Seismic Rays - Quantitative Plate Tectonics - page 252

r ® D Œ r A.r/ i¨A.r/ r T.r/e i¨.tT.r//
(9.4)

2 ® D r

2 A.r/ 2i¨ r A.r/ r T.r/

r

2 T.r/
¨ 2 A.r/ r T.r/ r T.r/ e i¨.tT.r//
(9.5)

i¨A.r/ r

Regarding the time derivative, we easily

Fig. 9.1 Wavefront propagation. At any arrival time t D
T 0 , the set of points T ( r )

obtain:

D T 0 is a 3-D surface and the
gradient of T is locally normal to the surface. The distance
—

@ 2 ®
@t 2 D A.r/¨ 2 e i¨ . tT . r //

—( t ) from the source, placed at the origin, can be used
to build a parametric seismic ray equation r

D

(9.6)

D r (—)

Substituting these expressions in one of the
seismic wave Eqs. ( 9.1 )or( 9.2 )gives:

This is known as the eikonal equation .Letus
introduce now the slowness s , which is simply the
reciprocal of a seismic velocity ' or “:

2 A.r/ 2i¨ r A.r/ r T.r/

r

2 T.r/ ¨ 2 A.r/ jr T.r/ j 2

1
c.r/

i¨A.r/ r

s.r/ (9.12)

A. r /¨ 2
c 2 .r/

D (9.7)

Substituting into ( 9.11 ) allows to express the
eikonal

equation in the following alternative

where c D c ( r ) is either the P -wave velocity ' or
the S -wave velocity “. Now we decompose this
equation into the real and imaginary parts. The
resulting equations are:

form:

2

D s 2 .r/

jr T.r/ j (9.13)

A. r / ¨ 2
c 2 .r/
(9.8)

The phase factor T D T ( r ) is a scalar field
having the dimensions of a time. According
to the eikonal equation, the amplitude of its
gradient locally coincides with the slowness s .
The physical interpretation of an arbitrary surface
T ( r ) D T 0 is simple. From ( 9.3 ), we see that for t
D T 0 the propagating anomaly (either in or )
associated with the wave assumes a unique value
along this surface. Therefore, a surface T ( r ) D T 0
can be interpreted as a wavefront and T assumes
the significance of travel time necessary to the
propagating wavefront to reach that location.
We know that the gradient of a scalar field is a
vector that is always normal to its iso-surfaces
(see Appendix 1 ) . Therefore, at any location r ,
r T is normal to the wavefront passing through r
(Fig. 9.1 ). Starting from a seismic source located
at r 0 and considering a sequence of wavefronts
with arrival times T D k T ( k D 1,2, :::)for
some small time interval T , we can build a

2 A.r/ ¨ 2 A.r/ j r T.r/ j 2

r D

2 T.r/ D 0 (9.9)

2 r A.r/ r T.r/ C A.r/ r

The second of these equations is called the
amplitude transport equation . For the moment,
we shall focus on the first equation only. Dividing
both sides by A ¨ 2 gives:

2 A.r/
A.r/¨ 2 c 2 .r/ D r

1

2

jr T.r/ j (9.10)

When the angular frequency of the wave is
sufficiently high, hence in the limit ¨ !1 ,
the term at the right-hand side can be ignored.
Therefore:

1
c 2 .r/

2

jr T.r/ j D (9.11)