A new angle on seismic imaging and inversion

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A new angle on seismic imaging and inversion
Sverre Brandsberg-Dahl
BP Upstream Technology Group Seismic
Imaging Houston, USA
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Applications

• Imaging in geologically complex areas
• True amplitude PP and PS depth images
• PP and PS angle common-image gathers (CIGs)
• AVA-analysis/interpretation
• P- and S-wave velocity estimation

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Two main approaches

• Ray tracing based methods
• Kirchhoff, inverse generalized Radon transform
• Wavefield downward continuation methods
• finite difference, phase shift, PSPI,
• generalized screen

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Ray tracing based methods

• Intuitive physics behind method
• Target oriented
• Separate wave-modes
• Good handling of amplitude
• Simple/cheap to construct gathers for AVA and
  velocity analysis

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Angle gathers from standard Kirchhoff migration

• Perform two Kirchhoff migrations (Bleistein,
  1987)
• Construct offset-to-angle map
• Convert offset CIGs to angle CIGs by simple
  stretching and re-sampling
• Cannot handle caustics (multi valued wavefields)

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Which arrival?
source/receiver offset
time
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Which arrival?
source/receiver offset
time
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Inverse generalized Radon transform (GRT)
(Beylkin, 1987)

• Ray tracing based, but more suited in complex
  media
• Re-parameterization of data/operator
• Direct inverse through construction of normal
  operator
• Born or Kirchhoff approximation

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Dip control and specular reflections
specular reflection
Migration dip
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Why angle coordinates?
Acquisition parameters
Angle parameters
Homogeneous TIV medium, PSV reflected wave
traveltime curves
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Angle common image gathers
Incomplete inverse GRT
amplitude
data, polarization
traveltime
integral over migration dip at the image point
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Imaging-inversion

• Least squares direct inversion nested inside the
  migration (all quantities calculated in
  background medium)
• True amplitude
• correct for illumination
• compensate for uneven fold

even sampling and coverage on surface even
coverage in subsurface (image)
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Focusing in dip
Focusing in dip select the dip cone
corresponding to the reflector normal.
Perform imaging with a range of dip cones
estimate geological dip by standard semblance
techniques
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What we really can do
map the wavefront from data space to model space
reflected signal
interfaces
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Pre-stack imaging

• Add one dimension to the imaging volume DIP
• Produce dip focused images by using both a
  velocity model and a dip model

dip
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Angle gathers Kirchhoff vs. GRT
1.0
2.0
3.0
4.0
K
G
K
G
5.0
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Two main approaches

• Ray tracing based methods
• Kirchhoff, inverse generalized Radon transform
• Wavefield downward continuation methods
• finite difference, phase shift, PSPI,
• generalized screen

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wave equation methods

• All migration approaches are based on the wave
  equation, so terminology is misleading
• Implementation extrapolation in depth
• one-way two-way
• Implementation extrapolation in time
• reverse time migration

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Reverse time migration of shot records the
ultimate migration
A simple example by Clifford Nolan (1997)
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Reverse time migration of shot records the
ultimate migration
but not without dip information
from Nolan, C. 1997
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Finite-difference reverse time migration
constant velocity and single dip
velocity lens several dips
from Nolan, C. 1997
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Imaging by downward continuation

• One-way propagation of wavefield
• Needs full wavefield (data)
• Implicitly accounts for multi pathing
• Common azimuth approximation (Biondi) speed!

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Wavefield downward continuation
imaging condition correlate wavefields at each
depth level
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Imaging by downward continuation

• Often lost accuracy with larger angle of
  propagation
• Amplitudes?
• Double square root equation
• source and receiver side downward continued
  simultaneously
• Shot record migration
• source and receiver side downward continued
  separately

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Angle gathers from DCM
s
r
Sava and Biondi, 2001.
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Angle gathers from DCM

• Migration outputs image gathers in offset or
• offset ray parameter
• Convert offset ray parameter to angle
• Post migration slant stack procedure applied to
  offset gathers

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Downward continuation migration vs. ray-tracing
based migration
Kirchhoff
DCM
offset
angle
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Angle stack
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Offset gather
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Angle gathers
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GRT/Kirchhoff

• Multi-arrival Kirchhoff
• Within the approximation, GRT includes all
  arrivals
• Target oriented and can treat different modes of
  reflection PP, PS
• True amplitude and easy amplitude manipulation,
  i.e. AVA compensation of images before velocity
  analysis
• Dip focusing increases S/N in CIGs and images

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Downward continuation methods

• Use full wavefield
• As used today, more accurate for structural
  imaging?
• Efficient implementation for common azimuth
• Hard to make true amplitude
• Accuracy?

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Angles and depth imaging

• Common image gathers (CIGs) parameterized in
  angle directly applicable for AVA analysis
• Angle CIGs unravel complex wave field
• We have to introduce dip as extra dimension in
  images and CIGs for all types of migration

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Acknowledgments
I would like to thank the following
FORCE and O.I. Barkved for invitation.
BP for permission to publish data.
John T. Etgen (BP) for providing the common
azimuth results.
Clifford Nolan for providing figures from his
Ph.D. thesis.