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“Exploring” spherical-wave reflection coefficients. Chuck Ursenbach Arnim Haase. Research Report: Ursenbach & Haase, “An efficient method for calculating spherical-wave reflection coefficients”. Outline. Motivation : Why spherical waves? Theory : How to calculate efficiently
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“Exploring” spherical-wave reflection coefficients Chuck Ursenbach Arnim Haase Research Report: Ursenbach & Haase, “An efficient method for calculating spherical-wave reflection coefficients”
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
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
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
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
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
Class I AVO application • Values given by Haase (CSEG, 2004; SEG, 2004) • Possesses critical point at ~ 43
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
Behavior of Wn qi = 15 qi = 0 qi = 45 qi = 85
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
Spherical effects Waveform inversion Joint inversion EO gathers Pseudo-linear methods
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
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
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.