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**Differential Semblance Optimization for Common Azimuth Migration**

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

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**Context of the project Prestack Wave Equation depth migration**

Wavefield extrapolation method Automating the velocity estimation loop (time-consuming)

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**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

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**Common Azimuth Migration**

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

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

For wrong velocity

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**Example: two reflectors data set**

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**True velocity common image gather**

Offset gather at x=1000 m

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**Example: two reflectors data set**

One gather at midpoint x=1000m

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**Differential Semblance Optimization**

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

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**Differential Semblance Optimization**

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

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**Gradient calculation The objective function : Gradient calculation :**

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

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**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

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**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

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**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

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**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)

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**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)

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**Wrap-around in the subsurface offset domain**

For wrong velocity Image Gather

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**Wrap-around in the subsurface offset domain**

Effect of padding and split-spread for wrong velocity h Image Gather

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**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)

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**Artifacts propagation**

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

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**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)

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**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)

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**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

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**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

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**Differential Semblance Optimization**

Start of the optimization: V=2300 Image Gather

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**Differential Semblance Optimization**

10 iterations: Right position Image Gather

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**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

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**Differential Semblance Optimization**

Top of salt : image x=5000

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**Differential Semblance Optimization**

Top of salt : one gather

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**Differential Semblance Optimization**

Plot of : localization of the energy of the objective function

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**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

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**Differential Semblance Optimization**

True velocity

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**Differential Semblance Optimization**

Starting velocity

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**Differential Semblance Optimization**

Starting image

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**Differential Semblance Optimization**

Optimized image

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**Differential Semblance Optimization**

True image

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**Differential Semblance Optimization**

Optimized velocity

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**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.

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**Acknowledgment Prof. William W. Symes Total E&P**

Dr. Peng Shen, Dr Henri Calandra, Dr Paul Williamson

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