Geoscience Reference
In-Depth Information
Eliminating d
θ
from Eqs. (A3.14) and (A3.15) yields an expression for d
s
,
r
d
r
d
s
=
(A3.19)
v
(
r
2
/
v
2
−
p
2
)
1
/
2
The travel time d
t
along this short ray segment d
s
is d
s
/
v
. Integrating this along the ray
between the surface (
r
=
r
0
) and the deepest point (
r
=
r
min
)gives an expression for
t
, the
total travel time for the ray path:
t
=
2
v
=
2
r
0
r
0
d
s
r
d
r
(A3.20)
p
2
)
1
/
2
v
2
(
r
2
/
v
2
−
r
=
r
min
r
=
r
min
Sometimes for convenience another variable
η
, defined as
r
v
η
=
(A3.21)
is introduced. When this substitution is made, Eqs. (A3.18) and (A3.20) are written as
2
p
r
0
d
r
=
(A3.22)
r
(
η
2
−
p
2
)
1
/
2
r
=
r
min
t
=
2
r
0
2
d
r
η
(A3.23)
r
(
η
2
−
p
2
)
1
/
2
r
=
r
min
These two integrals can always be calculated: the travel times and epicentral distances can be
calculated even for complex velocity-depth structures involving low-velocity or hidden layers.
In order to use the
t-
curves to determine seismic velocities, it is necessary to change the
variable in Eq. (A3.22) from
r
to
η
,which is possible only when
η
decreases monotonically
with decreasing
r
:
=
2
p
η
0
η
=
η
min
1
d
r
d
η
(A3.24)
d
η
−
p
2
)
1
/
2
r
(
η
2
The limits of integration are
η
0
=
r
0
/
v
0
and
η
min
=
r
min
/
v
(
r
=
r
min
). However, since by Eq.
(A3.8),
p
=
r
min
/
v
(
r
=
r
min
), the lower limit of integration
η
min
is in fact equal to the ray
parameter
p
for the ray emerging at epicentral angle
.
Now, at
r
=
r
1
,where
r
1
is any radius for which
r
0
≥
r
1
>
r
=
r
min
, let
η
and
v
have values
η
1
and
v
1
, respectively. Assume that there is a series of turning rays sampling only the region
between
r
0
and
r
1
with values of
p
between
η
0
(
η
0
=
r
0
/
v
0
), which is the ray travelling at a
tangent to the Earth's surface and hence having
=
0, and
η
1
(
η
1
=
r
1
/
v
1
), which is the ray
whose turning point is
r
1
. Multiplying both sides of Eq. (A3.24)by1
/
(
p
2
−
η
1
)
1
/
2
gives
1
1
/
2
η
0
2
p
1
d
r
d
η
(A3.25)
=
d
η
p
2
1
1
/
2
p
2
r
(
η
2
−
p
2
)
1
/
2
2
2
−
η
−
η
η
=
p
Now integrate Eq. (A3.25) with respect to
p
between the limits
η
1
and
η
0
:
η
0
η
0
1
1
/
2
η
0
d
p
2
p
1
d
r
d
1
1
/
2
d
p
=
d
η
(A3.26)
p
2
p
2
r
(
η
2
−
p
2
)
1
/
2
η
−
η
−
η
p
=
η
1
p
=
η
1
η
=
p
It is mathematically permissible to change the order of integration on the right-hand side of
Eq. (A3.26) from
η
first and
p
second to
p
first and
η
second:
η
0
η
d
p
d
η
0
2
p
d
r
d
(A3.27)
1
1
/
2
d
p
=
η
p
2
r
p
2
1
1
/
2
(
η
2
2
−
η
−
η
η
2
−
p
2
)
1
/
2
p
=
η
1
η
=
η
1
p
=
η
1
Integrating the left-hand side of Eq. (A3.27)byparts gives
η
0
cosh
−
1
p
η
1
η
0
η
0
d
d
p
cosh
−
1
p
d
p
d
p
=
=
η
1
−
(A3.28)
p
2
1
1
/
2
η
1
2
−
η
p
=
η
1
p
=
η
1
p
