Geoscience Reference
In-Depth Information
noise, and that sophisticated averaging and enhancement techniques must be used
to detect reflecting horizons.
4.4.2 A two-layer model
Consider the two-layer model in Fig. 4.34.Byapplication of Pythagoras' theorem,
the travel time
t
for the reflection path SCR is given by Eqs. (4.29) and (4.30)as
SC
α
1
+
CR
α
1
t
=
z
1
+
2
α
1
x
2
4
=
or
4
z
1
α
x
2
α
t
2
=
1
+
(4.65)
2
2
1
which is the equation of a hyperbola. At normal incidence (
x
=
0), the travel time
is
t
=
t
0
,where
2
z
1
α
1
t
0
=
(4.66)
This is the two-way normal-incidence time. At large distances (
x
z
1
) the travel
time (Eq. (4.65)) can be approximated by
x
α
1
t
≈
(4.67)
This means that, at large distances, the travel-time curve is asymptotic to the
travel time for the direct wave, as illustrated in Fig. 4.32(b).Inreflection profiling,
since we are dealing with distances much shorter than the critical distance, the
travel-time-distance plot is still curved. Notice that, with increasing values of
α
1
, the hyperbola (Eq. (4.65)) becomes flatter. If travel-time-distance data were
obtained from a reflection profile shot over such a model, one way to determine
α
1
and
z
1
would be to plot not
t
against
x
,but
t
2
against
x
2
. Equation (4.65)is
2
1
and an intercept on the
t
2
axis
then the equation of a straight line with slope 1/
α
of
t
0
4
z
1
/α
2
1
.
The normal-incidence reflection coefficient for P-waves is given by Eq. (4.62).
Since normal-incidence reflections have small amplitudes, it is advantageous to
average the signals from nearby receivers to enhance the reflections and reduce
the background noise. This averaging process is called
stacking
.
Common-depth-
point
(CDP)
stacking
,which combines all the recordings of reflections from
each subsurface point, is the method usually used. Common-offset stacking,
which combines all the recordings with a common offset distance, is less popular.
Figure 4.40 shows the layout of shots (or vibrators) and receivers used in CDP
reflection profiling. The coverage obtained by any profile is
=
number of receivers
twice the shot spacing
coverage
=
(4.68)
