2. Average Velocity This is simply the depth z of a reflecting surface below a datum
divided by the observed one – way reflection time t from the datum to the surface so
that.
( 7 – 22 )
If z represents the sum of the thicknesses of layers z1, z2, z3, …, zn, the average velocity
is defined as
( 7 – 23 )
The average velocity is used for time – to – depth conversions and for migration.
Interval Velocity if two reflectors at depths z1 and z2 give reflections having respective
one – way time of t1 and t2, the interval velocity Vint between z1 and z2 is defined
simply as (z2 – z1) / (t2 – t1) .
Instantaneous Velocity. If the velocity varies continuously with depth, its value at a particular
depth z is obtained from the formula for interval velocity by contracting the interval z2 – z1 until it
be comes an infinite simally thin layer having a thickness dz. The interval velocity computed by
the formula above becomes the derivative of z with respect to t, and we designate it as the
instantaneous velocity Vinst, defined as
( 7 – 24 )
1 2 3 1
1 2 3
1
av
n
k
n
av n
n
k
z
V
t
z
z z z z
V
t t t t
t
inst
dz
V
dt
3. Root – Mean – Square Velocity. If the section consists of horizontal layers with
respective interval velocities of V1, V2, V3, . . . , Vn, and one – way interval travel time
t1, t2, t3, . . . , tn, the root – mean – square (rms) velocity is obtained from the relation
( 7 – 25 )
The slope of this line in t2, x2 space, using the reflection from a flat bed, is shown in
Fig. 7.22. Kleyn15 offers a further discussion of velocity is obtained from the relation
Stacking Velocity Stacking velocity, Vst , is based on the relation
( 7 – 26 )
The best fit over all offsets of hyperbolic move outs derived from Eq. (7.26) to the
actual reflection events. In general.
2
2 2 2 2
1 1 2 2 3 3 1
1 2 3
1
n
k k
n n
rms n
n
k
V t
V t V t V t V t
V
t t t t
t
2
2 2
0 2
st
av rms st
x
T T
V
V V V
4. Mirror or fold plane, so that
the times ( T, T0) are two
way time.
T
½T½T½T0
½T0 ½T
T0
X
Offset
Source Reseive
r
2
2 2
0
X
T T
V
Figure 7 . 22. A simple derivation of the normal-
move out equation uses the zero offset time T0,
the time T tc a given offset s, a fold plane at the
reflector, the Pythagorean theorem Vst, the
stacking velocity, is the processing parameter
that achieves the best time alignment of a
reflection at an offset with the zero – offset
reflection time.
5. x
S
d
ShotZ
Q
D
Well
Detector
1
cos SD
Z d
T
Recording
Truck
16000
14000
12000
10000
8000
6000
4000
2000
20001000 3000 4000 5000 6000 7000
Depth ft
Velocityft/sec
Interval Velocity
Average overall
velocity, V̅
Detector
positions
Figure 7 – 23 Well shooting
arrangement with typical interval –
velocity and average – velocity curves
thus obtained.
6. t
A B
Z True locations
of reflecting
points
True locations
of reflecting
points
Apparent reflecting
points positions
when plotted
vertically below
reflection spreads
(a) (b)
Figure 7 – 39 (a). Tight syncline showing the true and apparent spatial locations of the
reflected doing energy. (b) The “bow- tie effect” that is the time response over the
syncline. A further discussion of this effect is found in Secs. 8-3 and 8-5.
7. The need for migration has been recognized since the first reflection surveys [
Jakosky30 (pp. 670 – 696)]. Figure 7 -40 traces the development of migration
techniques
Graphical Methods
Straight Ray
Curved Ray path – wave front Charts
Diffraction Overlays
Digital Computation
Ray Tracing
Diffraction Summation
Wave front Interference
Wave Equation (CDP Data)
Finite Differences
Frequency Domain
Kirchhoff (Summation)
Imaging in depth Before Sack
Figure 7 – 40 Development of migration techniques. ( From Johnson and French. 32)
8. When discussing migration topics, certain variables have widespread preassingned
meanings :
z = depth
t = one – way time
ω = frequency
x = spatial location (the midpoint axis)
y = the second spatial coordinate, orthogonal to x, used in 3 – d situations
v = velocity
Manual and Graphical Methods
Straight – Ray (Constant – Velocity) Manual Method . Manual approximate migration
can be performed by use of the correct average velocity overlying the dipping event,
on the assumption of stratified media. Using Claerbout ‘S14 formulation , we can
associatiate an apparent location (t0, x0) and an apparent time dip θa = dt/dx, with
each reflection on the CDP stack. We wish to find its migrated position (tm, xmo) and
dip, θm. From Fig. 7 – 41 we see that
( 7 – 34 )
Where v is the overlying velocity. The basic geometric equation linking the un
migrated and the migrated dips is
( 7 – 35 )
sin m
dt
v
dx
sin tanm m
9. 0 xm0 xm1 x0 x1
θmθa
θm
θa
VtM
VtD’ VtD
D
Vdt
Figure 7 – 41 . Diagram of terms used in development of hand – migration equations.
θm = migrated dip θa = apparent dip. The basic geometric equation linking the
unmigrated and migrated dips is sin θm = tan θa .
10. Which can be observed from triangles OX1D’ and OX1D.
( 7 – 36 )
The migrated time is
( 7 – 37 )
The lateral location after migration is
( 7 – 38 )
Where v sin θa is horizontal component of velocity. The migrated dip of the reflection
segment, ρm, is given by
( 7 – 39 )
2 2
0
2
0 0 0 0
2 2
tan
cos 1
sin
tan
1
a
dt
m a a dx
dt
mo a dx
a a
m
a
dt
v
dx
t t t v
x x t v x t v
p
v v
11. 0
1
2
3
4
-4 -2 2 4 km sec
( a )
-4 -2 2 4 km0
0
1
2
3
4
sec
0
( b )
Figure 7 – 42
(a). Diffraction overlay, also
known as a surface of
maximum convexity.
(b). Wave front chart,
showing the position of
seismic wave front at
100 – ms increments of
time, for a given
velocity function. (from
Kleyn. 15)
12. S1 S2
R2
R1
R’2
R’1
D
X
T
MAC
curve
Figure 7 43 . The use of the maximum convexity curve and a wave front curve to
migrate a dipping event, R1 to its correct location, R2, and determine its true dip. (from
Klen.15)
