Seismic Rays - Quantitative Plate Tectonics - page 254

We h ave :

dA
d— D

@A
@x i

dx i
d— Dr A

d d— Dr A r T

s D k r A
(9.20)

where k D k ( r ) Dr T / s is the versor tangent to the
seismic ray at r . Therefore:

s r A D sk r A D s dA
d—

(9.21)

Using ( 9.16 ), we can convert the derivative of
A with respect to — into a derivative with respect
to the travel time T .Wehave:

Fig. 9.2 Curvilinear coordinates on a wavefront and ray
tube at a point ( ¥ 0 , § 0 )

s r A D s dA
dT

dT

d D s 2 dA

(9.22)

dT

Let us assume that for any eikonal surface ˆ
is defined through fixed displacements d ¥ and d §
from (¥ 0 ,§ 0 ):

Substituting this result into ( 9.19 ) and express-
ing A as a function of T we obtain a first-order
ordinary differential equation along the ray:

ˆ.¥ 0 ;§ 0 / Df .¥;§/ W ¥ 0 ¥ ¥ 0 C d¥ I
§ 0 § § 0 C d§ g

2s 2 .r/ dA

2 T.r/ D 0

dT C A.T/ r

(9.23)

(9.25)

The set of rays crossing the wavefront through
ˆ forms what is known as a ray tube (Fig. 9.2 ).
The area dS of ˆ is clearly a function of T 0 ,as
shown in Fig. 9.2 . It can be calculated as follows:

To so lve ( 9.23 ), it is necessary to know the
Laplacian of the travel time T along the seismic
ray. Let us consider the surface of a wavefront
T ( r ) D T 0 (Fig. 9.2 ). The position of a point
on this surface can be specified through two
curvilinear coordinates (¥,§):
8
<

LJ LJ LJ LJ

d§ d§ LJ LJ LJ LJ

dr
d¥ d¥

d r

dS.T 0 / D

LJ LJ LJ LJ LJ LJ LJ LJ

x D x.¥;§/
y D y.¥;§/
z D z .¥;§/

dr
d¥

d r
d§

D d¥d§

(9.24)

:

J.T 0 ;¥ 0 ;§ 0 /d¥d§ (9.26)

For example, in the case of a spherical wave-
front propagating from the Earth's center, we
could use spherical coordinates of longitude and
latitude to identify the intersection of a seismic
ray with the propagating wavefront. A set of pairs
(¥,§) defines the intersection of a family of rays
and the eikonal surface T ( r ) D T 0 .Let(¥ 0 ,§ 0 )be
the coordinates of a point along this surface, and
consider the set of seismic rays that intersects the
wavefront in a neighbor ˆ of this point.

The quantity:

LJ LJ LJ LJ LJ LJ LJ LJ

d r
d¥

d r
d§

J.T 0 ;¥ 0 ;§ 0 / D (9.27)

is called geometrical spreading of the tube and
describes the focusing and defocusing of seismic
rays. It increases when the distance between rays
raises with the distance from the source. Con-
versely, if the rays converge to a point, then J