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Elastic and Acoustic Wave Phenomena

Elastic and Acoustic Wave Phenomena. by Phil Bording Memorial University of Newfoundland Swastika Sharma and Steve Butt Dalhousie University SIAM GEOSCIENCES - 2007. Acoustic Wave Equations. 1D 2D 3D 2.5D. Source – Causality ?. Uniform Grid. Nyquist Criteria.

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Elastic and Acoustic Wave Phenomena

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  1. Elastic and Acoustic Wave Phenomena by Phil Bording Memorial University of Newfoundland Swastika Sharma and Steve Butt Dalhousie University SIAM GEOSCIENCES - 2007

  2. Acoustic Wave Equations 1D 2D 3D 2.5D

  3. Source – Causality ?

  4. Uniform Grid

  5. Nyquist Criteria • Two grid points – support wave without aliasing • Dominate frequency at source • Does the earth act as a filter? • Yes, deeper has higher velocities and lower frequencies

  6. Stability Limit Courant – Nifty Mathematical Fellow– had a condition!

  7. Grid points per Wave length • Wave length = Velocity/Frequency • Nyquist requires 2 points to support .. • Using at least two points we can reconstruct the waves without aliasing • Modeling requires more than 2 to reduce the error in dispersion, numerical error

  8. 0 1 2 3 4 Kirchhoff PSDM 100% Velocity - inline 300 Faults Horizons approximated by stair steps! Marfurt, 2004 (Lazarevic, 2004)

  9. So just how many Grid Points per Wavelength do we really need??? Nyquist says………. 2 We can image to…. ¼ Lambda 4 Dispersion error requires too many…. 10+

  10. So just how many Grid Points per Wavelength do we really need??? The difference operators control everything! Error = D(spatial) – D(time) Taylor Series -- Weights(T=0) Optimal Weights – Weights(T=t) Spectral Methods -- D(spatial) = 0 Frequency Domain -- D(time) = 0

  11. Complexity – how long will it run?? n(x,y,z) are the spatial grid sizes n(t) is the number of time steps w(s) is the work per step n(s) is the number of shots

  12. Realistic Models

  13. Early Elastic Modeling • Amoco Group -- Kelly, Alford, and Treitel

  14. Elastic Staggered Grid Alan Levander – Rice P-Sv waves with higher Poisson’s ratio

  15. Elastic P-S Modeling Levander

  16. Elastic - Continued Levander

  17. Elastic Wave Equation Levander, 1988

  18. Elastic today • 4th order staggered grid - Veiiux • Topography Surface - Hestholm • Slow shear waves dominate the grid sizing (spacing is small) • 3D is at least 10X more computing • Visco-elastic schemes in development – another 10X more? -- Aldridge at Sandia

  19. Imaging Machine

  20. Wave Machine

  21. Acknowledgements • ACOA – AIF • Husky • IBM

  22. DATA TIME

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