BEI Chapter 3 - Stanford University

BEI Chapter 3

Monday, December 3, 12

Travel-time depth1500 m/s2000 m/s

2500 m/s3000 m/s

1000m

2000m

3000m4000m

z


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


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

⌧


Horizontal moving waves Offset

time


Horizontal moving waves Offset

Head-wave

Groundroll, guided waves, direct arrivals


LMO


LMO

Critical angle


Linear moveout


LMO


LMO

Head waves


LMO


LMO

Critical angle


Head wavesCritical angle


LMO


Mute


Muting

Input MutedMonday, December 3, 12

Dipping waves

z

x

ray

front

v

✓✓


Dipping waves

z

x

ray

front

v

✓✓

z0


Dipping waves

z

x

ray

front

vz = z0 � x tan ✓

✓✓

z0


Dipping waves

z

x

ray

front


z0 =

vt

cos ✓

✓✓

z0


Dipping waves

z

x

ray

front


✓✓

z0 z cos ✓ = vt� x sin ✓

t(x, z) =

z

v

cos ✓ +

x

v

sin ✓


Dipping waves

z

x

ray

front

✓✓

z0

f

⇣t� x

v

sin ✓ � z

v

cos ✓

⌘


Horizontal velocity: Constant velocity

v2

v1


Horizontal velocity: Wavefront in media

v2

v1


Horizontal velocity: Interface

v2

v1


Horizontal velocity: Reflection and transmission

v2

v1


Horizontal velocity: Reflection & transmission wavefronts

v2

v1


Stepout doesn’t change in v(z) media

v2

v1

@t0

@x


Phase velocity

@t0

@x

=sin ✓

v

@t0@z

=

cos ✓

v

Horizontal phase velocity

Vertical phase velocity

Snell wave


Phase velocity

@t0@z

=

cos ✓

v

p=Snell parameter, observable with surface measurements

Vertical phase velocity

Snell wave

@t0

@x

=sin ✓

v

= p


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


Velocity

v2

v1 Velocity can be described in turns

of v(z) or v’(p,t)


Velocity

v2

v1 Velocity can be described in turns

of v(z) or v’(p,t)

v̂ =

rx

t

dx

dt


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


RMS Velocity

v2

v1

x(p, t) =Z t

0v0(p, t) sin✓(p, t) dt


RMS Velocity

v2

v1

x(p, t) =Z t

0v0(p, t) sin✓(p, t) dt

sin✓p=v


RMS Velocity

v2

v1

x(p, t) =Z t

0v0(p, t) sin✓(p, t) dtsin✓

p v


RMS Velocity

v2

v1

x(p, t) =Z t

0dtp v0(p, t)2

dx

dtp=

v = dx

dtx

t


RMS Velocity

v2

v1

x(p, t) =Z t

0dtp v0(p, t)2

pv = x

t1


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


RMS Velocity

v2

v1

x(p, t) =Z t

0dtp v0(p, t)2

pv = x

t1


True vs estimated wavefront


NMO Correction


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�⌧


Interval from RMSV 2(1) = v2(1)

2V 2(2) = v2(1) + v2(2)

3V 2(3) = v2(1)v2(2) + v2(3)


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)


BEI Chapter 3 - Stanford University

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