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

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

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3 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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4 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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5 Subsurface Offset SR M R’=S’ M' SR M R’S’ h’ For true velocity For wrong velocity

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6 Example: two reflectors data set

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

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

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9 Differential Semblance Optimization Fromwe define the objective function : For Criteria for determining the true velocity !

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

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11 Gradient calculation The objective function : Gradient calculation : Adjoint-state calculation (Lions, 1971): code operator

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

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13 H H H *, B* Algorithm of the gradient calculation Wavefield p z Wavefield p z+ z Wavefield p z+2 z MIGRATION Gradient at depth z+2 z H -1 Gradient at depth z+ z H -1 Adjoint variables propagation Dp, Dc

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

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15 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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16 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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17 Wrap-around in the subsurface offset domain For wrong velocity h ImageGather

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18 Wrap-around in the subsurface offset domain Effect of padding and split-spread for wrong velocity h ImageGather

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19 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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20 Artifacts propagation Necessity to taper the data on both offset and midpoint axes and in time

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21 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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22 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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23 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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24 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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25 Differential Semblance Optimization Image Gather Start of the optimization: V=2300

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26 Differential Semblance Optimization 10 iterations: Right position GatherImage

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27 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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28 Differential Semblance Optimization Top of salt : image x=5000

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29 Differential Semblance Optimization Top of salt : one gather

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

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31 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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32 Differential Semblance Optimization True velocity

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33 Differential Semblance Optimization Starting velocity

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34 Differential Semblance Optimization Starting image

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35 Differential Semblance Optimization Optimized image

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36 Differential Semblance Optimization True image

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37 Differential Semblance Optimization Optimized velocity

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

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

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