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(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
 
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