Geoscience Reference
In-Depth Information
since
d
x
cosh
−
1
(
x
)
=
(
x
2
−
1)
1
/
2
The first term on the right-hand side of Eq. (A3.28)iszero because the epicentral angle
is
=
η
1
, cosh
−
1
(
p
zero when
p
=
η
0
, and, when
p
/η
1
)iszero. The second term simplifies to
cosh
−
1
p
η
1
d
0
−
=
1
or
1
cosh
−
1
p
η
d
1
=
0
where
1
is the value of
for the ray with parameter
η
1
(which has its deepest point at
r
=
r
1
) and
=
0, for the ray with parameter
η
0
. Thus the left-hand side of Eq. (A3.27)is
simplified to
η
0
1
cosh
−
1
p
η
1
d
d
p
(A3.29)
=
p
2
1
1
/
2
2
−
η
p
=
η
1
=
0
The right-hand side of Eq. (A3.27)ishandled by first performing the
p
integration:
η
p
p
2
)
1
/
2
d
p
p
2
1
1
/
2
(
−
η
η
2
−
p
=
η
1
On making the substitution
x
=
p
2
,weobtain
tan
−
1
x
−
η
1
/
2
η
η
2
2
1
1
2
d
x
x
)
1
/
2
=
x
1
1
/
2
(
η
2
−
x
2
−
η
η
2
−
x
=
η
2
1
2
1
x
=
η
2
tan
−
1
(
tan
−
1
(0)
(A3.30)
=
∞
)
−
=
(Reference works such as the
Standard Mathematical Tables
, edited by S. M. Selby,
Chemical Rubber Company, are invaluable in solving integrals such as these. Alternatively, if
we make the substitution
p
2
the integral simplifies to
π/
2
θ
=
2
1
sin
2
2
cos
2
=
η
θ
+
η
θ
0
d
θ
.)
The solution to Eq. (A3.27)isnow provided by Eqs. (A3.29) and (A3.30):
1
cosh
−
1
p
η
1
d
=
η
0
2
r
d
r
d
η
2
d
η
=
0
η
=
η
1
r
1
=
π
log
e
r
0
r
1
r
0
d
r
r
=
π
[log
e
r
]
r
0
(A3.31)
=
π
r
=
r
=
r
1
curves,
provided that certain conditions are met. As was shown in Eq. (A3.13),
p
is the slope of the
t-
This equation now allows the velocity at any depth to be evaluated from the
t-
curve, and d
t
/
d
is a function of
.For chosen values of
1
and
η
1
(the value of d
t
/
d
at
1
), the integral on the left-hand side of Eq. (A3.31) can be evaluated and
r
1
determined.
Repeating the calculations for all possible values of
η
1
means that
r
1
is determined as a
function of
v
, this determination means that the
seismic velocity has been determined as a function of radius.
Such an inversion (due to Herglotz, Wiechert, Rasch and others and dating from 1907) has
been invaluable in enabling us to evaluate the seismic structure of the interior of the Earth. It
is generally called the
Herglotz-Wiechert inversion
. The main limitations of the method stem
from the mathematical restriction that
η
1
. Recalling from Eq. (A3.21) that
η
=
r
/
v
must decrease with depth (i.e., increase with
increasing radius). Thus Eq. (A3.31) cannot be used in situations in which
r
η
=
r
/
/
v
increases with
