Transcript 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
•
•
•
•
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:
p
 ( )  Ai exp(it )  R ( p) J ( pr ) exp[i (h  z )] dp

• Computing the gradient yields displacements

PP
0
PP
0
u()   ()
• Integrate over all frequencies to obtain trace
u(t )  


f ( ) u() d
• Extract AVO information: RPPspherical
Hilbert transform → envelope; Max. amplitude; Normalize
Alternative calculation route
PP ( )

gradient
u ( )
 p -integration
u[{i }]
 inverse FFT
u (t )
 Hilbert transform
| u (t )  iH [u (t )] |
 locate maximum
envelope[u (t  tmax )]
 normalize
RPP


0



0

p

RPP  f ( )i e  it J 0 ( pr ) ei ( h  z ) d   dp
 0
 
analytic for f ( )   n exp(  | s |)
p
RPP I ( p, t ) dp




0



0

gradient
I ( p, t ) p
dp
R 
RPP
t  R /  ; normalize with RPP  1
RPP Wn ( p )
p

dp
numerical integration
spherical
RPP
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
VS (m/s)
879.88 1882.29
r (kg/m3) 2400
2933.33
2000
Test of method for
f (n) = n4 exp([.173 Hz-1]n)
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
i = 0
i = 15
spherical
RPP



0
RPP Wn ( p )
p
sin 

p

dp,
i = 45
i = 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
r
r
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 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.