Exploring spherical wave reflection coefficients
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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

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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


Alternative calculation route

Alternative calculation route


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)


Wavelet comparison

Wavelet comparison


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


Exploring spherical wave reflection coefficients

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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