Geoscience Reference
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i 0
i 0
r 0
Figure A3.2.
and
(A3.4)
OQ = OP 2 sin i 2 = r 2 sin i 2
where OP 1 =
r 1 and OP 2 =
r 2 . Thus, on combining Eqs. (A3.3) and (A3.4), we have
(A3.5)
r 1 sin j 1 = r 2 sin i 2
Multiplying Eq. (A3.1)by r 1 and Eq. (A3.2)by r 2 and using Eq. (A3.5) means that
r 1 sin i 1
v 1
r 1 sin j 1
v 2
r 2 sin i 2
v 2
r 2 sin j 2
v 3
(A3.6)
=
=
=
At this point we define a parameter p as the ray parameter:
r sin i
v
p
=
(A3.7)
where r is the distance from the centre of the Earth O to any point P, v is the seismic velocity
at P and i is the angle of incidence at P. Equation (A3.6) shows that p is a constant along the
ray. At the deepest point to which the ray penetrates (the turning point ), i is
π/
2, so Eq. (A3.7)
becomes
r min
v
(A3.8)
p
=
where r min is the radius of the turning point and v the velocity at the point. The value of the
ray parameter p is different for each ray.
Now consider two adjacent rays (Fig. A3.2). The shorter ray A 1 B 1 subtends an angle
at
the centre of the Earth, and the longer ray A 2 B 2 subtends
+
. The travel time for ray
A 1 B 1 is t , and the travel time for ray A 2 B 2 is t
+
t .Inthe infinitesimal right triangle A 1 NA 2 ,
the angle A 2 A 1 Nis i 0 and
A 2 N
A 2 A 1
(A3.9)
sin i 0 =
Assuming that the surface seismic velocity is v 0 ,
1
2 v 0
A 2 N
=
t
(A3.10)
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