Geoscience Reference
In-Depth Information
Figure 4.41.
(a) Reflections from an interface, recorded at distances
x
a
,
x
b
,
x
c
,
x
d
,
x
e
and
x
f
. Three travel-time curves (1, 2 and 3) are shown for two-way
normal-incidence time
t
0
and increasing values of velocity. Clearly, curve 2 is the
best fit to the reflections. To stack these traces, the NMO correction (Eq. (4.74)) for
curve 2 is subtracted from each trace so that the reflections line up with a constant
arrival time of
t
0
. Then the traces can be added to yield a final trace with increased
signal-to-noise ratio. (b) The power in the stacked signal is calculated for each value
of the stacking velocity and displayed on a time-velocity plot. The velocity which
gives the peak value for the power in the stacked signal is then the best stacking
velocity for that particular value of
t
0
. For (a), velocity 2 is best; velocity 1 is too low
and velocity 3 too high. (After Taner and Koehler (1969).)
Using Eq. (4.66) for
t
0
, the two-way normal-incidence time, we obtain an alter-
native expression for
t
NMO
:
x
2
2
α
t
NMO
=
(4.74)
2
1
t
0
This illustrates again the fact that the reflection time-distance curve is flatter
(
t
NMO
is smaller) for large velocities and large normal-incidence times. This
NMO correction must be subtracted from the travel times for the common-depth-
point recordings. The effect of this correction is to line up all the reflections from
each point P with the same arrival time
t
0
so that they can be stacked (added
together) to produce one trace. This procedure works well when we are using a
model for which
α
1
and
z
1
are known, but in practice we do not know them: they
are precisely the unknowns which we would like to determine from the reflections!
This difficulty is overcome by the bootstrap technique illustrated in Fig. 4.41.A
set of arrivals is identified as reflections from point P if their travel times fall on
ahyperbola. Successive values of
α
1
and
t
0
are tried until a combination defining
ahyperbola that gives a good fit to the travel times is found. These values
α
1
and
