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

2. Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

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

4. Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

5. Traveltime Linearization Under Rytov approximationUnder Rytov approximation = TraveltimeResidual ObjectFunction Wavepath (Woodward, 1992)

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

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

8. Partial Reconstruction Depth migration (Lecomte, 1998 SEG)

9. Wavenumber rsrsrsrs rgrgrgrg Source Geophone r K

10. Spatial Traveltime Resolution Limit Formula

11. source-receiver pairs where the wavepath visits r the wavepath visits r

12. Reflection Traveltime Tomography rsrsrsrs rgrgrgrg Source Geophone

13. Transmission Traveltime Tomography rsrsrsrs rgrgrgrg Source Geophone

14. Available Wavenumbers rsrsrsrs rgrgrgrg Transmission ReflectionSource Geophone

15. Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

16. Numerical Examples Crosswell Traveltime TomographyCrosswell Traveltime Tomography Refraction Traveltime TomographyRefraction Traveltime Tomography Global TomographyGlobal Tomography

17. Crosswell Traveltime Tomography X L (0, L/2) (X, L/2) (0, -L/2) (X, -L/2) r 0 (X/2, 0) Source Geophone

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

19. Crosswell Traveltime Tomography 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 traveltimeThe 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 limitsThe intersection area of the wavepaths at the scatterer defines the spatial resolution limits

21. Wavepath Intersection Transmission Example r s1 r g1 Fresnel Zone

22. Wavepath Intersection Transmission Example r s1 r g1 r g2 r s2

23. r s1 r g1 r s2 r g2 r s3 r g3 Wavepath Intersection Transmission Example

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

25. Numerical Examples Crosswell Traveltime TomographyCrosswell Traveltime Tomography Refraction Traveltime TomographyRefraction Traveltime Tomography Global TomographyGlobal Tomography

26. Refraction TomographyS V1V1V1V1 V2V2V2V2 R

27. Numerical Examples Crosswell Traveltime TomographyCrosswell Traveltime Tomography Refraction Traveltime TomographyRefraction Traveltime Tomography Global TomographyGlobal Tomography

28. 1Hz Global Tomography0 6000 12000 0 6000 12000 (km) 13.72 10.29 6.858 3.429 0 (km/s) Core Mantle Scatterer Wavepath

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

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

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

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

33. Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

34. Summary Used the inverse GRT to get the spatial traveltime resolution formulasUsed the inverse GRT to get the spatial traveltime resolution formulas We have Developed a practical means of estimating traveltime resolution limits for arbitrary velocity models and finite-frequency source dataDeveloped a practical means of estimating traveltime resolution limits for arbitrary velocity models and finite-frequency source data Obtained resolution limits of global tomo.Obtained resolution limits of global tomo.

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