jeff@sep.stanford.edu

Riemannian wavefield extrapolationof seismic data

J. Shragge, P. Sava, G. Shan, and B. Biondi

Stanford Exploration Project

S. Fomel

UT Austin

jeff@sep.stanford.edu

Overview

• Prelude– Remote sensing/Echo sounding– Seismic wavefield extrapolation

• Fugue – Riemannian wavefield extrapolation– Example

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

Echo soundings of the earth

Transmit sound-waves

into earth

Record echoesfrom earthstructure

Determine earthstructure that

created echoes

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

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)

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

Migration by wavefield extrapolation

• Robust, Accurate, Efficient• Current Limitations

– steep dip imaging– no overturning waves

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

Riemannian wavefield extrapolation

x

z

jeff@sep.stanford.edu

Overview

• Prelude– Remote sensing/Echo sounding– Seismic wavefield extrapolation

• Fugue – Riemannian wavefield extrapolation– Examples

jeff@sep.stanford.edu

Helmholtz equation

UsU 22

Laplacian

i j j

ij

i

UgU

g

g

1

(associated) metric tensor

)( kii x

Coordinate system

jeff@sep.stanford.edu

(Semi)orthogonal coordinates

i j j

ij

i

UgU

g

g

1

200

0

0

GF

FE

gij

2

2

0

0

J

gij

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

Dispersion relationR

iem

anni

anC

arte

sian

2222 skckickickc

2222 skk xz 1

0

cc

cc

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

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

jeff@sep.stanford.edu

interpolate

interpolate

jeff@sep.stanford.edu

Summary

• Riemannian wavefield extrapolation– General coordinate system

• Semi-orthogonal (3-D)

– Incorporate propagation in coordinates– Applications

• Overturning waves• Steeply dipping reflectors

jeff@sep.stanford.edu

Collaboration?

• Numerical development• Wave-based imaging

– Ultrasound– Sonar– Radar

• Applications– Medical imaging– Non-destructive testing– Marine navigation– Archaeology site assessment

jeff@sep.stanford.edu

distancede

pth

jeff@sep.stanford.edu

distance

dept

hRWE vs. time-domain finite differences

jeff@sep.stanford.edu

angletim

e

jeff@sep.stanford.edu

angletim

e

jeff@sep.stanford.edu

distancede

pth