Transcript Extension and Evaluation of Pseudo

Testing old and new
AVO methods
Chuck Ursenbach
CREWES Sponsors Meeting
November 21, 2003
CREWES 2003 Research Reports
I.
II.
III.
IV.
V.
Testing pseudo-linear Zoeppritz
approximations: P-wave AVO inversion
Testing pseudo-linear Zoeppritz
approximations: Multicomponent and joint
AVO inversion
Testing pseudo-linear Zoeppritz
approximations: Analytical error expressions
Using the exact Zoeppritz equations in
pseudo-linear form: Isolating the effects of
input errors
Using the exact Zoeppritz equations in
pseudo-linear form: Inversion for density
Outline
•
•
•
•
•
New Inversion Methods
Testing with error-free data
Analytical error expressions
Testing on input with errors
Density inversion
I, II
III
IV
V
Aki-Richards Approximation
1

 1 

2

 2  sin 

2
2 cos  
 2 
 
2
RPP
  (1   2 ) / 2

Depends on /

2   / 
sin 1
Snell’s Law: sin  2 
2   / 
2





Pseudo-Linear expression
1

 1 

2

 2  sin 

2
 2 
2 cos  
 
2
A R
PP
R
PL
RPP
2
2
 4 cos1 cos 2  
1

 1       
 

 2  sin1 sin 2
 1  
 


2
Q
2
cos

cos




2
2








 


1
2
  
  
Q  1 
 cos1  1 
 cos 2
2 
2 


Pseudo-quadratic expression
PQ
PP
R
4 cos1 cos 2

Q2

1

 2 
 2 sin 1 sin  2 2


 
 2 cos1 cos 2 
  2  
 
sin 1 sin  2


1



  2
  Q   2[1   /(2 )]cos cos 2  
  cos cos 2
  
2
sin 2 1
  3
sin 3 1 sin  2
 

4 2

2 2
3
3
 [1   /(2 )] Q   2[1   /(2 )] cos cos 2   
  2   / 
1 

2 
    2 
1 



  2  


  1  
   
sin 1 sin  2


1   2

Q


2
[
1



/(
2

)]
cos

cos


2 
 
 
  
  
Q  1 
 cos1  1 
 cos 2
2 
2 


Accuracy depends on /
Impedance
• IP = 
• IS = 
• IP/IP  / + /
• IS/IS  / + /
P-impedance
contrast is
predicted
accurately
Comparison of IS/IS predictions
Comparison of RPS inversion for IS/IS
Average %-errors
A-R P-L P-Q
RPP
8.6 13
2.0
RPS
8.2 3.2 .22
joint 7.2 3.6 .50
Section Summary
• Accurate Zoeppritz approximations can be
cast into an Aki-Richards form for convenient
use in AVO
• Errors in predicted contrasts are strongly
correlated with /
• Strong cancellation of error for / + /
• Strong cancellation of error for / + / in
Pseudo-quadratic method
• Pseudo-linear and Pseudo-quadratic
methods give superior values of IS/IS for RPS
and joint inversion
Analytical Inversion
• Observation: Inversion of 3 points of
noise-free data, ( = 0, 15, 30 ) gives
very similar results to densely sampled
data
• Conjecture: Inversion should be semianalytically tractable (with aid of symbolic
computation software [Maple])
• Remark: For inversion of PS data only two
points should be required ( = 15, 30 )
Method
• Leave /, /, /, / as variables
• Assume their value in coefficients is exact
• Evaluate necessary functions at :
 = 0, 15, 30  where
sin() = 0, 2( 3 1) / 4, ½
• Carry out inversion using Cramer’s rule
• Expand contrast estimates up to cubic
order in exact contrasts, and up to first
order in (/ - ½)
S-Impedance contrast error
PP
PP
 I S 
 I S 
3

  024  071  ( )




 I S  AR  I S  exact
 146  341  ( ) (  )
2
 0988   1 2  ( ) ( )
2
  111  272   ( ) ( ) 2
  00081  027   ( ) (  )
2
  085   042  ( )
3
  065   020  ( ) ( )
  052  0049   ( )
PP
PP
 I S 
 I S 
3

