1. Extending AVO Inversion Techniques
Charles Ursenbach
Robert Stewart

2. CREWES Report 2001
Ursenbach & Stewart, “Extending AVO inversion techniques” (Chapter 28)
Ursenbach, “A generalized Gardner relation” (Chapter 7)

3. Objectives Develop improved AVO linear inversion techniques
Develop an effective approach to assessment of AVO approximations

4. The Linear Approximation
Zoeppritz Equations:
Aki-Richards Approximation:

5. From 3 variables to 2 Aki-Richards Approximation:
- Convert to impedances
Use Gardner’s relation
- Discard remainder
Smith-Gidlow:
Fatti et al:

6. Smith-Gidlow approximation

7. Fatti et al. approximation

8. Comparison of Smith-Gidlow and Fatti et al.

9. Possible improvements
#1 Combine strengths of Smith-Gidlow & Fatti et al.
#2 Improve on Gardner relation
#3 Extract /
#4 Assess approximations efficiently

10. #1 How to combine best features of Smith-Gidlow and Fatti et al.?
The Full Offset approximation

11. #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

12. #2 How to improve on the Gardner relation?
Lithology independent, -(VP, VS)
The generalized Gardner relation

13. Density contrast approximations
#3 How to estimate /?
Density contrast approximations

14. #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

15. Another assessment issue
How does predicted result compare to exact value?
How does predicted result compare to Aki-Richards?
How does Aki-Richards compare to exact value?

16. Results for Aki-Richards
Quantity
(method)
/
(-,
-)
IP/ IP
(IP -)
/
-)
 IS / IS
(IS -)
/
(-,
shale/sst
1.18, 8.83
2.28
6.24, 6.87
10.25
155, 18.4
shale/
limestone
0.95, 14.4
2.33
22.9, 23.8
41.1
258, 11.2
dolostone
0.41, 22.1
0.41
14.7, 14.7
11.4
376, 13.0
anhydrite/
0.81, 16.0
3.02
20.1, 21.1
141
14.6, 0.51
anhydrite/dolostone
1.86, 58.1
4.99
40.1, 41.4
49.8
272, 8.64
Linear inversions have high potential accuracy

17. Results for Smith-Gidlow and Fatti et al.
%-error in contrast
/
(A-R, Smith-Gidlow)
IP / IP
(A-R,
Fatti et al.)
/
IS / IS
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
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

18. 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
2.33, 2.46, 2.30
41.1, 29.8, 32.8
dolostone
0.41, 0.31, 0.23
11.4, 14.5, 15.3
anhydrite/
3.02, 10.5, 10.43
141, 88.7, 83.2
4.99, 1.60, 1.04
49.8, 48.5, 47.8
Full Offset improves on Fatti et al. for IP

19. Results for generalized Gardner
No significant improvement in present form

20. Results for density inversion
%-error
(/)
VP- 
VS- 
IP- 
IS- 
Aki-Richards
-
shale/sst
512
4160
233
18.4
shale/
limestone
181
1344
118
11.2
dolostone
126
1477
143
13.0
anhydrite/
95.7
129
104
0.51
162
335
81.1
8.64
Still considerable potential for improvement

21. 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

22. Acknowledgments
The authors wish to thank the sponsors of CREWES for financial support of this research