riemannian wavefield extrapolation of seismic data
DESCRIPTION
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 PresentationTRANSCRIPT
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
