This document discusses different methods for calculating seismic velocity and summarizing seismic data. It describes: 1) Average velocity, which is used to convert reflection time to depth by dividing depth by one-way travel time. 2) Interval velocity, which is the velocity between two reflectors and is calculated by dividing the depth difference by the time difference. 3) Instantaneous velocity, which is the derivative of depth with respect to time for a continuously varying velocity. 4) Methods have evolved from manual graphical techniques to digital methods like wave equation migration to properly position seismic reflectors.

1. GEOPHYSICAL
PROSPECTING

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)