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Jianming Sheng and Gerard T. Schuster University of Utah February, 2000

Finite-Frequency Resolution Limits of Traveltime Tomography for Smoothly Varying Velocity Models. Jianming Sheng and Gerard T. Schuster University of Utah February, 2000. Outline. Objective Inverse GRT and Resolution Limits Numerical Examples Summary. Objective.

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Jianming Sheng and Gerard T. Schuster University of Utah February, 2000

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  1. Finite-Frequency Resolution Limits of Traveltime Tomography for Smoothly Varying Velocity Models Jianming Sheng and Gerard T. Schuster University of Utah February, 2000

  2. Outline • Objective • Inverse GRT and Resolution Limits • Numerical Examples • Summary

  3. Objective Develop a resolution method that • Estimates spatial resolution of traveltime tomograms • Accounts for finite-frequency effects • Is applicable for arbitrary velocity models

  4. Outline • Objective • Inverse GRT and Resolution Limits • Numerical Examples • Summary

  5. Traveltime Residual Object Function Wavepath Linearization • Under Rytov approximation =

  6. Linearization • Using geometrical approximation = It is related to the causal generalized Radon transform (Beylkin, 1985)

  7. Controls resolution and what model parts can be recovered Inverse GRT Partial Reconstruction

  8. Inverse GRT Partial Reconstruction

  9. K Wavenumber rs rg r Source Geophone

  10. Spatial Resolution Limits Formula

  11. source-receiver pairs where the wavepath visits r Spatial Resolution Limits Formula

  12. Reflection Traveltime Tomography rs rg Geophone Source

  13. Transmission Traveltime Tomography rs rg Source Geophone

  14. Reflection Transmission Available Wavenumbers rs rg Geophone Source

  15. Outline • Objective • Inverse GRT and Resolution Limits • Numerical Examples • Summary

  16. Numerical Examples • Crosswell Experiment • Refraction Tomography • Global Tomography

  17. L r0(X/2, 0) X Crosswell Experiment (0, L/2) (X, L/2) Source Geophone (0, -L/2) (X, -L/2)

  18. Crosswell Experiment A. Reflection Tomography the same as the migration-spatial-resolution limits for crosswell migration derived by Schuster (1996) in far-field approximation.

  19. Crosswell Experiment B. Transmission Tomography The results are similar to Schuster(1996) for traveltime tomography in far-field approximation

  20. Key Idea • The velocity anomalies within the first-Fresnel zone or wavepath affect the traveltime • The intersection area of the wavepaths at the scatterer defines the spatial resolution limits

  21. Fresnel Zone Wavepath Intersection Transmission Example rs1 rg1

  22. Wavepath Intersection Transmission Example rs1 rg2 rs2 rg1

  23. Wavepath Intersection Transmission Example rs1 rg3 rg2 rs2 rs3 rg1

  24. 72m 44.7m 400 (m) 200 (m) Wavepath Intersection Transmission Example C=3000 m/s f=300 Hz

  25. Numerical Examples • Crosswell Experiment • Refraction Tomography • Global Tomography

  26. Refraction Tomography S R V1 V2 The same as the result of Schuster (1995)

  27. Numerical Examples • Crosswell Experiment • Refraction Tomography • Global Tomography

  28. Wavepath Scatterer 1Hz Global Tomography 12000 0 (km) 6000 0 Mantle 13.72 10.29 6000 Core 6.858 3.429 0 (km/s) 12000 (km)

  29. 1Hz Global TomographyResolution Limits (Depth=100km) -100 Depth (km) 100 300 0 200 400 Horizontal (km)

  30. 1Hz Global TomographyResolution Limits (Depth=300km) 100 Depth (km) 300 500 0 200 400 Horizontal (km)

  31. 1Hz Global TomographyResolution Limits (Depth=400km) 200 Depth (km) 400 600 0 200 400 Horizontal (km)

  32. 1Hz Global TomographyResolution Limits (Depth=800km) 600 Depth (km) 800 1000 0 200 400 Horizontal (km)

  33. Outline • Objective • Inverse GRT and Resolution Limits • Numerical Examples • Summary

  34. Summary We have • Derived the inverse GRT and the spatial resolution formulas • Developed a practical means of estimating resolution limits for arbitrary velocity models and finite-frequency source data • Reexamined whole-earth tomograms

  35. Acknowledgment We thank the sponsors of the 1999 University of Utah Tomography and Modeling /Migration (UTAM) Consortium for their financial support .

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