Geoscience Reference
In-Depth Information
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)