“Exploring” spherical-wave reflection coefficients

1. “Exploring” spherical-wave reflection coefficients Chuck Ursenbach Arnim Haase Research Report: Ursenbach & Haase, “An efficient method for calculating spherical-wave reflection coefficients”

2. Outline • Motivation: Why spherical waves? • Theory: How to calculate efficiently • Application: Testing exponential wavelet • Analysis: What does the calculation “look” like? • Deliverable: The Explorer • Future Work: Possible directions

3. Motivation • Spherical wave effects have been shown to be significant near critical angles, even at considerable depth See poster: Haase & Ursenbach, “Spherical wave AVO-modelling in elastic isotropic media” • Spherical-wave AVO is thus important for long-offset AVO, and for extraction of density information

4. Outline • Motivation: Why spherical waves? • Theory: How to calculate efficiently • Application: Testing exponential wavelet • Analysis: What does the calculation “look” like? • Deliverable: The Explorer • Future Work: Possible directions

5. Spherical Wave Theory • One obtains the potential from integral over all p: • Computing the gradient yields displacements • Integrate over all frequencies to obtain trace • Extract AVO information: RPPspherical Hilbert transform → envelope; Max. amplitude; Normalize

6. Alternative calculation route

7. Outline • Motivation: Why spherical waves? • Theory: How to calculate efficiently • Application: Testing exponential wavelet • Analysis: What does the calculation “look” like? • Deliverable: The Explorer • Future Work: Possible directions

8. Class I AVO application • Values given by Haase (CSEG, 2004; SEG, 2004) • Possesses critical point at ~ 43

9. Test of method for f (n) = n4 exp([.173 Hz-1]n)

10. Wavelet comparison

11. Spherical RPP for differing wavelets

12. Outline • Motivation: Why spherical waves? • Theory: How to calculate efficiently • Application: Testing exponential wavelet • Analysis: What does the calculation “look” like? • Deliverable: The Explorer • Future Work: Possible directions

13. Behavior of Wn qi = 15 qi = 0 qi = 45 qi = 85

14. Outline • Motivation: Why spherical waves? • Theory: How to calculate efficiently • Application: Testing exponential wavelet • Analysis: What does the calculation “look” like? • Deliverable: The Explorer • Future Work: Possible directions

16. Spherical effects Waveform inversion Joint inversion EO gathers Pseudo-linear methods

17. Conclusions • RPPsph can be calculated semi-analytically with appropriate choice of wavelet • Spherical effects are qualitatively similar for wavelets with similar lower bounds • New method emphasizes that RPPsph is a weighted integral of nearby RPPpw • Calculations are efficient enough for incorporation into interactive explorer • May help to extract density information from AVO

18. Possible Future Work • Include n > 4 • Use multi-term wavelet: Sn An wn exp(-|snw|) • Layered overburden (effective depth, non-sphericity) • Include cylindrical wave reflection coefficients • Extend to PS reflections

19. Acknowledgments The authors wish to thank the sponsors of CREWES for financial support of this research, and Dr. E. Krebes for careful review of the manuscript.