Transcript Noise alignment in trim statics

Extending AVO Inversion
Techniques
Charles Ursenbach
Robert Stewart
CREWES Report 2001
• Ursenbach & Stewart, “Extending AVO
inversion techniques” (Chapter 28)
• Ursenbach, “A generalized Gardner
relation” (Chapter 7)
Objectives
• Develop improved AVO linear inversion
techniques
• Develop an effective approach to assessment of
AVO approximations
The Linear Approximation
Zoeppritz Equations:
R pp  f (VP ,1 ,VP , 2 ,VS ,1 ,VS , 2 , 1 ,  2 , angles )
Aki-Richards Approximation:
R pp  A( ,VP / VS )
VP , 2  VP ,1
(VP ,1  VP , 2 ) / 2
R pp
 B( ,VP / VS )
VS , 2  VS ,1
(VS ,1  VS , 2 ) / 2
 C ( ,VP / VS )
VS
VP

A
B
C
VP
VS

 2  1
( 1   2 ) / 2
From 3 variables to 2
Aki-Richards Approximation:
R pp
VS
VP

A
B
C
VP
VS

Use Gardner’s
relation
Smith-Gidlow:
R pp
VS
VP
 ( A  C / 4)
B
VP
VS
- Convert to impedances
- Discard remainder
Fatti et al:
R pp
I S
I P
A
B
IP
IS
Smith-Gidlow approximation
  aV
1/ 4
1 3 / 4
d  a V
dV
4
d 1 dV

 4 V
R pp
VS
VP
 ( A  C / 4)
B
VP
VS
Fatti et al. approximation
I ( V ) V  V V 




I
V
V
V

R pp
I S
I P

A
B
 (C  A  B)
IP
IS

R pp
I S
I P
A
B
IP
IS
Comparison of Smith-Gidlow and Fatti et al.
Possible improvements
• #1 Combine strengths of Smith-Gidlow
& Fatti et al.
• #2 Improve on Gardner relation
• #3 Extract /
• #4 Assess approximations efficiently
#1 How to combine best features of
Smith-Gidlow and Fatti et al.?
R pp
I S
I P

A
B
 (C  A  B)
IP
IS

1     1  VP   1 I P
 
  
  4


 5 
  5  VP
  5 IP

R pp
I S
C  A  B  I P

 A
B

5
IS

 IP
The Full Offset approximation
#2 How to improve on Gardner’s
relation?
• Lithology specific – Castagna et al., 1993
• Based on Vs – CREWES, 1997, 1998, 1999
• Laboratory data – Wang, 2000
- -VP and -VS
- shale, clean sandstone, shaley
sandstone, unconsolidated
sandstone, limestone, dolomite
#2 How to improve on the Gardner
relation?
• Lithology independent, -(VP, VS)
  aVP VS
b
R pp
c

VS
VP
b
c

VP
VS
VS
VP
 ( A  aC )
 ( B  bC )
VP
VS
The generalized Gardner relation
#3 How to estimate /?
VS
VP

R pp  A
B
C
VP
VS

R pp
VS 1  b VP


VS
c  c VP
VP

 ( A  (b / c) B)
 (C  (1 / b) B)
VP

Density contrast approximations
#4 How to efficiently assess the value
of an approximation?
• Choose lithology
• Generate a representative sampling of (,VP,VS)
values for upper and lower layers
• For each sample generate synthetic offset gather
for some angle range (spike wavelet)
• Using chosen approximation, find best fit value of
VP / VP, VS / VS, etc.
• Average over all samples for chosen lithology
Incorporate in user-friendly interactive program
Another assessment issue
How does predicted result compare to exact value?
R pp
Zoeppritz
VS
VP
 ( A  C / 4)
B
VP
VS
How does predicted result compare to Aki-Richards?
R pp
Aki  Richards
VS
VP
 ( A  C / 4)
B
VP
VS
How does Aki-Richards compare to exact value?
R pp
Zoeppritz
  

 C 
  
exact
VS
VP
A
B
VP
VS
Results for Aki-Richards
Quantity
(method)
/
(-,
-)
IP/ IP
(IP -)
/
(-,
-)
 IS / IS
(IS -)
/
(-,
-)
shale/sst
2.28
6.24,
6.87
22.9,
23.8
14.7,
14.7
20.1,
21.1
10.25
shale/
limestone
shale/
dolostone
anhydrite/
limestone
1.18,
8.83
0.95,
14.4
0.41,
22.1
0.81,
16.0
155,
18.4
258,
11.2
376,
13.0
14.6,
0.51
anhydrite/
dolostone
1.86,
58.1
4.99
40.1,
41.4
49.8
2.33
0.41
3.02
41.1
11.4
141
272,
8.64
Linear inversions have high potential accuracy
Results for Smith-Gidlow and Fatti et al.
%-error in
contrast
/
(A-R, SmithGidlow)
IP / IP
(A-R,
Fatti et al.)
/
(A-R, SmithGidlow)
IS / IS
(A-R,
Fatti et al.)
shale/sst
1.18, 38.8
2.28, 3.07
6.24, 35.1
10.25, 9.99
shale/limestone
0.95, 39.2
2.33, 2.46
22.9, 113
41.1, 29.8
shale/dolostone
0.41, 133
0.41, 0.31
14.7, 249
11.4, 14.5
anhydrite/
limestone
0.81, 130
3.02, 10.5
20.1, 168
141, 88.7
anhydrite/limestone
(0 to 50 degrees)
0.80, 137
14.5, 82.5
20.4, 239
53.6, 143
anhydrite/
dolostone
1.86, 39.4
4.99, 1.60
40.1, 69.4
49.8, 48.5
Fatti et al. superior; wide variance in values
Results for Full Offset method
%-error of
I/I
IP (A-R, Fatti et al.,
Full Offset)
IS (A-R, Fatti et al.,
Full Offset)
shale/sst
2.28, 3.07, 2.74
10.25, 9.99, 11.1
shale/
limestone
shale/
dolostone
anhydrite/
limestone
2.33, 2.46, 2.30
41.1, 29.8, 32.8
0.41, 0.31, 0.23
11.4, 14.5, 15.3
3.02, 10.5, 10.43
141, 88.7, 83.2
4.99, 1.60, 1.04
49.8, 48.5, 47.8
anhydrite/
dolostone
Full Offset improves on Fatti et al. for IP
Results for generalized Gardner
No significant improvement in present form
Results for density inversion
%-error
(/)
shale/sst
shale/
limestone
shale/
dolostone
anhydrite/
limestone
anhydrite/
dolostone
VP- 
VS- 
I P- 
I S- 
AkiRichards
-
512
512
4160
233
18.4
181
181
1344
118
11.2
126
126
1477
143
13.0
95.7
95.7
129
104
0.51
162
162
335
81.1
8.64
Still considerable potential for improvement
Conclusions
• The method of Fatti et al. is more reliable than
that of Smith and Gidlow
• Considerable variation in accuracy is possible
for a given lithology
• The Full Offset approximation yields a clear
improvement on the method of Fatti et al.
• The generalized Gardner relation as presently
formulated does not improve AVO
• The Aki-Richards approximation could provide
the basis for a suitable density inversion scheme
Acknowledgments
The authors wish to thank the
sponsors of CREWES for financial
support of this research