  085   042  ( )




 I S  PQ  I S  exact
  065   020  ( ) ( )
2
2
  050   038  ( ) 2( )
  0086  082   ( ) ( )
  000037  000071  ( ) 3
  020  039   ( ) 2
  025   040  ( ) ( ) ( )
2
  050   038  ( ) 2( )
  000037  071  ( )
3
P-impedance contrast error
PP
PP
PP
PP
 I P 
 I P 
1             
 

 

 



4             
 I P  AR  I P  exact
 I P 
 I P 
1      

 
  


4     
 I P  PL  I P  exact
2
m
n
l

 I P 
 I P 
         

 
  O 
 
 , l  m  n  4

         
 I P  PQ  I P exact
PP
PP
Section Summary
• Analytical inversion is tractable
• Cubic order formulae give reasonable
representation of error
• Potential use in correcting inversion
results
• Rigorous illustration of the superiority of
P-wave impedance estimates
Sources of AVO error
• Assumptions of the Zoeppritz equations
• Approximations to the Zoeppritz equations
• Limited range of discrete offsets
represented
• Errors in input – R (noise, processing),
background parameters (velocity model,
empirical relations, etc.), angles (velocity
model)
Exact Zoeppritz in Pseudo-Linear form
 1
 
 1     1   1  
 1   1  










RPP  (cos1  cos 2 )1 
 1  2  1  2   cos1  1  2  1  2   cos 2  
4


 4

 





 
1  1 
 1 
4  2 
 1  
 1 
1 
 cos1  1 
 2  
 2 

 1  
1 
 cos 2  
 2  

 1    cos1  cos 2 
1  


 

1 
 / 

2 

 4   


([1  RPP ] cos1  [1  RPP ] cos 2 )


 2 sin 1 sin  2 
 1   1   1  
1 
 cos1 cos1
 2
(
1

R
)
1 

PP 1 
 [1  ( / 2 ) 2 ] 
2

2

2





 1   1 
 (1  RPP )1 
1 
 2   2 
 
 1  
1 
 cos 2 cos 2 
 2  
 
3
sin 2 1



 3
(
1

R
)
cos

cos


(
1

R
)
sin

sin

cos(



)
PP
1
1
PP
1
1
2
2
 [1  ( / 2 )]2



 1    
 (1  RPP ) sin 1 sin  2 sin 1 sin  2
 1 


4   2  


2
 

 

 

 


/ = (/)exact + 0.2
Gaussian noise on R: magnitude 0.01
Section Summary
• AVO inversions can be carried out with the
pseudo-linear form of the exact Zoeppritz
equations
• Provides a means of examining the effect
of individual input errors
• Provides a guide to uncertainty
propagation
• Provides a guide to assessing the
significance of approximation errors
An exact expression quadratic in /
R
D 
Exact
PP
D

D
Exact
PP
R
D  D  0

    
1     








1

1

cos


1

1

cos


1
2







4  2   2  
 2   2  

    
  2 sin  1 sin  2 
    
 cos 1  1 
 cos 2   2

1 
1 
1 
2
2

2

2

2








  (1  [ / 2 ] )
      

      








1

1

1

cos

cos


1

1

1

cos

cos





1
1
2
2
 2  
 2  
2

2

2

2










 sin  1 sin  2 cos( 1  1 ) cos( 2   2 )   


 3
2

(1  [ / 2 ] )
  
2


 1    
 sin  1 sin  2 sin 1 sin  2
 1 

 4   2  

   


   


3

2
Least-squares determination of /
a  b( /  )  c( /  )  0
2
a, b, c are functions of
, , /, , R ()
2
3

 

   

2  
2    
a
b

2
a
c

b

3
b
c

2
c
 i i    i i  i  
 i i    
 i      0
i
 i
  i

 i

 i



/ = (/)exact + 0.2
Gaussian noise on R: magnitude 0.01
Section Summary
• The exact Zoeppritz equation can be
formulated to allow least-squares
extraction of / by solution of a cubic
polynomial
• The / errors from this method are
distinctly different from those of 3parameter inversion
• Random input errors seem to be controlled
very effectively in this method