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Wave-equation common-angle gathers for converted waves
Paul Sava & Sergey Fomel
Bureau of Economic GeologyUniversity of Texas at Austin
Imaging condition
Image
Source wavefield
Receiver wavefield
Wavefield reconstruction
Imaging sketch
S
R
Angle decomposition
Angle-dependent reflectivity
Wavefield reconstruction
sssmm dDGU sss ,,,, rrrmm dDGU rrr ,,,,
Source wavefield
Receiver wavefield
S
R
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}
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!
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
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
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
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)
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)
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
PP angle-gather PS angle-gathertan(0)
dept
htan(0)
dept
h
0 15 30 45 0 15 30 45
PP transformation
PP angle-gather PS angle-gatherde
pth
dept
h
0 15 30 45 0 15 30 45
PS transformation
tan(0) tan()
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):
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?