Riemannian Wavefield Migration: Wave-equation migration from Topography

1
Riemannian Wavefield MigrationWave-equation
migration from Topography

• Jeff Shragge, Guojian Shan, Biondo Biondi
• Stanford University
• Paul Sava, Sergey Fomel
• Bureau of Economic Geology
• UT Austin

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The Dossier WE Imaging Limitations
Computational Power
Incomplete Data
Physical Inaccuracy
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The Dossier WE Imaging Limitations
Computational Power
Incomplete Data
Physical Inaccuracy
4
The Dossier WE Imaging Limitations
5
The Dossier WE Imaging Limitations
Migration Physics decoupled from Geometry
Propagator Inaccuracy (Illumination)
Difficult Steep Dip Imaging
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Steep Dip Imaging

• Accuracy of wavefield extrapolation decreases as
  propagating waves tend to horizontal

Extrapolation Direction
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The Dossier WE Imaging Limitations
Migration Physics decoupled from Geometry
Propagator Inaccuracy (Illumination)
No use of Overturning waves
Difficult Steep Dip Imaging
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Using Overturning Waves

• Currently do not use potentially useful
  information provided by overturning waves

Extrapolation Direction
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Proposed Solution

• Coordinate system conformal with wavefield
  propagation

Extrapolation Direction
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The Dossier WE Imaging Limitations
Migration Physics decoupled from Geometry
Propagator Inaccuracy (Illumination)
Topographic Surface Limitations
No use of Overturning waves
Difficult Steep Dip Imaging
Extrapolation from complex free-surface
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Free Surface Topography

• How to define extrapolation surface orthogonal to
  free surface?

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The Dossier WE Imaging Limitations
Migration Physics decoupled from Geometry
Propagator Inaccuracy (Illumination)
Topographic Surface Limitations
No use of Overturning waves
Extrapolation from deviated boreholes
Difficult Steep Dip Imaging
Extrapolation from complex free-surface
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VSP Deviated Well Topography

• Receiver wavefield acquired in well deviated in
  3-D
• How to define wavefield extrapolation from
  borehole surface?

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Proposed Solution

• Coordinate system conformal with borehole
  geometry

15
The Dossier WE Imaging Limitations
Migration Physics decoupled from Geometry
Propagator Inaccuracy (Illumination)
Topographic Surface Limitations
No use of Overturning waves
Extrapolation from deviated boreholes
Difficult Steep Dip Imaging
Extrapolation from complex free-surface
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A SolutionSummerized
Couple Migration Physics with Geometry
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Migration from Topography I
Make wave-equation imaging conformal with
acquisition geometry










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Migration from Topography II
Make wave-equation imaging conformal with
acquisition geometry
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Migration from Topography III
Make wave-equation imaging conformal with
acquisition geometry
Two requirements i) Propagating wavefields
in generalized coordinate systems ii)
Creating the specific topographic coordinate
system
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Agenda

• Riemannian Wavefield Extrapolation
• Coordinate System Generation
• Example Migration from Topography
• Imaging Husky

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Agenda

• Riemannian Wavefield Extrapolation
• Coordinate System Generation
• Example Migration from Topography
• Imaging Husky

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RWE in 2-D ray-coordinates
Riemannian
Cartesian
Extrapolation Direction Orthogonal Direction
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RWE Helmholtz equation
Sava and Fomel (2005)
(associated) metric tensor
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RWE (Semi)orthogonal coordinates
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RWE Helmholtz equation
Sava and Fomel (2005)
Ray-coordinate Interpretation a velocity
function J geometric spreading (i.e. Jacobian)
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RWE Dispersion relation
Riemannian
Cartesian
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RWE Dispersion relation
Riemannian
Cartesian
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RWE Kinematic dispersion relation
Riemannian
Cartesian
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RWE Kinematic dispersion relation
Riemannian
Cartesian
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RWE Wavefield extrapolation
Riemannian
Cartesian
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Extrapolation Work Flow
CANONICAL DOMAIN
PHYSICAL DOMAIN
MODEL DOMAIN
WAVEFIELD DOMAIN
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Agenda

• Riemannian Wavefield Extrapolation
• Coordinate System Generation
• Example Migration from Topography
• Imaging Husky

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Conformal Mapping Technique
O
O
O
O
O
O
O
O
O
O
PHYSICAL DOMAIN
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Conformal Mapping Technique
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
PHYSICAL DOMAIN
CANONICAL DOMAIN
g f-1
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Conformal Mapping Technique
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
PHYSICAL DOMAIN
CANONICAL DOMAIN
g f-1
O
O
O
O
O
O
O
O
O
O





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Conformal Mapping Technique
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
PHYSICAL DOMAIN
CANONICAL DOMAIN
g f-1
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O





g-1f
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Conformal Mapping Technique

• Shot Profile Migration
• use same grid for both source and receiver
  wavefields

O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O





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RWE Shot-profile migration
Riemannian
Cartesian
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Single shot Imaging Work Flow
CANONICAL DOMAIN
PHYSICAL DOMAIN
MODEL DOMAIN
IMAGE DOMAIN
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Agenda

• Riemannian Wavefield Extrapolation
• Coordinate System Generation
• Example Migration from Topography
• Imaging Husky

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Husky data set Topography

• 10x Surface Topography Exaggeration

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Husky data set Time Migration Geology
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TopoWEM
TopoWEM Image
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TopoWEM
TopoWEM Image
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TopoWEM Angle Gathers
TopoWEM Image
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TopoWEM Angle Gathers
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Kirchhoff from Flat Datum
Kirchhoff Image
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Kirchhoff from Flat Datum
Kirchhoff Image
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Kirchhoff surface offset gathers
Kirchhoff Image
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TopoWEM vs. StaticsDatum Kirchhoff
TopoWEM Image
TopoWEM Image
Kirchhoff Image
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TopoWEM vs. StaticsDatum Kirchhoff
TopoWEM Image
TopoWEM Image
Kirchhoff Image
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Summary

• TopoWEM produced good image
• Resolved near-surface structures and dips
• Kirchhoff migration from flat datum produced good
  image
• Better at mid-to-basement depths but could not
  resolve near-surface
• Bottom line statics migration from datum is
  less accurate
• Kinematic errors in static assumptions
• TopoWEM is more expensive
• Requires Taylor series expansion about 2
  coefficient parameters
• Imaging from acquisition coordinate works

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Acknowledgements

• Anatoly Baumstein
• Tom Dickens
• Ernestine Dixon
• Dave Hinkley
• Matt Hong
• Linda Price
• Mike Rainwater
• Peter Traynin
• Joe Vanderslice
• Kim Wilmott

• Husky Oil
• Talisman Oil
• Friday Lunch Folks
• John Sumner
• No doubt many others