bei chapter 3 - stanford university
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BEI Chapter 3
Monday, December 3, 12
Travel-time depth1500 m/s2000 m/s
2500 m/s3000 m/s
1000m
2000m
3000m4000m
z
Monday, December 3, 12
Travel-time depth1500 m/s2000 m/s
2500 m/s3000 m/s
Two-way travel-time1.331.00
0.80.670.8
z1000m
2000m
3000m4000m
Monday, December 3, 12
Travel-time depth1500 m/s
2000 m/s2500 m/s3000 m/s
1s
2s3s
Two-way travel-time1.331.00
0.80.670.8
4s⌧ =
2z
v
⌧
Monday, December 3, 12
Horizontal moving waves Offset
time
Monday, December 3, 12
Horizontal moving waves Offset
Head-wave
Groundroll, guided waves, direct arrivals
Monday, December 3, 12
LMO
Monday, December 3, 12
LMO
Critical angle
Monday, December 3, 12
Linear moveout
Monday, December 3, 12
LMO
Monday, December 3, 12
LMO
Head waves
Monday, December 3, 12
LMO
Monday, December 3, 12
LMO
Critical angle
Monday, December 3, 12
LMO
Critical angle
Monday, December 3, 12
Head wavesCritical angle
Monday, December 3, 12
LMO
Monday, December 3, 12
Mute
Monday, December 3, 12
Mute
Monday, December 3, 12
Muting
Input MutedMonday, December 3, 12
Dipping waves
z
x
ray
front
v
✓✓
Monday, December 3, 12
Dipping waves
z
x
ray
front
v
✓✓
z0
Monday, December 3, 12
Dipping waves
z
x
ray
front
vz = z0 � x tan ✓
✓✓
z0
Monday, December 3, 12
Dipping waves
z
x
ray
front
vz = z0 � x tan ✓
z0 =
vt
cos ✓
✓✓
z0
Monday, December 3, 12
Dipping waves
z
x
ray
front
vz = z0 � x tan ✓
✓✓
z0 z cos ✓ = vt� x sin ✓
t(x, z) =
z
v
cos ✓ +
x
v
sin ✓
Monday, December 3, 12
Dipping waves
z
x
ray
front
✓✓
z0
f
⇣t� x
v
sin ✓ � z
v
cos ✓
⌘
Monday, December 3, 12
Horizontal velocity: Constant velocity
v2
v1
Monday, December 3, 12
Horizontal velocity: Wavefront in media
v2
v1
Monday, December 3, 12
Horizontal velocity: Wavefront in media
v2
v1
Monday, December 3, 12
Horizontal velocity: Interface
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection and transmission
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection and transmission
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection & transmission wavefronts
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection & transmission wavefronts
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection & transmission wavefronts
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection & transmission wavefronts
v2
v1
Monday, December 3, 12
Horizontal velocity: Reflection & transmission wavefronts
v2
v1
Monday, December 3, 12
Stepout doesn’t change in v(z) media
v2
v1
@t0
@x
Monday, December 3, 12
Phase velocity
@t0
@x
=sin ✓
v
@t0@z
=
cos ✓
v
Horizontal phase velocity
Vertical phase velocity
Snell wave
Monday, December 3, 12
Phase velocity
@t0@z
=
cos ✓
v
p=Snell parameter, observable with surface measurements
Vertical phase velocity
Snell wave
@t0
@x
=sin ✓
v
= p
Monday, December 3, 12
A little math for later@t0
@x
=
sin ✓
v
= p
@t0
@z
=
cos ✓
v
=
s1
v(z)
2� p
2
t0(x, z) =
sin ✓
v
x +
Z z
0
cos ✓
v
dz
t0(x, z) = p x +
Z z
0
s1
v(z)
2� p
2dz
Monday, December 3, 12
Velocity
v2
v1 Velocity can be described in turns
of v(z) or v’(p,t)
Monday, December 3, 12
Velocity
v2
v1 Velocity can be described in turns
of v(z) or v’(p,t)
v̂ =
rx
t
dx
dt
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =
Z t
0v
0(p, t) sin ✓(p, t) dt = p
Z t
0v
0(p, t)2 dt
v̂ =
rx
t
dx
dt
vRMS =
s1
t
Z t
0v0(p, t)2 dt
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0v0(p, t) sin✓(p, t) dt
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0v0(p, t) sin✓(p, t) dt
sin✓p=v
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0v0(p, t) sin✓(p, t) dtsin✓
p v
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0dtp v0(p, t)2
dx
dtp=
v = dx
dtx
t
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0dtp v0(p, t)2
pv = x
t1
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0dtp v0(p, t)2
pv = x
t1
vRMS =
s1
t
Z t
0v0(p, t)2dt
Monday, December 3, 12
RMS Velocity
v2
v1
x(p, t) =Z t
0dtp v0(p, t)2
pv = x
t1
Monday, December 3, 12
True vs estimated wavefront
Monday, December 3, 12
NMO Correction
Monday, December 3, 12
Vrms discrete form
vRMS =
s1
t
Z t
0v0(p, t)2 dt
V (it) = vRMS(it) =
vuut 1
it ⇤�⌧
itX
0
v(it)2�⌧
Monday, December 3, 12
Interval from RMSV 2(1) = v2(1)
2V 2(2) = v2(1) + v2(2)
3V 2(3) = v2(1)v2(2) + v2(3)
Monday, December 3, 12
Interval from RMS
3V 2(3) = 2V 2(2) + v2(3)
v2(3) = 3V 2(3)� 2V 2(2)
V 2(1) = v2(1)
2V 2(2) = v2(1) + v2(2)
3V 2(3) = v2(1) + v2(2) + v2(3)
Monday, December 3, 12
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