Riemannian wavefield extrapolation of seismic data

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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

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Overview

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

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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

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Echo soundings of the earth
Transmit sound-waves into earth
Record echoes from earth structure
Determine earth structure that created echoes
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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

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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

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Wavefield Extrapolation
Wavefield extrapolation
Wave phenomena
Wave-equation
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One-way wavefield extrapolation
Want solution to Helmholtz equation
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Migration by wavefield extrapolation

• Robust, Accurate, Efficient
• Current Limitations
• steep dip imaging
• no overturning waves

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One-way wavefield extrapolation
Steep Dip limitation
Advection solution on Cartesian grid
Overturning wave limitation
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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

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Riemannian wavefield extrapolation
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Overview

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

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Helmholtz equation
Laplacian
Coordinate system
(associated) metric tensor
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(Semi)orthogonal coordinates
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Helmholtz equation
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Dispersion relation
Riemannian
Cartesian
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Dispersion relation
Riemannian
Cartesian
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Wavefield extrapolation
Riemannian
Cartesian
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Summary

• Riemannian wavefield extrapolation
• General coordinate system
• Semi-orthogonal (3-D)
• Incorporate propagation in coordinates
• Applications
• Overturning waves
• Steeply dipping reflectors

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Collaboration?

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

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RWE vs. time-domain finite differences
distance
depth
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