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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.

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riemannian wavefield extrapolation of seismic data

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
Overview
  • Prelude
    • Remote sensing/Echo sounding
    • Seismic wavefield extrapolation
  • Fugue
    • Riemannian wavefield extrapolation
    • Example

jeff@sep.stanford.edu

why seismic imaging
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
Echo soundings of the earth

Transmit

sound-waves

into earth

Record echoes

from earth

structure

Determine earth

structure that

created echoes

jeff@sep.stanford.edu

seismic imaging similarities
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
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
Wavefield Extrapolation

Monochromatic frequency-domain: Helmholtz equation

Recorded wavefield U(x,y,z=0) Want U(x,y,z)

Wavefield

extrapolation

Wave phenomena

Wave-equation

jeff@sep.stanford.edu

one way wavefield extrapolation
One-way wavefield extrapolation

Wave-equation dispersion relation

Wavefield propagates by advection - with solution

Want solution to Helmholtz equation

jeff@sep.stanford.edu

migration by wavefield extrapolation
Migration by wavefield extrapolation
  • Robust, Accurate, Efficient
  • Current Limitations
    • steep dip imaging
    • no overturning waves

jeff@sep.stanford.edu

one way wavefield extrapolation1
One-way wavefield extrapolation

Wave-equation dispersion relation

Steep Dip

limitation

Advection solution on Cartesian grid

Overturning

wave limitation

jeff@sep.stanford.edu

migration by wavefield extrapolation1
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
Riemannian wavefield extrapolation

x

z

jeff@sep.stanford.edu

overview1
Overview
  • Prelude
    • Remote sensing/Echo sounding
    • Seismic wavefield extrapolation
  • Fugue
    • Riemannian wavefield extrapolation
    • Examples

jeff@sep.stanford.edu

helmholtz equation
Helmholtz equation

Laplacian

Coordinate system

(associated) metric tensor

jeff@sep.stanford.edu

semi orthogonal coordinates
(Semi)orthogonal coordinates

jeff@sep.stanford.edu

helmholtz equation1
Helmholtz equation

2nd order

1st order

1st order

2nd order

jeff@sep.stanford.edu

dispersion relation
Dispersion relation

Riemannian

Cartesian

jeff@sep.stanford.edu

dispersion relation1
Dispersion relation

Riemannian

Cartesian

jeff@sep.stanford.edu

wavefield extrapolation1
Wavefield extrapolation

Riemannian

Cartesian

jeff@sep.stanford.edu

slide20

interpolate

interpolate

jeff@sep.stanford.edu

summary
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
Collaboration?
  • Numerical development
  • Wave-based imaging
    • Ultrasound
    • Sonar
    • Radar
  • Applications
    • Medical imaging
    • Non-destructive testing
    • Marine navigation
    • Archaeology site assessment

jeff@sep.stanford.edu

slide23

distance

depth

jeff@sep.stanford.edu

rwe vs time domain finite differences
RWE vs. time-domain finite differences

distance

depth

jeff@sep.stanford.edu

slide25

angle

time

jeff@sep.stanford.edu

slide26

angle

time

jeff@sep.stanford.edu

slide27

distance

depth

jeff@sep.stanford.edu