“Exploring” spherical-wave reflection

coefficientsChuck 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

• Spherical-wave AVO is thus important for long-offset AVO, and for extraction of density information

See poster: Haase & Ursenbach, “Spherical wave AVO-modelling in elastic isotropic media”

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

00( ) exp( ) ( ) ( ) exp[ ( )]PP PP

pAi i t R p J pr i h z dp

• Integrate over all frequencies to obtain trace

• Extract AVO information: RPPspherical

( ) ( ) ( )u t f u d

( ) ( )u

Hilbert transform → envelope; Max. amplitude; Normalize

Alternative calculation routePP

max

PP

( )

gradient

( )

-integration

[{ }]

inverse FFT

( )

Hilbert transform

| ( ) [ ( )] |

locate maximum

envelope[ ( )]

normalize

i

u

p

u

u t

u t iH u t

u t t

R

( )PP 00 0

PP0

PP0

PP

PP0

( ) e ( ) e

analytic for ( ) exp( | |)

( , )

gradient

( , )

/ ; normalize with 1

( )

nume

i t i h z

n

n

pR f i J pr d dp

f s

pR I p t dp

I p t pR dp

R

t R R

pR W p dp

sphericalPP

rical integration

R

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

Upper Layer

Lower Layer

VP (m/s) 2000 2933.33

VS (m/s) 879.88

1882.29

(kg/m3)

2400 2000

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

Wavelet comparison

Spherical RPP for differing wavelets

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

sphericalPP

PP0( ) ,

sin

n

R

pR W p dp

p

i = 0

i = 85

i = 45

i = 15

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

Waveform

inversion

Joint

inversion

Spherical effects

EO gather

s Pseudo-linear

methods

Conclusions• RPP

sph 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: n An n exp(-|sn

• 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.