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Wave-equation common-angle gathers for converted waves

Paul Sava & Sergey Fomel

Bureau of Economic GeologyUniversity of Texas at Austin

paul.sava@beg.utexas.edu

Imaging condition

Image

Source wavefield

Receiver wavefield

Wavefield reconstruction

Imaging sketch

S

R

Angle decomposition

Angle-dependent reflectivity

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Wavefield reconstruction

sssmm dDGU sss ,,,, rrrmm dDGU rrr ,,,,

Source wavefield

Receiver wavefield

S

R

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Imaging condition

,, * mmm sr UUR

,,, * hmhmhm sr UUR Rickett & Sava (2002)

Biondi & Symes (2004) Sava & Fomel (2005)

Claerbout (1985)

Space shift: h={hx,hy,hz}

Location: m={x,y,z}

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Angle decomposition

,,, mhm RR

Reflection angle

Azimuth angle

Space shift: h={hx,hy,hz}

Location: m={x,y,z}

Message: images obtained by space-shift imaging contain sufficient information for converted-wave angle decomposition!

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Angle decomposition

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PP reflection geometry

ps

pr

2pm

2ph

srh

srm

pp2p

pp2p

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PS reflection geometry

pspr

2ph

2pm

2

s

2

rhm

rs

2

r

2

s

2

h

rs

2

r

2

s

2

m

pppp

cospp2ppp

cosppppp

4

θ24

θ224

sv

sv

s

p

1

1

r

s

p

p

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PS reflection geometry

pspr

2ph

2pm

14

θ2214

θ2214

22

22

22

s

s

s

hm

2

h

2

m

pp

cosp

cosp

hh

mm

kp

kp

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PS reflection geometry 14

θ2214

θ2214

222

222

222

s

s

s

hm

2

h

2

m

kk

cosk

cosk

3 relations, can eliminate 2 variables:

,,,zyx hhh kkk

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PS transformation 2222

2222

2

11

11θtan

hm

mh

kk

kk

2

2

02 θtan

m

h

k

k

Example: eliminate and .hm kk

14

θ2214

θ2214

222

222

222

s

s

s

hm

2

h

2

m

kk

cosk

cosk

3 relations, can eliminate 2 variables.

1

Sava & Fomel (2005)

paul.sava@beg.utexas.edu

PS transformation (2D)

Example: eliminate and .zhk

14

θ2214

θ2214

222

222

222

s

s

s

hm

2

h

2

m

kk

cosk

cosk

3 relations, can eliminate 2 variables.

xh

xh

zz

kkb

kka

abkk

a

x

x

11

11

142

1θtan

222

z

h

k

kx0θtan 1

Weglein & Stolt (1985) Sava & Fomel (2003)

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0

222

20

222

θtan11

1θtan1θtan

θtan,

θtan,

tanθ,

,

,

0

0

m

m

k

kk

hm

m

hm

R

R

R

R

R

2

2

02 θtan

m

h

k

k

Angle decomposition algorithm

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Example 1

0

15

3045

distance

dept

h

vP=2 km/svS=1 km/s

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PP data PS datasurface offset

time

surface offset

time

0

15

30

45

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PP imagedistance

dept

h

0

15

3045

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PS imagedistance

dept

h

0

15

3045

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PP offset-gather PS offset-gatherspace-shift

dept

hspace-shift

dept

h

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PP angle-gather PS angle-gathertan(0)

dept

htan(0)

dept

h

0 15 30 45 0 15 30 45

PP transformation

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PP angle-gather PS angle-gatherde

pth

dept

h

0 15 30 45 0 15 30 45

PS transformation

tan(0) tan()

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Example 2distance

dept

h

• acquisition• shots: 51 at 0.2km• receivers: 401 at 0.025km

1

1

0

0

1.0

1.0

/2

2

sg

sg

skmv

zgxgvv

v

v

z

x

zx

s

p

Modified from Baina et al. (2005):

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PP data PS datasurface offset

time

surface offset

time

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PP image PS imagedistance

dept

h

distance

dept

h

Uneven amplitude

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PP offset-gathers PS offset-gathersde

pth

dept

h

space-shift space-shift

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PP angle-gathers PS angle-gathersde

pth

dept

h

angle angle

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PP angle-gather PS angle-gatherangle

dept

hangle

dept

h

PP transformation

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PP angle-gather PS angle-gatherangle

dept

hangle

dept

h

PS transformation

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PP angle-gathers PS angle-gathersde

pth

dept

h

angle angle

Normal polarity

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PP angle-gathers PS angle-gathersde

pth

dept

h

angle angle

Reversed polarity

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PP stack PS stackdistance

dept

h

distance

dept

h

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Conclusions

• Angle decomposition for converted-waves

• Space-shift imaging condition– Independent of extrapolation method– Contains all required information

• Real challenge: what are the velocity models?