A.T/ d

1

D d—d¥d§

dT C dT .s.T/J.T// D 0

(9.32) @x

@§

@y
@§

@ z
@§

This is a first order differential equation with
separable variables, with solution:

D.—;¥;§/dd¥d§
D D.T;¥;§/dTd¥d§

c.¥;§/
p s.T/J.T/

wherewehaveused( 9.16 ) to transform deriva-
tives with respect to — into derivatives with respect
to T . On the other hand, from ( 9.26 )wehavethat:

A.T/ D (9.33)

where c is a constant not depending from T ,
which can be expressed in terms of seismic ray
parameters ¥ and §. This solution shows that
the amplitude decreases when the geometrical
spreading

dV D dSd— D J.T;¥;§/d¥d§d—

1
s.T/ J.T; ¥ ; § /d ¥ d § dT (9.29)

D

increases. More precisely, the