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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

“Exploring” spherical-wave reflection coefficients

Chuck Ursenbach

Arnim Haase

Research Report: Ursenbach & Haase, “An efficient method for calculating spherical-wave reflection coefficients”


Outline
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
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


Outline1
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
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



Outline2
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
Class I AVO application

  • Values given by Haase (CSEG, 2004; SEG, 2004)

  • Possesses critical point at ~ 43


Test of method for f n n 4 exp 173 hz 1 n
Test of method for f (n) = n4 exp([.173 Hz-1]n)



Spherical r pp for differing wavelets
Spherical RPP for differing wavelets


Outline3
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 w n
Behavior of Wn

qi = 15

qi = 0

qi = 45

qi = 85


Outline4
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
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
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
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.


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