Geoscience Reference
In-Depth Information
ds
dr
rd
θ
r
r
d
θ
O
Figure A3.3.
and
1
2
r
0
A
2
A
1
=
(A3.11)
Substituting Eqs. (A3.10) and (A3.11) into Eq. (A3.9)gives
v
0
t
r
0
sin
i
0
=
(A3.12)
Comparison with Eq. (A3.7) means that, in the limit when
t
,
→
0,
d
t
d
p
=
(A3.13)
The ray parameter
p
is therefore the slope of the curve of travel time versus epicentral angle
(Fig. 4.16) and so, for any particular phase, is an observed function of the epicentral angle.
Let d
s
be the length of a short segment of a ray, as shown in Fig. A3.3. Then, using
Pythagoras' theorem on the infinitesimal triangle, we obtain
(d
s
)
2
(d
r
)
2
)
2
=
+
(
r
d
θ
(A3.14)
However, from Eq. (A3.7)wehave
r
sin
i
v
r
v
r
d
d
s
(A3.15)
p
=
=
Eliminating d
s
from Eqs. (A3.14) and (A3.15)gives an expression for d
θ
,
r
4
(d
)
2
p
2
v
2
θ
(d
r
)
2
r
2
(d
)
2
=
+
θ
(A3.16)
which, upon rearranging, becomes
p
d
r
d
θ
=
(A3.17)
−
p
2
)
1
/
2
r
(
r
2
/
v
2
Integrating this equation between the surface (
r
=
r
0
) and the deepest point (
r
=
r
min
)gives an
expression for
:
=
2
p
r
0
d
r
(A3.18)
p
2
)
1
/
2
r
(
r
2
/
v
2
−
r
=
r
min
