Differential Semblance Optimization for Common Azimuth Migration

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Differential Semblance Optimization for Common Azimuth Migration
TRIP Annual meeting Differential Semblance Optimization for Common Azimuth Migration Alexandre KHOURY

Context of the project Prestack Wave Equation depth migration
Wavefield extrapolation method Automating the velocity estimation loop (time-consuming)

Motivation of the project
Encouraging results in 2D for Shot-Record migration (Peng Shen, TRIP 2005) Efficiency of the Common Azimuth Migration in 3D enables sparse acquisition in one direction very economic algorithm Goal of the project: Implement DSO for Common Azimuth Migration in 3D after a 2D validation

Common Azimuth Migration
Wavefield extrapolation in depth: “survey sinking” in the DSR equation h M Subsurface offset Variable used for Velocity Analysis : Subsurface offset

Subsurface Offset S M R S M R h’ M' M' R’=S’ R’ S’ For true velocity
For wrong velocity

Example: two reflectors data set

True velocity common image gather
Offset gather at x=1000 m

Example: two reflectors data set
One gather at midpoint x=1000m

Differential Semblance Optimization
From we define the objective function : For Criteria for determining the true velocity !

Differential Semblance Optimization
Plot of the objective function with respect to the velocity c=ctrue

Adjoint-state calculation (Lions, 1971): code operator

Migration: Structure of the Common Azimuth Migration
DSR equation: Wavefield at depth z Phase-Shift in the Fourier domain H1 H2 Lens-Correction in the space domain H3 General Screen Propagator or FFD in the space domain Imaging condition Wavefield at depth z+Dz Image at depth z+Dz

Wavefield pz Gradient at depth z+Dz MIGRATION H H-1 H*,B* Wavefield pz+Dz Gradient at depth z+2Dz H H-1 H*,B* Adjoint variables propagation Dp, Dc Wavefield pz+2Dz

Velocity representation on a B-spline grid: B-Spline transformation Fine grid B-Spline grid LBFGS Optimizer Adjoint B-Spline transformation Gradient calculation respect to B-Spline grid Gradient calculation respect to Fine Grid

Several critical points
- Avoid wrap-around in the subsurface offset domain -Avoid artifacts propagation by tapering the data -Constrain the optimization to keep the velocity in a specified range -Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

Several critical points
- Avoid wrap-around in the subsurface offset domain -Avoid artifacts propagation by tapering the data -Constrain the optimization to keep the velocity in a specified range -Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

Wrap-around in the subsurface offset domain
For wrong velocity Image Gather

Wrap-around in the subsurface offset domain

Several critical points
- Avoid wrap-around in the subsurface offset domain -Avoid artifacts propagation by tapering the data -Constrain the optimization to keep the velocity in a specified range -Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

Artifacts propagation
Necessity to taper the data on both offset and midpoint axes and in time

Several critical points
- Avoid wrap-around in the subsurface offset domain -Avoid artifacts propagation by tapering the data -Constrain the optimization to keep the velocity in a specified range -Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

Several critical points
- Avoid wrap-around in the subsurface offset domain -Avoid artifacts propagation by tapering the data -Constrain the optimization to keep the velocity in a specified range -Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

Differential Semblance Optimization
Tests on different data sets: -Test on flat reflectors with a constant background velocity -Test on the top of a salt model -Test on a 4-Reflectors model

Differential Semblance Optimization
Tests on different data sets: -Test on flat reflectors with a constant background velocity -Test on the top of a salt model -Test on a 4-Reflectors model

Differential Semblance Optimization
Start of the optimization: V=2300 Image Gather

Differential Semblance Optimization
10 iterations: Right position Image Gather

Differential Semblance Optimization
Tests on different data sets: -Test on flat reflectors with a constant background velocity -Test on the top of a salt model -Test on a 4-Reflectors model

Differential Semblance Optimization
Top of salt : image x=5000

Differential Semblance Optimization
Top of salt : one gather

Differential Semblance Optimization
Plot of : localization of the energy of the objective function

Differential Semblance Optimization
Tests on different data sets: -Test on flat reflectors with a constant background velocity -Test on the top of a salt model -Test on a 4-Reflectors model

Differential Semblance Optimization
True velocity

Differential Semblance Optimization
Starting velocity

Differential Semblance Optimization
Starting image

Differential Semblance Optimization
Optimized image

Differential Semblance Optimization
True image

Differential Semblance Optimization
Optimized velocity

Conclusion Migration is critical and has to be artifacts free.
Is the DSR Migration precise enough for optimization of complex models ? Can we deal with complex velocity model ? Next: test on the Marmousi data set and on a 3D data set.

Acknowledgment Prof. William W. Symes Total E&P
Dr. Peng Shen, Dr Henri Calandra, Dr Paul Williamson