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An automatic wave equation migration velocity analysis by differential semblance optimization

An automatic wave equation migration velocity analysis by differential semblance optimization. The Rice Inversion Project . Objective. Simultaneous optimization for velocity and image Shot-record wave-equation migration. Theory. Nonlinear Local Optimization Objective function

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An automatic wave equation migration velocity analysis by differential semblance optimization

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  1. An automatic wave equation migration velocity analysis by differential semblance optimization The Rice Inversion Project

  2. Objective • Simultaneous optimization for velocity and image • Shot-record wave-equation migration.

  3. Theory • Nonlinear Local Optimization • Objective function • Gradient of the objective function • Remark: • Objective function requires to be smooth . • Differential semblance objective function is smooth.

  4. Differential semblance criteria z x offset image angle image z z h h

  5. Objective function I : offset domain image c : velocity h : offset parameter P : differential semblance operator || ||: L2 norm M : set of smooth velocity functions

  6. Gradient calculation Definitions: Downward continuation and upward continuation S0 R0 gradient derivative cross correlate* down down SZ RZ DS* DR* cross correlate up up S*z R*z image cross correlate reference field

  7. spline Vmodel gmodel spline* M : set of smooth velocity functions Gradient smoothing using spline evaluation Vimage I gimage migration differential migration*

  8. Optimization BFGS algorithm for nonlinear iteration • Objective function evaluation • Gradient calculation loop • Update search direction coutIout

  9. Synthetic Examples • Flat reflector, constant velocity • Marmousi data set

  10. Experiment of flat reflector at constant velocity x Ccorrect = 2km/sec z

  11. Offset image Angle image Initial iterate: Image (v0 = 1.8km/sec) Image space: 401 by 80 Model space: 4 by 4

  12. Offset image Angle image Iteration 5: Image

  13. Iterations v5: Output velocity at iteration 5 vbest - v5

  14. Marmousi data set

  15. Marmousi data set

  16. V

  17. Initial iterate: Image (v0=1.8km/sec) Image space: 921 by 60 Model space: 6 by 6 Offset image Angle image

  18. Iterate 5: Image Offset image Angle image

  19. v5: output velocity at iteration 5 vbest: best spline interpolated velocity v5 - vbest iterations

  20. Low velocity lense + constant velocity background Vbackground = 2 km/sec

  21. Seismogram Shot gathers far away from the low velocity lense Shot gathers near the low velocity lense

  22. Iteration 1 Start with v0 = 2km/sec Iteration 2 Iteration 3 Iteration 4

  23. 1.0 1.5 2.0 2.5 3.0

  24. Conclusions • Offset domain DSO is a good substitute for angle domain DSO. • Image domain gradient needs to be properly smoothed. • DSO is sensitive to the quality of the image. • Differential semblance optimization by wave equation migration is promising.

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