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

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
Upper Layer
Lower Layer
VP (m/s)
2000
VS (m/s)
879.88
r (kg/m3)
2400

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. Pseudo-linear methods
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.