Geoscience Reference
In-Depth Information
which is, after all, like Eq. (4.65), a hyperbola. The value of the constant
c
0
is
given by
n
t
m
2
c
0
=
(4.78)
m
=
1
where
t
m
=
α
m
is the two-way vertical travel time for a ray in the
m
th layer.
The two-way normal-incidence travel time from the
n
th interface,
t
0,
n
,isthe sum
of all the
t
m
:
2
z
m
/
n
n
2
z
m
α
m
t
0
,
n
=
t
m
=
(4.79)
m
=
1
m
=
1
Equation (4.78) can therefore be more simply written as
c
0
=
(
t
0
,
n
)
2
(4.80)
The second constant of Eq. (4.77),
c
1
,isgivenby
m
=
1
t
m
n
n
c
1
=
(4.81)
m
=
1
α
m
t
m
2
n
We can define a time-weighted root-mean-square (RMS) velocity
α
as
m
=
1
α
n
2
m
t
m
2
n
α
=
(4.82)
m
=
1
t
m
n
With these expressions for
c
0
and
c
1
, Eq. (4.77) becomes
x
2
α
t
2
=
(
t
0
,
n
)
2
+
(4.83)
2
n
This now has exactly the same form as the equation for the two-layer case (Eq.
(4.65)), but instead of
t
0
we have
t
0
,
n
, and instead of the constant velocity above
the reflecting interface
α
n
, the time-weighted RMS velocity
above the
n
th interface. This means that NMO corrections can be calculated
and traces stacked as described in Section 4.4.2. The only difference is that, in
this multilayered case, the velocities determined are not the velocity above the
interface but the RMS velocity above the interface.
Figure 4.42 shows a typical velocity-spectrum display for real data. By stacking
reflection records and using such a velocity-spectrum display, one can estimate
both
t
0,
n
and
α
1
,wenowhave
α
n
is not enough;
to relate these values to the rock structure over which the reflection line was shot,
we have to be able to calculate the thicknesses and seismic velocities for each
layer and to obtain an estimate of the accuracy of such values.
Let us suppose that the RMS velocity and normal-incidence times have
b
een
determined for each of two successive parallel interfaces (i.e.,
t
0,
n
−
1
,
t
0
,n
,
α
n
for each reflector. However, determining
t
0,
n
and
α
n
−
1
