Automatic%20Wave%20Equation%20Migration%20Velocity%20Analysis

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Automatic Wave Equation Migration Velocity
Analysis

• Peng Shen, William. W. Symes
• HGRG, Total EP
• CAAM, Rice University
• This work supervised by Dr. Henri Calandra at
  Total EP
• Thank to Dr. Scott Morton at Amerada Hess Corp.

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Velocity Analysis

• The coefficients of wave equation (relevant to
  imaging) are separable
• Long scale
• Short scale
• Challenges
• Nonlinear effect
• Coupling of long scale and short scale
• Multiple

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Methods of Velocity Analysis

• Data domain objectives
• Waveform inversion
• Stereotomography
• Image domain objectives
• WE-Migration forward, ray tracing inverse
• WE-Migration forward, WE-Migration inverse

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Outline

• Theory
• Objective function
• Gradient
• Calculation
• Physical meaning
• Smoothing
• Aliasing
• Examples
• Angle Offset
• Reconstruct short scale and large scale variations

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Generalized Born Modeling
Reflection occurs instantaneously with no
separation in space.
Reflection occurs instantaneously separated by a
finite distance.
Do not require to use the true velocity.
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Subsurface Image Measure
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Example
Data generated with caustic, migrated using
correct and background velocity.
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Differential Semblance
Offset domain
Angle domain
The objective function is smooth in velocity and
is suitable for automatic velocity updating
(Stolk Symes, 2003).
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Gradient Calculation
Offset
Angle
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Gradient Physical Meaning
Single scattering, constructive interference
occurs at zero offset.
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Gradient Physical Meaning
Two scatterings, constructive interference
occurs on ray segments.
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Smoothing

• Problem
• The raw gradient is singular with full data
  bandwidth.
• Solution
• Confine the velocity model to the space of
  B-splines.
• Controlled degree of smoothness
• Compactly supported basis
• Implication
• Projection to B-spline model space.
• B forward interpolation - sparse
  dense
• B adjoint projection - sparse
  dense

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Example
Flat reflector, constant velocity
Projected gradient using BB for one shot
The gradient B-spline projection reconstructs
wide ray paths which are controlled by the degree
of smoothness supplied.
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Optimization on B-spline space
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Aliasing

• We care not only the image in zero offset but
  also its move-out in non-zero offsets.
• There are many non-zero offset aliasing effects.
• Data pre-conditioning.
• Acquisition edge effect.

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Kinematics
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Kinematics of Image in Offset
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Examples
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Anti-aliasing
Aliasing reduced but loose some image
Strong aliasing
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Examples

• Born data
• Full data with rough model
• Initial model construction
• Optimization starting with v(z)

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Born Data Examples
Smooth Marmousi velocity model, singular
reflectivity, one-way wave simulation,
acquisition full spread, receiver dense on
surface.
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Starting Model
Starting model, large horizontal scales, assumed
to be obtainable through conventional velocity
analysis tools. Optimization 150m x 200m
B-spline grid.
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Initial Image
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Optimized Image
Optimized image using angle domain DSO.
Optimized image using offset domain DSO.
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Optimized Velocity
Optimized using angle domain DSO.
Optimized using offset domain DSO.
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Initial Angle and Offset Gathers
Top offset gathers, bottom angle gathers
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Optimized Gathers (angle driven)
Top offset gathers (not used in the
optimization), bottom angle gathers.
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Optimized Gathers (offset driven)
Top offset gathers, bottom angle gathers (not
used in the optimization)
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Velocity Difference
Difference between optimized velocity and the
projected true velocity (optimized by angle DSO).
Difference between optimized velocity and the
projected true velocity (optimized by offset DSO).
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Rough Marmousi Model
Data generated using full wave equation
simulation, acquisition split spread, receiver
spacing 25m, receiver array across entire
surface. Optimization offset driven, B-spline
grid 120m by 22m.
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Initial Velocity and Image
Initial velocity model, corresponds to B-spline
grid 900m by 300m.
Initial image
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Optimized Velocity and Image
Optimized velocity at 99th iteration
Optimized image at 99th iteration
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Optimized Velocity and Image
Optimized velocity at 49th iteration
Optimized image at 49th iteration
The optimization is stable and convergent.
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Obtain a Starting Model
A v(z) starting model. Optimization run up to
10Hz, coarse B-spline grid 800m by 400m, 500m by
200m.
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Optimized Velocity
800m by 400m, DSO optimized.
Projected from the true model.
500m by 200m, DSO optimized.
Projected from the true model.
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Null Space
Optimized image at 20th iteration.
Optimized image at 49th iteration.
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Starting with v(z) Velocity
Start from v(z) velocity model, increase
frequency and spatial resolution in two steps.
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Conclusions

• The angle domain DSO is not superior to offset
  domain DSO.
• The velocity analysis within migration is a
  promising direction to pursue.
• The adjoint-differential-migration provides an
  ideal platform for AWEMVA.