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

Gradient calculation The objective function : Gradient calculation : 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

Algorithm of the gradient calculation 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

Algorithm of the gradient calculation 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 Effect of padding and split-spread for wrong velocity h Image Gather

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