Geology Reference
In-Depth Information
(a)
(b)
x
v
1
v
2
z
V
rms
, n
v
3
Actual curve
v
n-
1
Hyperbolic curve
t
2
=
t
0
2
+
x
2
V
rms
v
n
t
Fig. 4.3
(a) The complex travel path of a reflected ray through a multilayered ground, showing refraction at layer boundaries. (b) The
time-distance curve for reflected rays following such a travel path. Note that the divergence from the hyperbolic travel-time curve for a
homogeneous overburden of velocity
V
rms
increases with offset.
Thus at small offsets
x
(
x
<<
z
), the total travel time
t
n
of the ray reflected from the
n
th interface at depth
z
is
given to a close approximation by
4.2.3 Dipping reflector
In the case of a dipping reflector (Fig. 4.4(a)) the value of
dip
q
enters the time-distance equation as an additional
unknown. The equation is derived similarly to that for
horizontal layers by considering the ray path length
divided by the velocity:
12
(
)
t
=+
x z V
2
4
2
cf. equation (4.1)
n
rms
and the NMO for the
n
th reflector is given by
12
(
x
2
++
4
z
2
4
xz
sin
q
)
t
=
cf. equation (4.1)
2
x
Vt
V
D
T
ª
cf. equation (4.7)
n
2
2
rms,
n
0
The equation still has the form of a hyperbola, as for
the horizontal reflector, but the axis of symmetry of the
hyperbola is now no longer the time axis (Fig. 4.4(b)).
Proceeding as in the case of a horizontal reflector, using
a truncated binomial expansion, the following expres-
sion is obtained:
The individual NMO value associated with each reflec-
tion event may therefore be used to derive a root-mean-
square velocity value for the layers above the reflector.
Values of
V
rms
down to different reflectors can then
be used to compute interval velocities using the
Dix
formula
. To compute the interval velocity
v
n
for the
n
th
interval
(
2
)
x
+
4
2
xz
Vt
sin
q
t
ª+
t
(4.9)
0
2
12
0
2
2
Vt
-
-
V t
È
Í
˘
˙
rms,
nn
rms,
n
n
-
1
-
1
v
=
n
t
t
Consider two receivers at equal offsets
x
updip and
downdip from a central shot point (Fig. 4.4). Because of
the dip of the reflector, the reflected ray paths are of dif-
ferent length and the two rays will therefore have dif-
ferent travel times.
Dip moveout
D
T
d
n
n
-
1
where
V
rms,
n
-1
,
t
n
-1
and
V
rms,
n
,
t
n
are, respectively, the
root-mean-square velocity and reflected ray travel times
to the (
n
- 1)th and
n
th reflectors (Dix 1955).
is defined as the