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Riemannian wavefield extrapolation of seismic data. J. Shragge, P. Sava, G. Shan, and B. Biondi Stanford Exploration Project S. Fomel UT Austin. Overview. Prelude Remote sensing/Echo sounding Seismic wavefield extrapolation Fugue Riemannian wavefield extrapolation Example. - PowerPoint PPT Presentation
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Riemannian wavefield extrapolationof seismic data
J. Shragge, P. Sava, G. Shan, and B. Biondi
Stanford Exploration Project
S. Fomel
UT Austin
Overview
• Prelude– Remote sensing/Echo sounding– Seismic wavefield extrapolation
• Fugue – Riemannian wavefield extrapolation– Example
Why seismic imaging?• Applied seismology
– Hydrocarbon exploration – “Easy” targets already located– remaining large fields located in
regions of complex geology
• 3-D seismic imaging– Delineate earth structure – property estimation and prediction– improve probability of finding oil
Echo soundings of the earth
Transmit sound-waves
into earth
Record echoesfrom earthstructure
Determine earthstructure that
created echoes
Seismic imaging - Similarities
• Related methods– Acoustic wave methods
• Ultrasound
• Sonar
– EM wave methods• Radar
• X-ray
• Related applications– Medical imaging– Non-destructive testing– Marine navigation– Archaeology site assessment
Seismic imaging - Differences
• Complex earth structure – Velocity
• V(x,y,z) – 1.5 – 4.5 km/s
• Strong gradients
– Material properties• heterogeneity
• anisotropy
• Wave-phenomena– Multi-arrivals, band-limited– Frequency-dependent illumination– Overturning waves
• Ray theory cannot capture complexity
Wavefield Extrapolation
Wave phenomena Wave-equationWavefield
extrapolation
Uz)y,v(x,
ωΔU
2
2
Monochromatic frequency-domain: Helmholtz equation
Recorded wavefield U(x,y,z=0) Want U(x,y,z)
One-way wavefield extrapolation
Want solution to Helmholtz equation
2x2
2
z k- z)y,v(x,
ω±=k
Wave-equation dispersion relation
zikxx
ze ω)z,,U(k=ω)z,z,U(k
Wavefield propagates by advection - with solution
Uz)y,v(x,
ωΔU
2
2
Migration by wavefield extrapolation
• Robust, Accurate, Efficient• Current Limitations
– steep dip imaging– no overturning waves
One-way wavefield extrapolation
2x2
2
z k- z)v(x,
ω±=k
Wave-equation dispersion relation
zikxx
ze ω)z,,U(k=ω)z,z,U(k +
Advection solution on Cartesian grid
Steep Diplimitation
Overturningwave limitation
Migration by wavefield extrapolation
• Robust, Accurate, Efficient• Current Limitations
– steep dip imaging– no overturning waves
• Our solution– Change coordinate system to be
more conformal with wavefield– Riemannian spaces
Overview
• Prelude– Remote sensing/Echo sounding– Seismic wavefield extrapolation
• Fugue – Riemannian wavefield extrapolation– Examples
Helmholtz equation
UsU 22
Laplacian
i j j
ij
i
UgU
g
g
1
(associated) metric tensor
)( kii x
Coordinate system
1st order 2nd order2nd order 1st order
Helmholtz equation
UsU
JJ
UJ
J2211
UsU
J
U
JJ
UJ
J
U 222
2
22
2
2
1111
UsU
cU
cU
cU
c 222
2
2
2
Dispersion relationR
iem
anni
anC
arte
sian
2222 skckickickc
2222 skk xz 1
0
cc
cc
Dispersion relationR
iem
anni
anC
arte
sian
sk
cc
cc
o
1
0
22
2
k
c
ck
c
cik
c
cik o
222xz ksk
Wavefield extrapolationR
iem
anni
anC
arte
sian
sk
cc
cc
o
1
0
zikxx
ze ω)z,,U(k=ω)z,z,U(k
τΔγγ
τττΔτ ike ω),,U(k=ω),,U(k
Summary
• Riemannian wavefield extrapolation– General coordinate system
• Semi-orthogonal (3-D)
– Incorporate propagation in coordinates– Applications
• Overturning waves• Steeply dipping reflectors
Collaboration?
• Numerical development• Wave-based imaging
– Ultrasound– Sonar– Radar
• Applications– Medical imaging– Non-destructive testing– Marine navigation– Archaeology site assessment
